CONSTANTIN BÂNICÂ
and
OCTAVIAN STÂNÂSILÂ
ALGEBRAIC
METHODS
in the
GLOBAL
THEORY
COMPLEX
SPACES
i
EDITURA ACADEMIEI
Bucureçti
©
JOHN WILEY & SONS
London-New York-Sydney Toronto
1976

ALGEBRAIC
METHODS
in the
GLOBAL
THEORY
of
COMPLEX
SPACES

Copyright © 1976, by John Wiley & Sons, Ltd.
All rights reserved.
No part of this book may be reproduced by any means, nor
translated, nor transmitted into a machine language without
the written permission of the publisher.
Library of Congress Cataloging in Publication Data
Bânicâ, Constantin.
Algebraic methods in the global theory
of complex spaces
Rev. English version of Metode algebrice in teoria globalâ
a spatiilor complexe, published in 1974.
Bibliography: p.
1. Analytic spaces. 2. Homology theory. 3. Duality theory
(Mathematics) I. Stânâsilâ, Octavian, joint author. II. Title.
QA331.B25413 515'.7 76-5823
ISBN 0 471 01809 0
This edition is the revised English version of the Romanian book
METODE ALGEBRICE ÎN TEORIA GLOBALÂ A SPATIILOR COMPLEXE'
published in 1974
by EDITURA ACADEMIET, Calea Victoriei 125, BUCHAREST.
All rights reserved.
PRINTED IN ROMANIA

Preface
The theory of functions of several complex variables has been significantly
developed in the last decades. The study of complex spaces, i.e. the
geometric models associated to this theory, involves methods of
analysis, algebra, differential equations, geometry and algebraic topology.
The author intends to present some global problems of complex spaces,
where the emphasis falls on algebraic methods.
Each chapter is preceded by a paragraph on preliminaries and by an
introduction which summarises the main results of the text. These
introductory passages will give the reader a rough picture of the book.
The proofs, except some of those in chapter seven, are complete apart
from the assumption of basic results on complex spaces, algebra, sheaf
theory, topological vector spaces, etc.
The book is intended mainly for experts in complex spaces who are not
fully acquainted with the algebraic aspect, and for experts in algebraic
geometry who wish to be introduced to the theory of complex spaces.
The authors would wish to express their profound gratitude to the
Institute of Mathematics of the Romanian Academy.
Bucharest, October 1973. C. BÂNICÂ
O. STÂNÂÇILÂ
We are under obligation to our colleagues V. Brînzânescu and Manuela Stoia
for their valuable assistance in the preparation of the English edition.

To the Diary of Theodor Pallady

Contents
Chapter I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES . . 9
§ 1. Preliminaries 10
§ 2. Duality on Stein manifolds 23
§ 3. Dimension and depth of a coherent analytic sheaf 37
§ 4. Applications 42
Chapter II. ANALYTIC LOCAL COHOMOLOGY 51
§ I. Preliminaries 52
§ 2. The singular sets of the coherent sheaves 62
§ 3. The vanishing theorem 63
§ 4. The finiteness theorem 70
§ 5. Absolute local cohomology 75
§ 6. The separation theorem 81
Chapter III. PROPER MORPHISMS OF COMPLEX SPACES 91
§ 1. Preliminaries 92
§ 2. The finiteness theorem 99
§ 3. The comparison and the base change theorems 112
§ 4. The semicontinuity and continuity theorems. The invariance of
Euler-Poincaré characteristic 123
Chapter IV. PROJECTIVE MORPHISMS OF COMPLEX SPACES 137
§ 1. Preliminaries 138
§ 2. The behaviour at + oo of the sheaves S-(m) 142
§ 3. The behaviour at — oo of the sheaves &(m) 150
§ 4. Two criteria for ampleness 155
Chapter V. FLAT MORPHISMS OF COMPLEX SPACES 163
§ 1. Preliminaries 163
§ 2. Algebraic and topological properties of the flat morphisms . . . 177

8
§ 3. A noetherianity theorem with respect to Stein compacts . . 182
§ 4. The flatness locus of a morphism 187
Chapter VI. THE FORMAL COMPLETION OF A COMPLEX SPACE WITH
RESPECT TO A SUBSPACE 193
§ I. Preliminaries 194
§ 2. Definition and elementary properties 199
§ 3. A finiteness theorem 205
§ 4. The comparison theorem 218
Chapter VII. DUALITY ON COMPLEX SPACES 227
§ 1. Preliminaries 228
§ 2. The construction of the dualizing complex 251
§ 3. Theorems of absolute duality 259
§ 4. Duality on complex manifolds 271
§ 5. The dualizing sheaves 289
BIBLIOGRAPHY 293

Chapter I
Cohomology with compact supports
on Stein spaces
Introduction
We recall the following two theorems from the function theory of several complex
variables :
"If D is an open subset of (£,", n ^ 2 and K <= D is a compact such that D\K
is connected, then any holomorphic function on D \ K can be extended to a holo-
morphic function on D" (Hartogs' theorem, [41], Ch. II, § 3).
"If X is a Stein manifold of dimension $s 3, and U <= X is a relatively compact
Stein open subset, then the additive Cousin problem has a solution on X \ U; if in
addition H2(X\ U; Z) = 0, then the multiplicative Cousin problem has also a
solution on X \ U" (Serre's theorem, [74]).
The generalizations suggested by the first theorem are of the following nature :
— the substitution of D for a complex manifold or, more generally, for a
complex space;
— the consideration of some entities more general than functions (sections in a
sheaf, cohomology classes, divisors, meromorphic sections, subspaces, coherent sheaves).
All of these are subsumed to the problem of the extension of analytic entities
defined out of a compact.
The second theorem belongs also to a general problem in connection with the
previous one, namely, the study of the properties of the complementary of a compact
(or a relatively compact open subset) in a complex space.
Let X be a topological space, K cr X a compact and 3r a sheaf of abelian
groups on X. The exact cohomology sequence
... -► H"(X, ®) -► H"(X \ K, &) ^HqK+\X, §9 -► ...
shows that the elements of the cohomology group with supports in K, H^+1(X, SF)
are just the obstructions for the extension to whole X of the cohomology ^-classes
on X \ K with coefficients in S\
If U <= X is a relatively compact open subset, then we have the exact sequence
. . . -* H"(X, ST) -► H%X \U,S:)-> W+\V, &) -► .. . .
Therefore, it results that the elements of the cohomology group with compact
supports //«+1([7, &) represent the obstructions for the extension to whole X of the
cohomology ^-classes on X \ U with coefficients in 3\

10 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Thus one can notice the prominent part of the invariants H'c, H'K in the
investigation of the above stated problems. Our aim in this chapter is to study these
invariants on Stein spaces and, in particular, to find vanishing theorems for them.
The first paragraph contains preliminaries of sheaf theory, topological vector
spaces and local algebra. In the second one we prove two duality theorems (2.1 and
2.9) on Stein manifolds, which allow us to express the invariants H'c, H'K in terms of
global invariants Ext" (X; ...), Ext' (K; ...). The third paragraph deals with the
algebraic notions of depth and dimension for a coherent analytic sheaf. Under the
assumption that X is a Stein space and K is a holomorphically convex compact,
the global Ext's determine completely the local Ext's; this fact will allow us, via the
duality proved in § 2, to reduce the invariants H'c, H'K to invariants Ext^.... (6X is
the local ring at x e X). So we will make the connection with the depth and the
dimension and will obtain the cohomological characterizations 3.6 and 3.7 for them:
the cohomology groups with compact supports vanish out of the interval
[prof, dim] and are <£0 at the ends of this interval. The fourth paragraph contains
applications (in the frame of the above considered problems): results of type Hartogs
or Cousin, properties of the boundary of a Stein space, applications regarding the
compact analytic spaces. For instance, when X is a Stein space and K <= X is a
holomorphically convex compact, the following assertions are proved:
— The restriction map T (X, 6) -► T (X \ K, 6) is bijective if and only if
prof Qx $s 2 for all xeK (corollary 4.4).
— If prof <3X ^ 3 for any x e K, then the additive Cousin problem has
solution on X \ K; if in addition H2 (X\K, 7L) = 0, then the multiplicative Cousin
problem has also solution on X \ K (corollary 4.5).
— If prof 6X $s 2 for all x e K, then X is connected if and only if X \ K
is (corollary 4.8).
§ 1. Preliminaries
(a) We recall some facts in sheaf theory [26], [35].
Let X be a topological space. A family $ of closed subsets of X is called
a family of supports if any closed subset of an element of $ belongs to $, and if
any union of two elements of $ is still an element of $. Let SF be a sheaf of abelian
groups on X (we will briefly write ^ e Ab (X)). Denote by T® (X, &) the subgroup
of T (X, SF) of the sections whose supports belong to $. Thus, one obtains a left
exact additive functor & \-> T^, (X, &) from the category Ab (X) to the category
Ab of abelian groups. Its right derived functors are denoted by H^{X, *) and
are called the cohomology groups with supports in <f>. These invariants can be
calculated by means of resolutions with flabby sheaves ([26], Ch. II, § 4; [35],
Ch. Ill, § 3). We have Hg (X, §0 ^ r0(X, %=). In particular, if <D is the family of
all closed sets of X, we obtain the usual cohomology groups H' (X, *).
Next suppose X paracompact. If $ is the family of compact subsets, one
obtains the invariants H'C(X, *), the cohomology groups with compact supports.
For their calculation (and for the invariants H' (X, *)), one could make also

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
use of resolutions with soft sheaves ([26], Ch. II, § 4). We have natural
isomorphisms
lim HXU, Sr) =* HXX, SO,
where the inductive limit is taken with respect to all open subsets U of X (or over
a cofinal part of them).
If $ is the family of closed subsets of a compact K <= X, then the obtained
invariants are denoted by H'K(X, *) and are called the cohomology groups with
supports in K. For any open neighbourhood U of K there are canonical
isomorphisms
#^(£7, 8F)=> Hk(X,$=) (the excision property!).
We also have canonical isomorphisms
lim H&X, ST) =; H;(X, Sr),
K
where the inductive limit is taken on all the compact subsets K of X (or over a cofinal
part of them).
Let 0 —> 8F —> 3' be an injective resolution of SF. One thus obtains an exact
sequence
o -► rxcsr, a-) -► r(x, a-) -► rtsr \ a, a*) -> o
and passing to cohomology, an exact sequence
... -► H"(X, ®) -► ««(A" \ K, ®) -► tf&+1(*, ®9 -► • • • •
This sequence shows that the invariants H'K(X, S-) represent the obstructions for
the extension to whole X of the cohomology classes from X \ K. In the same
manner one obtains the exact sequence
... -► H"C(X \K,Sr)-+ H%X, Sr) -► H"{K, Sr) -► Hpx(X \K,Sr)^> ... .
In particular, if X is compact and K a closed part of it, it will result the
exact sequence
... -► m{X \K,8r)-> H"(X, §0 -► H"(K, &) -> tf?+1(* \ A, 8F) -►
Let now £/ be a relatively compact open subset of X. From the exact sequence
o -► rc(u, r) -> r(x, s-) -> t(x \ u, a-) -► o
we deduce the exact cohomological sequence
. .. -► m{U, §9 -► Hi{X, SF) -► H"(X \U,3r)-+ H?+\U, Sr)-+ ... ,

12 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
hence the invariants H'C{U, SF) are the very obstructions for the extension of the
cohomology classes on X \ U to X.
Consider now (X, (9) a ringed space. For any two (S-modules SF, § denote
Homo.e (8% <?) = r^ (X, Home (SF, <?)) and this correspondence defines an exact
left additive functor whose right derived functors are denoted by Ext'u,,^*, *),
and are called the Ext's with supports in O. In particular, Ext&.^ûF, §) ^
cz Homj,,^, <$). If SF is a free (9-module of finite rank, then from the natural
isomorphisms W ®e § -> Homa(W, §) we get isomorphisms
where &• = Home(3r, (9) is the dual of SF (the proof follows easily since for any
injective (9-module 3 and any (9-module 3f, Hom0QX, 3) is a flabby sheaf).
Recall now the definition of the functors Ext. Denote by ExtQ{*, *) the
derived functors of the functor (*, *) i-»- Hom&(*, *). One can easily check that
for any two (9-modules SF and <$, Exte{9, <J) is the sheaf associated with the pre-
sheaf U t-+Ext&u (V; 8F| U, <f | U) (for convenience, we sometimes write Extjg (£/; SF, <$)
instead of Ext3lU(V; Sf\ U, §\U)). We have Ext%{$, <f) =i Hom3{§, <f). One
can easily deduce that Ext3(&, <$) = 0 for q ^ 1 and SF a locally free (9-module
of finite rank: accordingly, the Exfs can be calculated by means of resolutions
with free sheaves of finite rank in the first argument (in case that such resolutions
do exist!). If SF is locally an (9-module of finite presentation (locally there is an
exact sequence of the form &" —>■ £" -> SF -» 0), then there are natural
isomorphisms
Home(&, C)x =* Homex(arx, qx)
for any (9-module (J and all xe X. Moreover, if the stalks of the structural sheaf
are noetherian rings, then the stalks of any injective (9-module are injective modules
over these rings. Under these supplementary conditions on (9 and SF, we obtain
natural isomorphisms
Ext3(®,q)x^Exf3x($x,qx)
for any point xeX and any (9-module <f.
We also recall the following property: if (9 is a coherent sheaf of rings and
SF, <f are coherent (9-modules, then the (9-modules Ext3(W, §) are coherent too.
This fact can be easily proved by induction on q: the case q = 0 represents a well-
known property of coherent sheaves and, for the general induction step, one can
use the local existence of some exact sequences of the form 0 -» SF' -» 6" -» SF -» 0.
For details with respect to the functors Ext, Ext, one can see ([26], Ch. II,
§ 7; [35], Ch. IV).

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
13
(b) Let #„ #2, ^3 be abelian categories, cê1 and #2 having enough injective
objects (for any object M there is a monomorphism M -» 7, /an injective object).
J 7" ...
Let Ç?!—^.^2—^^3 be left exact additive functors. Suppose in addition that S
transforms any injective object in a 7-acyclic object (an object M of Çf2 is called
T-acyclic if RqT(M) = 0 for any 9 ^ 1). Then there is a spectral sequence ol term
E?2'q=(RpT)(R9S) which converges to the derived functors of the composition
Rp+i(TS) ([35], 2.4.1). Recall the construction of lateral morphisms
R"(TS) = E" -> Ef n = T(R"S), (R"T)S = El ° -> E" = R"(TS).
Let M be an object of cë1 and 7" one of its Cartan-Eilenberg resolutions ([16],
XVII, 1.2). By the left exactness of 7 we get isomorphisms Kei(TS(r)->TS(I"+1))^>
^ 7(Ker(S(7") -> S(7"+1))). From the commutative diagram
TSQ"-1) -► TS(1")
T(B")
where B"= \m{S{In~1) -» S(I")), we derive a morphism (in fact a monomorphisme)
ImiTSil"-1) -► 75(7")) -► 7(Im(S(7"-1) -► 5(7"))). Then we get morphisms
R"(TS)(M) = H\TS(I')) -> T(H"(S(I'))) = T(R"S)(M).
Their independence on the choice of the resolution 7* and the functoriality in M
can be proved. One proceeds similarly for the other lateral morphisms.
We will use the following property : if for an object M of ^ RPTCR" S(M)) = 0
for all q and any p ^ 1, then the natural morphisms R"(TS)(M) ->■ T(R"S(M)) are
isomorphisms for any integer n ^ 0. This is an immediate consequence of the
general properties of spectral sequences [16], [26], [35]; however, one can
straightforwardly prove it as follows
s t
Lemma 1.1. Suppose f€^ —> ^2 —> ^3as above, and let M be an object of ^
If R"T(R'S(M)) = Oforp ^ 1 and q ^ 0, then the natural morphisms R\TS){M)^
—► T(R"S(M)) are isomorphisms for any integer n ^ 0.
Proof. Let 0 ->■ M ->■ 7° ->■ 71 ->■ ... be an injective resolution for M.R'S(M)
are the cohomology objects of the complex S(l') and R'(TS)(M) the ones of TSQ').
Denote Z" = Ker(S(7") -> S(/"+1)), Bn= Im^/""1) -> S{1"% 'Z" = Ker(7S(7") ->
-► 75(7"+1)) and 'Bn = lm{TS{In-^) -► 75(7")). Obviously, Z° = 5(A/). By the
left exactness of functor 7 we get isomorphisms 'Z" -> T(Z"). If the exact sequences
0 -► Z" -► 5(7") -► £"+1 -► 0 (n > 0),
O^fi'^Z"-» R"S(M) -► 0 (« =s 1),

14
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
and the ascending induction on n are used one can verify that
RPT(Z") = RpT(Bn+1) = 0(p =s 1) and 'B"+1 •=* T(B" + 1).
The proof will be concluded by means of the exact commutative diagram:
0 -► <Bn -► 'Z" -► R"(TS)(M) -► 0
4- 4- *
0 -► r(5") -► TXZ") -► T(R"S(M)) -► 0.
Remark. By the same argument one may establish the following result: if
rpT(R"S(M)) = 0 for/? ^ 1 and 0 < # < n0, then the natural morphisms R"(TS)(M)
->■ T(R"S(MJ) are isomorphisms for each integer «, 0 < n < «0 + 1.
We will also use the following result, which is sometimes called "the de Rham
abstract theorem".
T
Lemma 1.2. Let <£—>^" be a left exact additive functor between abelian
categories, # having enough infective objects. Let M be an object of <& and V a
resolution of M with T-acyclic objects. Then there are natural isomorphisms H"(T(l'))->
-> R"T(M), n>0.
Proof. We proceed by induction on n. The case n = 0 will follow by the left
exactness of T. Put M' = Ker (I1 -> P). From the exact sequence 0 ->■ M ->■ 7° ->■
->■ M' -> 0 we get the exact sequence 0 ->■ T(M) ->■ T(I°) -> T(M') ->■ R^TiM) -> 0
and isomorphisms R"T(M') ^ Rn+1T(M), n > 1. Anyhow, by the exact sequence
0 ->■ T(M') ->■ rC/1) -► T(72) the general induction step can be concluded.
We now consider two examples. Let (X, 6) be a ringed space and O a family
of supports on X. Fix an (9-module and consider ^ = ^ = the category of
à-modules, ^3 = Ab, S = HomQ(Sf, *) and T = r<j, (1", *). Functor 5 carries injective
objects in T-acyclic objects, since for any injective (9-module 9, the sheaf Homa(&, S)
is flabby. The following equality TS = Hom$, s(Sr, *) holds. Therefore, we get a
spectral sequence of term E%i—H§,(X, Exfy(&, *)) which converges to Extg+| (X; SF, *).
In particular, there is a spectral sequence of term Egq = H" (X, Ext% (&, *))
which converges to Extg+? (A"; S\ *). In this case, the lateral morphisms
Ext^(A"; Sr, *) ->■ T(X, Extg(9, *)) coincide with the morphisms given by passing
from the presheaf to the associated sheaf.
Let now (X, 6X)—>(y, QY) be a morphism of ringed spaces. Consider c£x~
= the category of (9x-modules, <të2 = the category of (9y-modules, #3 = Ab, S ==f^
(the direct image of sheaves) and T — T(Y, *). Functor S carries injective objects
to T-acyclic objects since any injective sheaf is flabby and/,,, preserves the flabby
sheaves. On the other hand, the equality TS~T(X, *) holds. Thus, there is a spectral
sequence of the term ££"? = H"(Y, R"f*(*)) which converges to Ep+q{X, *). In this
case one can show too that the lateral morphisms H"(X, *) -» T(Y, i?"/*(*)) enjoy
remarkable interpretations. If in addition X and Y are paracompact and / is a
proper map, then the equality of functors rc(Y,f#(*)) = TC(X, *) generates a spec-

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
15
tral sequence of term £?■« = HPCÇÏ\ R«j'*(*)) which converges to Hpc*i(X, *). The
spectral sequences associated to the morphism / are called the Leray's spectral
sequences.
In both previous examples, one can obtain remarkable consequences from
lemma 1.1.
(c) In this section we recall some facts of topological vector spaces
([34], [68]), the basic field being the complex field C-
A locally convex topological vector space, in addition metrisable and complete
is called a Fréchet space. Such a space will be called of type Schwartz if for any
neighbourhood of zero U there is a neighbourhood of zero V such that: for any
n
8 > 0 there are xit . • ■, x„ e V with the property V cr U (x; ~\- sU). Denote by FS
(respectively DFS) the spaces Fréchet-Schwartz (respectively the strong duals
of such spaces). These spaces are reflexive and have remarkable properties of
stability, any closed subspace of an FS space is FS and the quotient of an FS space by
a closed subspace is also FS and similar properties hold for DFS. We will often
make use of the Banach theorem: a surjective continuous (£,-linear map between
two FS spaces (or DFS) is open.
Recall that a linear continuous map u: E -> F between two topological vector
spaces is called strict (or topological homomorphism) if the quotient topology on
u(E) coincides with that induced from F. The following result which is called "Serre's
duality lemma" [75] will often be used in this chapter.
U V
Lemma 1.3. Let E—>F—>G be linear continuous maps between locally con-
iS v'
ex spaces, such that v is strict and vu = 0. Denote by E' <— F' <— G' their
transposition by topological duals. Then there is a natural algebraic isomorphism
Ker w'/Im v' =► (Ker v/lm «)'.
Proof. Let L e Ker u' c F'. If eeE, then L(u(e)) = u'{L){e) = 0, hence
L|Im u = 0. Denote by L the functional determined byL on Ker v/lm u. If Lelm v',
then L|Ker v=0, hence L = 0. In this way we get a C-linear map Ker u'/lmv' —>■
-» (Ker v/lm «)' and we will prove its bijectivity. Let L e Ker «' be such that L = 0.
Then L factorizes by a linear continuous functional L':F/Keiv-+ C- Since v is
strict, the topology of the quotient F/Ker v^ Imv coincides with the topology
induced from G. By Hahn-Banach theorem L' extends to a linear continuous
functional L": G ->■ C- Since v'{L") =L, then L e Im v'. It remains only to prove the
surjectivity of the previous map. Suppose T e (Ker v/lm «)'. By composition Ker v-+
T
-> Ker r/Im u —> C we get a linear continuous functional on Ker v and let L : i7-»- C
be an extension of it. We have (w'(L))(<?) = L(u(e)) = T(u(e) mod Imu) = 0, eeE
an arbitrary element, hence L e Ker «'. It is easy to check that L = T.
Corollary Ï.4. If ... -> £/_1 ->■ E! -> £; + 1 -> ... is an exact sequence of F S
spaces (or DFS) where the maps are linear and continuous, then the transposed
sequence ... <- (E1'-1)' <- (£'')' <- (£;+1)' <- • • • is also exact.

16 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. Since Im (E^1 -► E!) = Ker (E! -> E!+1) is a closed subspace of E\
the differentials of the given complex are strict. The conclusion follows by lemma 1.3.
We will make also use in this chapter of the following facts :
— Any Banach space of at most countable dimension is finite-dimensional
(consequence of Baire theorem).
— Any locally convex topological vector space, countable inductive limit of
Fréchet spaces such that the maps of the inductive system are compact and injective,
is separated, hence DFS taking into account the assertion which follows (recall
that a linear continuous map w.E^F between locally convex topological vector
spaces is called compact if there is a neighbourhood U of zero in E so that u(U)
is relatively compact in F).
— Any separated locally convex topological vector space, countable inductive
limit of Fréchet spaces such that the maps of the inductive system are compact,
is DFS.
— If E = lim En is an inductive limit (in the category of locally convex
topological vector spaces) of FS spaces such that the maps En -» En+1 are compact and
in addition E is separated, then any bounded subset of E is the image of a bounded
subset of some E„ (theorem of Raikov-Silva).
To conclude with, recall some things on complex manifolds. Let A" be a
complex manifold of dimension n. Denote by § the sheaf of germs of C°°-differentiable
functions on X. The space &(X) = T(X, §), endowed with the topology of the
uniform convergence of functions and all their derivatives, is an FS space. Denote by
§"' '(respectively Si"'q) the sheaf of germs of differential forms of the type (p,q)
with coefficients in B (distributions, respectively). The convergence of the coefficients
of the forms defines FS topologies on the spaces &"'1(X) = r(X,ê"'i). The space
rc(A\ $(,"•1) coincides with the topological dual of §"-?• "-i(X), hence, endowed with the
strong dual topology, it becomes a DFS space. The differential §"• i(X) —> §>"' i+ 1 (X)
is continuous with respect to these topologies and its transposition (modulo the
sign) is the differential TC(X, Si"-"'"-i-1) -^ TC(X, 31"-"- "-<?).
Denote by 6 the sheaf of germs of holomorphic functions on X. The space
Q(X) = T(X, 0), endowed with the topology of the uniform convergence on
compacts, is an FS space, more precisely a closed subspace of &(X). If U is a relatively
compact open subset of X, then the restriction map &{X) -> Q{U) is compact. Let
K be a compact of X and (C/„)„>0 a fundamental system of neighbourhoods of K
such that U„+1 <=<=£/„ for any n > 0. Suppose in addition that any connected
component of each U„ does intersect K. The restriction maps 6(U~) -> &(Un+1)
are compact and injective. It will result that the spice O(K) = lim <9(t/„), endowed
'vith the topology of inductive limit in the category of locally convex topological
vector spaces, is DFS; its topology is independent on the above considered
fundamental system. Moreover, it is a space of LF type (a separate locally convex
topological vector space is called LF if it is a countable inductive limit of Fréchet
spaces).
Any bounded subset of @(K) is the image of a bounded subset of some G(Un).
Similar considerations can be also made for the space Q.(K) = T(K, Q), where Q

1. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
17
is the sheaf of germs of differential forms of the type («, 0) on X, with analytic
coefficients.
(d) In this section we recall the most basic definitions and properties of the
depth and of the dimension of a module ([10], [37] Ch. 0m, [79]).
Let A be a ring (commutative and unitary). By Spec A one denotes the set
of the prime ideals of A. If a is an ideal of A, then put V(a) = {p|p e Spec A, p => a}.
The sets by the form V(a) are the closed sets of a topology on Spec A, called the Zariski
topology. A basis of open sets is given by the subsets D(f) = {pe Spec A,f £ p},
f or / e A.
Let M be an A-module. The set Supp M = {p|p e Spec A, M^ # 0} is called
the support of M.
Subsequently, we will assume that A is a noetherian ring and M an ^-module
of finite type. We have Supp M=F(Ann M) where Ann M is the annihilator of M.
An ideal p e Spec A is said to be associated to M if there is a monomorphism of
^-modules A/p -> M. The family of the prime ideals associated to A is finite and is
denoted by Ass M. Obviously, Ass M <= Supp M. The set of zero divisors for M is
just (J P- The minimal elements of Ass M, Ass (^/Ann M) and Supp M are the
VeAssAf
same and one has
dim M = dim(^/Ann M) = sup dim(^/p).
pgAssM
We also would like to mention the following remarkable fact: there is a
composition series 0 = M0 cM^.., cM, = M such that each factor M; bl/Mi
is isomorphic to A/p for some p e Supp M. For all the above considerations one
could find details in ([79], Ch. III).
A sequence of elements xlt ..., xr of A is called a regular M-sequenee if any
i- 1
X; is nonzerodivisor in M I V xsM (the hypothesis with respect to xx says that
/ 7=1
it is nonetheless a zerodivizor in M).
Suppose now A is a local ring and let in be its maximal ideal and k = Aim
the residual field. Here is one further required result: "For any integer r > 0,
Ext^(A:, M) = 0 for i < r if and only if there is a regular A/-sequence formed by r
elements of in". This can be seen in ([79], IV, prop. 6; [37], Ch. 0IV, 16.4.4) or in the
second chapter of this book, where a more general case will be considered. Denote
prof M = sup {r\ there is a regular A/-sequence formed by r elements
of m} = inf {/ | Extfi(fc, M) # 0},
that is called the depth (or profondeur) of M. If M = 0, it results that prof M = oo.
Lemma 1.5. If x em is a M-regular element, then
dim M/xM = dim M — 1.
Proof. The inequality dim M/xM ^ dim M — 1 is always true in virtue of
the definition of the dimension via systems of parameters. If x is A/-regular, then it
is not contained in any ideal p e Ass M. The inequality dim M/xM < dim M — 1
2-c. 2398

1 8 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
easily results making use of the equality dim N = sup dim A/p, for N = M
peAssTV
and for N = MjxM.
Proposition 1.6. Let Abe a noetherian local ring, m its maximal ideal, and
M an A-module of finite type. Then:
(i) prof M = 0 if and only if in e Ass M.
(ii) If xe M is M-regular, then prof MjxM = prof M — \.
(iii) // M # 0, then prof M < dim M\ accordingly, prof M < oo for M # 0.
(iv) All maximal regular M-sequences of elements of m have the same length,
namely prof M.
Proof, (i) prof M = 0 if and only if any element of m is zerodivisor for M,
hence if and only if in <= (J p and the conclusion results by use of ([10], Ch.IV,
PëAssM
§ 1, n= 1, cor. 2, prop. 2).
(ii) If (xlt • ■ •, xr) is a regular A//xA/-sequence formed by elements of in,
then (x, xlt ..., xr) is a regular M-sequence. Thus, prof MjxM < prof M — 1. Let i
be an integer such that Ext^(A:, MjxM) ^ 0. We will prove that i + 1 ^ prof A/
and the proof is completed. Suppose the contrary, prof M > /+ 1. Then
Ext^(A:, M) = Ext^+1(A:, M) = 0. From the exact sequence
0 -► M ^ M -► A//xA/ -► 0
we obtain the exact sequence
... -► Ext^Ofc, M) -► Extj^fc, M/xtW) -► Ext^Â:, A/) -» ...
and hence a contradiction.
(iii) If prof M = 0, then the inequality is clear. Assume prof M > 0 and let
x e in be an A/-regular element. By lemma 1.5, dim MjxM = dim M — 1 and the
assertion will be concluded in accordance with (ii) by induction.
(iv) Again by (ii).
p
Proposition 1.7. Let A -> B be a finite local morphism of noetherian local rings.
If M is a B-module of finite type, then
prof^ M[Pj = profB M.
(A/[Pj is M, regarded as an ^4-module by means of p).
Proof. For M = 0 the conclusion is obvious, so we can suppose M # 0.
Let n = prof,, A/[p) and (Xj)i«.'<« be a maximal regular A/[p)-sequence, all x; e mA,
mA being the maximal ideal of A. The elements p(x;) belong to the maximal ideal
inB of B and constitute a regular A/-sequence. Denote N — M S] p(x;) A/. Clearly,
/ i = l

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
19
profB./V = profB M — n, Nlp] = M]p] / J] x{Mlp] and piofANlPi = 0. We thus reduce
the problem to the case n = 0. Then mA e Ass,, A/[p), hence there is an injective
morphism A/mA -> A/[p). Let M' be the submodule of M generated by the image of
this morphism. Since the morphism p is finite, some power of the ideal mB is embedded
in mAB and therefore annihilates M'. Then mB e Ass M' <= Ass M, hence
profB M = 0.
Proposition 1.8. Let Abe a noetherian local ring which is supposed to be normal
of dimension ^ 2. Then prof A ^ 2.
Proof. Let in be the maximal ideal of A and x an element of m, x # 0. Since
A is an integral domain, x is a nonzerodivizor. For any p e Ass(A/xA), htp = 1
([79], III, prop. 9). The hypothesis dim A $s 2 implies that there is ;; e trt which does
not belong to any ideal of Ass(^/x^4). Hence y is a nonzerodivisor in AjxA and the
conclusion follows.
We recall that a module M (of finite type over a noetherian local ring) is called
a Cohen-Macauley module if prof M — dim M. A noetherian local ring A is called
a Cohen-Macauley ring if it is a Cohen-Macauley ^4-module, that is prof A = dim A.
Corollary 1.9. Any normal noetherian local ring, of dimension 2, is a
Cohen-Macauley ring.
We now turn our attention to the characterization of the depth of the modules
over regular rings. Recall some considerations of homological algebra ([16]; [79],
IV C; [37], Ch. 0nI, 17.2).
Let A be a commutative unitary ring and M an ^-module. The smallest n
(integer or +oo) such that there exists a left resolution of M which is projective and
of length n, is called the projective (or homological) dimension of M and is denoted
by dimproj M (or dhM). Equivalently, dhM is the smallest n such that for any
^-module N, ExtA(M, N) = 0 for i > n, or only for i = n + 1. The smallest n
(integer or + oo) such that dhM < n for any ^4-module M, is called the global
cohomological dimension of the ring A and is denoted by dimcoh A. This n results to
be the smallest number such that for any two ^-modules M, N, Ext^ (M, N) = 0
for i > n or i = n + 1 only.
Suppose now A a noetherian ring and M an ^4-module of finite type. Under
these assumptions, in the definition of dhM we can consider projective resolutions
with modules of finite type only. Consequently, dhM < n if and only if Ext^(M, N) =
= 0 for i $: n + 1 (or i = n + 1 only), N being any ^4-module of finite type (by
using induction on the number of generators one can consider here monogene
modules N only). Since any such module N admits a composition series 0 = N0 <=
<= ... c Nk = N such that the succesive quotients Ni/Ni_1 are isomorphic to
modules of the form A/p, p e Spec A, we derive: dhM ^ n if and only if
Ext^(M, A/p) —,0 for any i > n + 1 (or / = n + 1 only) and all p e Spec A.
We now/restrict ourselves to a more particular case, namely when A is a
noetherian local/ring of maximal ideal in. An ^4-module of finite type M is then free if
and only ^f it is projective or equivalently, Tor^(A/, A/m) = 0 ([10], Ch. II, §3,
n = 2, cor. 2, prop. 5; [79], IV, prop. 20). Therefore, one gets dh^ M^n if and
only if Toii+1(M, A/m) = 0 (M being an ^-module of finite type) and dimcoh A < n

20
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
if and only if Tor^+1 (A/m, A/m) = 0 (for the proof we need only to interpret the
definitions of dh and dimcoh by means of projective resolutions; for details, we send
for instance to [79], Ch. IV).
Recall that a noetherian local ring A of maximal ideal m and residual field
k = A/m is called regular if dim^ = dim^ m/tn2 (the dimension theory gives rise
to dim A < dim*. rrt/rrt2). Henceforth we will make essential use of the following result
("Hilbert-Serre theorem" [79], IV, th. 9 or [37], Ch. 0m, 17.3.1): "Let A be a
noetherian local ring. A is regular if and only if its cohomological dimension is finite
and in such case dimcoh A = dim A".
We would point out some consequences of these facts concerning the notion
of depth.
Lemma 1.10. Let A be a noetherian local ring, m its maximal ideal, let M
be an A-module oj finite type, and x an element of m. If x is M-regular, then
dh A//xA/ =dh M + 1.
Proof. From the exact sequence 0 -» M -» M -> M/xM -> 0 induced by the
homothety given by x, we get the exact sequence
Toif(M, A/m) -► Toif(M, A/m) -► Toif(M/xM, A/m)
-+ TorJ^A/, A/m) -► Toif_i(M, A/m),
where the first and the last arrow are both the homotheties given by x. Since
x e m, these two morphisms are null; hence we obtain the exact sequence
0 -► TorftA/, A/m) -► Toif (M/xM, A/m) -► TorfL^A/, A/m) -► 0
and the conclusion easily follows.
Proposition 1.11. Let A be a regular noetherian local ring of dimension n.
Then for any A-module M # 0 of finite type,
prof M + dhA/ = n.
Proof. We proceed by induction on r = prof M. If r = 0, then there exists
an ,4-submodule N of M isomorphic to A/m (1.6, i). The exact sequence 0 ->■ N ->■ M
-> M/N -> 0 yields the exact sequence
Tori+1(M/N, A/m) - Tor^(7V, A/m) -> Tor^(A/, A/m).
The hypothesis on A shows that Tor*+1(M/N, A/m) = 0 and Tor^(A^, A/m) a
c^ Tor^(/4/irt, A/m) # 0. Therefore, Tor^(A/, A/m) # 0, hence dhA/ ^ «; but on
the other hand dhA/ < dimcoh A = n, hence dhA/ = n.

1. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
21
Assume now r > 0 and let x be an A/-regular element, which belongs to m.
As prof MjxM = r — \ in accordance with (1.6, ii) and dh MjxM = dh M + 1
(cf. 1.10), the general induction step is now clear.
Remark. For modules over regular rings, the depth is also called cohomological
codimension and is denoted by codh. The above proposition asserts that dh + codh =
p
= dim A and thus it justifies that terminology. If B -» A is a surjective morphism
of noetherian local rings, B regular of dimension n, then for any ^4-module M of
finite type, M # 0, prof^A/ = n — dhBM\Pi. This fact will be often used.
Corollary 1.12. Any regular noetherian local ring is a Cohen-Macauley ring.
Corollary 1.13. Suppose A is a regular noetherian local ring and 0 -> N -»
-* L -+ M -> 0 an exact sequence of A-modules of finite type where L is free. If M
is not free, then
prof M = prof N — 1.
Proof. We first remark that M, N are different from zero. The hypothesis
on M then implies dhM = dhN + 1 and the proposition can be applied.
We must give, for future use, the characterization of the depth in terms of
Ext^*, A).
Lemma 1.14. Let A be a regular noetherian local ring, M an A-module of
finite type and q > 0 an integer. Then dhM ^ q if and only if Ext^(M, A) = 0 for
any i > q.
Proof. We proceed by descending induction on q. Since dhM < dimcoh A =
= dim A, the assertion of the lemma is obvious for q > dim A. For the step
q + 1 \-+q, assume Ext^,(M, A) — 0 when i > q and prove that dhM < q (the other
implication is clear). It follows dhM < q + 1 If a module N would satisfy
ExtqA+1(M, N)^0, then take an exact sequence 0-+P-+L-+N-+0(L free) and got
a contradiction.
Corollary 1.15. Let A be a noetherian local ring, which is regular of dimension
n. Then for any A-module M, M # 0 and for any integer q, we have prof M > q
if and only if Ext^M, A) = Ofor i ^ n — q.
We are also going to give a characterization of the dimension in terms of
ExVA(*,A).
Lemma 1.16. Suppose A is a noetherian local ring, in its maximal ideal, M
and N are two A-modules of finite type such that the support of N contains only in,
and q is an integer. Then prof M ^ q + 1 if and only if ExtA(N, M) = 0 for any
i < q.
Proof. Choose a composition series 0 = A^0 cr N1 <= .. . <= Nr = N, with
the factors of the form A/p, p e Supp N. Since Supp N = {tn}, all factors are in fact
isomorphic to A/m. If prof M $s q + 1, then Ext^/m, M) = 0 for i < q and the
direct implication follows by applying the exact sequences of the Ext's associated to
the exact sequences 0 -> Nj-> NJ+1 -> NJ+1/Nj ->■ 0 (0 < j < r — 1).
Now we will prove the converse by induction. Look first at the case q = 0:
if HomA(N, M) = 0 we derive prof M ^ 0; for otherwise there exists a monomorphism
0 -» ^/m -> M, hence a monomorphism 0 -» HomA(N, A/m) -» HomA(N, M),

22 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
hence HomA (N, A/m) = 0; this fact is a contradiction because there is a surjection
N = Nr -* NJNr+1 ^> A/m. We now assume q > 1 and the assertion proved for
q — 1. Then prof M > q > 1 and fix x e in an A/-regular element. From the exact
sequence 0 -» A/ —> A/ -» MjxM -> 0 we derive Ext^TV, M/xM) = 0 for z < g — 1,
hence prof(A//xA/) > </. So, prof A/ = prof(A//xA/) +1 ^ q + 1.
Let us fix a regular ring A and a prime ideal p of ^. By the Hilbert-Serre
theorem it follows that the ring Apis regular too (as dimcoh Ap < dirndl). In the
proposition below we will use the relation
dim A = dim Ap + dim A/p.
This formula is proved in ([79], IV, th. 6), in ([37], Ch. 0m, 17.1.3 and 16.5.10)
or in the case when A is a convergent power series ring (anyhow the only case when
we apply 1.17), in Chapter II, cor. 1.28.
Proposition 1.17. Let A be a noetherian local ring, regular of dimension n.
Then for any A-module of finite type M and for any integer q, dim M < q if and only
if Ext^Af, A) =0 for i < n -'q.
Proof. Suppose Ext^(M, A) = 0 for i < n — q. Let p be an ideal, minimal
within Supp M. We have
ExtAp(Mp, Ap) o^ Ext'A(M, A)p.
The ring Ap is regular, hence a Cohen-Macauley ring and prof Ap = dim^0 = dim A
— dim A/p. One can see easily that Supp A/p (in Spec Ap) contains onlyp^p;
by lemma 1.16 we conclude that prof Ap ^ n — q, hence dim A/p < q. Accordingly,
dim M < q.
We are now going to prove the converse and proceed by induction on q.
The case q = 0 follows by lemma 1.16, as Supp M contains at most the maximal
ideal of A and prof A = dim A = n. Let now q > 1 and the assertion is already
verified for q — \. Consider a composition series 0 = M0 c Mx <= ... c Mr = M
such that the factors M}\MJ_1 are of the form A/p, p e Spec A. It is enough to prove
that ExtiA(Mj/MJ_1, A) = 0, for i < n — q and 0 <./ < r; so we can assume M
of the form A/p (note that dim A/7-/A/7-_! < dim M ^ q). If p = in, then the
conclusion occurs by the very definition of the depth. Otherwise, there exists an element
x of in, which is A/-regular. By induction hypothesis, the first morphism of the exact
sequence Ext^M, A) -» ExtA(M, A) -» Ext'A+1(M/xM, A) is surjective for i < n — q.
The proof will be achieved by applying Nakayama lemma.
To conclude this section, we should mention a remarkable class of rings
obtained from the regular rings. A ring is called complete intersection if it is
isomorphic to a ring of the form A I J] x,A, where A is a regular noetherian local ring
and (Xj, ..., xr) is an ^-regular sequence.
Proposition 1.18. Any complete intersection ring is a Cohen-Macauley ring.

I. COHCMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
23
Proof. Suppose .8 == A I J] x,A as above. According to 1.5 and 1.6, dim B
= dim A — r and prof B = prof A — r. The conclusion follows from 1.12.
(e) Let X, Y be two separated topological spaces. We recall that a continuous
map/: X -> y is said to bt finite provided that it is a closed map (T closed in X =>
=>/(T)closed in y) and its fibers/_1(y), j> e Y, are finite sets.
Lemma 1.19. Letf:X-+ Y be a finite map and y a point of Y. If'°f is a
fundamental system of neighbourhoods of y then /_1(°?) is a fundamental system of
neighbourhoods of the fiber f~1(y).
The proof is a simple exercise of topology.
Corollary 1.20. Let /: X->Y be a finite map between two separated topological
spaces.
(i) If SF e Ab(A"), then the canonical morphism
urn,-* n sfx
m-y
is bijective for any y e Y.
(ii) The functor /*: Ab(A") -► Ab(Y) is exact.
(iii) // tp is a morphism in Ab(A") such thatf*(y) is an isomorphism then <p itself
is an isomorphism.
Proof. The assertion (i) immediately follows by the preceeding lemma and
by the definition of /*. The assertions (ii) and (iii) are obvious consequences of (i).
Lemma 1.21. Under the same assumptions as in 1.20, the canonical morphisms
H\Y,U9)) - H\X, S=), HXY,USSr)) - H'C(X, Sf)
are bijective.
Proof. If we choose a flabby resolution 3' for 3% then /*(9') is a flabby
resolution for f^{S-). The conclusion follows by using the equalities :
T(Y, Uh')) = T{X, S*), TC(Y, /*(&•)) = TAX, '»).
§ 2. Duality on Stein manifolds
The first result of the paragraph is the following:
Theorem 2.1. Let (X, 6) be a paracompact Stein manifold of dimension n and
Q. the sheaf of germs of differential forms of the type (n, 0) on X with analytic
coefficients. Then, for any coherent analytic sheaf & on X and for any integer
q > 0, E\tne~q{X; SF,Q) has a natural structure of Fréchet-Schwartz space with the
topological dual algebraically isomorphic to H%(X, &).

24 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
To prove it we need some preparations. Let (X, 6) be a complex space. We
shall denote by Coh(A") the category of the coherent analytic sheaves on X. If
W e Coh(A"), then T(À',Sr) has naturally a structure of topological vector space,
which is even FS if X has a countable topology (a countable basis of open subsets);
see [38], Ch. 8 and Chapter VII. Recall the definition of this structure. We
confine ourselves to the case when X has a countable topology and consider a countable
Stein open covering \l of X, which is "sufficiently small". For any U e If, there is
a closed immersion U -» V, V being a Stein open subset of some numerical space.
We also assume the sheaf f#(SF|£/) is the quotient of some sheaf <2£ by a coherent
subsheaf § c Opv. In virtue of theorem B, T(U, §9 = T{V, /*(§0) ^ T(V, OC)j
jT(V, q). If we consider on T(V, QÇ) the topology of uniform convergence on compact
subsets, then one obtains an FS structure. By a theorem of Cartan, T(V, §) is a
closed subspace of T(V, QÇ). In this way, we can endow T(U, &) with an FS structure.
r(A", SF) is a closed subspace of H F(U, &) and endowed with the induced TVS
UeV.
structure, T(X, &) has even an FS structure. This is independent on all choice. Thus
§• becomes a Fréchet-Schwartz sheaf, that is a sheaf such that for any open subset U,
T(U, &) has an FS structure and for any two open subsets V <= U, the restriction
maps T(U, SF) -► T(V, S-) are continuous.
If S- -► q is a morphism of coherent analytic sheaves on X, then the map
r(A\ cF) -» r(A", q) is continuous in the above constructed topologies.
Lemma 2.2. Let X be a Stein space and IF -> ty a morphism of coherent analytic
sheaves on X. Then the map T(X, &) -> F(X, ($) is strict.
Proof. Denote % = Coker (^ -► ($). From the exact sequence 8F -► q ->. 2C,
by applying theorem B, we get an exact sequence of linear continuous maps of FS
spaces r(X,®)-+r(X,q)-+r(X,%). Then the image of r(X,S=) in r(A\ 3K) is
a closed subspace and in order to complete the proof we must make use of the Banach
theorem.
Here is the convention we intend to use further. If A" is a complex space, K <= X
' i
a compact, & e Coh(A"), U a Stein neighbourhood of K, U -> V a closed immersion,
where V is a Stein open subset of a numerical space, L a compact in V such that
i~\L) = K and £pv -> z'*(S0 is a sheaf epimorphism, then by taking the supremum
on L, a seminorm on F(V, 6f) is so defined. By the surjection T(V, Qpv) ->■
->■ r(F,z+(Sr))= r([7, &) a seminorm on r(C7, S=) is induced; the seminorm on F(X, &)
obtained by means of the restriction T{X, B?) ->■ F(U, &) will be called "a seminorm
of type 'sup' on r(A\ &)". Such a seminorm has the following property: any Cauchy
K
sequence with respect to "sup" remains Cauchy by restriction to T(D, SF), whenever
k
D is an open subset such that D <= K and /(£>) cr L.
The family "sup", K compact in X, is a family of seminorms defining the
topology of T (A", SF).
Lemma 2.3. Let X be a Stein space and (C/r)r^i an exhaustion by relatively
compact Stein open subsets for X. Suppose in addition that the maps T{X, 6) -*■

1 COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
25
-» T (Ur, 6) are dense for any r^l. Then for any S^ eCoh(X), the canonical
morphism
lim {T{Vr,9)y ^T{X,&)'
r
is bijective (the accent means as usual, the topological dual).
Proof. We first prove that the restriction maps r(A\ §F) -» T{Ur, &) are dense.
Let r > 1 be an integer. By theorem A there is a sheaf morphism Gpx -> SF, which is
surjective on Ur. The conclusion follows from the commutative diagram
r(*,<s£)-r(*,so
l l
r(c/r,ej)-r(c/r,ao,
where the map r(£/r, £§.) -»r(£/r, SF) is surjective by theorem B. Then it results
that the maps T(Ur, SF)' -> T{X,&)' are injective, hence the asserted morphism is
injective.
Next we will prove its surjectivity. Let L: T(X, SF)-» C be a linear continuous
functional. There is a constant a > 0, a compact K <= X and a seminorm ^K of
the type "sup" on T(X, SF), such that |Z-(j)| < &Pk(s) f°r anY ■* e F(^> ^0- F°r '*
K
sufficiently large, one easily derives that L factorizes by any T{U„ SF) (again
according to the fact that the maps T(X, 8r) ->■ T(Ur, SF) have dense images!).
Lemma 2.4. Let X be a Stein space and S\ <$ e Coh (X). Then the canonical
morphism
Ext§cr;ff,<?)-rcr,£xfS (§%<?))
« bijective, for any q > 0.
Proof. There exists a spectral sequence which converges to Ext^A"; SF, <|)
for which E% i = Hp (X, Ext% (S\ §)), the morphisms from the statement being just
the lateral morphisms E> -> E\,q. Since the (9-modules Exîq(&, §) are coherent, we
obtain by theorem B, EP' i = 0 for any ^ ^ 1. The conclusion follows from the
properties of the spectral sequence.
We remark that one could avoid the considerations of spectral sequence by
applying directly lemma 1.1 (according to section (b), § 1).
This lemma allows us to consider FS topologies on ExtJ (X; S\ §), q $s 0
arbitrary integer. In particular, one thus obtains the topologies required by the
theorem.
Lemma 2.5. Let X be a Stein manifold of dimension n and SreCoh(A'). Then
Hic(X, 8r) = 0forq > n.
Proof. Let (Ur)r^!l be an exhaustion of X by relatively compact Stein open
subsets. We have h{(X, SF) ^ lim H«{Ut, &). It suffices to show that H%U, &) = 0
r
for q > n and U <= c X Stein open subset. The sheaf Sr admits on U a finite
resolution by locally free sheaves of finite rank. By induction on the length of such a
resolution, we are able to restrict the problem to the case Sr locally free.

26 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Let
o -»• e -► g -^ §»■ ! -%■... ->• §c' " -► o
be the Dolbeault resolution by forms C10. By tensoring ®e § we get a resolution
of Sr
0-> Sr ->Sr (g^g-*^ ®c-,g0>1-> ... ->• S? ®aë°-" ->-0.
The sheaves SF®a g0-* are soft (as modules over the sheaf of germs of Cx functions
on X). We thus conclude the lemma since H'C(X, Sf) are the cohomology groups of
the complex rc(X,& ®aS°- *).
Lemma 2.6. Let (X, 6) be a Stein manifold of dimension n and let £ be a free
sheaf of finite rank. Then Hf.{X, £) = 0 for q # n and H"(X, £) is isomorphic to the
topological dual of Homa (£, Q) where Homô (£, Q) has the F S topology defined
on the module of global sections of a coherent analytic sheaf, Homa(£, Q) =
= T{X, Hom@{£, Ù)). Moreover, the duality isomorphisms agree with the G-morphisms
between free sheaves of finite rank.
Proof. Let
„ if , if if
o -► q -►g"-° —»ê"-1 —>... —»&"■" ->•o
and
0 -»• G -> SC°-° -^ Si0'1 —> ... -%■ Si^" -»• 0
the Dolbeault-Grothendieck resolutions of Q (6, respectively), by differential forms
with coefficients C°° (distributions, respectively). Since £ is of the form 6", we then
obtain the resolutions
0 -► Home (.?, Q) ->
-► Hom@ (l, g"-°) -^ Home (£, g"-1)-^ . . . -^ Hoire (£, $"■") -► 0,
0 -► £ -> £ ®a SC°>° -%■ £ ®a St0-1 -%■... -^-> £ ®a 5t°'" -► 0.
The sheaves Home (£, §"•*) and £ ® a 3(0-* are soft (as modules over the sheaf of germs
of C°° functions on X). It will then result that the invariants H'(X, Home(£, Q))
are the cohomology groups of the complex r(X, Homô(S, g"-*)) = Homa(£, §"■*)
and the invariants H'C(X, £) are the cohomology groups of the complex
FC(X, £ ®aaro-*). The complex F(X, ê"-*) is FS and its topological dual is isomorphic
to Tc (X, SC0-*) (the isomorphism changes the degree q in n—q). Since £ is of the
form 6", one can easily deduce that Homa(2, §"■*) is an FS complex and its
topological dual is isomorphic to TC(X, £ ®a Si0-*).
The FS topology obtained on Homa (£, Q), as the kernel of the map
Homa(£, $n>°)—>Homa(£, ê"-1), coincides with that defined at the beginning of the

1. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
27
paragraph (it is obviously finer and both of them are Fréchet topologies!). The
conclusion of the lemma easily follows by theorem B and by 1.3.
Proof of the theorem 2.1. We consider on the invariants Exte (X; Sf, fi) the
FS topology given by lemma 2.4 and prove that it verifies the theorem. We first
make the supplementary hypothesis that SF admits a resolution (not necessarily
finite !)
... _> £2 _> £i _> £0 _> g; _> o
by iree sheaves of finite rank. Consider the complex
(*) ...-►#? (X, £a) -► Hc" (X, £!) -» Hnc (X, £°) -► 0.
Since H$(X, £') = 0 for any i and q # n (by 2.6), H$(X, SF) is just the homology
group in dimension n—q of this complex: one can easily verify this assertion by
descending induction on q, by using the sheaves SF' = Coker (£'+1 -> £'), / ^ 0.
Consider now the complex
(**) ... <- Homa (£2, Q) <- Homa (£\ Q) <- Home (£°, Q) <- 0.
Its cohomology groups are equal to Ext^A"; SF, Q). By lemma 2.6, (**) is an FS
complex and its topological dual is algebraically isomorphic to (*). By lemma 2.2
the maps of (**) are strict. Then, by 1.3 we derive that H%(X, SF) is algebraically
isomorphic to the topological dual of Exta"'? (X; SF, Q) taking on the latter the FS
topology deduced from (##).
The homology sheaves of the complex
... <- Home (£2, Q) <- Home (£\ Q) <- Home (£°, Q) <- 0
coincide with Ext® (&, Q.). By applying lemma 2.2 it will easily result that the topology
of Extjè,-' (X; SF, Q) deduced from (**) is identical with that deduced from
ExtQ~q(^', Q) by means of lemma 2.4.
In this way, the theorem is proved in the supplementary hypothesis on SF.
We now pass to the general case. Let (C/r)rSïl be an exhaustion of X by relatively
compact Stein open subsets such that the restriction maps T{X, 6) -* T(Ur, 0)
are dense, r ^ 1. We have natural isomorphisms
m(X, SF) ~ lim H?(Ur, SO,
r
q ^ 0 being an arbitrary integer. According to lemmas 2.3, 2.4, we have also natural
isomorphisms
(Extg-9 (X; SF, a))' =i lim (Ext"@-i ([/,; SF, Q))'.
r
Since every Ur is a relatively compact Stein open subset, SF|[/r admits a resolution
by free sheaves of finite rank.

28 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
To conclude with, it is enough to prove that the isomorphisms already built
hs(v„ ar) « (Extent/,; s, o))'
agree with the morphisms given by the inclusions U, <= Ur+1. We must only show
that the isomorphisms
constructed in the supplementary hypothesis on SF, do not depend on the resolution
used. Let £* -> SF and £" -> & be two such resolutions. Since X is Stein, one easily
derives that £* and £" are homothopic (over SF). The conclusion then follows by
lemmas 1.3, 2.6. The theorem is completely proved.
Corollary 2.7. Let (X, Ô) be a Stein manifold of dimension n, and & a locally
free sheaf of finite rank on X. Then E? (X, S;) = 0forq^n and H?{X, &) is alge-
V
braically isomorphic to the topological dual of Homa (SF, Q) ^ T(X, & ®a Q).
Corollary 2.8. Let (X, 6) be a Stein space and U <= X a Stein open such that
the restriction map T{X, &) -* T{U, &) is dense. Then, for any & e Coh(A") and for
any integer q ^ 0, the natural maps
HHU, SF) - m{X, 9)
are injective.
Proof. Let (£/r)r>i be an exhaustion for U by relatively compact Stein open
subsets, such that the maps r(U, 6) -»• F(U„ 6),r^\, are dense. Since H^(U, ®)<^
~ lim H${Ur, S\), we have only to prove that every map H$(Ur, &) -»• H§(X, §0 is
r
injective. In other words, we can assume from the very beginning that U is relatively
compact.
Suppose now (£/r)r>i is an exhaustion of A" by relatively compact Stein open
subsets such that the maps r(X, &)-> T(Ur, 0), r $s 1, are dense and U-i = U.
Suffice it to prove that every map //?(£/, SQ -► H%(Ur, &) is injective. In this way,
we can assume in the statement of the corollary that there is a closed immersion
X -» Y, Y being a Stein manifold. Again, using a suitable exhaustion of U, we can
suppose in addition that there is a relatively compact Stein open V in Y such that
V n i(X) = /(£/) and the map HT, 6Y) -► T{V, 0Y) is dense.
So we have reduced the proof to the case U <= cr X and X a Stein manifold.
The conclusion follows by theorem 2.1, making use of the fact that the maps
Ext'(X; &, Q) = r(X, Ext\$, fi)) -»• Exf(C/; 3\ Q) = T(U, Exf(®, Q))
are dense.
Remark. By their construction, the isomorphisms from the theorem do result
functorial in SF, and are compatible with the short exact sequences and with the
restrictions given by inclusions U <= X, U Stein open. As a matter of fact, in

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
29
Chapter VII, which is dedicated to the analytic duality, we will show that the duality
isomorphisms are naturally defined by means of the Yoneda map and of the "trace
map". The statement 2.1 will be sufficient for the applications given in this chapter.
The following result refines 2.1.
Theorem 2.9. Let X be a complex manifold of dimension n, K <= X a Stein
holomorphically convex compact, and Q. the sheaf of germs of differential forms of
the type (n, 0) with analytic coefficients. Then for any coherent analytic sheaf SF and
for any q~^0, ExtQ~q(K; 8% Q.) has a natural DFS structure and its topological dual
is algebraically isomorphic to HqK{X, SF).
(By definition, Ext^ (K; S% Q) = Ext^ K (K; &\K, Q\K)).
In order to prove this, we need some preparations. Recall that a compact
subset of a complex space, admitting a fundamental system of Stein neighbourhoods
is called a Stein compact (e.g. any holomorphically convex compact in a Stein space).
Suppose A" is a complex space, K <= X a Stein compact and cFeCoh (X) and choose
a countable fundamental system of Stein neighbourhoods Ur forK . Thus, T(K, SF) =
= lim r(Ur, S?)and consider on T{K, SF) the topolgy of inductive limit in the category
r
of locally convex topological vector spaces. We will introduce another topology
in the following way. We may clearly replace X by some neighbourhood of K such
that there is a closed immersion X -*■ Y, Y being a Stein open subset of a numerical
space such that there exists an epimorphism of <9y-modules 6$ -> /* (SF)- By theorem
B one finds a surjection r(i(K),6§.) -> T(i(K), /*(SF)) = r(K, ^). But r(i(K), 6pr)
has a natural DFS structure (§ 1, (c)) and we can endow F(K, S-) with the quotient
topology. It can be easily verified that this one coincides with the inductive limit
topology defined above, in virtue of the definitions of these topologies and the fact
that the topology on every r(Ur, SF) coincides with the quotient topology from
r(i(Ur), <9f ).
Proposition 2.10. Let X be a complex space, K <= X a Stein compact and
qF e Coh (X). The topology constructed above on T(K, SF) is LF and DFS. Moreover,
it is independent on all choice.
Proof. Fix x e X an arbitrary point. For any integer n ^ 0, S;Jm"S:x is
an (SymJ-module of finite type, hence it is of finite dimension over the complex
field. Consider on it the Haussdorf topology of topological vector space. Let U
be an open set containing the point x. We are going to prove that the
composition
r(U, SF) ^ &x -, &Jm&x
is continuous. We can assume that U is a Stein open set such that there is a
closed immersion U —> V, V being a Stein open subset of a numerical space
and, furthermore, i^\U) is the quotient of some £pv. Finally look at the
commutative diagram
T{V, Cft - T(V, i*(?\U)) = HU, SF) - 0
I I

30 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
and notice that r(U, fr) has the quotient-topology from T(V, Of,) and the first
vertical arrow is obviously continuous.
In this way, we have proved that the maps r(U, 8F) -> S:x/m"!¥x are continuous,
for any xeK, n^\. Accordingly, the maps T{K, SF) -> S?Jm"S?x are also
continuous. By the Krull theorem, the continuous map
HK, so- n (n^/msfj
is injective, hence, F(K, §0 is separated. The topology on T(K, &) turns out to
be LF and DFS by the very way it has been defined. Its independence on the
choice already made readily results by using the open map theorem for DFS
spaces (or for LF spaces). The proposition is thus proved.
A remarkable case is provided by the reduction of K to one point. The
topology thus obtained is called the canonical topology or the topology of the
uniform convergence on neighbourhoods [33], [43].
Lemma 2.11. Let X be a complex space, K <= X a Stein compact, and S" -> <$
a morphism in Coh (X). Then the map T(K, §) -> F(K, §) is continuous and strict.
Proof. The first assertion can easily derive from the continuity of r(U, Sr) -*■
-* r(U, §), U being a Stein neighbourhood of K. In order to prove the second
assertion one can proceed as in lemma 2.2.
Lemma 2.12. // (X, 6) is a complex space, K <= X a Stein compact and
S\ ($ e Coh (X), then for any q ^ 0 there exists a canonical isomorphism
ExtS(K; SF, <2)-r(tf, Ext%{9, <?)).
Proof. We claim that the canonical morphism
Exte{®, q}\ K - Ext&K(¥\K, q\K)
is an isomorphism: one can prove this by passing to stalks or computing the functors
Ext by means of resolutions with free sheaves of finite rank for & around K,
the existence of such resolutions being assured under the hypothesis. The morphism
asserted in this lemma is given by this isomorphism and by the morphism
Ext|(K; 3% q) = Ext9m(K; 3f\K, q\K)^T{K, Ext%iK{&\K, q\K))
of passing from presheaves to associated sheaves. There is a spectral sequence
by the term £|.« = H"(K, Ext"Q K(S=\K, § \K)) =- Hr(K, Ext^i®,§)), which
converges to ExtgIK(K; &\K, q\K). Since the sheaves Ext^(®,§) are coherent,
theorem B shows that the sequence degenerates and the conclusion follows.
This lemma allows us to define a DFS topology on every ExfJK; 5% ($).

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
31
Lemma 2.13. Let X be a paracompact topological space with countable
topology, K cr X a compact and 8f a sheaf of abelian groups on X. Then the canonical
morphism
H"{X\K, ®0-> I'm H«(X\U, SF)
<
UZ)K
is an epimorphism for any q. If, moreover, the restriction maps Hq~\X, SF) ->
-> H9~1(X\U, SF) are surjective for any open U which belongs to a fundamental
system of neighbourhoods of K, then this morphism is an isomorphism.
Proof. Consider a resolution SI' oî %r\X\Kby injective sheaves (or soft ones!).
This allows us to calculate the invariants H\X\K, SF) and H'{X\U, SF), U
neighbourhood of K. Choose a countable fundamental system of neighbourhoods
{Ur)r>i for K (and for the second assertion, a system verifying the supplementary
hypothesis!). The maps r(X\Ur + 1, STf)->■ T(X\U, SU') are surjective, r ^ 1.
The conclusion actually follows by an elementary reasoning of projective systems
by using a suitable diagram.
The proof of the theorem 2.9. The problem being around K, we can assume A"
Stein. On the invariants Extà(K; S% fi) we shall consider the DFS topology given
by lemma 2.12 and proposition 2.10.
There is a canonical morphism (given by restriction)
(*) HqK{X, 8F)^ lim #*(£/, SF),
<
<JZ)K
9^0 being an arbitrary integer. Since K is a Stein compact, the projective limit
can be taken over a countable fundamental system of relatively compact Stein
neighbourhoods UofK. According to theorem 2.1, every C-linear space Ext@~ 1(U; SF, Q)
has a natural FS structure, such that its topological dual is algebraically
isomorphic to H%(U, SF). Moreover, these isomorphisms agree with the maps
induced by inclusions V <= U. We thus obtain a natural isomorphism
lim H%U, SF) ~ lim (Ext£-«(C/; 8r, O))' m (Extn&-i(K; SF, Q))',
< <
UZ)K UZ)K
the last isomorphism being a consequence of lemmas 2.3, 2.4, 2.12 and of the
manner of considering the topologies. By composition with (#) one thus obtains
the following morphism
(**) H%(X, 8F) - (Ext^-i(K; SF, Q))',
q > 0 being an arbitrary integer. These morphisms are functorial in S- and
compatible with short exact sequences. The maps (##) are bijective if and only if
the maps (#) are. Thereby we can assume X a numerical space.
We first prove that (*#) is an isomorphism for & locally free. In this case
it is enough to prove that (*) is an isomorphism. The case q = 0 is obvious. For
q = 1 we apply the exact sequences
o -► rK(x, sf) = o -► r(x, sf) -► r(*\A:, sf) -► Hk(x, &) -»• o
o -» rc(u, sf) = o -» r(x, &) -» r(x\ u, &) -» h1c{u, sf) -» o

32 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(rK(X, S=) and TC(U, 9) are subspaces in rc(A", §•), which is null by corollary 2.7).
Passing to projective limit (this can be considered countably indexed), one gets
the exact sequence
0 -► T(X, SF) -»• lim T(X\U,®)^ lim H^U, &) ^0.
< <
UZ)K UZ)K
Since the restriction map T(X\K, SF)-> lim r(X\U,SF) is bijective, the conclusion
<
follows for q = 1.
Suppose now q^-2. We have the exact sequences
... -► Hi~\X, SF) -► Hi-\X\K, ®) -► #£(*, ^0 -»• #'(*, SF) -► ...,
...-»• Hi~\X, SF) -► Hi-\X\U, SF) -»• ««(C/, SQ -»• W(JT, SF) ->• ...
For q >fc n -\- \, the conclusion follows by lemma 2.13, making use of corollary 2.7.
It only remains to show that Hk+\X, &) = 0. To this aim it is sufficient to prove
that the space H%+1(X, 6) =* H"(X\K, 6) is null. At this point we use the
properties of the Laplace operator (B. Malgrange, Ann. Inst. Fourier, 6, 1955):
let co = a dzi A • • • A dz„ be a form of type (0, n) on X\K with C°° coefficients;
d2S
there is a C°°-function (3 on A"\A: such that 1/4 A(3 = V —~ = a, hence the
_ . dS O
form co = V (— 1)'+1 —— dziA • • • Adz;... Adz„ verifies the relation d"co = co.
Another argument for proving that H"(X\K, <9) = 0 can be given by using
the theory of hyperfunctions (P. Schapira, Lecture notes, Vol. 126, 1970).
We now prove that (**) is an isomorphism in the general case Sr e Coh(A").
We may replace A" by a suitable neighbourhood of K (the invariants of interest
do not change), thus we can assume that SF admits a finite resolution by locally free
sheaves of finite rank. By induction on the length of such a resolution, all that
remains to end the proof is to show the following assertion : if0->SF->^->3C->0
is an exact sequence in Coh(A") and the morphism (**) is an isomorphism for SF
and §, then it is an isomorphism for 3f too. But we have the exact sequences
.. . - Hl(X, 8=) - H%{X, <Sf) - H%{X, 3£) - HJt\X, SF) - Hj?\X, <?)-...,
... «- Extg-i(K; SF, Q) «_ Extnô-i(K; §, O) «- Ext£-«(tf ; 9f, ÎÎ) «-
«- Ext'^-i-^K; SF, Q) «- Ext$-i-\K; éf, £i) «- ...

1. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
33
The maps of the second sequence are strict by 2.11 and 2.12. In virtue
of 1.4, one obtains a new exact sequence passing to topological duals. The
proof ends from the lemma of the five homomorphisms.
By the above proof it results that the dualities of 2.1 and 2.9 agree with the
natural maps
H'K(X, SF) -► HC\X, 80, Ext'3(K; Sr, Q) «- Ext^A"; SF, Q).
Remark. If A'is an analytic set in some (£,", then one can prove that any Stein
compact of AT is a Stein compact in <£.". By using this fact, in the statement of
the theorem as well as in the following results (except 2.17) we can assume K
to be Stein only.
In the meantime let us give some immediate consequences of theorems 2.1
and 2.9.
Corollary 2.14. Let X be a complex manifold of dimension n, K <= X a
holomorphically convex compact and SF a locally free sheaf of finite rank on X.
Then H^X, 8F) = 0 for q ^ n and H%(X, §F) is algebraically isomorphic to the topo-
V
logical dual of the space T(K, 8r ®q£1).
Corollary 2.15. Suppose (X, 0) is a complex manifold of dimension n,
SF e Coh X and x e X. For any integer q ^ 0, the topological dual of the analytic
module Ext0~q(&x, Ox), which is endowed with the topology of the uniform
convergence on neighbourhoods, is algebraically isomorphic to H%{X, SF) (H^(X, SF) =
= H{xy(X, 8F) are the cohomology groups of SF with supports in {x}).
Corollary 2.16. Let X be a Stein space, KcX a holomorphically convex
compact and SF e Coh(Af). Then the canonical morphism
HKX, SF) -► lim HRU, HF)
UZ)K
is an isomorphism for any integer q > 0.
Proof. If AT is a manifold, we proved this fact in the course of the proof of 2.9
(and it is easy to be obtained a posteriori from the dualities given by 2.1 and 2.9).
The general case may be reduced to manifolds, putting instead of AT a Stein
neighbourhood of K and making use of a suitable immersion.
Corollary 2.17. Let X be a Stein space, K <= X a holomorphically convex
compact and SF e Coh (A"). Then the canonical morphisms
HKX, SF) - WC(X, SF)
are injective, q ^ 0.
Proof. If AT is a Stein manifold, the theorems 2.1 and 2.9 give the
isomorphisms
H?(X, SF) ~ (Ext"e-i(X; SF, Q))', h&X, 9) ~ (Ext$-%K; SF, £i))\
Since Ext£f <?(A"; F, Q)~r(X, Ext"^^, Q))and Ext"&-i(K; 9,G)s±T{K, Ext%-i(&, O)),
the corollary then follows by the fact that the map T{X, %) -> T(K, 30 is dense for
any % e Coh (X).
3-c. 2398

34 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
For the general case, consider an exhaustion (C/r)r3îl of X by Stein relatively
compact open sets containing K, each Ur being embedded in some numerical
space. Obviously, H%(X, SF) ~ lim H%(Ur, SF), hence it is enough to prove the
r
corollary for U/s; by immersions we reduce the proof to the above treated case
of manifolds.
Let now X be a complex space with countable topology and & e Coh (A").
Recall how the invariants H\X, SF) can be topologized ([38] or Ch. VII). Let V,
be a countable Stein open covering of X. We have a canonical isomorphism
H'ÇW, SF) ~ H'(X, SF) and if we consider on any T(U, SF), U an arbitrary open
set in X, the usual FS topology, then C'(% SF) becomes an FS complex and so,
by passing to cohomology, one gets a topology on H'(X, SF), generally nonseparated.
Its independence on the covering %t can be shown. If U is an open set in X, then
it can be proved that the maps H'(X, &) -> H'(U, SF), deduced by restriction,
are continuous. If SF->(?is a morphism of coherent sheaves, then the maps H'(X, SF)
-> H\X, (?) are continuous too. Now, consider an exact sequence 0 -> SF' ->
_> sf _> §;"_►() in Coh (X); then the "coboundary" maps H%X, SF") -> W+1(X, SF')
are also continuous (the proof easily follows in virtue of the exact sequence
of FS complexes 0 -> CÇU, SF') -> C'(% SF) -► C\% SF") -► 0 and by the
construction of the coboundary map, where ''LE is a countable Stein open covering
of X).
Lemma 2.18. Let X be a complex space with countable topology, W -> (?
a morphism in Coh (X) and q ^ 0 an integer. If the map Hi(X, SF) -> Hi(X, (?)
is surjective, then it is strict.
Proof. Take a countable covering ''LE of X, by Stein open subsets and consider
the commutative diagram
z<?cw., s?) © a^cu, sf) -► zi(\i, (?)
1 1
Hi(X, SF) ~ Hi^t, SF) -► Hi(V., (?) ~ Hi{X, (?).
The notations are the usual ones and the morphisms are obtained as follows:
the first horizontal arrow is that given by the sum of the canonical maps Zf(yt, SF)->-
-► Zi(*li, (?) and Ci^CM, (?) -► Zi(<M, (?), the first vertical arrow is the
composition of the projection on ZiQU, SF) with the canonical surjection Zi(fU, SF)-»^^^);
finally, the maps Z<?(ie, (?) -> Hi(% (?) and H%V., SF) -► H%%t, (?) are the canonical
ones. By hypothesis the first horizontal arrow is surjective (and obviously
continuous!), hence by Banach theorem it is strict. The conclusion easily follows.
Theorem 2.19. Suppose X is a Stein space, K <= X a holomorphically
convex compact and SF a coherent analytic sheaf on X. Then the topological vector
spaces H'(X\K, &) are separated.
Proof. Since T{X\K, SF) is separated, it only remains to prove the separation
of spaces Hi(X\ K, SF), q ^ 1. We have the exact sequence
.. . -► Hi{X, SF) -► H%X\ K, SF) -► H3c+1(X, SF) -► Ht+\X, SF) -> .. . .

1, COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
35
hence we get a canonical isomorphism Hi(X\K, SF) —» H&+1(X, &) for any q > 1.
If U is a relatively compact Stein neighbourhood of K, then we also derive a
canonical isomorphism W(U\K, SF) ~ Hj?\U, 8Q. SinceHp\U, SF) ~ H^\X, ¥),
we deduce the bijectivity of the natural map
W{X\K, SF) -► H%U\K, SF),
q ^ 1, which, moreover, is continuous. Therefore, if the spaces Hq(V\K, SF) were
separated, the same would be true for Hq(X\K, SF).
So it suffices to prove the theorem for V and further, by a suitable
immersion, we reduce the problem to the case of manifolds.
We can suppose henceforth X a Stein manifold. Denote n = dim X. Then
Extjg-'-1^ SF, Q) has a DFS structure such that its topological dual is
algebraically isomorphic to H]C+1(X, SF). We shall consider on H%+1(X, SF)the strong
topology on duals which is an FS topology (as FS spaces are reflexive). We are going
to prove the continuity of the canonical map H%X\K, SF) -> H%+1(X, SF); since
it is bijective for q ^ 1 and H^\X, SF) is separated, the separation of H%X \K, SF)
follows as required.
We first consider SF locally free. In a Stein neighbourhood U of K, consider
an exact sequence of the form O-^cf-x^-^-^O. The map H%X\K, SF) -»•
->• HqK+\X, SF) is just the composition of maps W(X\K, SF) -► H%U\K, SF) -►
-»• HZ+^U,®) ~ H%+\X,&) and thus we can replace X by U, that is we can
suppose the above exact sequence defined on whole X. Since it splits in any point,
the sequence 0 -»• Horn (SF, q) -► Horn (SF, 6") -► Horn (SF, &)-> 0 is exact. By
theorem B, H\X, Hom{S:, q)) = 0, we have therefore an exact sequence
0 -► Horn (SF, q) -► Horn (SF, 6?) -> Horn (SF, f)-f0
and deduce the existence of a morphism SF -> 6" such that its composition with
the map 6" -* SF is the identity. Thus, in order to prove the continuity of the
maps H\X\K, SF) -> H^+1(X, SF) we could suppose, without loss of generality,
SF a direct factor in some &". Then Hi^(X, SF) is a closed subspace of H^iX, Op).
From the following commutative diagram given by the morphism SF -> 6?
H%X\K, SF) -► tftfHJT. SF)
1 1
Hi(X\K, 6") ->• H1£XX, 0p)
it only remains to prove the continuity of the maps Hq(X\K, SF) -> HqÊXX, SF)
for SF = 0' and by additivity for SF = 6. But tfj^, <9) = 0 for z ^ « and #£(^> <9)
is the topological dual of the space Q(K). Hence what remains to demonstrate
is the continuity of the map
H"-\X\K, 0) -► HnK(X, 0).

36 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
If V is a Stein open in X, then H"(U, <S>) is algebraically isomorphic to the
topological dual of the space T(U, Œ); endowed with the strong topology, H%U, 6)
becomes an FS space. Let (£/r)r>i be a fundamental system of Stein
neighbourhoods of K such that Ur + 1 <= <= Ur, for any r. Then we have an isomorphism
of locally convex topological vector spaces r(K, Q) -P lim r(Ur, Q). Passing to
r
the strong topological duals we get a topological isomorphism
H"-\X, <9)^lim H?(Ur, <9)
r
since any bounded subset of T(K, Q) is the image of a bounded subset of some
T(Ur, Q). Therefore, we must show that the canonical morphism
H"-\X\ K, 6) ^ Hnc(U, 0),
is continuous for a relatively compact Stein neighbourhood U of K. We shall
explicit this morphism by means of Dolbeault resolution. Consider two open
sets V, U" such that K <= V <= <= u" <= U and choose 9 e C*(X) such that
9=0 on U' and 9 = 1 on X\U". If ln is a sequence of cycles from T(X\K, S0'""1)
which tends to a cycle \ e T{X\ K, g0'""1), we have to show that 8(£„) -► 0(1)
where " A " denotes the associated cohomology class. The elements d"((p^n), d"(9?)
belong to T(X, $0,n) and their supports are contained in the closure of JJ" ; hence
they belong to TC(U, &0'"). The morphism S is the composition H"~1(X\ K, 6) -»•
-> Hn~\X\ U, 0) -> H"(U, <S), where the first map is given by restriction and
the second by the coboundary map. The latter can be constructed by means
of the exact sequnce of complexes
0 -> rc(u, a°' *) -► r(x, a°' *) -> r(x\ u, ë0- *) -► o
which connects the invariants H'C(U, &), H'(X, &), H'(X\ JJ, &). It will then result
that 8(Ç„), S(Ç) are just the cohomology classes associated to the elements d"(9E„),
d"(9^) and the required result easily follows.
We analyse now the general case, namely we shall prove that for any sheaf
& e Coh (X), the maps
H%X\ K, S?) -► HffXX, SF)
are continuous. That will be done by induction on the number prof Sr =
= inf(prof, S'S). The case prof = n is already treated. We can write an exact
sequence of the form 0 -► § -► &•<> -> SF -»• 0. By 1.13, prof § = prof ^ + 1
(prof & < n). The diagram
... -+H"(X\ K, 6"»)^Hq(X\ K, S:)^H"+1(X\ K, q)^>H*+1(X\ K, 6"°)-+. . .
(*) I || I
... ^H£+1(X, <9"o)-*H%+ XX, 9) -► HqK+ XX, §) -► H1}? XX, <9"°) -► .. .

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
37
is commutative (modulo the signs) and the horizontal arrows arc continuous,
those of the line below being even strict (this line is an exact sequence of FS
spaces). Let q < n — 2, then by 2.14, H^+1(X, €"«) — 0 and the assertion follows
by means of the diagram (*) by induction hypothesis. Finally, if q $s n — 1, then
H%+1(X\K,§) ~ H£+2(X,§) = 0 and the assertion follows again from the
diagram (*) by using 2.18. This completes the proof.
Corollary 2.20. Let X be a Stein space, K cr X a holomorphically convex
compact and O^Sf-^^^%^0 an exact sequence in Coh(X). Then the sequence
...-> H"(X\ K, SO -► H"(X\ K, <f) -► H"(X\ K, 3Q -> H"+1(X\ K, Sr) -► ...
is exact in the category of all topological vector spaces.
Proof. The maps of the sequence are continuous; since the corresponding
spaces are FS, these maps are strict.
Corollary 2.21. Let X be a Stein space, x any point of X and & e Coh(A').
Then the topological vector spaces H'(X\ {x}, 8F) are separated (hence Fréchet-
Schwartz).
§ 3. Dimension and depth of a coherent analytic sheaf
Let (X, <9) be a complex space and SF e Coh(A'). Following Andreotti and
Grauert [2] we will denote
prof 5F = inf profaa; & x
(we will often write prof 5F_V instead of prof^ Srx). This number is called the depth
(or profondeur) of the sheaf 3\ Recall some of its simple properties:
— prof W = oo if and only if SF = 0.
' • . . . _
— If X -> Y is a closed immersion of complex spaces and if Sf e Coh(A'),
then prof SF = prof i?(&) (cf. 1.7).
— If X is a manifold of dimension n and & e Coh(A'), then prof SF= n— dhîF,
where dhSF = sup dhsx&x(ci. 1.11).
X£X
Recall that a complex space is said to be perfect space if the stalks of the
structural sheaf are Cohen-Macauley rings. Special cases of such spaces are the
locally complete intersections, the stalks of the structural sheaf being then complete
intersection rings. If ^ a perfect space and & e Coh(A') is locally free of positive
rank, then the formula prof 8rx = prof Qx = dim 6X, x any point in X, allows
us to compute prof S\
Theorem 3.1. Let X be a Stein space, K cr X a holomorphically convex
compact, ¥ e Coh(A') and N $s 0 an integer. Then
(a) prof &x > N + 1 for any x e K if and only if H&X, &) = 0forq < N;
(b) there is a neighbourhood U of K such that prof (oF|C/\ K) ^ N -f- 1 if
and only if the vectorial C -spaces H^X,^) are finite-dimensional for any q <iV.

38 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof, (a) We first suppose X a manifold of dimension n. According to
theorem 2.9, lemma 2.12 and Cartan's theorem A, we have the equivalences
WK{X, 80 = 0 o Extnô-*(K; S=,Çî)=0o Ext^ÇSF, Q)\K = 0 o Ext"fxi (& x,Çlx) =
= 0 for any xe Ko Extg,"'^, Ox) = 0 for any x e K. Since the rings Qx_x are
regular, the conclusion is clearly forthcoming by 1.15.
For the general case, eventually replacing Iby a suitable Stein neigh-
i
bourhood of K, we can set up a closed immersion X —> (£." for some p. The
image K' = i(K) is a holomorphically convex compact in <£.". If we put &* = ^(SF),
then Hk(X, §0 ^ H'K,{(£.", ^*) and prof Srx = prof SF*^, for all x e X. The question
is reduced to the case of manifolds.
The assertion (b) can be similarly proved by applying to the sheaves Ext
the following
Lemma 3.2. Let X be a Stein space, K <= X a compact and % e Coh(X).
Then dim^ F(K, 3C) < oo if and only if there exists a neighbourhood U of K
such that %\U\K=0.
Proof. Let Ube a neighbourhood of Aland CK\U\K = 0. Then Supp(2£|£/)cr
<= K is a finite set, be it {xlt..., xp}. By Nullstellensatz, any stalk %Xi is
annihilated by a power of the maximal ideal mXi, hence dim %Xt < oo. Since T(K, 3t) —
= n %*s then dimŒr(A:, T) < oo.
k = l
We now prove the converse implication. For any point x e K denote by
m(x) the assigned maximal ideal of T(X, &). The hypothesis Aim(iT{K, %) < oo
implies the existence of an integer r such that m(x)r T(K, %) = m(x)r+1r(K, T) =
= ... By theorem A we get mx'Xx = mx+1^(x = ..., hence by Krull's theorem
mrJK,x = 0. Then the points of K n Supp 3C are isolated. Hence one can find
a neighbourhood U of K so that %\U\K = 0.
Corollary 3.3. Let X be a complex space, xeX, SF e Coh(T) and N an
integer. Then
(a) prof &x > N + 1 if and only if HX(X, SF) = 0 for q < N;
(b) there exists a neighbourhood U of x such that prof (SF|£/\ {x}) ^ N + 1
if and only if the spaces H%\X, SF) are finite-dimensional for q -^ N.
Remark. If U is neighbourhood of £, then H^V, W) ~ #^<T, SF). From
the exact sequence of cohomology
... -► H£(U, SF) _► #<?(£/, 80 -► #<?(£/\ A!, 8F) -+ i/l+1(C/, SF) -► . ..
one obtains an interpretation of the assertion (a) of 3.1, in terms of extension of
cohomology classes.
For the next theorem we need two lemmas.
Lemma 3.4. // X is a Stein space and % e Coh(A'), then dim^F^, T) < go
if and only if Supp 5Î is a finite set.
The proof is analogous to that for 3.2.
Lemma 3.5. // X is a Stein space and 5Ï e Coh(A'), then the topological
dual of the Fréchet space F(X, 3£) has at most countable complex dimension if
and only if Supp % is a discrete set.

t. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
39
Proof. Let us fix (Ur)r>1 an exhaustion of X by relatively compact Stein
open subsets such that the restriction maps F(X, &) -* T(C/r, <9) are dense, r ^ 1.
If Supp % is discrete, then the previous lemma shows that every T{Un 3f) is finite-
dimensional. Since T(X, X)' ~ lim T(Ur, %)', an implication is immediate.
—*
r
Conversely, by lemma 3.4 we should prove only that dimar(Ur, %) < <x>
for all r. For some fixed integer r, denote Ur = U and let K be a holomorphically
convex compact, U <= K. Take p a seminorm on T(X, 3C) of type "sup", enjoying
K
the property that any Cauchy sequence with respect to p remains Cauchy by
restriction at T{U, ?C). By T{X, T)A we mean the completion of the space r(X, Tj/p'^O)
in this seminorm. T{X, 3()A is a Banach space and since the map T(X, 2C)->
->• r(X, X)A is dense, the map (T(X, X)A)' -► T(X, %)' is injective. But {T{X, ?C)A)'
has a structure of Banach space, and by hypothesis it has at most countable
dimension. Accordingly, it is finite-dimensional and dim HX, 5£)A < oo. On the other
hand, by the choice of K and p, we easily deduce that the map T(X, 3f) -> T(U, %)
factorizes by T(X, 3£)A -> T{U, 5f) and that the latterhas dense image. Therefore
dim^nc/, X) < co and the lemma is proved.
Theorem 3.6. Let X be a Stein space, & a coherent analytic sheaf on X
and N > 0 an integer. Then
(a) prof & ^ N + 1 if and only if WC{X, 3Q = 0 for q < N;
(b) there is a finite set A c X such that prof (S^X A) ^ N + 1 if and only
if the spaces H%X, SF) are finite-dimensional for q < N;
(c) there is a discrete set A cr X such that prof (SFIA'X A) > N + 1 if and
only if the spaces H$(X, SF) have at most countable dimension for q < N.
Proof, (a) The nonsingular case follows exactly as the assertion 3.1 (a),
by using the duality theorem 2.1. Actually, the general case may be reduced to
this case by a suitable exhaustion and by immersions in Stein manifolds.
(b) We first consider the particular case when A' is a manifold and let
n = dim X. Let A elbea finite set such that prof (&\X\ A) ^ N + 1. It follows
Ext"ô-J(&x,Çlx) = 0 for any x $ A, q < N. Hence Ext$-i(®, Ù)\X\ A =0(q < N)
and Supp (Ext$-1(&, Q)) will be a finite set. By 2.4, 3.4, Extn0~i(X; &, O) is a
C-linear space of finite dimension and the assertion follows from 2.1. The converse
implication can be proved in the same way using again 3.4. Suppose now X
is an embeddable Stein space; this means there is a closed immersion X —> <£,"
for a suitable p. If we put ^* = i*(&), then H'C{X, §0 ~ H'c{(£.p, 8F*) and for all
xeX, prof &x = prof $*x); we thus reduce the proof at the nonsingular case.
We finally consider the general case. We first prove the direct implication.
Let A <= X be a finite set such that prof (S^IA'X A) ^ N + 1 and U a relatively
compact Stein open set containing A, chosen such that the map T(X, 6) ->
-»• r(U, 6) has a dense image. We shall show that the map H%U, ®) -»• H%X, ®0
is bijective (q < N) and the conclusion will follow from the above considered
case of embeddable Stein spaces. We must only show the following: for any
relatively compact Stein open subset V ■=> U, the map H%(U, SF) -> H%V, &~) is bijective.
Let F-> C" be a closed immersion. We have H%{y, SF) ~ Hl{(£.p, z'*(8F)) ^

40
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
a (Extg-9 (C"; i,(S0, &©■)' « II (ExtT«. (S\v, ©a?..,,»,))'» 1<N that proves the
assertion.
Next look at the converse implication. Denote A = {x e X\ prof &x ^N}.
Choose a relatively compact Stein open set U in X such that the map T(X, 6) ->
-> r(£/, (9) has a dense image. The maps H'C(U, 0) -> #C'(X, <9) are injective.
From the above considered case of Stein embeddable spaces we deduce that
A n U is a finite set. If ([/,.),.j>i is a Runge exhaustion of A'by relatively compact
Stein open sets, then H%(X, $9 ^ lim H%(Ur, SF) and exactly as above, i n [/,=
—*
= ^ n Ut+1 = ..., as soon as the maps H%U„ 8) -»• HRUr+1, ®), H?(Ur+l, SF) -►
-> H$(Ur+2, $% • • are bijective, 9 < A?. So ^ is a finite set.
The assertion (c) can be proved as (b) by using lemma 3.5.
Remark. The theorem may be proved straightforwardly by means of 3.1,
which is a consequence of the duality theorem 2.9. The above given proof shows
that it is but a consequence of 2.1.
We shall now give some results in connexion with the dimension of sheaves.
Let (X, 6) be a complex space and SF e Coh(A'). Denote
dim SF = sup dim^ S\.
xex
This number is to be said the dimension of the sheaf S\ If Ann cF is the ideal-
sheaf annihilator of SF, then (Ann S% = Ann 8X where Ann &x is the annihilator
of the (9^-module §x. Accordingly, dim § = dim (Supp $9. If 8 ^ 0, then prof § <
< dim 8.
Remark. One can easily construct examples where the above inequality
is strict. A less trivial example is the following [71].
Let X c (C4 be the reduced complex space which corresponds to the
analytic set
{Zl = z2 = 0} U {z3 = z4 = 0}.
Obviously, X is connected, of pure dimension 2 at any point and it has a unique
singularity in origin. The local ring at origin is
A = <L{^i,Z2,z3>Zi}/(z^z2) n (z3, z4).
The class of z1 — z3 in A is a nonzerodivisor, since z± — z3 does not belong to
any of the prime ideals (z1; z2), (zs, z4). Accordingly
prof A = prof A\{z^ — z3)A + 1.
The element z± has a nonnull image in B = Aj(z1 — z3)A and one easily verifies
that z^e^, z2) fl (z3, z4)) 4- (zx — z3), i = 1,2,3,4. Thus, all the elements
of the maximal ideal of B are zerodivisors, hence prof B = 0. So prof A = 1 and
therefore
prof <9* = 1 < 2 — dim <9^.
Theorem 3.7. Le? X be a Stein space, 8 e Coh(A') and N ^ 0 an integer.
Then the following assertions are equivalent:

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
41
(i) dim SF < N;
(ii) W{X, SF) = 0 for q > N;
(iii) the complex vectorial spaces H%{X, SF) are finite-dimensional jor q > N;
(iv) the complex vectorial spaces H%X, SF) have at most countable dimension
for q > N.
Proof. We proceed exactly as for theorem 3.6. It is sufficient to prove the
implications (i) => (ii) and (iv) => (i).
(i) => (ii). Let (Vr)r>1 a Runge exhaustion of A'by relatively compact Stein
open sets. By 2.8 it follows that the maps H'c{Ur, SF) -> H'C{X, &) are injective.
We must only prove the implication for all Ur. By using immersions in numerical
spaces we can furthermore suppose X a manifold. The conclusion then follows
by duality theorem 2.1, lemma 2.4, theorem A and by the characterization of the
dimension of modules of finite type over regular local noetherian rings, given in 1.17.
(iv) => (i). As shown above we reduce the question to the case X a Stein
manifold. Put n = dim X. Then the topological duals of the spaces Ext^%X; SF, Q) =
= T(X, Extg'ifë, Q)) have at most countable complex dimension for q>N.
By 3.5 there exists a discrete set A <= X so that Ext^'i (SF, ÇÏ)\X\ A = 0
for q > N. For all x é A, Ext^SF^, 6x) = 0 for q > N, hence dim SFX < A'.
Therefore, dim (S^X A) < N and, since A is discrete and N $s 0, it follows
dim SF < A'.
By virtue of 3.6 and 3.7, the cohomology groups with compact supports
for coherent analytic sheaves on Stein spaces vanish out of the interval [prof, dim].
For the extremities of this interval the following two corollaries hold:
Corollary 3.8. Let X be a Stein space and SF e Coh(^). Then H»ro!®(X, SF)
has a finite {resp. at most countable) complex dimension if and only if the set
{x\ prof S\. = prof 8F} is finite {resp. discrete).
This derives from theorem 3.6.
Corollary 3.9. Let X be a Stein space and SF e Cob(^). Then Hfa^{X, SF)
has at most countable complex dimension if and only if dim SF = 0.
Proof. If dim SF = 0, then SF has discrete support and H^(X, SF) =
= rc(A\ SF) has at most countable dimension. If dim SF ^ 1, then the conclusion
follows from 3.7, applied for N = dim SF — 1.
Recall that a coherent analytic sheaf SF ^ 0 is said to be a Cohen-Macauley
sheaf if prof SF = dim SF.
From corollaries 3.8, 3.9 it results:
Corollary 3.10. Let X be a Stein space and SF ^ 0 a coherent analytic
sheaf on X. Then
{a) SF is Cohen-Macauley if and only if there is an integer N ^ 0 such that
H%X, SF) = 0 for any q ¥= N;
{b) there exists a finite set A c X so that &\X\ A is Cohen-Macauley if
and only if there is an integer N S= 0 such that the spaces H%X, SF) are finite-
dimensional for q & N;
(c) the assertion {b) is still true if A need not be a finite but a discrete set.
This corollary gives particularly a cohomological characterization of the
perfect spaces.

42 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
§ 4. Applications
(a) The interpretation of the depth in small dimensions: results of the Hartogs
and Cousin type.
The first two corollaries characterize the coherent analytic sheaves (on Stein
spaces) of depth ^ 1 or, respectively, $s 2.
Corollary 4.1. Let X be a Stein space and SF e Coh(A'). Then prof 3= $: 1
if and only if any section of T(X, SO with compact support is null.
The proof follows from 3.6. (a), for A/ = 0.
Corollary 4.2. Let X be a Stein space and & e Coh(A'). Then the following
assertions are equivalent:
(i) prof & ^ 2;
(ii) for any relatively compact Stein open set U <= X, the restriction map
r(X, SO -► r(X\V, SO is bijective;
(iii) for any holomorphically-convex compact K <= X, the restriction map
r(X, SO -»• r(X\K, §0 " bijective.
Proof. It is sufficient to apply 3.1 and 3.6 for N — 1 by using the exact
sequences
o -► rc(u, so -»• HA", so -»• r(x\u, so -»• hku, so -»• h\x, so
o -► rK(x, so -> r(x, so -► r(^r\A:, so -> #£(a\ so -► ^(X ®0-
The implication (i) => (ii) does not require the hypothesis X Stein.
The following two corollaries are results of Hartogs type.
Cokollaky 4.3. Suppose X is a Stein space, K <= X a holomorphically convex
compact and %r e Coh(A'). Then the restriction map F(X, 8F) -> F(X\K, SO is bijective
if and only if prof & x ^ 2 for any x e K.
Proof. Apply 3.1 for N = 1.
Cokollaky 4.4. Suppose (X, 6) is a reduced Stein space and K a compac
such that X\K has no relatively compact irreducible components (branches) in X.
If prof 6X ^ 2 for all x e K, then the map T(X, 6) -»• T(X\K, 6) is bijective.
Conversely, if X is Stein and K <= X a holomorphically convex compact such
that the map T(X, 6) -»• T(X\ K, 6) is bijective, then prof Qx S* 2 for all xeK.
Proof. The second assertion follows by the previous corollary.
We are going to prove the first assertion. Let 9 e T(X, 6), null on X\K.
The set of zeroes of 9 is a closed analytic subset of X and since it is closed
in K, it will be finite. As X is reduced and of positive dimension, it follows
<p = 0. We then consider a holomorphically convex compact K' containing K
and <p' eF(X\K, Ô). By the previous corollary there is (peT(X, 0) so that
<p|;r\A: = <p'|^\-K'- The function 9 - 9' e f(X\K, 6) vanishes on X \ K'
and therefore is by hypothesis null. This completes the proof.
Remarks. 1) Corollaries 4.2, 4.3, 4.4 (as well as those from the next
section) may be applied to the structural sheaf of a normal space of dimension > 2

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
43
(respectively normal of dimension > 2 in the points of K), because in this case
prof ^ 2 (by 1.8). We can also consider the case of locally free sheaves of finite
rank over perfect spaces (for instance locally complete intersections) of
dimension > 2.
2) Let Jfbe a paracompact, noncompact topological space. An open subset £/f
is called neighbourhood of the boundary of X if A'X U is compact. If W is a shea-
in Ab(A') we denote T{dX, &) = lim F(U, SF), where lim is taken over the neigh
u
bourhoods U of the boundary ([38], Ch. VII.D).
Suppose now A' is a Stein space, W e Coh(T) and prof W > 2. For any
A
compact/K there is a compact K' (for instance K' = K, the holomorphically convex
envelope) such that any section s e F(X \K, 8r) admits a unique extension
s'e T(X, Sr) and s'=s on X\K'. This follows from 4.3. In this way we have proved
the following assertion : if X is a Stèin space, & e Coh(A') and prof W > 2, then
the canonical morphism T(X, 8P) -> T(dX, 8r) is bijective.
3) As we have seen in the extension of Hartogs' theorem to the case of
complex spaces, the condition dim ^ 2 must be substituted by prof > 2. The
following example due to Harvey [40] points out this phenomenon.
Let X be the image of the morphism h: C2 -»• C4, h(x, y) = (x2, x3, y, xy).
^ is a subspace of dimension 2 of C4 with a single singularity in the origin and,
moreover, Af\{0} is connected. The function/(z), zel\{0}, equal to z2jz1
if Zj ^ 0 and to zjz3 if z3 ^ 0, is a holomorphic function on A^\{0}. It can not
be extended to whole X. For, otherwise, the holomorphic map X -* <£2, z = (zlt z.,,
z3» zù l-> C/(z)> zs)> is an inverse of h: C2 -> X, hence a contradiction. In this
example X is irreducible and
1 = prof 6X < dim 6X = 2.
The next applications regard the Cousin problems. First of all, restate
them on complex spaces (not necessarily reduced).
Let (X, 6) be a complex space. For any open set U cr X denote Sv =
= {/e T(U, &x)\fx is nonzerodivisor in 6X, for all x e U}. S^ is a multiplicative
system and the sheaf DU defined by the presheaf which assigns to an open subset
(7cX the ring r(C7, 0x)sv (the ring of quotients with denominators in Sv) is
called the sheaf of germs of meromorphic sections on X. There is obviously a
canonical inclusion of sheaves 6 <= $11. Let x e X. If sx e &x is nonzerodivisor,
then the multiplication by sx defines an injective morphism 6X -> 6X. Pick
a section s e F(U, 6), U being a neighbourhood of x, such that its germ in x
is sx. The morphism 6\U -î><9|£/ is injective in x and since <9 is coherent^
there is a neighbourhood V <= [/ of x such that <9|F —> &\V is injective. Thus
^|Fe Sv. Accordingly, one will be able to verify that §i\lx is canonically identified
with the total ring of quotients of 6X. If A' is reduced, the above multiplicative
systems can be thus characterized: Sv = {/eT{U,&)\f does not vanish on any
nonempty open set in £/}.

44 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The Cousin problems can be formulated exactly as for manifolds. The
additive (respectively multiplicative) problem consists in giving conditions of
surjectivity of the map HT, SHI) -► r(^, SHI/6) (respectively r(X, S>li*) -►
r(X, SHl*/0*)). As usual, 0* (resp. SÏÏL*) is the subsheaf of 0 (resp. SHI) formed
by the invertible sections.
Cokollaky 4.5. Let X be a Stein space and K <= X a holomorphically
convex compact, If prof 0X > 3 for all x e K, then the additive Cousin problem
has solution on X\K. If moreover H2(X\ K, 7L) = 0, then the multiplicative
Cousin problem has also solution on X\K.
Proof. We have an exact sequence
Hl{X, 0) -► H1(X\K, 0) -► H%{X, 0).
By 3.1 we deduce H1(X\K, 0) = 0 and the first assertion follows from the
exact sequence
r(x\K, mi) -► r(x\K, shi/0) -► hi(x\k, 0).
For the second part of the corollary consider the exact sequence
H1(X\K, 0) -► H1(X\K, 0*) -► H2(X\K, 71),
associated to the exact sequence
O^>7l^0^0*->O
(recall that 7L is the simple sheaf of stalk the ring of integers and that the mor-
phism 0 -»• 0* is given by 9 h-> e27ti:?). Then there follows H\X \ K, 0*) = 0
and the exact sequence
r(x\ k, ail*) -^ r(x\K, sni*/0*) -» «^xa:, £>*)
completes the proof.
Cokollaky 4.6. Suppose X is a Stein space and U <= X a relatively
compact Stein open set. If prof (0 \ U) ^ 3, then the additive Cousin problem has
solution on X\U. If in addition H2(X\U, Z) = 0, then the multiplicative Cousin
problem has also a solution on X\U.
The proof is similar to that of the previous corollary.
(b) Some topological properties of the boundary of a Stein space.
A complex space X (supposed paracompact but noncompact) is called
connected at boundary if for any compact K <= X the complementary in X\K
of the union of relatively compact connected components of X\K is connected
([38], Ch. VII. D).
Recall the notion of germ of analytic set around the boundary: it is a pair
(U, S) where U is a neighbourhood of the boundary and S a closed analytic
subset in U; we shall identify two such pairs (JJU SJ, (U2, S2) whenever there
is a neighbourhood of the boundary [/cf/ifi C/2 for which S1 n U = S2 n U.

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES 45
Denote by dX the germ defined by any neighbourhood of the boundary.
We shall say that a complex space X (paracompact, noncompact) is irreducible at
boundary if the germ dX can not be written as a union of two germs distinct
of it (the unions, intersections, etc. of germs as above are naturally defined).
Any such space is connected at boundary: indeed, for any compact K we have
X = K u (U Fj) U (U Ua), where Vh Ua are the connected components of
i a
X\ K, Vt are those relatively compact, and Ux the others. For any compact
K' => K and any a, Ux fi (X\ K') # 0; by irreducibility it results that the
index-set {a} contains only one element.
Corollary 4.7. // (X, 6) is a connected Stein space such that prof <9 $s 2,
then X is connected at boundary.
Proof. Let K be an arbitrary compact and X = K U (U vd U ( U Ua) as
above. Since for any a, K n Ua *£ 0, it is sufficient to show that A^\ K is
connected. If this happens, then the index-set {a} would be reduced to one element
and therefore X is connected at boundary. So we need
Corollary 4.8. // (X, 6) is a Stein space and K <= X a holomorphically
convex compact such that prof 6X ^ 2 for all x e K, then X is connected if and
only if X\ K is connected.
Proof. By 4.4 the map r(X, 0) -»• T(X\ K, Ô) is bijective. The corollary
will be concluded by the following
Lemma 4.9. A ringed space in local rings (X, 6) is connected if and only
if the ring T(X, 6) can not be written as product of two nonnull rings
{commutative and unitary).
Proof. If X = UU V, U and V disjoint open sets, then T(X, 6) =T{U, 6) X
X T(V, 0). Conversely, suppose T(X, 6) = A X B (A, B commutative unitary
rings). The unity of T(X, 6) can be written 1 = ex + e2, ex e A, e2 e B. Let U =
= {xe X\e^x) ^ 0} and V = {xe X\e2(x) ^ 0}. As usual, for a section / e r(X, 6),
f(x) désignes its image by the composition T(X, 6) -> &x -* @Jmx (mx is the
maximal ideal of 6X). The condition/(x) ^ 0 means fx $ mx, hence it is equivalent
to the fact that / is invertible in x. It will result that U and V are open sets.
On the other hand, it is not difficult to see that X = UW and U 0 V = 0.
Corollary 4.10. // X is an irreducible Stein space of dimension ^ 2, then X
is irreducible at boundary and in particular, connected at boundary.
Proof. We need to show that X\ K is an irreducible space for any
holomorphically convex compact K cr X. Let X' be the normalization of X and K'
the inverse image of K by the normalization morphism. The space X' verifies
the conditions of the previous corollary and K' is holomorphically convex. Hence
X'\ K' is connected. The conclusion follows since X'\ K' is the normalization
of JT\ K.
Let (X, 6X) be a complex space, not necessarily reduced. The algebra of the
global sections T(X, 6X) has a natural structure of topological <£, -algebra. If X
has a countable topology, then V(X, 6X) is a Fréchet algebra. If (X, 6X) -> (Y, 6r)
is a morphism of complex spaces, then the corresponding morphism T(Y, 6Y) -*
-> T^, 6X) is a morphism of topological algebras. We shall use the following
result of Forster [22], whose proof given below belongs to the authors:

46
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Theorem 4.11. // Y is Stein space, then the canonical map
Hom (X, Y) - Homtop Œ_alg (r(7, 0Y), T(X, ®x))
is bijective for any complex space X.
Proof. Denote by 6 the map we are concerned with; two cases may be
distinguished.
(1) Y embeddable. Let i: Y -> C" be a closed immersion and 3 <= <9Œb the
coherent ideal-sheaf defining i(Y). We first prove 6 is injective. Let /, g e Hom (X, Y)
be such that 6(/) = 6(g). To show that / = g it is sufficient to show i.f — i.g,
that is we can suppose Y = <C". Hereupon the maps /, g are completely determined
by the systems of maps (pr,-./)^^,,, (pry. g)1<J<n, where pry: <£"-►€ are
the canonical projections [15]. But the system (pry./) is identified with the system
of images in T(X, 6X) of the coordinates z1;..., z„ e r(C\ ^Œ") through the
map 6(f). A similar assertion holds for the system (pr,-. g). Therefore / = g.
We now check the surjectivity of 6. Let a: T(Y, 6r) -> T(X, 6X) be a
continuous morphism of topological C-algebras. By theorem B the following sequence
0 ->• r(C", 3) -»• HC", <9<p.) -»• T(7, 6Y) -► 0 is exact. The composition [3: r(C", <9Œ«)
-»• T(7, (9y) -»• T{X, 6X) gives rise to n elements p^),-.., (3(zn) e r(^, 6X), hence
yields a morphism /: A' ^-C"- The morphism 6(/): T(C", <9<e») -»• T(X, 6X) which
corresponds to / coincides with p, because both morphisms coincide on
coordinate functions zu..., zn and by continuity, on all entire functions. We are going
to prove that/ factorizes by 7 and the surjectivity of 6 will follow. In this respect
we must verify the equality f*(S)&x = 0, that is 3/{x)6XiX = 0 for all xeX. But
by theorem A, 3/w is generated as 6qp.f(x) -module by r(C"> 3) and from the
definition of p and the commutative diagram
r(cn,<9c»)^r(x^)
I I
<9<e»./M -»• ex,x
it will follow that r(C\ 3) has null image in &x>x and hereof the conclusion.
(2) The general case. We check the injectivity of 6. Let /, g e Hom (X, 7)
be such that 6(/) = 6(g). In order to prove that / = g, it is enough to find, for
any x e X, a neighbourhood of x on which the two morphisms coincide. Choose
an embeddable Stein open subset V of 7 containing/(x), g(x) and such that the
map T(7, 6r) -»• T(V, 6r) is dense. Then U =f~1(V) î\g~\V) is a neighbourhood
of x. Denote by /', g' e Hom (U, V) the morphisms deduced from /, g,
respectively. Since 6(f) = 6(g), 6(f) = 6(g') hence /' = g' by the particular case (1).
We now prove the surjectivity of 6.
Let a e Homtop.j.aig (T(Y,6Y), T(X, 6X)). Since 6 is injective, it suffices to
find for any x e X a neighbourhood U and a morphism U -* X such that the
corresponding morphism T(7, 6Y) -> F(U, &x) coincides with the composition
restriction
r(Y,6r)^r(X,6x) —> r(U,Gx). Fix xeX and K a holomorphically

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
47
convex compact in a neighbourhood of x such that x e K. Denote by || ||K
a suitable seminorm of type "sup" on T(X, &x). By the very definition of the
K
topology on T(Y, 6Y), the seminorms || \\L of type "sup", L holomorphically
L
convex compact in Y, constitute a cofinal family. By the continuity of a it then
follows there is such a seminorm || ||t and a constant C > 0 such that
\Hs)\\k < C. \\s\\L for all seT(Y, 0Y). Denote U = £ Let V be an embeddable
Stein neighbourhood of L such that the map F(Y, 6Y) -> T(V, 6Y) is dense.
If (sn)n is a sequence of elements of F(Y, 6Y) which converges in T(V, 6Y), then
the sequence (cc(sn))n of elements of T(X, 6X) is Cauchy with respect to the
seminorm || ||K, hence it converges in T(U, &x). Thus we get a continuous map
T(V, 6Y) -> F(U, 6X) which agrees with a. The corresponding morphism U^V
deduced by the particular case (1) and composed with the inclusion V -*■ Y,
verifies the assertion and the theorem is completely proved.
Corollary 4.12. Let X, Y be Stein spaces. If prof 6X $; 1 {respectively ^ 2),
then the natural map
Horn (X, Y) -+ Horn (dX, Y)
is injective (bijective, respectively) where Horn (dX, Y) = lim Horn (U, Y) with
u
respect to the neighbourhoods U of the boundary of X.
Proof. Let A! be a holomorphically convex compact in X and U = X\ K.
Consider the commutative diagram
Hom(X,Y) —> Horn (U, Y)
I I
Horn (T(Y, 6Y), T(X, 0X)) ^ Horn (r(7, <9y), T(U, 6X)\
the vertical arrows being bijective by the theorem. If prof 6X ^ 1 (resp. > 2),
then by 3.1 the undermost horizontal arrow is injective (resp. bijective). The
conclusion follows because in any Stein space the complementaries of the holo-
morphically-convex compacts give a cofinal system of neighbourhoods of the
boundary.
(c) Applications to compact complex spaces.
Theorem 4.13. Let X be a compact complex space, Y <= X a closed subset
such that X\ Y is a Stein open subset and & e Coh (X). Then prof (&\X\ Y) ^
$s N + 1 if and only if the canonical map H"(X, SF) -> H"(Y, SF) is bijective for
q ^ N — 1 and injective for q = N.
Proof. The assertion follows from theorem 3.6 by means of the exact
sequence
... -► H&X\ Y, SF) -► Hq(X, 5) -► H%Y, SF) -► H^\X\ Y, SF) -» ...

48 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Corollary 4.14. Let X be a compact complex space and Y a closed subset
of X such that X\ Y is Stein. If prof 0X > 2 for all xeX\Y, then X is
connected if and only if Y is connected.
The proof is immediate by theorem and lemma 4.9.
Remark. These facts can be applied to the case when J is a projective
complex manifold and Y a hypersurface of it.
Similar results can be obtained if Y is substituted for a holomorphically
convex compact in a neighbourhood of it; for instance, X — \Pr and Y a
holomorphically convex compact in an affine cart.
Proposition 4.15. Let X be a complex space such that it has a covering
by d + 1 Stein open subsets. If & e Coh (X), N 3= 0 is an integer and prof S > N + 1,
then H%X, SF) = 0 for any q < N - d.
For the proof we proceed by induction on d when using 3.6 and the following
result of Mayer-Vietoris type:
Lemma 4.16. Suppose X is a paracompact space and SF a sheaf of abelian
groups on it. If X is a union of two open subsets U, V then there exists the
following exact cohomology-sequence
...-»• Hqc{u n v, §0 -»• h«{u, 59 © Hi(y, 39 -► Hqc{x, so ->
-»• h9c+1(u n v, sf) -> ...
Proof. For any sheaf of abelian groups 3 onl we have an exact sequence
0 - rc(u nv,3)^ rc(u, 3) © rc(v, s) X rc(x, 3)
where the second arrow is induced by the diagonal map and the third is defined
by the formula Q(s, t) = s — t, seTc(U,3), teTc(V,&) (we use in both cases
the trivial extension of sections).
We now suppose 3 flabby. Let v e TC{X, 3) and K = Supp v. Consider the
compact sets K± = Kf] (X\ U), K2 = K n (X\ V). They are obviously disjoint
and one can choose Ul5 U2 as disjoint neighbourhood of theirs. We put
_ JO on (X\ K) U Ut
\v on C/2
and thus obtain a section of 3 over the open (X\ K) U V1 U U.2. If s' e T(X, S)
is an extension of its then s'\Ue rc(U, 3) and if we put^i = s'\U, s2 = (s' — v)\V,
it is easy to check that s2 e Fc(v> 3) and si — ^2 = v, hence 6(^i, ^2) = v- Therefore
the map 6 is surjective.
Let now 3' be a flabby resolution for 3\ We then get an exact sequence
of complexes
0 - rc(u n v, s-) - rc(u, s') ® rc(v, s*) -» rc(^, s') -. 0.
The lemma follows by passing to cohomology.

I. COHOMOLOGY WITH COMPACT SUPPORTS ON STEIN SPACES
49
For a compact complex space X, in conformity with Andreotti-Vesentini [3],
we shall denote by d(X) the minimal number of Stein open subsets which can
cover together the space X.
Corollary 4.17. Let (X, 6) be a compact complex space. Then
d{X) > prof 6+ 1.
In particular, X can not be covered by less than prof 6 + 1 Stein open subsets.
Proof. Otherwise, T(X, 6) = TC(X, 0) = 0, hence a contradiction.
Corollary 4.18. A normal compact complex space of dimension ^ 2 can
not be covered by two Stein open subsets.
The assertion follows since the depth of the structural sheaf is > 2 (by 1.8).
(d) The invariance of the depth with respect to finite morphisms.
f . .
Recall that a morphism of complex spaces X -* Y is said to be finite if the
corresponding topological map is finite.
Proposition 4.19. Let X -> Y be a finite morphism of complex spaces and
WeCoh(X). Then
prof & = prof (/*(S0).
Proof. The problem being local on Y, we can suppose Y Stein. By lemma
1.21 and theorem B we get H\X, §) ~ H%Y,f *(<f)) =0 for any q ^ \ and
($eCoh(X), /*(<?) being also coherent [15], [56]. Accordingly, X is Stein. Then
we shall use the characterization of the depth on Stein spaces in terms of
cohomology with compact supports given in 3.6. Since the canonical morphisms
H'C(Y,f *($0) — H'(X, S-) are isomorphisms, the conclusion follows.
The particular case when/ is the normalization morphism was proved without
cohomological methods in [94].
Bibliographical indications
The idea of using cohomology with compact supports in the problems recalled at the beginning
of the introduction of this chapter, as well as the informations about this cohomology by means
of duality, appear in Serre's papers [74], [75].
Theorem 2.1 belongs to the authors [8] and it is a particular case of the general Ramis-
Ruget duality on complex spaces [61].
Theorem 2.9 for X domain of holomorphy, K polycompact (product of compacts of the
complex plane) and ^==19 was proved by Frenkel by means of Laurent expansions [24]. The
general case (for locally free sheaves) was proved by Martineau [55], by using Malgrange's result
on the triviality of the cohomology in the highest dimension on noncompact manifolds. The
extension to coherent sheaves was done by Harvey [40] and by the authors (the proof given here is
that from [8]); in [40] the result applies for a class of compacts larger than Stein compacts.
4-o. 2398

50 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The separation theorem 2.19 was proved by Trautmann in [92] for the case when the compact
is reduced to one point (2.21); the proof given hereby means of duality is due to the first author [5].
Theorem 3.1 belongs to Harvey [40] and the authors [8]. Its particular case 3.3 was
previously proved by Trautmann [90].
The inverse implication from theorem 3.6 (a) is a simple case of the Andreotti-Grauert
finiteness theorems [2]. The implication (i) => (ii) of theorem 3.7 is a particular case of a result
of Reiffen's [64]. The characterizations of the depth and dimension of coherent analytic sheaves
on Stein spaces in terms of cohomology with compact supports (the direct implication of 3.6 (a)
and the implication (ii) => (i) of 3.7) were obtained by Y. T. Siu in [81] by means of Thimm's
gap sheaves; the characterization of the depth was also independently found by the authors [8].
The proofs given here, which make use of duality, are those from [8], [40]. The results 3.6
and 3.7 can be fairly exposed by means of the dualizing sheaves introduced by Andreotti and Kas
([4] and Ch. VII, § 5 of this book).
Paragraph 4 is conceived following papers [74], [8], [9] and [40], the results being
straightforward consequences of the previous paragraph.

Chapter II
Analytic local cohomology
Introduction
An important problem of the theory of complex functions is the extension of
the analytic entities defined beyond an analytic set. Recall some results in this
context:
"Let Q be an open set in <£." and A <= Q an analytic subset of codimension
> 2. Then any holomorphic function on Q \ A extends uniquely to Q" (der 2.
Riemannsche Hebbarkeitssatz [56]).
"The additive Cousin data in C3\ {0} extend uniquely to C3" This result
of H. Cartan [13] was generalized for superior dimensions by Rothstein [67],
namely:
"If G is an open in a numerical space and A <= G is an analytic subset
of codimension ^ 3, then the additive Cousin data on G\ A extend to whole G."
To these examples of extensions of functions and Cousin data one can add
examples of extensions of cohomology classes, analytic subsets, sheaves,...
Let X be a topological space, A <= X a closed part and SF a sheaf of abelian
groups on X. The exact sequence
... -► m(x, sf) -► m (x\ a, §o -► ha+'(x, sf) -► ...
shows that the cohomology group with supports in A, HqA+1 (X, fr) stands for the
obstruction to the extension to whole X of cohomology ^-classes from A^\ A.
Beside the global invariants HÀ(X, 8r)one can consider local invariants: the sheaves
%"A 5F associated to the presheaves U m» H'Anu (.U,Sr). This chapter is dedicated
to the study of the invariants H'A (X, 8r) and %'A & (X complex space, A analytic
subset, fir e Coh (X)) and in this way, to the extension of cohomology classes.
The first paragraph contains some preliminaries of local cohomology and
algebra (the depth of a module with respect to an ideal). In the second one, to
any coherent analytic sheaf we associate a family of analytic subsets, which play
a decisive part in the formulation and the proof of the results in this chapter.
In the third paragraph we introduce the notion of depth of a coherent sheaf with
respect to an analytic set and prove the vanishing theorem: 3C^SF =0 for i < q
if and only if prof A& > q + 1- Among the applications one can mention:
— If X is a perfect complex space and A an analytic subset of
codimension $s 2, then the restriction T(X, 6) -> r(X\ A, 6) is bijective.

52 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
— Let Jf be a perfect reduced complex space. Then X is normal if and
only if the codimension of its singular locus is $s 2.
— If X is a normal complex space and A an analytic subset of codimension
^ 2, then the restriction T(X, 6) -► T{X\ A, 6) is bijective.
In the fourth paragraph we give necessary and sufficient conditions
regarding the coherence of the sheaves 3(.'^S\ The next paragraph deals with the absolute
local cohomological invariants : the sheaves "K'd ¥ associated to the presheaves
U i-> lim {H'A (17, Sr)\ dim A < d}. A vanishing theorem and a finiteness theorem
are proved for them. In the last paragraph some topologies on the spaces H'A(X, S?)
are defined and the following separation theorem is proved : if X is Stein and
the sheaves %lA W are coherent for / < q, then the spaces HA{X, Sf) are separated
for / < q + 1.
§ 1. Preliminaries
(a) Some elements of local cohomology ([36], SGA 2). Let us consider a
topological space X. A part A of A' is said to be locally closed if there exists on open subset
U of X such that A is a closed subset of U. Let & e Ab(A') be a sheaf of abelian
groups on X. Denote by TA(X, SF) the subgroup of the elements of T(U, SF) whose
supports are contained in A. One easily verifies that TA(X, 5F) is independent on U.
If SF -> (| is a morphism of sheaves of abelian groups, then a natural map VA(X, 3=)
-> rA(X, <Pj) is induced. An additive functor
TA(X, *) : AbOQ -»• Ab
is thus obtained and its right derived functors are denoted by HA(X, *) and are
called the groups of local cohomology with supports in A. Since the functor TA(X, *)
is left exact, HA(X, *) ~ TA(X, *). If ¥ e Ab (X) and
is a resolution of SF by injective sheaves in the category Ab (X), then HA (X, 8r)
are the cohomology groups of the complex of abelian groups
o - rA(x, 3°) - rA(x, a1) - rA(x, a2) - ...
Lemma 1.1. (excision theorem). If Y is an open subset of X and A is
a closed subset of Y, then
H'A (X, Sr) ^ H'A (Y, & | 7), 9 e Ab (X).
The proof follows by definitions since the restriction of an injective sheaf to
an open subset remains injective ([35], prop. 3.1.3.).

H. ANALYTIC LOCAL COHOMOLOGY
53
In particular, it follows by this lemma that H'A{X, SF) ^ H'(A, SF) for any
open subset A of X.
Lemma 1.2. // 0 -► SF' -► SF -► SF" -► 0 « a« exact sequence in Ab(Af) and A
is locally closed in X, then we have the exact long cohomology sequence
0 -* H°A (X, SF') -► HA (X, SF) ->■ HA (A^, SF") -> H\ (X, SF') -»• H\ (X, SF) -► ...
The proof follows in virtue of general properties of derived functors.
Let A be a locally closed subset of A^ and A' a closed subset in A; A" =
— A\ A' is also locally closed in X.
Proposition 1.3. For any JeAb (X) the sequence
0 - HA, (X, S>) - #3 (A-, 9)^H%. (X, SF) - tf}, (AT, SF) - ...
is exact, junctorial in SF and agrees with the short exact sequences in Ab(Af). The
maps HA,(X, &) -»• H'A (X, SF) and H'A (X, SF) -»• HA,.(X, SF) are the natural ones; the
first is given by the inclusion A' cr A and the second HA(X, SF) -> HA\A,(X\ A', SF) =*
^ HA,,(X, SF) is obtained by restriction.
Proof. If 3 is a sheaf on A', then we have the following exact sequence
given by inclusion A' <= A and by restriction
o - ta,(x, 3) - r, (x, 3) - r^.(at \ ^', a) = rv (at, 3).
If S is injective (hence flabby!), then the map
r^(AT, 3) = TA(U, 3) - TA„ (X, 3) = TA(U\ A', 3)
is surjective (£/ is an open such that A is a closed subset of U). Let 3' be an
injective resolution for SF. We then derive the exact sequence
0 ->• TA. {X, I') ->• r^(AT, S") ->• r> (AT, 3') -»• 0
and the assertion follows.
Corollary 1.4. // A is a closed subset of X and SF e Ab (X), then we get
a natural exact sequence
0 -»• H°A {X, SF) -► H°{X, SF) -► H° (X\ A, SF) -► HA (X, SF) -► T/^AT, SF) -► ...
From this sequence we can remark that the invariants HA+1(X, SF) are just
the obstructions concerning the surjectivity of the restriction maps H'(X, SF) ->
-»• H'(X\ A, SF).
Corollary 1.5. Let A be a locally closed subset of X and SF a flabby sheaf
on X. Then HA {X, SF) = Ofor any i ^ 1.
Proof. If U is an open such that A is a closed subset of U, then the sheaf
SF|C7 is flabby on U. By substituting X for U we can assume A closed in X. The

54 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
map H°(X, SF) -► H°(X\ A, SF) is surjective and H'(X, SF) = H'(X\ A, SF) = 0 for
/ ^ 1. The conclusion follows by the previous corollary.
Corollary 1.6. The groups of local cohomology with supports in a locally
closed subset can be calculated by means oj resolutions by flabby sheaves.
Apply 1.5 in accordance with a general property of derived functors
("de Rham's abstract theorem", I. 1.2).
Remark. In the investigation of the invariants HA(X, *) as well as the other
invariants of this chapter, we can avoid the injective sheaves. For a sheaf SF e Ab(T)
we can consider the flabby resolution of Godement ([26], Ch. II, 4.3)
o -> w -> <s° -> e1 -> e2 -►...
and define H'A{X, SF) as the cohomology groups of the complex
o -+ r,(x, <s°) -+ r,(x, &) -+ rA (x, <s*) -+ ...
Starting from this definition we obtain again the preceeding properties of local
cohomology. By corollary 1.5 there results that in order to calculate the invariants
HA(X, SF) we can replace Godement's resolution by any flabby resolution.
To conclude with the global cohomological considerations, we introduce
a new invariant. If O cr W are two families of supports in X (not necessarily para-
compactifying), then we denote
rT/<D (x, sf) = rv (x, &)/r9 (x, sf)
for any sheaf SF e Ab(X).
One thus obtains a functor and its right derived functors are denoted by
7/4-/a> (^' *)• Generally, rY/a, (X, SF) is not isomorphic to H§-/<5> (X, SF). From the
properties of derived functors we get the following
Lemma 1.7. Under the above conditions, there is a naturally exact sequence
0 -+ Hl{X, SF) -+ H^{X, SF) -+ H^lo(X, SF) -+ Hi(X, SF) -+ H^.(X, SF) ^ .. .
We are now going to study the local cohomological invariants, defined by
Grothendieck. Preserve the above notations; hence ^ is a topological space,
A <= X a locally closed subset of X and SF e Ab(X).
For an integer i denote by %'$ the sheaf associated to the presheaf
U m* HA(U, SF), U <= V m* the natural map HA(U, SF) <-HlA(V, SF)
(by H'JJJ, SF) we mean HA(?U(U, SF|[7)...). A family of functors (3(^(*));îïo is thus
obtained and one can easily check that this is just the family of the derived functors
of the functor W h->- %AW. The invariants Sf^SF are called the sheaves of local
cohomology with supports in A associated to SF. If A is a closed subset of X, then

II. ANALYTIC LOCAL COHOMOLOGY
55
%A& is the subsheaf of 8F, formed by the sections whose support is in A; to be
more precise, for any open U of X,
r(u, 3c^so = rA(u, sf) c r(u, hf).
It is easy to see that Supp %A& <= A for all i.
Lemma 1.8. // 0 -> SF' -> SF _► SF" -> 0 w cr« exac? sequence in Ab(X),
//ze« we Aave /&e /o«g cohomology sequence
o -»• 3£^' -► %A® -► gfe^gF" ->■ s^sf' -► s^sf -►...
The proof follows by lemma 1.2.
By using the proposition 1.3 we deduce
Proposition 1.9. // A is a locally closed subset of X, A' a closed subset
in A, A" = A \A' and _8F e Ab(A"), then we have an exact sequence
0 ->• ^A-fr -► 3^SF -► "XA..® ->• 3^-f ->• 3QSF ->• . . .
enjoying the same properties as the sequence from 1.3.
We shall now consider the case when A is a closed subset of X. Denote
by Sl'^oF the sheaf of abelian groups defined by the presheaf
£/-► H!(U\A,B;).
The sheaves <St^oF are called the gap-sheaves of SF with respect to A. By the above
notations, $lAér = %'X\^A^. These sheaves coincide also with the direct image
sheaves given by the inclusion A^\ A <= X.
Since the inductive limit is an exact functor and since for all xeX, lim H'(U, !¥) =
us &
= 0 (z ^ 1), we get by 1.4 the following
Corollary 1.10. Let A <= X be a closed subset and ¥ e Ab(A"). Then we have
the exact sequence
0 -»• %°AW -> SF -»• .StOjSF -»• %AW -► 0
and, for any i > 1, we have the isomorphisms
3lAW ~ 3^+1SF.
Corollary 1.11. // A cr X is a locally closed subset of X and SF a flabby-
sheaf, then %A§ = 0 for i ^ 1. // in addition A is closed, then %°A SF is flabby.

56 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. Only the second assertion is not trivial. Let U be an arbitrary open
subset of X and seT{U, 3e°,SF) = TA{U, SF) c r(U, SF). Consider
,. _ j 0 on X\A
\ s on U.
We have 1 e F(U U (X\ A), SF); since SF is flabby, 7 can be extended to a section
seF(X, &). As î|^\/4 =0 and s\U=s, the conclusion follows.
Corollary 1.12. The sheaves of local cohomology with supports in a locally
closed subset can be calculated by means of flabby resolutions.
i
Corollary 1.13. Let X -> Y be a closed immersion of topological spaces,
A cr X a closed subset and SF e Ab(X). Then
i"*(3l^)) * Tl{A)(i*(?)).
Proof. Apply the previous corollary by virtue of the exactness of ;'„. and
of the fact that i*(3) is flabby for any flabby sheaf 3 on X.
Before closing our considerations on local cohomology we shall establish
the connexion between the local and global invariants.
Let A be a closed subset of X. We have the equality of functors TA(X, *) =
= F(X, %%(*)). By corollary 1.11, the functor "KA transforms flabby sheaves (so
much the more injective) into flabby sheaves, hence T(X, *)-acyclic. Applying ([26],
Ch. II, 2.4) we get the following:
Proposition 1.14. There exists a spectral sequence of term EP-i = H"(X,CKA&)
which converges to HPA+\X, 8f), SF e Ab(JT).
Recall that this spectral sequence can be built in the following way: consider
an injective (or flabby) resolution 0 -> & -* 3'. For the complex 0 -> 3(^(5°) ->
-> "X^iS1) -> 3Q(32) -> ..., we consider again an injective (or flabby) resolution 3'".
The spectral sequence we look for is one of the spectral sequences attached to
the bicomplex (T(X, 3P- i))Pi q. From the properties of spectral sequences there follows :
Corollary 1.15. Let A be a closed subset in the topological space X and Sr
a sheaf of abelian groups on X. Let q ^ 0 be an integer such that H'(X, fflfî) = 0
for j < q and i ^ 1. Then the natural morphism
H'A(X, SF) -► r(X, 3£'»
is an isomorphism for any i < q + 1.
Remark. The spectral sequence of the proposition is used only to prove this
corollary. By applying 1.1.1 to the composition TA(X, *) = r(^, %%*)), we obtain
a straightforward proof of the corollary and the considerations of spectral sequences
can thus be avoided (one can also see [84], lemma 0.6).
By the above corollary and by 1.4 we get:

H. ANALYTIC LOCAL COHOMOLOGY
57
Corollary 1.16. Let A be a closed subset in the topological space X and
S- e A\*(X). For the integer q ^ 0 the following three conditions are equivalent:
(a) 1i'A9 = 0 for i < q.
(b) For any open subset U of X, HA(U, #) = 0 for i < q.
(c) For any opzn subset U of X, the restrictions
H'(U, §9 -»• Hi(U\A, ^)
are bijective for i < q and infective for i = q.
We also remark that if 6 is a sheaf of rings on X and SF is an (9-module,
then the sheaves %'AW and Sl^SF have a natural structure of (9-modules.
(b) The depth of a module with respect to an ideal ([36], SGA 2).
Theorem 1.17. Let A be a noetherian ring, a an ideal of A, M an A-
module of finite type, and q an integer. Then the following three conditions are
equivalent :
(a) Ext^ (N, M) = 0 for any A-module of finite type N so that Supp N cr V(a),
and for any i < q.
(b) There exists an A-module of finite type N with Supp N = V(a) such that
Ext^ (N, M) = 0 for all i < q.
(c) There exists a regular M-sequence formed by q elements of the ideal a.
In order to prove the theorem we need two lemmas.
Lemma 1.18. Let A be a noetherian ring and let M, N be two A-modules
of finite type. Then Ass (Horn,, (TV, M)) = Supp N n Ass M.
Proof. Obviously, Ass {UomA{N, A/))c Supp (Horn,, (N, A/)) <= Supp N. If we
consider an epimorphism Ar -> TV -> 0, we get thereby an exact sequence 0 ->
-»• Hoin^/V, A/)-»- Hom^OT, M)=Mrand, therefore, Ass {UomA{N, A/))cAss A/r=
= Ass M. We have thus proved the inclusion Ass (Hom^ (N, Af))<=Supp N n Ass M.
The inverse inclusion is more difficult. Let p e Supp N. We first show that
there exists a non-null morphism of ^4-modules N-*■ A/p. Indeed, Nv/pNv is a
linear Av/pAv-space ^ 0 (by Nakayama lemma) and it is finite-dimensional. Let
v:Nv/pNv -* Ay/pAp be a non-null /4p/p^4p-morphism. Choose an element a e Ajp c
<= ^4p/p^psuch that the morphism av satisfies av{NvjpN^)<=Alp. The composition
av
N -* Np -* Nv/pNy —> Ajp is a non-null ^-morphism. Let now p e Supp N n Ass M.
Then there is an ^-monomorphism Ajp -* M and by composition a morphism
u: N -*■ M is thus determined. One can easily check that p = Ann u, hence p e
e Ass (HomA(N, A/)).
Lemma 1.19. Let A be a noetherian ring, a an ideal of A, and M an
A-module of finite type. The following assertions are equivalent:
(a) HomA(N, M) = 0 for any A-module of finite type N such that
Supp N c V(a).
(b) There exists an A-module of finite type N such that Supp N = V(a)
and HomA(N, M) = 0.
(c) The ideal a is not contained in any ideal of Ass M.
(d) The ideal a contains a nonzerodivisor of M.

58 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. The implication (a) => (b) is utterly trivial. The implications (b) => (c),
(c) => (a) easily follow from the above lemma (the set Ass associated to a module
of finite type is empty if and only if the module is null!). The equivalence (c) o(d)
is clear since the set of zerodivisors of M is just the union of the ideals of Ass M
and apply ([10], Ch. IV, § 1, n = 1, cor.2, prop. 2).
The proof of the theorem. The implication (a) => (b) is patent.
(b) =>(c). The case q < 0 is clear; hence we can assume q > 0. As
Hom(./V, M) = 0, there is, by 1.19, an element/ e a which is M-regular. We then have
the exact sequence 0 -> M—> M -* M If M -* 0. From the hypothesis and the exact
sequence of Ext's it follows that Ext'A(N, M/fM) = 0 for / < q — 1. One easily
deduces the assertion by induction on q.
(c) =±> (a). Again we proceed by induction on q. The case q < 0 is clear and
so we can suppose q > 0 and prove the general induction step.
Let/e a be a M-regular element. From the exact sequence 0 -> M—>M ->
-> M/fM -> 0 and the induction hypothesis, we deduce ExtA(N, M/fM) = 0 for
any ^-module of finite type N such that Supp N <= V(a) and for all / < q — 1.
Accordingly, the homotethy given by /, Ext^(iV, M) —> Ext'A(N, M) is injective.
Since f e a, some power of it annihilates N, hence Ext^(./V, M) too. It will follow
Ext^(iV, M) = 0 for i < q; this completes the proof.
Let A, a, M be as in the statement of the theorem. The largest number q
which satisfies the equivalent assertions of 1.17 is called the depth of M with
respect to a and it is denoted by prof aM. If A is a local ring with maximal
ideal a, one finds again the usual notion of depth.
Corollary 1.20. All maximal regular M-sequences formed by elements of a
have the same length, namely prof aM.
Corollary 1.21. //fea is M-regular, then
profa M = profa M/fM + 1.
The proof is in the same manner as 1.1.6, (ii).
Corollary 1.22. prof aM = inf (profA Mp).
peK(a)
Proof. If the elements Jv.. .,fq of a give a regular A/-sequence, then for
any p e V(a) their images in Aç are in pAp and form a regular A/p-sequence.
Consequently, prof aM < inf (prof Mv).
peK(a)
In order to prove the inverse inequality we shall first consider the case when
prof aM = 0. It will follow that a is formed by zerodivisors for M. Then we
have peAssAM such that a <= p. Since pAç belongs to Ass^Mp, piofAi?Mp=0
(I. 1.6) and the conclusion follows. Consider now the case when prof aM > 0 and
let fea be an M-regular element. By 1.21 and I. 1.6 we have prof a(M/fM) =
= prof aM — 1 and pvof(Mv/fMv) = prof Mv — 1, p e V(a). To conclude the
proof we apply induction.
Corollary 1.23. prof aM < oo if and only if V(a) n Supp M # 0.
Apply I. 1.6 (iii) and 1.22.

H. ANALYTIC LOCAL COHOMOLOGY
59
Corollary 1.24. Let A be a normal ring and a <= A an ideal of height > 2.
Then prof CA > 2.
Proof. Let p e V(a). The ring ^p is normal, of dimension ^2. By I. 1.8,
prof Ap ^ 2 and to conclude we must apply 1.22.
The assertion may be proved straightforwardly as in I. 1.8.
We close this section by the following lemma.
Lemma 1.25. Suppose A—>B is a surjective morphism of noetherian rings,
a <= A an ideal and M a B-module of finite type. Then
prof9(a)(M)=profaA/[ç1.
Proof. Let/!,... ,fq be elements of a. They constitute a regular M[9l-sequence
if and only if their images in B form a regular M-sequence.
(c) Further on we shall use the following
Lemma 1.26. Let A be a regular noetherian local ring, n = dim ,4, and M
an A-module of finite type. Then for any integer q,
dim (Ext"A(M, A)) < n — q.
Proof. LetpeSupp (Ext%(M, A)). Suppose the contrary, dim A/p > n — q.
Since A is regular, we can write
dim A ■= n = dim Ajp + dim Ap.
The ring Av is regular and dim Av < q. By applying the Hilbert-Serre theorem
([79], IV, th. 9), we get (Ext^(A/ , A))p ~ Ext/p(Mp, Av) = 0, a contradiction.
Remark. We shall apply the lemma for a ring of convergent power series.
The relation dim A = dim Ay + dim A/p will be proved for this case in the next
section (1.28).
(d) The lemma of normalization for analytic algebras. Recall some well-known
facts about analytic algebras [15], [33].
A C -algebra isomorphic to a non-null factor-ring of a convergent power
series ring is called an analytic algebra. The analytic algebras are noetherian local
rings of residual field C. the complex field. If A and B are two analytic algebras,
then any morphism of C-algebras A -> B is local. We will use the following result:
for any analytic algebra A and for any integer n, the map
n times
HomŒ.,|g.(C{A'i,. ■ -,X„}, A) -»• m^ X ... Xitij,^ (<?(Xd)i*zKn
is bijective (nt^ is the maximal ideal of A).
Consider a morphism of analytic C"algebras <p: A -> B. The morphism 9
is called_/z«zYe if B[tfi is an A -module of finite type. The morphism 9 is called quasi-
finite if B/mAB is a finite-dimensional linear space over A/mA ^ C \ this means that
mAB contains some power of the maximal ideal mB of B. We shall also use the
following form of Weierstrass preparation theorem: a local morphism of analytic
algebras is finite if and only if it is quasi-finite.

60
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Let A be an analytic algebra and xu.. .,x„ a system of parameters for mA.
From the above, it follows that the correspondence Xti-*- xt, 1 < / < n defines a
finite morphism C{'^i>- ■ -, X„} -> A. As dim A = n, by the first Cohen-Seidenberg
theorem ([79], III, prop. 3) this morphism is injective. In this way, we have shown
how the lemma of normalization can be obtained : for any analytic algebra A there
is a finite injective morphism C{^i>- ■ ■> Xn} -> A (n = dim A).
Next, by using this lemma of normalization we prove a more precise result.
Theorem 1.27. Let A be an analytic algebra and
Û! C 02 C . . . C Up
a sequence oj proper ideals oj A. Then there exists a finite and injective morphism
oj (£.-algebras (£{XU.. ., X„} -> A and, jor any i, 1 < / </?, a number h(i),
0 < h(i) < n, such that
û,nc{4-,x»} =(*!,■■■, AW <C{*i,..., xn},
jor any i = 1,.. .,p.
Prooj. We proceed exactly as in the case of polynomials ([79], III, th. 2).
There exists a finite injective morphism C(zi>- ■ ■> zn} ~* ^ and thus it is sufficient
to prove the theorem for A = (£, {zlt..., z„}.
We first consider the case p = 1 and ctj a non-null principal ideal a1 = x^A.
Since A is an integral domain, Xj is nonzerodivisor and can be completed to
a system of parameters xu x2,..., xn for A ([79], Ch. III). The morphism C {X} -> ^4,
A'i h> Xj, X = (Xu..., A^,), is finite and injective. We need the equality ax n
n C{X} = A'jClX}. An inclusion is obvious. For the other, let xea1r\ <C{X},
a = XjS, where Se/ Since the element S is integral over the ring C{X} and
lies in its quotient field, there results that Se C{X}(C{X}is a normal ring!),
therefore a e i'iClX}. The theorem is thus proved in this particular case, if we put
h{\) = 1.
Further, we will make induction on p. In the case p = 1 use induction on n.
If n = 1, then the ideal <xx is principal; hence we have the same previous situation.
For the general induction step let us fix Xj e alt x1 # 0. There exists a finite
injective morphism C{^i> ^V ■■» Y„} -> A such that xxA n C{^i. ^2.---, ^„} =
=^iC{^i! ^2>- • ■> i'n}- The induction hypothesis applied to the ideal ax n <£{Y2,...
..., Y„} gives rise to a finite injective morphism (£,{X2,..., Xn} -> (£,{Y2,..., 7„}
and to a number h! such that ctj n (£{X2,. ..,X„} = {X2,..., Xh')<£,{X2, ...,X„}.
It is easy to check that the morphism C{X} -> <C{^i, Y2,..., 7„} is finite and
injective; the composition C{X} -> C{^i> ^2>- ■ ■■ Yn} -> A has the same property.
Moreover, ctj n C{X} = (A^, ..., A*') C{X} and the case /j = 1 is over.
Suppose the theorem proved for p — 1 and prove it for p. There exists a finite
injective morphism Cl^i,---, ^B}—^ and some numbers h(i), 1 </</> — 1
which satisfy the assertion of the theorem with respect to the sequence of ideals
ûj c ... c ap_v Denote d = h(p — 1). Then we can find a finite injective morphism

II. ANALYTIC LOCAL COHOMOLOGY
61
<C{Xd+1,. ■ -, X„} —> <E{Yd+1,. ■ ■» Yn} and a number r ^ d + 1 such that ap n
<\ <L{Xi+l,.. .,Xn} ={Xi+1,..., Xr) <Z{Xi+1,.. .,Xn}. The morphism £{*„...
...,^, jrd+1,...,jrII}-»c{i'i,-.., ^ yd+1,...,yII},-Jr,H>F, for 1 </<</,
A',- h> y(Afj) for d -f 1 < f < /? is finite and by composition with u it determines
a finite injective morphism C{^i,- ■ -, X„} -»• -<4- It only remains to put/z(/>) = r.
This completes the proof.
Remark. The hypothesis that the analytic algebras are considered over the
complex field C is not esential; (£ can be replaced by an arbitrary complete valued
field.
Corollary 1.28. Let A be an integral analytic algebra and p a prime ideal
of A. Then
dim A = dim A/p + dim Af..
Proof. There exists a finite injective morphism <C{X} = <C{^n- ■ ■ > ^B} -»• A
and a number A, 0 < h < «, such that p n <C{X} = (A^, **)<C{X}. Bv
applying the Cohen-Seidenberg theorems ([79], Ch. Ill), we get dim ,4 = dim C{X} = n,
dim A/p = dim (C{X}/p n C{X}) = dim <£{Xh+1,..., X„} = n — h and dim Av =
= htp =ht(p fl C{X}) = A.
Corollary 1.29. Let A be an analytic algebra and p <= q prime ideals of A.
Then all maximal chains of prime ideals which link p with q have the same length.
This fact follows by the preceding corollary, applied to A/p and q/p.
(e) We shall use the following result: "If X—> 7 is a finite morphism of
complex spaces and if & e Coh (X), then/+(Sr) 6 Coh (Y)".
For a proof see [15] or [56], Ch. IV, th. 7, or the next chapter, where the more
general case of proper morphisms is considered.
We shall also use the following result: "If A' is a complex space and x
is one of its points, then there is a neighbourhood U of x and a finite morphism
U -* V, V being an open subset of a numerical space ([15], exp. 19)".
Proposition 1.30. Let X be a complex space and & e Coh(Af). Then any
ascending chain of coherent subsheaves of & is locally stationary.
Proof. As the problem is of local nature, we may assume X
finite-dimensional. We proceed by induction on n = dim X. The case n < 0 is utterly trivial,
since X is void. Let n > 0 and let S:1 <= SF2 c ... be a sequence as in the
statement of the proposition. We must show that any point x e X admits a
neighbourhood where the above sequence is stationary. Let x e X. There exist a
neighbourhood U of x, an open F in a numerical space and a finite morphism n : U -*
-*■ V. The sheaves iz*(&), 7t*(^i) are coherent. Moreover tc^SS) <= ^(SF^c.
... c kJ^) and any equality -k*($j) = ■**(&]) implies 3^ = 8F. (by I. 1.20).
We have thus reduced the problem to the case when Af is a manifold, which
can also be supposed connected. As & is locally the quotient of a sheaf of the
form 6r, it is not difficult to reduce the proof to the case & = 6r, hence to the case
8r = 6. Here, SF£ are coherent ideal-sheaves. If SF4 = 0 for all /, we have nothing to
prove. Otherwise, choose an index i such that SF£ # 6. The sheaf SF£ defines a
subspace Xt of X, which is distinct of X and, consequently, of dimension </?—!,

62
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The sheaves 8rj,j ^ 1 then correspond to ideal-sheaves on Xj. The conclusion follows
by the induction hypothesis.
The above given proof is taken from [80]; another proof can be found in [29],
th. 8.
§ 2. The singular sets of the coherent sheaves
Let X be a complex space and 0 its structural sheaf. For any sheaf &■ e Coh (X)
we make the notations
prof W = inf prof SFX and dim W = sup dim 3rx{= dim Supp oF).
xex xeA
For an integer m denote
5M(ff) = {xejr|profffx</n}.
These sets are called the singular sets of the sheaf 3\ The main result in this paragraph
is the following theorem of Scheja [71]:
Theorem 2.1. Let X be a complex space and & an analytic coherent sheaf
on X. The singular sets Sm(8r) are closed analytic subsets of X and dim Sm{&) < m
for any integer m.
Proof. The problem is local in nature; thus, we may assume that there exists
a closed immersion X—>V where Fis an open subset of some numerical space.
By I. 1.7, Sm(8F) = S^i^Sr)); thus, we should consider only the case X a manifold.
Let n = dim Jf. By I. 1.15,
Smm= U Supp(£x^(SF,3)).
The sheaves Exf are coherent and vanish for r > n, hence the first assertion
is proved.
In order to prove the second assertion, we must only prove that dim
Ext'^, 6) < n — r for any integer r, that is dim (Extr3(^, 6)x) < n — r for all x e X.
But we have the isomorphism Extre{W, 6)x ~ Extr3x(^x, &x) and in virtue of 1.26,
we can conclude.
If A <= X is a closed analytic subset, then codim A = inf coding,4 where
X£X
codinv* is the height of the ideal defined in 6X by A (if x £ A we put codim^ = oo).
Generally, for xe A, codim^ ^ dim^ — dim^, but the following remarkable
fact is anyhow true: there is a neighbourhood U of x such that codim,./* =
= inf (dim X - dim A) ([15]).
yeu y
If A c B are two closed analytic subsets, then codim {A, B) is the codimension
of A in the reduced analytic subspace of X given by B (the codimension of A

II. ANALYTIC LOCAL COHOMOLOGY
63
in X and in XKà is the same and this definition of the codimension agrees with
the previous one).
For any sheaf 8F e Coh(A') and any point xelwe set
{ dim ¥x - prof &x if &x ^ 0.
For any integer m denote Dm(&) = {x e X\ def/SF) > m}. These sets are called the
deject sets of S [15], [71]. If m > 1, then Dm(&) c Supp SF and D^Sr) =
= {xe A^dim 8rx # prof Srx} coincides with the set of the points of X where the
stalk of SF is not Cohen-Macauley.
Corollary 2.2. The sets Z),B(SF) are closed analytic subsets of X and, for
any m > I, codim (£>„,( §F), Supp SF) $: m.
Proof. For any m < 0, DJ&) = X. Replacing A' by (Supp W, 6/Ann SF|Supp SF),
we can suppose Supp SF = X. The problem being local we may also assume dim X =
= d < oo. If m > d, then Dm(%r) is empty and the stated assertions are obvious.
Suppose thereby 1 < m < d. Then we can write the decomposition
A»(S0 = U Ak where Ak={xe Z|dim^ ^ k} n ^_m(S^).
Since {x\ dimxX^k} is the union of all irreducible components of A'of dimension >
$s k, by the theorem, Dm(5F) are analytic closed subsets. Moreover, dim,.^4fc <
< dim.,. Sk..m(&) < k — m < dim^ — m for any k and whatever x e Ak. Then
it will result that dimxDm(aF) ^ dim.,!' — m, for all x e X. Therefore codim Dm(SF) ^ m
and the corollary follows.
Corollary 2.3. Le? X be a complex space and & e Coh(A'). The set of all
points xe X, such that &x is not a Cohen-Macauley 6x-module, is analytic and closed.
Corollary 2.4. For any complex space X, the set of all points of X where X
is not perfect is a closed analytic subset of codimension ^ 1.
§ 3. The vanishing theorem
Let Ibe a complex space, 6 the structural sheaf, A <= X a closed analytic subset,
and 61 = 3(A) the maximal ideal-sheaf defining A. For a sheaf oF e Coh(A') we denote
Prof^.x^ = profa^C* e A) and prof^ = inf prof^jJC§\
By 1.23, prof^>X<F = co if and only if Wx = 0. The number prof^ is called the
depth of SF with respect to A.
Lemma 3.1. The function x h> prof^^SF is lower semi-continuous on A.
Proof. The assertion is obvious in the points xe A where SF^ = 0. Assume
xe A and prof^^ = q < co. Then there are elements (f^)x,..., (fq)xe &x which

64 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACBS
give a regular §vsequence. Let/j,- ..,/,£ T(U, <9) be sections in some
neighbourhood U of x, which represent these germs. The sheaf-morphisms on U
I 3=1 / J=l
defined by multiplication by/;(l < / < q) are injective in x; hence they remain
injective in a neighbourhood V of x. For any y e V C\ A, (f1)y ,. .., (fq)y are elements
of <9y and form a regular Sysequence, therefore prof^ $■ > q.
Proposition 3.2. Le? X be a complex space, A a closed analytic subset
of X, and & e Coh(A'). Then the sheaf 31^ is coherent and is equal to the subsheaf
of all sections of & annihilated by suitable powers of the ideal £1 = 904).
Proof. Let &x denote the subsheaf of & formed by those sections which are
locally annihilated by suitable powers of GL. If &k is the subsheaf of & of all sections
annihilated by âk, we get an ascending chain
§=! c §v.. c & and ^a, = U ®k-
fc=i
Every lFk is coherent, as the kernel of the canonical map
& -»• Home(âk, §=), defined by si->(f >->fs).
By 1.30 the above chain is locally stationary; hence S^ is coherent. The proposition
will follow if we can prove that "XPaŒ =&<». Obviously IF,» <= <K°A^. Conversely,
let s e FA(U, &), U an open subset of X. Let a& denote the kernel of the map 0\U ->■
-> SF|£/,/ >-»/.?; oB is the annihilator sheaf of s and Supp s is the associated analytic
subset V(So). Since F(oB) <= A r\ U, the conclusion easily follows.
Proposition 3.3. Let A be a closed analytic subset of a complex space X
and SF e Coh(A'). For a point xeA the following two conditions are equivalent:
(a) prof^JF > 1;
(b) (K®)x = 0.
Proof, (a) => (b). Letfx ed^bea nonzerodivisor for IFx{& = 3(A)). Consider
an arbitrary element sx e OiA^)x- By the above proposition, &xsx = 0 for some
integer k. In particular, fkxsx = 0, hence sx = 0.
(b) => (a). On the contrary, if profarBrx = 0, then all elements of &x will
be zerodivisors for 9x, hence there exists an ideal p 6 Ass &x such that £LX <= p.
The ideal p is the annihilator of some element sx ^ 0 in Sx. We have âxsx = 0,
hence &s = 0 for a representing section s of sx in some neighbourhood U of x.
We then derive Supp s <= A n U, therefore sxe (*»„ contradiction.
We shall use the following particular case of the Frenkel's lemma [24].
Lemma 3.4. Let A <= <£" be an open polydisc centred in the origin, d > 0
an integer and A = A n {zd+1 = -. - = z„ = 0}. 77z<?« ?/?<? canonical map H"(A, 6) ->■

II. ANALYTIC LOCAL COHOMOLOGY
65
-> //°(A \ A, 0) is infective jor d < n — 1, bijective jor à < n — 2 and, in addition,
H'(A \ A, 6) = 0 for 1 < i < n - d - 1. '
Prooj [69]. Denote Ut = {z e A| z; ^ 0} for I = d + \,...,n. Thus one
obtains a Stein open covering Llf of A — A, therefore
H'(A \A, 6) ~ H'(% 6).
Let U = Ud+1 fl ... fl Un. Any element / e Q(U) can be expanded in Laurent
series
/ = E/ccOl, ■■■, Zd)zdd+Y- •• ZÏ" « = («<l+l> • • -, «n)>
a
ja being holomorphic functions. For S c {d + I,..., n} denote by 6S(/) the
sum of terms whose negative exponents appear in the very variables that have
indexes from S. In this way we get an endomorphism
6S : 3(t/) - 0(U).
It can be easily checked that the sum of all of these is just the identity
If j e <9(£//0. ..i,), then 6SC/) e &(Ui0 ;,.) and so any endomorphism 6S can be
extended to cochains. We also notice that whenever 1$ S and je &(Un0 z,.^,
6s(/)e3(t/,0...,,_,).
Let now 1 < i < n — d — 1 ; we are going to prove that any cocycle.
je Z'ÇW, Ô) is a coboundary. It is sufficient to prove that any 6S(/) is a cobound-
ary. The condition / < n — d — 1 shows that 6S(/) = 0 for S — {d + 1, ..., «}.
Thus we may assume that there is an index / e {d + \,...,ri\ so that I $ S.
Define g e C>~l{M, 6) by the formula
We have
(8s)i„.../. = i,(-i)kgh..x..,tt =
fc=o
= Qs ( tQ (- I)*/». • • ■ /Afc -.i.) = GsC/i.... I.)-
For the last equality we have applied the identity theorem for holomorphic
functions. Therefore, H'(A \ A, 6) = 0 for 1 < i < n - d - 1.
The other assertions of the lemma can be easily proved.
Remark. The above lemma can be easily deduced by studying the coho-
mology of the complementary of a point (I. 3.3), by means of topological tensor
products (a simple argument of Kunneth type').
Proposition 3.5. Let D <= C" he a domain, A <= D a closed analytic
subset oj dimension d and 8F e Coh (D). Ij prof SF ^ d + q, then %'A& = 0 jor
0 < i < q.
S - c. 2398

66 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. We first prove that %A6 = 0 for 0 < i < n — d. Consider in the
beginning the case A nonsingular. Let x be a fixed arbitrary point of A. By
a biholomorphic transformation around x we can assume x = 0 and A n V =
= {zi+1 = ... — zn = 0}, U being some neighbourhood of x. For any open
polydisc A which is contained in U, and which is centred in x, by Frenkel's
lemma and theorem B, we get H'A(A, 0) = 0 for i < n — d. Whence ÇKA&)X = 0
for i < n — d.
We now consider an arbitrary analytic subset A and proceed by induction
on d = dim A. The case d = 0 follows from above. For the general induction
step, let A' be the singular locus of A and A"=A\ A'. We can write the exact
sequence, given by (1.9)
...->• %A,6 -► %A6 ->• 3^„<S> -► ...
Since dim A' < d — 1, the induction hypothesis shows that %'A. (9 = 0 for i < n —
- d + 1. Let D' = Z>\ ^'. By the first part of the proof, 3^„<S> =0 on D' for
z < n — d. Let C/cD be an arbitrary open set. By 1.1 and 1.16, H'A,,(U, 0) —
= HA,,(D' fl U, 0) = 0 for i < n - d. Hence 3^,(9 = 0 for / < n - d; it follows
that %'Ae = 0 for / < n - d.
We shall now prove the assertion of the statement and we shall proceed
by descending induction on prof S\
If prof & > n, then W — 0 and the assertion is clear. For the general induction
step, let x be an arbitrary point of D. If &x is free, then & is free in a
neighbourhood of x; it will follow q < n— d, hence (%!A&)X = 0 for 0 < i < q by the above.
Suppose now &x is not a free (9^-module. Then n > d -\- q. In some neighbourhood
[/ of x we have an exact sequence of the form
O-^-x^ltZ-^lt/^o.
We deduce the inequality prof ($ S* profile/) + 1 $: prof & + 1. By induction
hypothesis, %A§ =0 for 0 < / < q + 1. Since n ^ d+ q + I, %AG = 0 for
/ < q + 1. From the above exact sequence we derive that %!AW = 0 for / < q
and the proposition is proved.
Remark. The proposition remains true if we assume that A is an analytic
subset which is not necessarily closed. Indeed, let D' c D be an open subset
where A is closed. We have %A& = 0 on D' for / < q. Let U be an arbitrary
open set in D. By applying 1,1 and 1.16 we obtain HA(U, $=) = HA(D' n £/, §0 = 0,
therefore %'A& = 0 for / < q.
The main result of the paragraph is the following vanishing theorem for
local cohomology of Scheja and Trautmann [71], [90], [84],
Theorem 3.6. Let A' be a complex space, A <= X a closed analytic subset
and & a coherent analytic sheaf on X. Then for any integer q ^ 0 the following

II. ANALYTIC LOCAL COHOMOLOGY
67
four conditions are equivalent:
(a) prof^ > q+l.
(b) dim Of fl Sfc + „+i(S0) < k for any k.
(c) %AW = 0 for i < q.
(d) For any open subset U of X, the restriction maps
H'iU,®)^ H'(U\A, SO
are bijective for i < q and injective for i = q.
Proof. The equivalence (c) <*■ (d) follows from 1.16.
(a) => (b). Let & be the ideal-sheaf given by A and x an arbitrary point
of A. Since prof^S^ ^ q + 1, there is a neighbourhood £/ of x and sections
fi,---Jq+i °f r(C/, <3) such that for any y e U fl A, the germs (/,)y, 1 < * <
< q + 1, form a regular S^-sequence (by the proof of lemma 3.1). By applying
1.21 one obtains
A n unsk+q+1($)=An UnsJs /"^JtA-
Then we derive by theorem 2.1, dimx {A fl Sk + q+1(S:J) < k.
(b) => (c). We use induction on dim A. Let dim ,4=0 and x be an arbitrary
point of A. We have dim (A fl Sq(&)) < — 1, hence the intersection (/ fl Sq(a?)
is empty for some neighbourhood U of x. Then prof (S!\U)'^q + I. In virtue
of 3.5 (and by the assertions of 1.13, 1.25!), we get 3t^|C7 = 0 for i < q.
We now prove the general induction step. Let d = dim A and suppose the
implication proved for analytic subsets of dimension < d. Put A' = A fl Sq+d(S:)
and A" = A\ A'. By 3.5, %'A..® = 0 for i < 9. The set A' has dimension < J - 1
and satisfies the condition (b), hence by induction hypothesis, %A, SF = 0 for / < q.
The conclusion follows from the exact sequence
, .. -»• %!A, % -»• °HA^ -»• 3^,,2F -»• ...
(c) => (a). We will prove a more precise result; if {^}AW)X = 0 for a point
x e A and for / < q, then profAmX& ^ q + 1. By 3.3 we can find a neighbourhood
U of x and an element / e r(U, <3) such that the sequnc:
O-»- &\U -> ®\U -> S^/f^lU ^0
is exact; the arrow &\U-> &\U is the multiplication by/. We then obtain the
exact sequence on U
... -► 3^ -► WA^IJ®) -> %lAL1® -»•...
hence 3*^(37/^0* = 0 for 1 < 9 — 1. To conclude the proof we make use of
induction on 9 by applying corollary 1.21.

68 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Remark. For q = 0 we obtain some necessary and sufficient conditions for
the maps H°(U, SF) -> H°(U\ A, SF) to be injective. For q = 1 we obtain
conditions equivalent to the fact that the maps H°(U, SF) -> H°(U\ A, SF) are bijective
and H\U, ®) -»• H\U\ A, SF) injective.
In the following we shall draw some consequences of the vanishing theorem.
Corollary 3.7, Let x be a point of A such that Q^A^)X = 0 for i < q.
Then there exists a neighbourhood U of x such that %A3'\U = 0 for i < q.
Apply the proof of the implication (c) => (a), lemma 3.1 and the very
equivalence (a) <*■ (c),
Corollary 3.8. Let (X, 6) be a complex space, f eT(X, 6), V(J ) =
= Supp (<9//<9) be the set of the zeros of f and SF e Coh(X). Then the following
conditions are equivalent:
(a) dim V(j) n Sk+1(&) < k for all k;
(b) fx is nonzerodivisor for Wx, whatever xeX.
Proof, (a) => (b). Apply the theorem for A = V(f) and q = 0. Then
profk(/)SF > 1. Let now x be a point in V(J). There exists an element gx e 3(V(J))X
which is nonzerodivisor for &x. One of its convenient powers is fxhx, hx e 6X.
It follows that fx is nonzerodivisor for SF^. If x does not lie in V(J), then fx is
an inversible element and (b) holds too.
The inverse implication is an immediate consequence of the theorem.
We now reformulate 3.5 for complex spaces as a consequence of the theorem.
Corollary 3.9. Let X be a complex space, A <= X a closed analytic subset,
SF eCoh(J0, and q an integer. If prof SF ^ dim A + q + 1, then 3^8F =0
for any i < q.
Another consequence can be obtained if we consider that A is an analytic
set of dimension zero in the theorem.
Corollary 3.10. Let X be a complex space, x a point of X, S? e Coh(A'),
and q an integer. Then prof SF^ ^ q + 1 if and only if for any neighbourhood V
of x the canonical maps
H\V,Sf)^HXV\ {x}, S?)
are bijective for i < q and injective for i = q.
Corollary 3.11. Suppose (X, 6) is a complex space and A a closed analytic
subset of X such that prof A& > 2. Then X is connected if and only if X \ A is.
The proof follows from I. 4.9.
From the above we obtain remarkable results for perfect spaces and
implicitly for manifolds, for locally complete intersection spaces, for the pure (n — 1)-
dimensional analytic subsets of «-dimensional manifolds.
In particular, we get the following result of Thimm and Scheja [88], [70].
Corollary 3.12. Let (X, 6) be a perfect complex space and A a closed
analytic subset of codimension > 2. Then the restriction map
r(x,e)^r(x\A,G)
is bijective-

II. ANALYTIC LOCAL COHOMOLOGY
69
For the next corollary suppose that ^ is a reduced complex space and
let A be one of its closed analytic subsets. A holomorphic function / on X \ A
is called bounded at the points of A if for any point xe A, there exists a
neighbourhood U of x such that the function f\ U \ A is bounded.
Corollary 3.13. (Scheja [70]). Suppose X is a perfect reduced analytic
space. Then X is normal if and only if codim S(X) $s 2 (S(X) is the singular
locus of X).
Proof. The assertion X normal => codim S(X)^ 2 is a general property
of complex spaces [1], [15], [56]. Suppose now conversely, codim S(X) ^ 2.
Let x e J be an arbitrary point and fjgx an element of the total ring of quotients
of Qx, integral over Ox. Then there are a neighbourhood U of x and holomorphic
functions /, g, au. ■ ■ ar on U such that the germs of/, g in x are just fx, gx
and the following equality is true
uigy + a1uigy-l +... +or = o.
fig is a holomorphic function on U\ A where A is the set of all zeros of g.
By restricting eventually U, one derives that//g is bounded at the points of A.
From the first Riemann extension theorem [56], it will follow that there exists
a holomorphic function h on U\S(X) such that h =f/g on (U\ S(X))\ A.
By corollary 3.12, this function extends to a holomorphic function on U; hence
fxls.x e ®x> and ®x is normal ring.
Corollary 3.14. (Scheja [70]). Let (X, 6) be a normal space and A a closed
analytic subset of codimension > 2. Under these assumptions, the restriction map
T(X, 0)^T(X\A,tG)
is bijective.
According to the theorem by means of 1.24.
Recall now that an (9-module SF is- said to be reflexive if the canonical
morphism
SF -► (8F)V = Hom&{Home(S, G), 0)
is an isomorphism.
Corollary 3.15 (Serre [80]). Suppose X is a normal space, A <= X a closed
analytic subset of codimension ^ 2, and & e Coh(A') a reflexive sheaf. Then
prof^oF ^ 2 ; in particular, the canonical map
r(x,®)^r(x\A,&)
is bijective.
Proof. Denote by i the inclusion X\AcX. Let § = Home(W',&),
hence SF = Hom0(§, 6). We have /*SF = Hom{i*q, Gx\a), hence /*/*8F =
= i*(Hom(i*§, 6X\A)) = Hom($, i*Gx\A) = Hom(§, Ox) = S: (the second equality
follows by the adjoint functors i# and /* and the equality i*6x\A — ®x holds

70 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
by the previous corollary). From the exact sequence
0 -»• %°A& -> ^ -»• ij*& -> %A^ -> 0
we get ?f^8F = 3C^gF = o, and the required conclusion follows now from the
theorem.
§ 4. The finiteness theorem
We still use the notation from the preceding paragraphs, Denote by Sm(S!\X\ A)
the topological closure in X of Sm(8r\X\ A). It is easy to check that this coincides
with the union of the irreducible components of 5m(Sr)which are not contained
in A.
The main result of the paragraph is the following finiteness theorem for
local cohomology of Siu and Trautmann [83], [92], [84], which is analogous to
a theorem of Grothendieck in algebraic geometry.
Theorem 4.1. Let X be a complex space, A <= X a closed analytic subset
and oF e Coh(A'). For an integer q ^ 0, the following two conditions are equivalent :
(a) dim A n Sk+q + 1(S;\X\ A) < kforaflk;
(b) the sheaves "X'A8r are coherent for 0 < / < q.
In order to prove the implication (a) => (b) we need some preliminaries,
wherefrom we derive that in condition (a), the sheaves 3C^(0 < / < q) are locally
annihilated by some powers of & — 3(A). Considering the fact just stated,
the proof of the implication follows easily by induction on q.
Lemma 4.2. Let D be a domain in C", A <= D a closed analytic subset
oj dimension < d and S e Coh(Z>). For 0 < q < n — d, let 3q be the maximal
ideal-sheaf which defines 5^+lJ^1(Sr). Then, for any open subset X <= <= D whose
topological closure is holomorphically convex, there exists an integer k such that
for any open U of X and for all i < q,
T(U, 3q)"HA(U, SO = 0.
Proof. We first consider the case q = n — d. According to the hypothesis
on X, there is an exact sequence of the form
. .. _► 6ifl -»• .. . -»• ... G5° -> SF* -»• 0
on a neighbourhood of X, where 6 is the sheaf of germs of holomorphic
functions in C" and ff* the dual of S\ By applying the functor of passing to the
duals, we obtain the following sequence of morphisms between (9-modules
(I)
0JtW^3*°-^6^

II. ANALYTIC LOCAL COHOMOLOGY
71
where <p_x is the composite map Br -> S^** -> (9s». On Z>\ S„_1(S;) the sheaf &
is locally free; therefore the sequence (I) is exact on X \ S„_1(Si).
We define for / ^ — 1 the sheaves
<&i = Im (p,,!, S>; = Ker <p; and £,- = %>,•/$,-.
From above, Supp £; <= 5„_1(âî). By Nullstellensatz we can find an integer /
such that S'q£i\X = 0. Therefore
(II) T(U, 3q)'HkA(U, £,)=0
for any open subset £/ of A" and for all A: $s 0.
We are going to prove the following assertions by descending induction
on j ^ 0:
f There is an integer / > 0 such that for any open set U of X,
* ; j T(U, 3,)'tfi(C/, S,) = 0 for i < q - j.
f There is an integer / $s 0 such that for any open set U of X,
** J \ r(U, 3qyH'A(U, 8Sj) = 0 for / <q-j.
Forj > q both assertions are automatically fulfilled. From the exact sequence
0 -> 3>j -> % -> S,j -> 0 we get the exact sequence
H'aKU, £,) - //i(C/, «,) - //<,(£/, &,),
wherefrom derive, by means of (II), the implication (*),-=>(**),■. Suppose j $s 1.
s
By using the exact sequence 0 -> S?,-_i -> (9 J_1 -> ofc,- -> 0 and the equality
H'A(U, <9) — 0 for i < 9 (q = n — d), we get the isomorphisms HA~X{U, $_,.) ~
ci HA(U, %_i) for j < 9 — _/ + 1; hence the implication (**); => (*)j-iU ^ 1).
Thereby the assertions (*) and (**) are proved.
From the exact sequence 0 -> %_1 -*■ W -» $„ -> 0 we deduce the exact
sequence
... - //i(c/, %_,) - //^(c/, so - #j(c/, «„)-...
But S> _x = £_x and the conclusion of the lemma easily follows from (II) and (**)„.
The lemma being proved for q = n — d, we shall now study the general
case and make use of descending induction on q. Let q < n — d. Under the
assumption on X, there exists an exact sequence
()->■<$->■ (9*->8F->0

72 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
on a neighbourhood of X. Then we get the equality Sq+i-1(Si) = S(g+1)+d_1(G)
and both sets have 9q as maximal ideal-sheaf. By 3.5 and 1.16 we deduce the
isomorphisms
ha(u, ff) « h'/\u, q)
for any open subset U of X and for all i < q. Hence, for i < q, H'A(U, SF) is
annihilated by some power of T(U, 3g) if and only if HA+1(U, éj) is. We conclude
the proof by induction hypothesis.
Lemma 4.3. Let D be a domain in <£,", A <= V two closed analytic subsets
of D, &■ e Coh (D), and q < n an integer. Denote S^+q = Sk+q{S:\D\ V) and
suppose dim A fl Slc+q < k for all k. Let 3 be the maximal ideal-sheaf of V and
let X be a relatively compact open subset of D whose topological closure X is
holomorphically convex. Then there exists an integer / ^ 0 such that for any-
open subset U of X and for all i < q,
T(U, 5)< H'A(U, Sr) = 0.
Proof. For an integer k $s 0, we set Ak = A fl Sk+q and Ak+1 = Ak+1\
\ S/c + q- We will establish by induction on k the existence of an integer / ^ 0 such
that t(U, 3)' H a (U, Sr) = 0 for i < q and U an open subset of X and the lemma
will follow since for k = n — q, Ak= A.
First of all, consider the case k = 0. Then by hypothesis, dim A0 < 0 and
our assertion follows in virtue of the previous lemma since in some neighbourhood
of A0, we have S^^ïï) <= V; hence 3 c 3(5^(30).
We now pass to the general induction step. Since 59+(t+1)_1(oF) n (X\
\ Sq+iç) <= V and (X\ Sq+k)~ = X (otherwise, V would contain on open, hence
V = D, 3 = 0, etc.), we deduce by 4.2 that
T(U\ SUq, 3)1 HAk+(U\ Sq+k, SF) = 0
for / < q and for some integer / ^ 0 (which depends on X only). By the excision
lemma l.l,H'A (U\ S'+k, ff) = HA» (U, S-). Accordingly, T(U, $)lHA>> (U, Sr) =
= 0 for / < q. By the exact sequence
. .. - Htf (V, S?) - HA(U, SF) -. HA {V, Sf) - HlA» (U, SF) -. ...
Iz + 1 fc k + 1 k + 1
the assertion for Ak + l follows from that for Ak.
Here is the proof of the theorem.
The proof of the implication {a) => (b). We proceed by induction on q.
The case q = 0 follows by 3.2. Assume now the implication proved for integers
smaller than q and prove it for q.
The problem is local in nature, hence we can consider X a relatively compact
open subset of a domain D <= (£", X holomorphically convex and SF defined on D

II. ANALYTIC LOCAL COHOMOLOGY
73
(by 1.13 and 1.25, the problem agrees with the immersions!). By lemma 4.3
(applied to A = V and q + 1 instead of q), there is an integer / $s 0 such that
r(t/,3)' HtA(V,&)=0
for all i < q and any open subset U of X where 3 = 3(A). Hence 3'%'^ = 0.
We now notice that K0^/%°ASr) = 0 and %A(8r) = 3^(gF/3l'<W) for z ^ ]. Since
3f^8F is coherent we can assume however S^SF = 0 by replacing 8? by 3r/%A$r.
Let .v be an arbitrary point of A. By 3.3 there exist a neighbourhood U of x
and a section / e r((/, 9) such that the sequence
0-, SF|(/->gF|(/->(SF//gF)| (/-►()
is exact where the morphism SF| (/ -» SF| (/is the multiplication by /'. Since 3'%AW =0,
we get the exact sequence
0 -». %'Ai® -». Sf^SF//^) -». 31'^ -»• 0
for i < q. Since Sk+q(ar/f'S) cr 54+(t + 1(^), we have
dim ,4 n 5(4_1)+/c+1((SF//gF)|(/\/4) < jfc, for all Jt.
By induction hypothesis, the sheaves W^fâ/fSr) and %'A~lS; are coherent on U
for z < 9 and by the above exact sequence, the coherence of 2C%oF around x follows.
The proof of the implication (b) => (a). If q = 0, then SJi^X \ A) cr
<= S^SF/SC^gF). Since ?Q(S73£° SF) = 0, the implication will follow by the vanishing
theorem.
If q = 1, then we have the exact sequence
0 -»• %°A® _► SF -»• $°,aF -». 3C^ -»• 0,
where oJt^cF is the sheaf U <-^T(U\ A,^) defined in § 1. As 3('^gF is supposed
coherent,3lA3f is coherent and the conclusion follows by theorem 3.6, since 3(^(J>° âr)=
= 0 for i = 0, 1 and 5,„(SF|A"\ A) cr Sm(3{A$).
The general case q^-2 will be proved by induction on q. Assume the
implication already established for integers smaller than q and prove it for q. We have
dim ,4 fl ^+,(^|*\ A) < k, for all k.
On the contrary, assume that there is a point xe A such that
âimxA[\Sk+q+1(9\X\A)=k+l

74
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
for some k. Then there is a point y e A r\ Sk+q+1(W\X\ A) such that y <£
ft Sk+q(8r\X\A) and
<*) dimyA (\~Sk+t+1(S\X\A)=k+l.
For a suitable neighbourhood U of y we have U fl ~St+4(3r|A'\ ^) =0. By (*)
and since any component of 5t+9 + i(ot;|A'\ A) is not contained in A, we get the
inequality dimy Sk+q+1(&\X\ A) ^ k + 2. By restraining eventually U, we can
find a closed analytic subset B in U r\'sk+q + 1(^\X\ A) such that dim B =
= k + 2, dim ^ fl fi = k + 1 and fi = (B\ A)~. Since 9 ^ 2 and by shrinking
once more U, we can choose f eT(U, 6) such that the analytic set V(f) of all
zeros of/ contains B but does not contain any (/ + q + l)-dimensional
irreducible component of S,+4+1(3r|A'\ A) for / ^ k. So we have
dim V(f) fl S,+€+1(ff|*\ /Q < / + q, for / ^ At.
Since £/ fl Sk+q(&\X\ A) = 0, we get
dim V(f) n 5m + ,(S:) fl (U\ A) < m for each w.
In virtue of 3.8, fi is nonzerodivisor of fti for all ze £/\ A. Then we obtain an
exact sequence on U\ A
0 _> g; _». g; _». §;//§; _». 0
where the morphism SF -> ff is the multiplication by /. From the definition of
the sheaves SHA (§ 1) we derive an exact sequence on U
... -»• §i'A & -»• ji^ -»• si'A&lf®) -»• ji^+1sf -»•
Since q ^ 2 and J^SF are coherent for z < 9 — 1 (by hypothesis and by corollary
1.10), it follows that the sheaves cH'^ïï/f&) are coherent for i < q — 2. Accordingly,
the sheaves 9^(SF//SF) are coherent for i < q — 1. By induction hypothesis,
dim A fl Sl+q((S:/fS:)\U\A) < /, for all /.
Since B\A c Sk+q+1(S:\X\A) and prof ,(87/80 = prof, # - 1 forzefi\,4cr
<= F(/), we deduce fi\ A c Sk+q((®/fg:)\U\ A). Therefore, dim ^ fl fi < A:,
which contradicts the very choice of B.
The implication (b) => (a) and the finiteness theorem are proved.
Corollary 4.4. Let X be a complex space, A <= X a closed analytic subset,
and SF e Coh(A"). // for any point x e A there is a neighbourhood U such that
prof (8-|£/\ A) ^ q + dim., A -\- 1, then the sheaves S^SF are coherent for i < 9.
If SF is a coherent analytic sheaf on a complex space A" and Supp SF contains
at most a point x, then by Nullstellensatz, one gets dim<E &x < 00. Since
Supp ^^ <= ^, we obtain the following corollary from theorem 4.1:
Corollary 4.5. Let X be a complex space, x e X, SF e Coh(A") and q an
integer. Then the linear spaces H'X(X, SF) are finite-dimensional for i < q if and
only if there is a neighbourhood U of x such that prof (&\U\ {x}) $s q + 1.

II. ANALYTIC LOCAL COHOMOLOGY
75
§ 5. Absolute local cohomology
Let J be a complex space, 6 its structural sheaf and d ^ 0 an integer. For
any open subset U of X denote by 21/cV) the family of closed analytic subsets
of U, of dimension < d. Let & be an c?-module. Whenever A <= B lie in %d(U),
HA(U, §•) c Hg(U, &•). These inclusions are functorial and for derived functors
we obtain natural maps H'A(U, 8) -► H'B{U, SF). If U c V and A e 1ld(V), then
A n V 6 91/(7). The restriction map defines natural maps H'A{V, W) -> HA^u(U, W).
We thus obtain the presheaves
U^\im{HA(U,W)\AeSlld(U)}.
The associated sheaves are called the sheaves of d-absolute local cohomology and
are denoted by Xd& [84]. If Sf eCoh(X), then "%%& is equal to the subsheaf of &
of all sections whose supports are of dimension < d.
The sheaf 3^ is also denoted by W [d]; it is called the d-sheaf of Thimm [89]
and it enjoyes remarkable properties (prop. 5.3).
In the same manner, one can define the d-absolute gap sheaves 3Cd(&), by
means of the presheaves
U h> lim {H-(U\ A, §9| A e VL^U)}. [84]
The sheaves %'d& and S{'dW have a natural structure of (9-modules. By the exactness
of the inductive limit, we derive from 1.10 the exact sequence
0 -► ftd® -► 8 -► 8Ld® -> %d& -»• 0
and for i ^ 1, the isomorphisms
&d® ~ ïi+1f.
The results given below will be stated in terms of the sheaves 3£jSF; by
the above relations one can then deduce results for the sheaves JljSF.
Theorem 5.1. Let X be a complex space, d > 0 an integer and & e Coh(A").
For an integer q ^ 0 the following conditions are equivalent:
(a) dim Sfc+8 + 1(SF) < &:/or a«>> k < d.
(b) For any locally closed analytic subset A of dimension < d, %'A& = 0
for i < q.
(c) Xa® = 0 /or i < 9-
Proof. The implication (a) => (6) easily follows by 3.6. Prove now (b) => (a).
Suppose on the contrary that the assertion (a) is not true for some k < d. Then
we can find an open subset U and a closed analytic subset A in Sk+q+1(S:) fl U
such that dim A — k + 1 < J. By the assertion (6) and theorem 3.6, we have
k + 1 = dim ^ = dim ^ fl S/t+^+iW < k, hence a contradiction.

7G ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The implication (b) => (c) follows straightforwardly from the definitions.
It only remains to prove the implication (c) => (b). We use induction on q. The
case q = 0 follows if we notice that %°A&\U is. a subsheaf of 3^1 £/ whenever
A and U are as in (b).
Let us suppose the assertion (b) is verified for indexes /, / < p < q and
we will actually verify it forp + ]. Let (/be an open subset of X and A e 1ld(U).
Consider an arbitrary point x of U. We are going to prove that ("X^ 1gF)x = 0.
Let V be a neighbourhood of x contained in U and let £ e HA+1(V, <F).
Since (3f5+1Sr)I =0, there exist a neighbourhood W of x in F and Be%(^)
such that i fl WcB and £ has the image null through the composition of natural
maps
H»/\V, SF) - HA+nlw(W, §) - EPB+\W, SF).
Let B'=B\A. Since 3(^=0 for * <A we have hb- (W, $)=H"b(W\ A,W) =
= 0. Hence the map Hfaw(W, ^) ->■ H%+1(W, âF) is injective and the image of ç
in HfâyiW,®) is null. Thus the germ defined by £, in (Xyl âF)^ is zero and the
proof is completed.
Corollary 5.2. Let X be a complex space, d ^ 0 an integer and SF e
e Coh(A"). The following conditions are equivalent
(a) dim Sk + 1(&) < k for any k < d.
(b) 3(^gF = 0 for any locally closed analytic set A of dimension < d.
(c) c°d% = 0.
(d) gF has no sections ^ 0 whose supports are of dimension < d.
(e) For any point x e X, the &x-module &x has no associated prime ideals of
dimension < d.
Proof. The equivalences {a) o (b) ■**■ (c) are a consequence of the theorem.
The equivalence (c) <=> (d) follows from the definition of the sheaf 3(J}âF. We now
prove the implication (d) => (e). Let p be an ideal of Ass Wx. Then there is a
neighbourhood U of x and a non-null section s of T(U, âF) such that p = (0: sx).
Let ^4 be the analytic set defined by p in a neighbourhood V of x. By shrinking V
we may assume dim A < dim <3Jp. We have Supp s <= A, hence dim (<9x/p) > d.
We must show only that (e) => (a"). If s e &x is the germ of a section s, then
the annihilator sheaf GL of j is coherent and &.x is contained in some prime ideal
of Ass âF,.. Since Supp s = V(â), we derive dimx Supp s > d + 1.
Let us pass now to analysing the coherence of the sheaves %'d.
Proposition 5.3. Let X be a complex space, d ^ 0 an integer and âF e
e Coh(A"). Then the sheaf %%& is coherent and equal to 3Câd(S;)âF. Moreover, the
irreducible components of dimension d of Supp 3C^gF and S/âF) are the same.
Proof. Since dim SJ&) < d,%ïsdW)® c %%$. Let * be a point of *\ Sd(&)
and U one of its neighbourhoods such that U fl S^âF) = 0. By 5.2, 3('>|t/ = 0.
Hence Supp SC^SF <= S/.&) and Sf^âF coincides with %sd(§)& and in particular is
coherent (3.2).
We prove now the last assertion. Since Supp "Jl^âF cr tS/SF), we have to show
that any irreducible component S of Sd(8r) of dimension d is an irreducible cqm-
ponent of Supp3CS?- This assertion will be a consequence of the inclusion S c

II. ANALYTIC LOCAL COHOMOLOGY
77
<= Supp lltdiaF). Thus, it remains to check this inclusion. Let x be arbitrary in S.
Suppose on the contrary that (2frfSF).* = 0. Since %%& is coherent, %%& = 0 in some
neighbourhood U of x. By applying 5.2 we deduce that dim (S/gF) fl [/)<
< </ — 1. But 0^Sn C/c S/SF) fl £/, hence a contradiction.
Corollary 5.4. Supp (3l^ ^V^S-i^) /■? '»e ««/on o/ a// irreducible components
of dimension d of Sd(&).
If M is a subset of X we say it is of dimension < d if for any x e M there
is a neighbourhood J7 of x and ^4 eSL/t/) such that M fl J7 <= ^.
Lemma 5.5. // the sheaf 3fj/cF is of finite type, then
dim Supp 3C^aF < d.
Proof. Let xeX,U one of its neighbourhoods and ^,.. .,sr eF(U, 3C^)
generating 5fj,cF | C/. By shrinking J7 we can find ^ e 21/(7) and elements ^,...,^6
e HlA{U, SF), which represent j^.. .,jr. It follows that (Supp 3C>) n U c A and
the lemma is proved.
Theorem 5.6. Let X be a complex space, gF e Coh(A") and d, q > 0 two
integers. The following conditions are equivalent:
(a) dim Supp 2i j+ XSF < J /or / < q.
(b) dim 5d+,+1(ff) < d.
(c) 3f'rfcF /j coherent and equals ^'sd + q+1(&)(&) for i < q + 1.
(c') 3CJ/SF fa coherent for i ^ q + I.
(d) 3f'rf+poF fa coherent and equals 'K.'d(&) for i ^ 9 + 1 — p.
(^') 3C'rf+pâF fa coherent for i < 9 + 1 — p.
/V00/. (a) => (A). Let S = Ù Supp 3Crf+1SF. We have 3C'rf+1âF =0 on X\ S
; = i
for/<9.By5.1,dim(5lJ+4+1(SF)n(A'\ 5*)) <J. Since dim 5 < rf, dim 5,j+4 + 1(aF)<J.
(A) -* (c). Let 5= 5,j+4+1(SF), hence dim 5 < d. Since prof (SF|A"\ 5) ^ J +
+ q + 2, it follows by 4.4 that 3f^cF is coherent for / < q + 1. Let [/ be an
arbitrary open set and AeM^U). By 3.9, X^aF|£/\ S = 0; hence 3C£aF = 3^UiSaF
on £/. Accordingly, 5C^«F = WjW for / s= q + 1.
(c') => (b). Let 5; = Supp 3l ^gF. By the previous lemma, 5; is an analytic
set of dimension < d for / < q + 1. Put S = (J SV Since SK^IA'X 5 = 0 for
1=1
/ < q + 1, by 5.1 the following inequality may result: dim 5iJ+4+1(gF|A'\ S) <
< d — 1 < d. But dim 5 < d, hence dim 5,(l+8+1(S:) < d.
The implication (6) =* (d) can be proved exactly as (b) => (c), by writing
Sd + q + l(&) = 5(rf+p) + (,-p) + l(GF).
The implications (d) => (d') => (c'), (d) => (a) are obvious.

78 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Corollary 5.7. The following conditions are equivalent:
(a) dim Supp SCâ+iW < d.
(*) %«d^w = grsff.
(c) dim S„+1(ff) < d.
(d) $2^ w coherent.
The proof follows by 5.3 making use of the exact sequence
0 ->• W$ ->• SF -»• <&> -► 3^SF -► 0.
Application. We are going to construct a natural complex of sheaves
for any analytic space. The construction will be done in several steps.
(a) Let V be an open subset of a numerical space (£"• For an integer q
and an open subset U of V, we denote <S>"{U) = 2I„_4(£/), hence $«(£/) is the
family of all closed analytic subsets of codimension ^ q in £/.
Let W be an analytic sheaf on V. Denote by 9C3>s(39 (respectively ^Si/oi + i^))
thesheaf associated to the presheaf U h> #£(£/, SF) (respectively U*->H^ v)/a,,+ i((7
(£/, SF)), /> and 9 being arbitrary integers. Under the above notations,
^.(so = *;_.*•
By means of 1.7 we deduce for an integer p the exact sequence
• • • - K^ - x'w+m - 3CUS9 - • • •
For the index p + I, we have the exact sequence
• • • - km§) - **:1"/<W.+.(S9 - ^.co - • • •
From these exact sequences we get, by composition, the morphism
If we write the above exact sequences for p = 0, the morphism
is obtained. It is easy to check that in this way we get a complex
o- K°,^) - *W*) - ^/o^) - • • •

II. ANALYTIC LOCAL COHOMOLOGY
79
which is called the Cousin complex associated to the sheaf 8r, together with the
augmentation morphism W -> W^A^)-
We now consider the particular case & = 6.
Lemma 5.8. The Cousin complex associated to the structural sheaf <3 is
a resolution for <3; in other words the sequence
is exact.
Proof. We shall prove that Wm(®) = 0 for q ^ p; then the lemma easily
follows from definitions only.
Since the assertion is obvious for q — 0, we may assume q ^ \. Obviously,
%%<,(?) = K-„e- If P < <?> then ^_q6 =0 by theorem 5.1. Let us consider an
arbitrary point x e V and let U be a neighbourhood of x. For any A e 9I„_4([7)
we will prove that by shrinking U around x there exists B e 9I„_4(£/) such that
A c B and HPB{U, &) = 0 for p > q. Then it will result that (jHpn-q&)x = 0 for
p > q and therefore the conclusion. By shrinking eventually U we can further
find an element C in 9I„_4(£/) such that A <= C and dim.,. C = n — q.
In this way we can assume, for the assertion we are interested in, that dimx^ =
= « — 9. Apply the normalization lemma 1.27 for Qx and for the ideal 3(A)X.
There exists a finite injective morphism C{^"i>- • -, %„} -*■ ®x such that /_1(3(^)JC) =
= (A-!,..., Xq). Passing to germs of analytic spaces, and shrinking again V, we
can find an open polydisc U' in C" which is centred in origin, and a finite morphism
F
U -> V such that F(x) = 0, and the morphism induced by F in x between the
corresponding local rings is/. Let us denote by xu...,xn the coordinates in V.
If we restrain U and V we can assume that F is surjective (as a consequence
of the finiteness and of the equality dim U = dim V), that F#(6^ admits on U'
a finite resolution with free sheaves of finite rank and moreover, the functions
X;F(i = 1,..., q) vanish on A. We have A <= F~l(A') where A' = U' n {*i = • • •
... = xq = 0}. If we get B — F"\V) then one will get dim B = n — q and
F(B) = A'.
If W is a sheaf of abelian groups on U, then H%(\J, §9 ^ H°A,(U, F*(&)).
By using this and the exactness of F* we derive canonical isomorphisms
Hf/JJ, 6) =, H'AW, FM).
To conclude with, it is sufficient to show that H"A, (V, ^(f1^)) = 0 for p > q.
By hypothesis on F^{QV) we have only to prove that HPA,{V, t'v.) = 0 for p >q.
But we have the natural isomorphisms Hp-l{U'\A',eu,) ~ HpA.{U',^v) ip>q>\)-
If we put £/,' = V n {x; = 0}, 1 < / < q, we get a Stein covering of U'\ A'
by q Stein open subsets and the required conclusion follows.
We shall denote by £'y the resolution of 6V given by the lemma, and by S(v
the complex (£v®sv Qv) [n] where Qv is the sheaf of germs with holomorphic
forms of maximal degree, and the bracket [n] means that the complex is translated
to the left by n steps.

so
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(b) Now, we study the behaviour of the previously defined complexes
with respect to immersions.
Lemma 5.9. For any immersion V -* W with V, W as open subsets in
numerical spaces, there exists a natural isomorphism
J: Si'v ~ j!>Hom@tVU*®v$Uv)-
Moreover, the correspondence f h>/ agrees with the composition of immersions.
Proof. If / is an open immersion, the construction of/ is obvious.
Let us pass to the case of the closed immersions and consider the following
situation: the ideal-sheaf given by V in W is t6w for some teT{W,6w). Then we
have the exact sequence
o -► ew U ew ^f*{ev) -► o.
We will establish an isomorphism
Aî:~îï*~f* Hom3 Jf+ey, &1).
for any integer q. The above exact sequence yields the exact sequences
0 - HomeM^v, K« „>«+'(<2W) - K< /o'+'OV) ^ K< w+Wwl
W w
q-1 q-1
^ m ■ ' ^ t\r k t*r> ^ tit-
Vf' W
w ' w w' w
Since %^q (6W) = 0 for p*£q(see the proof of lemma 5.8), it follows j "that
w
I q Ciq+i(6w) =0. Then we deduce an isomorphism
w' w
**« „,+.(/*<SV)^ Horn er(f*&y, £■%■)■
If § is a sheaf of abelian groups on V, then
r 0<v-\r-wnv)(f-\i>) n r, <?) = r(1,^(z))(i), /*<?)
for any integer 9 and any open subset D of W. Whence the natural isomorphisms

II. ANALYTIC LOCAL COHOMOLOGY
81
According to the above said, the construction of the isomorphism A? becomes clear.
We then define the isomorphisms
by means of A" and of the natural morphism Çlw ->/!|<(fiI0, which assigns the form
2m^)\V to any form <\> A at. If we replace t by another parameter t', then A,- =
= (t/t'\V) A,. It is easy to check that the difference between the morphisms Qw ->
->/.,.(fiK), associated to t' and t is just a multiplication by t'/t\V. Then the
morphisms J" are independent on the parameter t and the family / = (/«) is an
isomorphism of complexes.
/
If the immersion V -> W is decomposed into a finite number of immersions
of the above type, then the construction of / can be made recursively, by
compositions. One can verify that / does not depend on the considered decomposition.
In particular, this fact allows us to construct / for an arbitrary immersion/
(locally, this is of the above type; the morphisms defined locally can be glued
together,. ..) and also to verify the functorial character of the association/ h>/!
(c) We now pass to the general case of a complex space X. For an open
subset U of X such that there is an immersion U -* V, V open subset of a
numerical space, we have
&l = <v*HomeM*®u, K),
where 3C". is the complex previously defined. If V is another open set enjoying
the same property, the immersions obtained from <p, <p' for U fl V can be refined
by means of a third one. By the above lemma we get an isomorphism
t„.^:K;.|C/ n U'c*3t'u\U n V.
If we consider a covering of X by open sets as above, then the
isomorphisms t satisfy the customary relations of compatibility (apply again 5.9) and thus,
the complexes 3t'v can be glued together and define a complex 3{.'x-
§ 6. The separation theorem
We use the notation from the previous paragraphs and assume in addition X
with countable topology. We have the exact sequence
a'"1 . « .
... -► H'-\X\ A, 30 > H'A (X, 50 -► H'(X, 3F) -► ...
The invariants H'(X,8r), H'(X\ A,S-) have natural structures of topological
vector spaces, quotients of FS spaces (Fréchet-Schwartz). For / ^ 1 we endow
6 - c. 2398

82
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
H'A{X, &) with the finest topology such that S'_1 becomes a continuous map. The
map a results continuous, as aS!_1 = 0. The invariant HA(X, &) can be endowed
with the topology induced from r(A", 8F) and then it is an FS space, as kernel
of the map r(X, 8) -> T(X\ A, SQ.
If X is Stein, then the spaces HA(X, &) and Hi~1(X\ A, §9 have the same
topology for i ^ 2 and H\{X, §) has the quotient-topology T{X, ®)/r(X\ A, &).
The main result in this paragraph is the following separation theorem
for local cohomology of Siu and Trautmann [84]:
Theorem 6.1. Let X be a complex space, A <= X a closed analytic subset,
q an integer and §• a coherent analytic sheaf on X such that the sheaves %'A&
are coherent for i < q. Then for any Stein open subset Q. of X, HA(Q, SF) is a
Fréchet-Schwartz space for i < q.
In order to prove this theorem we need some preparations.
Lemma 6.2. Let A be the unit polydisc in <£," and A = A ft {zd + 1 = ... —
= z„ = 0}, where 0 <(/<«— 1. Then HA~d(A, 6) is separated.
Proof. Let U, = A n {z, ^ 0} for d + 1 < i < ». Then nf = {Ui+1, ...,U„}
is a Stein covering of A\ A. Consider the case d < n — 2. Since HA~d(A, 6) ^
~ H"-d-1(A\ A, 6), we must prove that H"-d-l(\t, 6) is separated. Let U* =
= H C/;. Since Z"'^1^, 6) = T{U*, 6), every cycle \ e Zn~d~\M, 6) is a
Laurent series
V<J+i Vn= -00
where the coefficients aVd+1 v„ are holomorphic functions in the unit polydisc
of <Zd. It is easy to check that leff,-d~\M,6) if and only if aVd+1...v„ =0 for
v,j + 1 < — 1, ..., v„ < — 1. By Cauchy's integral formula, it follows that
B"~d~1(i\l, 6) is a closed subspace of T{U*, 6) (this or.e endowed, as usual, with
the topology of uniform convergence on compacts). Then H"~d~l('\l, 6), hence
HA~d(A, 6) is separated. The case d = n — 1 is similar.
We now return to the general situation (X, A, gF). For an open subset U of X
denote by N'A(U, &) the topological closure of zero in H'A(U, gF). If V is an open
subset of U, then the restriction map H!A(U, gF) -> H'A(V, oF) is continuous. It results
that $(NA(U, §0) <= •A^(P, SF). Denote by S^(âF) the sheaf associated to the presheaf
U h> NA(U, &), U => F h> the restriction map
.&&(£/, ff)-W,ff) given by p.
Lemma 6.3. Let D be a domain in C" and A <= D a closed analytic subset
of dimension d. Then aH'A(6) = Ofor i < n — d and HA~d(D, Ô) is separated.
Moreover if D is Stein, then H1~d{D, 6) is just an FS space.
Proof. Since %lA6 = 0 for i < n - d, HnA-\V, 6) = r(V, K"A-d6) for any
open subset U of D (corollary 1.15).

11. ANALYTIC LOCAL COHOMOLOGY
83
For i < it — (/the equality 3lA(6) = 0 is a consequence of the equality 3C^<9 =
= 0. If oZA~d(6) = 0, then by the commutative diagram
N^-\D, 6) —> r(Z>, 8H"A-d6)
I I
H\-\D, 6) = F(D, ï(nA-d6)
we can deduce the equality NA~d(D, &) = 0, whence HA~d(D, 6) is separated.
Suppose in addition D is Stein. For d = n, H°A(D, 6) is an FS space. If
d = n — 1, the separated space HA(D, 6) is isomorphic to F(D, <3)/r(D\ A, &), hence
itisFS. For d^n — 2, HA~d(D, 6) is topologically isomorphic to H"-d-\D\A, 6)
and again it will be FS.
So the lemma is proved as soon as 8!lA~d(&) = 0. Let A' be the set of the
singular points of A and A" = A \ A'. Let x be a point of A". There exists
a neigbourhood U of x contained in Z>\ A', which is isomorphic to the unit
polydisc in C" s°ch that the analytic set A n U corresponds to the set A fl
n {zd+1 =...=-„ = 0}. By the preceding lemma, HA~d(U, <9) is separated.
Therefore SflA~d€\D\ A' = 0. Taking into account the antecedence we derive
the separation of HA~d(D\ A', 6). Since dim A' < d, HA7d(D, 6) = 0, we have
the exact sequence
0 -» HA~d(D, <S>) -► HA'd(D\ A', <S>)
where the second map is continuous. We thus conclude that NA~d(D, 6) = 0.
By substituting for the domain D an arbitrary open set of D, it follows àlA"d6 = 0.
Lemma 6.4. Let D be a domain in <£.", A cr D a closed analytic subset of
dimension < d, and SF e Coh(Z>). For 0 < q < n — d, let 3q be the maximal ideal-
sheaf associated to Sq+d_1(S:). Then for any open set X cr cr D such that the
topological closure X~ is holomorphically convex, there exists an integer I ^ 0 so that,
for any open subset U of X,
F(U, 3jNA{U, 9) = 0.
Proof. We use the notation considered in the proof of lemma 4.2. Suppose
q = n — d. As in lemma 4.2 we can prove by induction the following assertions
. [There is an integer / > 0 such that, for any open subset U of X,
* J 1 F(U, 3q)> HA-'\U, £,) =0 (j > 1).
, [There is an integer / ^ 0 such that, for any open subset U of X,
** j\f(u, sqy ha-\u, a,) = o u > i).
In particular, there is an integer / such that F(U, 9,)' H^\U, $>i) = 0 for
any open set U of X. Then from the exact sequence
0 - H"A\U, «,) X HA(U, So) A HA{U, 0'")

84 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
and by the previous lemma, we obtain ol{N1a{U, 2>0)) = 0. Hence N%U, 8>0) is
contained in Im è and is thus annihilated by T{U, 3q)1. By means of the exact
sequences
... - H%-\U, *o) - H%U, So) - //?,(£/, So) - ...,
... - tfS(C/, £_,) - HA(U, 9) - //^(C/, So)
and taking into account that Supp £; <= 5B_1(S:) (which gives that some powers
of 3? = 3(,S'„_i(gF)) annihilate the sheaves £;, and the associated cohomological
invariants), the conclusion of lemma in the case q = n — d follows easily.
Consider now the case q < n — d. We have an exact sequence
0 - HA(U, ff) - H'A+1(U, <?)
and since the coboundary map is continuous, we get N9A(U, gF) <= N"A+1(U, §).
The conclusion follows from the case q = n — d, by descending induction on q.
Lemma 6.5. Let D <= (£" be a domain, V a closed analytic subset of D,
and gF e Coh(Z>) such that 5f^cF = 0. Let 3 be the maximal ideal-sheaf of V. For
any holomorphically convex compact K of D there exist a neighbourhood Y and
sections fv .. ,fp e I"(7, 3) such that (fi)x is nonzerodivisor of Wx for all x e 7(1 <
</</>) and, for some integer m ^ I, 3"'\K is contained in the ideal-sheaf generated
on K by f !,..., fp.
Proof. There exist a neighbourhood Y of K and sections gx,..., gq e I"( Y, S)
such that gu...,g9 generate 3\Y. Since %°y® = 0, dim V n Sk + 1(&) < k for all k.
Denote by Sk+1 the union of all irreducible components of dimension k + 1 of
7 n Sk+1(&). Choose a countable set A = {xv}, dense in l_J St+1\ F such that
k
A n 5fc+1 is dense in St+1\ V. For each xv, (gx(xv),..., gq(xv)) 9e 0. Then, in the
space C? with coordinates zlt..., z4, there exists a linear form V a;z;(a;e C)such
< = i
that J] a^,(xv) 7^ 0, for each v. Let A = £ aig; e T(7, 3). Since A0O ^ 0 for
;=i <=i
any v, dim F(/i) H 5t+1(GF) < k for all &:. By corollary 3.8 (A)* is nonzerodivisor
of SFx for all x e Y. If F(A) = V, the lemma is over by Nullstellensatz. Otherwise,
replacing A by a countable dense subset B of ((J ^t+i) U V{fù\ V whoseinter-
k
section with Sk + 1 is dense in St+1\ V and whose intersection with every irreducible
component W of V(Jl)\ V is dense in W\ V, we can find an element/2 e T(7, 3)
with AC*) t^ 0 for each xe B. Again, using 3.8, we obtain that (f2)x is nonzerodivisor
of &x for all x e Y. We also have
dim(F(A) n V(f,)\ V) < dim(F(/1)\ V).
If P(A) n ^TAO = ^ then (A1/2) satisfies the lemma. Otherwise, we go on and choose
the elements fu.. .,fp e T(Y, 3) such that any (A)* is a nonzerodivisor of §\x for
x e 7 and
dim (nLi^ao\ n < dim (ni-i*u)\ n 2 < ,■ < />.

11. ANALYTIC LOCAL COHOMOLOGY
85
For a sufficiently large p (for instance p ^ n + 1), one obtains V = PlJui^CA)-
By Nullstellensatz, for sufficiently large m, S'"\K is contained in the sheaf generated
by /i,. ..,/„•
Lemma 6.6. Let Dc(£n be a domain, A <= V two closed analytic subsets
of D and & e Coh (D). Suppose q ^ 1. Let S'k+q be the union of all irreducible
components of Sk+q (oF) which are not contained in V. Assume that dim A n Sk+q < k
for all k. Let 3 = 3 (V) and J ce D an open subset such that the topological closure
X~ is holomorphically convex.
Under these assumptions, there is an integer I ^ 0 such that for any Stein
open subset U of X
r (U, 3)' N% (U, 9) = 0.
Proof. Let § = SF/?C°.5F. We have ^+,(SF) = Sk+q(§) (SF and § are equal
beyond V) and %°vq = 0. For some / ^ 0, 31 "X^ = 0 on a neighbourhood of X,
hence T (U, 3)! H%(U, 3C^SF) = 0. From the above said and the exact sequence
... - H%(U, 3t'°30 - H%(U, 9) - H%(U, <?)-...
it follows that SF satisfies the lemma as soon as <$ satisfies it.
Thus, we may assume that 3(^SF = 0. Like in the proof of 4.3, let Ak = A n
n 5^ + , and A'k+1 = Ak+1 \ Sk+q. Since (X \ S'+k)~ = X~ (otherwise V would
contain an open set, hence V = D, 9=0,...) and Sq+k (SF) \ 5^+A <= V, by
lemma 6.4 it follows that r (U \ S'k + q, S)'i N9Ak+1(U \ S'kJrq, 3F)=0, where lx is
a suitable integer independent on U.
Now, consider the commutative diagram with the exact horizontal row:
h9^1 (u, sf) 4 //I (c/, so 4 m (c/, ff) -» # «, (#» *)
X ^
Hlk + i{U\S'k + q,®)
Since (3 is continuous, we get T (U, S)'> -Nj (U, SF) <= Ima. Increasing if necessary
the integer /j, we may assume, by 4.3, that
r(C/,3)'.//71 (C/,SF)=0.
Like in the proof of lemma 4.3 we will check, by induction on k, the equality
r (u, sy N9Ak(u, sf) = o
for / sufficiently large. For k = 0, dim A0 < 0 and the equality follows from lemma
6.4, since Sq_1 (SF) <= V in some open neighbourhood of A0.

86
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Suppose now there is an integer 4^0 such that F (U, S)1sNa (U, SF) = 0
for any Stein open subset U of X. We are going to prove the equality
r (u, 3)h+^h N%k+i (u, sf) = o
for U open set. So the induction will be complete and as for sufficiently large k
we have Ak = A, the lemma follows easily.
We apply the previous lemma for K = X~ and let Y, flt .. .Jp, m be the
entities so obtained. We may assume that 3 coincides on X with the sheaf generated
by A, •••,/p, since the annihilation by suitable powers of T (U, 3) is equivalent
to the annihilation by some powers of T (U, (fly .. .,fp) <9), U open set of X. For
U Stein we have the isomorphisms
mk(U, SF) ~ T (£/ \ Ak, SF)/r (U, SF), mk(U, SF) ~ H'-1 (U \ Ak, SF), q > 2.
Let It be a Stein covering of U \ Ak and °f be a Stein covering of U \ Ak+1 which
refines the restriction of 11' to U \ Ak+1. Consider the map
6: Z'-1 (%l, SF) © C-2 ("?, SF) -► Z'-1 ("?, &),
defined by
6 {I © ri) = I I V + 8rt
where C"1 (% SF) = T(U, SF) and C^1 (°?,_SF) -t Z° ("?, SF) is the restriction map
r(U, S=) -»• T(C/ \ ^t+1, SF). Denote by fl»"1 (f, SF) the topological closure of
£«-!(% SF) in Z'"1 (°?» S?)- Since r(£/, 3)'i A^+i (£/, SF) c Im a, we have
gî^O9, SF) c Im 6, for any ger(U,S)'.
Let g = /,-, ... fi;. Then, for any xeX, gx is a nonzerodivisor of SF^ and
hence the map
Cq-i (y^ gr) _> C«-i (^ g;)
defined by multiplication by g is a strict monomorphism. Consequently gBq~\"%, SF)
is a closed subspace, hence FS. By Banach theorem the map induced by 6
6"1 (g B9'1 (°f, SF)) -► g B*-1 (°f, SF)
is open. If £ e i^1 (% SF) and {£v} is a sequencejn &->- ("?, SF) converging to Ç,
then we can find a sequence {Çv © ty} in 6"1 (g B*'1 ("f, SF)) converging to some
element Ç © tj, such that
6 (Çv© t)v) = ^v e 5"-1 C?, »0 and 6 (Ç © •/)) = g£.

II. ANALYTIC LOCAL COKOMOLOGY
87
Since g£v e B^1 Çf, SF), it follows that /zÇv e B^1 (% SF) for all h e r(U, 3)\ as
the class in H\k{U, SF) defined by Çv lies in the image of 8. Hence KQ e B*'1 (nf, SF)
and by induction hypothesis h'KQ e B^1 (nf, SF) for any h' e T (U, 3)k Therefore
we have KhgleB4'1 (°f, SF). Since jx \U, ...JP\U generate r (U, 3) it follows
that
T (U, 3)l*+2h N\k + i (U, SF) = 0 (U Stein open set).
The proof of lemma 6.6 is completed.
Proof of theorem 6.1. In virtue of the isomorphisms
the map NA(Q, SF) -> T(0, SlA SF), i ^ q is injective. Then it is sufficient to show that
8KA8r = 0 for i < q; in this case the spaces HA (O, SF) are consequently separated
and from the exact sequences
. . . -► H'-1 (Q, SF) -> H'-1 (Q \A,Sr)^ H'A(Q, SF) -► ...
they result even FS. The problem is local in nature. By a suitable embedding in a
numerical space (the topologies on HA(U, SF) agree with such embeddings), we may-
assume X a relatively compact open subset of a domain D <= <£", such that the
compact X~ is holomorphically convex, A <= Z> is a closed analytic subset and
SF e Coh (£)). By means of the finiteness theorem,
dim A n Sk+q (SF | D \ A) < k, for all fc.
Let 3 = 3 (^). We prove by induction on q the following assertion: for any Stein
open subset U of X, the space HA(U, SF), z < q, is separated, and any F(U, 6>)-ho-
momorphism F (U, <5)r -> HA (£/, SF) whose image is annihilated by some power
of r (£/, 3) has closed image. In this way, the proof of the theorem will be concluded.
For q = 0 this assertion is trivial. Let q ^ 1, and prove the general induction
step q — 1 h> q. Let U be a Stein open subset of X. By lemmas 4.3 and 6.6, there
exists an integer / ^ 0 (dependent on X) such that T (U, 3)! HA(U, SF) = 0 for i < q
and T (U, 3)1 NA(U, SF) = 0. For the problem we are interested in, we may assume
that 3f°SF = 0 when replacing SF by Sr/ïKjBr. In accordance with 6.5, we can find a
neighbourhood Y of X~ and geT(Y, 3)1 such that gx is a nonzerodivisor of SF^
for any x e Y. We thus get an exact sequence
(*) ht1 (u, sf) I* //r1 (i/, svsso ^ #S(t/, so C ha (u, sf).
By induction hypothesis HA~X (£/, SF/gSF) is an FS space. Since 'KA^1S: is coherent,
we have a sheaf epimorphism on X
<b:<3s-> SC^SF.

88 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Since HA~l (£/, §0 = F (U, WAX §F), + induces a F(U, <S)-epimorphism
<\>*:F(U,6y ^HA^{U,%).
Since Im oc<p* = Ima is annihilated by F(U,3)1, by induction hypothesis Imoc is
closed. We get N9A(U, ®) <= Im8 because g*NA(U, S=) = 0.
V
By considering Cech cohomology, as in the proof of lemma 6.6, we deduce
that the map
8~HNA(U, &)) ^ NA(U, ®)
is open. Hence NA(U, 8r) is isomorphic to the FS space 8^(NA(U, ^))/Ima. It follows
that NA(U, &) = 0; hence HA(U, &) will be separated. In virtue of induction
hypothesis the separation of the spaces HA(U, SF) for i < q follows.
Let now
9:r(u,ey^HA(u, so
be a T(U, <9)-morphism such that Im<p is annihilated by some power of F(U, 3). We
may assume without loss of generality that F (£/, 3)! Im<p = 0, where / is the integer
which has appeared previously. From (*) we obtain Im<p <= Im 8. There exists a
T (£/, 6)-morphism
such that <p = S<p. Define now the morphism
Y: r(u, ey © f(u, ey - ha^(u, &/g&y
by the formula
Y(a ® b) = a<\>* (a) + $(b).
Since F (U, 3)1 8 (Im <p) = 0, Imy is annihilated by some power of F (U, 3) and,
in accordance with the induction hypothesis, is closed. The surjective map
H9^1 (U, &/g&) X ImS of FS spaces (Im S = Kerg*) is strict and S (Imy) = Im <p.
Since S^1 (Im <p)= Im y and Im y is closed, it follows that Im <p is closed. Thereby
the induction is complete and the proof of the theorem is over.
Corollary 6.7. The canonical isomorphisms
are homeomorphisms for i < q, where T (Q, % ^SF) is endowed with the natural F S
structure given by the coherent sheaf %'A8r.
Proof. By the open mapping theorem, it is sufficient to show that the inverse
of the isomorphism from the statement
\':r(a,3iA9)^HA(a,ar)
is continuous. According to the closed graph theorem, we have only to prove that
X' has a closed graph. Let sv -*■ s in F (Q, 3fj*5F) and let Ev -»• \ in HA(Ù, SF) such
that x'(Jv) = Çv If ff c c fi is a Stein open set, then %A& \ Q' is generated by a

II. ANALYTIC LOCAL COHOMOLOGY
89
finite number of sections tlt ..., tm e F (Q.', 3C^SF). Since sv | O' -> s \ Q', there are
holomorphic functions/^' onfi' converging to some holomorphic functions/^ on Q'
such that
m
S, I «' = S /£> t»
(we apply the Banach theorem for the map T (Q', &'") -► T (Q.', %A SF)).
Let Çpi, e //^ (Q', §F) be the image of ^ under the canonical isomorphism
m
Vfi.: T (fi', 31» - //i (£i\ §F). Then Çv | O' = Xh-Ov |0') = £ Z^, where Çv|£i'
is the image of £v under the map H'A (Q, &) -> H'A (Q', SF). We obtain £ | fi' =
= J /^ = xj,.(j | O') =X''(j) i O'. Since «i (fi, SF) ~ r (fi, 3f^ 3=) for i < q and Q'
is arbitrary, we conclude that i; = X' (s) and hence the graph of X' is closed for i<q.
Corollary 6.8. Let X be a complex space, A <= X a closed analytic siéset and
S" e Cob. (X). For any Stein open subset Q of X, HA(Q, SF) is an FS space.
This follows by the theorem and 3.2.
Bibliographical indications
This chapter follows Gap sheaves and extension oj coherent analytic subsheaves of Y. T. Siu and
G. Trautmann.
The proof given above for theorem 2.1 is that from [86]; other proofs can be found in
[15], [84]. Another proof for corollary 2.2 is in [15] as well as in [50]. Scheja's paper [71]
underlines the role of the defect sets £>m(SF) in extending the analytic entities defined beyond
an analytic set.
The proofs of the vanishing theorem 3.6, the finiteness theorem 4.1 and the separation
theorem 5.1, which are the main results of the chapter, are those from [84]. Corollary 3.10
is proved in a different way (by using duality) in [8], [40] (cf. I. 3.3). The position of
corollaries 3.12 and 3.13 in the framework of some results of Abhyankar, Oka, Rothstein, Thimm is
indicated in [70]. Other proofs for thefiniteness theorem 4.1 canbefoundin [83], [92]. A
straightforward proof for corollary 4.4 is in [91]. Corollary 4.5 is proved by means of duality in [8]
(cf. I. 3.3).
The results of §5, except the given application, maybe found in the book [84]. The study
of relative gap sheaves, the connection with Lasker-Noether decompositions, the position of these
with respect to some results of Thimm, as well as other results in this context can be found in the
same book.
The complex Si' is built in § 5 on the analogy of the algebraic case [39] and of the
dualizing complex of Ramis and Ruget [61] (cf. ch. VII). Recently, F. Fouché has shown that Si'
is in fact a dualizing complex!
The hypothesis from the statement of the separation theorem 6.1 makes the problem be
of local nature and thus, one can use the sheaves gfl'A. This hypothesis is however sufficient for
the separation of the invariants H^,, but not at all necessary (cf. I. 2.21).

Chapter HI
Proper morphisms of complex spaces
Introduction
Let A" be a compact Riemann surface and let D = £ «,• P-t be a divisor on X. Consider
the complex vectorial space
L (D) = {9 e SXL(X) I 9 = 0 or (9) + D ^ 0}
(Sll (A") stands for the set of the meromorphic functions on X and (9) is the divisor
associated to 9). By a classical result, L (D) is finite-dimensional.
Let now AT be a compact complex manifold and V a holomorphic fiber bundle
on X. Then the space of all global sections of V is also of complex finite dimension.
These two results are particular cases of the following finiteness theorem of
Cartan and Serre [17]: "if AT is a compact complex space and W a coherent analytic
sheaf on X, then the complex vectorial spaces H" (X, 8r) are finite-dimensional".
Our aim in this chapter is to present the extension of this result to the relative case
of morphisms, and to give some applications.
If/: X -* 7 is a morphism of complex spaces, then the natural hypothesis of
compactness required is the following: the inverse image of a compact set under/
is a compact set. Such a morphism is called proper. The relativization for morphisms
of the cohomology groups is given by the generalized direct images R9j*. The
main result which is obtained is the following finiteness theorem of Grauert: "if
/: X -> 7 is a proper morphism of complex spaces and & e Coh (X), then the sheaves
-R?/*($0 are coherent". If the space y is a point, then one refinds the theorem of
Cartan-Serre.
Among the direct applications we recall the Remmert's projection theorem
and the Stein decomposition of a morphism. The next problem is to give informations
and formulae for computing the coherent sheaves R"f* (SO- If 7 is a point of Y, then
the comparison theorem 3.1 gives a formula for computing the completion of the
©y-module of finite type R"/^ (^)y in the m^-adic topology.
By making use of the comparison theorem one gets necessary and sufficient
conditions for the sheaves Rqj* (§0 to agree with changes of basis Y' -> Y (theorem
3.4 of base change). In particular one can obtain conditions for the natural maps
to be bijective, conditions for the sheaves R9f*(&) to be null, locally free...

92 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
§ 4 deals with the functions
y m* dim H"(J^ (y), SF/m^SF)
and
y ^S (-if dim H"{j^(y), 37^30,
and establishes results of semicontinuity and continuity for them (Grauert's theorem
4.12); such results extend those obtained by Kodaira and Spencer [49].
§ 1. Preliminaries
(a) A continuous map between two separated topological spaces is called
proper if the inverse image of any compact subset is compact.
Lemma 1.1. Any proper map between locally compact spaces is closed.
A A
Proof. Let/: X^ Y be such a map. If X and Y are Alexandroff compacti-
fications of X and Y, then/ extends to a continuous map/: X -* Y. Let T be a
closed subset otX. If J is its closure in X, then/ (T) =/ (T) n Y. Since T is compact,
the lemma follows.
Corollary 1.2. Let f: X -* Y be a proper map. If L is a closed subset of Y
and °fL is a fundamental system of neighbourhoods for it, then f^1 (°fi) is a
fundamental system of neighbourhoods for f^1 (L).
Proof. Let U be an open subset which contains/^1 (L). Then Y \f (X \ U)
is an open subset containing L, hence there exists V e "?L such that V<= Y\f(X\U).
Obviously,/-1^) c/-1 (Y \f(X\ U)) c U.
Let/: X -> y be a continuous map of topological spaces and SF a sheaf of
abelian groups on X. Denote by Rqf* (Sr) the sheaf associated to the presheaf on Y
the restrictions being naturally defined.
The sheaves R"f* (SO are called the generalized direct images of SF. In
particular, ^%(SF) equals the direct image /*<?)• Morphisms like Rqf*{&) -> R"JJ§)
correspond naturally to any morphism SF -> § in Ab (X) and these correspondences
are functorial.
If
0 -». gF' -»• SF -». ®" -> 0
is an exact sequence in Ab (A7), then one obtains the exact sequence
o -/„(&') -/,($) -/.(ff") - *7*(*') - tf1/*^) - • • •

III. PROPER MORPHISMS OF COMPLEX SPACES
93
In fact one can easily prove that R'f% are the right derived functors of the functor
/,: AbCT)-Ab(y).
If V is an open subset of Y and/K:/^1 (V) -* V is the restriction of/, then we
obviously get isomorphisms
*7„(80| v * R-ftWf-HV)).
Lemma 1.3. Let J: X -* Y be a proper map between locally compact spaces
and SF e Ab (X). For any point y e Y and for all q,
The proof follows from 1.2 and ([26], Ch. II, 4.11.1).
We now consider a morphism of ringed spaces
/: (X, 6X) - (Y, 0Y).
If §F is an (9^-module, then the sheaves R9/^) display naturally a structure o
(9y-modules; in particular, if/ is a morphism of complex spaces and 8F an analytic
sheaf on X, then R9/.^) are analytic sheaves on Y.
Recall that 8= is called j-flat or flat with respect to Y if the 6f(x) -modules ^x
are flat for all x e X (generalities on flatness can be found in chapter V, § 1).
Suppose X, Y are paracompact, SF is an (9^-module and 111 an (9y-module.
V
For any open subset V of Y, we get natural maps by using Cech cohomology,
HV"1 (V), 30 ®r(K. srt T (V, M) - H%f^ (V), §F ®ôx/*(Slt)).
Whence, by passing to associated sheaves, we derive morphisms
tf"/*(S0 ® Sy^l -* Rqf* (? ® eJ* W),
which are functorial in SF and otl.
Consider now a commutative diagram of ringed spaces
and an «Sj-module &. For any open subset V of Y we have natural morphisms
#'(/-1 (n. 30 - & (g'-y-1 (V), g'* (so) = # « (/'-^W, s'* (*0).

94 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
From these and the morphisms #"(/'_1 gT1 (P),£'*(S0) -► rfe-1 (V), R%(g'%Sf))),
we deduce the sheaf morphisms
*7*(80 - s„ (*Y*fe'*(^))X
hence by adjunction the morphisms
It is easy to check their functorial character.
(b) Let (Y, Oy) be a complex space and & a coherent 6Y-algebra. In [15]
one proved the existence of a complex space over Y,
q
Specan 61 -> Y,
which is called the analytic spectrum of & and a morphism
<3 -" <7* (^Specan a)
such that the following universal property holds: for any complex space X over
p
Y, X -*■ Y, the natural map
Homy(J, Specan 61) -> Hom^-aigO^,/)* (<3X))
is bijective. One can also show that the structural morphism q is finite and that
the morphism 61 -> ^(^specan a) is an isomorphism.
If 61 -> â' is a morphism of coherent (9y-algebras, then we obtain a y-mor-
phism Specan 61' -> Specan 61. This correspondence is functorial; moreover, if
the morphism 61 -> 61' is an epimorphism, then the associated morphism is a closed
immersion.
g
We also recall the property of base change: for any morphism Y' -> Y there
exists a canonical isomorphism
Specan g*(&) ~ (Specan d) xY Y'.
(c) Let A be a ring and M an ^-module. On the direct sum A © M one can
consider the multiplication
(«i, mt) . (a2, m2) = fea,, %ma + a^n^.
Thus we obtain a ring which is called the Nagata ring associated to M. A © 0
becomes a subring of ^ © M which is isomorphic to A, and 0 © A/ is an ideal
(with null square) in A © M which is isomorphic as an A © 0-module to M.

HI. PROPER MORPHISMS OF COMPLEX SPACES
95
This construction has functorial properties and it can be easily extended to
sheaves.
Proposition 1.4. [22]. Let (X, el) be a complex space and §\l e Coh (61). Then
the ringed space (X, d © cfll) is a complex space.
Proof. The problem is local in nature, so we can assume X an analytic subset
of the open unit polydisc centred in origin P"in a numerical space C"and
furthermore, we can assume d = 6/3 \ X, where 6 is the structural sheaf of P" and 3 an
ideal-sheaf such that Supp (6/3) = X. We can suppose in addition that Sll = dk/S\,
where Si is a coherent submodule of dk.
We first prove that the ringed space (X, d © dk) is a complex space. To do
this let P"+k = P"xPk be the open unit polydisc centred in origin in <Z"+k and denote
by Xj, ..., x„; yu .. .,yk the coordinates. Let & be the sheaf of germs of holo-
morphic functions in Pn+k and 3' be the ideal of 6' generated by 3 and by the
functions ya.y&, 1 < a, (3 < k. Denote X' = Supp (6'/3') and d' = 6'/3' \ X'. Obviously,
X' = X x 0. Define a morphism
(<p,*<p):(.r,a©a*)-cr, a')
as follows: <p (a) = (a, 0) for a e X. Further on, any element/ e d\a< 0) can be
represented by a series in &'(Q: 0) of the form
GO
I, l'„ = 0
CX=1 /j I*„ = 0
Denote by g and respectively gx, the elements of da defined by the series
QO 00
Y, c'\:-dxi - "i)'1 ■■■ (xn — a„)'" e &a, respectively ^ ci^...,,,,
I",,..., I„ - 0 I*,,..., /„ = 0
(Xi — aO''i ... (x„ - a„)'" e (9fl. Then we put *<pfl (/) = (g; g1} ..., gk) e da © dk.
It is not difficult to check that an isomorphism
($,*<p):(.X,£L® dk)^(X',d')
of ringed spaces over the complex field is so obtained, and thus (X, d © dk) is
obviously a complex space.

96 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The sheaf Si determines a coherent ideal Si* = 0 © Si c a © âk. The set
of zeros of Si* coincides with X and therefore (X, â © âk/Si*) is a complex space.
Since
a. © ak/si* ^ a © (ak/Si) = a © su,
the proof is concluded.
We also remark that there exists a morphism of complex spaces
(x, a © &K) -► (jr, a)
obtained from the identical map X -* X and the inclusion a <= & © SIL
(d) Let (X, <£) be a ringed space in local rings. For a point xel, denote as
usual the maximal ideal of 6X by m, and by k (x) = @Jmx the residual field in x.
If SF is an (9-module, then we set SF (x) = SF^/m^ S^ ~ S;x ®Qx k(x). For a morphism
©: SF -> c| of (9-modules, denote by <p (x): Sf (x) -> § (x) the induced morphism.
If U is an open subset of X and/ e T (U, SF), then one denotes by f(x) the image
of/ under the composite map r (£/, SF) -> SF^ -> SF (x).
If SF is an (9-module of finite type, then for any point xe X, Sr(x) is a vectorial
k (x)-space of finite dimension; according to the Nakayama lemma, its dimension
equals the minimal number of generators of S-x over 6X.
Lemma 1.5. (i) For any 6-module SF of finite type, the function
x m* dimta) SF(x)
is upper semicontinuous.
(ii) For any morphism of 6-modules d: &" -* C, the function
x ^rgk(x) d(x)
is lower semicontinuous.
Proof, (i) Let sf, ..., s„ be a minimal system of generators of the (9^-module
SF^. There exist a neighbourhood U of x and sections slt .. .,sneT(U, SF) such
that (s1)x = sf, ..., (s^)x = s*. The conclusion follows if we remark that, by
shrinking eventually U, the germs (s^y, . . ., (s^y generate the (S^-module SFy for any
y e U.
d
(ii) The exact sequence 6" -> 6q -> Coker d -*■ 0 yields the exact sequence
6"(x) "™& (x) -► (Coker d) (x) -> 0
and apply (/).
We recall that a ringed space in local rings (X, 6) is said to be reduced if, for
any open subset U of X and any section/ e F (U, <9), the relations/ (x) = 0 for all
x e U, imply / = 0.

111. PROPER MORPHISMS OF COMPLEX SPACES
97
Lemma 1.6. Let (X, 0) be a reduced ringed space in local rings.
(i) // SF is an Q-module of finite type such that the function
x i-> dimfcU) SF(x)
is locally constant, then &• is locally free.
(ii) // d: £p -* <Sq is an Q-morphism such that the function
x ^ rgta) d(x)
is locally constant, then Coker d is locally free.
Proof, (i) Let xeX and n = dimt(;c) &(x). There exist a neighbourhood U
of x and an epimorphism <p : 6" \ U -> 3 \ U. By shrinking eventually U, we can
assume n = diml(J)) &(y) for any point y e U. We will show that <p is injective. Let
V be an open subset of U and / = CA, .. .,/„) e T(V, Ker <p). For any y e V we
have/,(>') =0, 1 < / < «; for otherwise the minimal number of generators of SFy
would be smaller than n. Hence/ = 0.
(ii) By means of the exact sequences 6"(x) -> <9?(x) -> Coker d(x) -* 0 we
deduce that the function x m» rgt(ï) Coker d(x) is locally constant.
Proposition 1.7. Let (X, 6) be a ringed space in local rings and £• a complex
of free Q-modules of finite rank. Then
(i) For any q, the function
x w dimk(x)H"(£'(x))
is upper semicontinuous.
(ii) //' X is reduced and If the function
x h> dimk(x)W(£.'(x))
is locally constant for a fixed integer q, then Coker dq~x and Coker d" are locally free
sheaves.
(iii) // £' is bounded, then the function
x^Il(-l)"dimk(x)H%l-(x))
is locally constant.
Proof. Denote by r the rank of £«.
(i) We have dim H"(&'(x)) = dim (Ker d"{x)) - dim (Im d"'l(x)) = rq -
— Tgd"(x) — rg dq-\x) and apply 1.5.
(ii) We have dim H"(S.'(x)) =rq — rg d"(x) - rg ef-1 (x). Since the functions
x m* rg rf?_1(x), x m* rg £?«(x) are lower semicontinuous, we derive by the hypothesis
that they are locally constant. The conclusion follows from 1.6.
"-c. 2398

98 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(iii) Clearly, £ (- 1)? dim H"(&-(x)) == S (- \)q dim l"(x) = £ (- \)qr".
q q q
(e) Let u: A' -*■ B' be a morphism between two complexes of abelian groups
(or modules over a ring, or sheaves...). The cone of the map u is by definition the
complex C"(= C'(u)) built as follows: C = B" ® A"+1, and the differentials
C" -> C"+1 are given by formula (b, a) m» (dB-(b) + u(a), —dA.(a)). The natural
inclusions and projections define an exact sequence of complexes
0-.-.B* -► C -> A'[\] -»• 0,
where A'[l] is the complex obtained by translating the complex A' one step to the
Jeft, together with the change in the sign of the differentials. One gets the long exact
sequence
...->• H\B') ->• H"(C) -► H**\A') ->• H"+1(B-) -► ...
It can be easily seen that the boundary morphism 77î(^,[l])=77î+1(^') -»• H"+l(B')
coincides with the morphism H9+1(u) induced by u.
We recall that the morphism u is said to be a quasi-isomorphism if all mor-
phisms H"{ii): Hq{A') -* H"(B') are isomorphisms. The morphism u is called
n-quasi-isomorphism if the morphisms Hq(u): H9(A') -* H"{B') are isomorphisms
for q > n and H"{u): H"(A') -> H"(B') is an epimorphism.
From the previous assertions one obtains
Lemma 1.8. (/) Let n be an integer. The morphism u is an n-quasi-isomorphism
if and only if H"(C) = 0 for q > n.
(ii) The morphism u is a quasi-isomorphism if and only if Hq(C) = 0 for any q
(i.e. if and only if the cone of u is acyclic).
u cp
Lemma 1.9. Let A' ->£"-> M' be morphisms of complexes such that <p is-
a quasi-isomorphism. Under these assumptions, the canonical morphism C'(u) -* C'(<pu)
is a quasi-isomorphism.
Proof. It follows from the canonical commutative diagram
...->• H"(A') ->• H"(B-) -► H"(C-(u)) -► Hq+\A-) -► H"+1(B-) -► ...
jid j! 1 |id j!
...->• H"(A') -> H"(M-) -> H"(C'(<fu)) -> H"+\A') -> Hq+\M-) -> ...
and from the lemma of the five.
Corollary 1.10. For any q the sum of the maps of the diagram
Z"(C-(u))
dC-(<tu) 4*
C-^u) > Z"(C-((pu))
is a surjective map (Z means as usual the group of cycles).

III. PROPER MORPH1SMS OF COMPLEX SPACES
99
(f) Proposition 1.11. Let A be a ring and
0 -► M'q ^ Mq ^ M'q' -► 0, 9 > 1
a projective system of exact sequences of A-modules. If the modules (Mq)q>1 are
artinian, then the sequence
0 ->• lim M9' -»• lim Mq -► lim A/^' ->• 0
f\s exact.
Proof. The exactness of the sequence
0 -»• lim M'q -> lim Mq -> lim M'q
follows from the general properties of the projective limits. It only remains to
prove the surjectivity of the map lim Mq -* lim M'q. By the hypothesis that all M'q
are artinian, it results that for any q there is an integer n{q) such that
Im {M'n{q)+i -► M'q) = Im {M'n{q) -► Mq), i > 0.
We may suppose that for any / ^ 0, q > 1 we have
Im (Mq+i -► Mi) = Im (MJ + 1 -► M'q),
passing eventually to a subsequence.
Let now m" = (m'q')q be an element of lim M'q. Choose m2 e M2 such that
+2(^2) = m'2' and denote by w^ the image of m2 in M±. Then ^(Wi) = mî'. Let
a e A/3 be such that <j>3(a) = m'z'. There exists (3 e A/2' so that <p2((3) = w2 — Im «•
Since Im (A/3 ->■ A/{) = Im {M'% -> A/(), there exists y e A/3 such that the images of
(3 and y in A/i are equal. Denote m3 = <p3(y) + a and let w2 be the image of m3
in A/2. Then one can easily check that 4*2(^2) = m2 and tne image of wz2 in Mx
equals mx. By the same reasoning, we will find an element m = (mq)q of lim Mq
such that its image in lim M'q is just m".
§ 2. The finiteness theorem
The main result in the paragraph is the following finiteness theorem of Grauert [29]:
Theorem 2.1. Let X -> Y be a proper morphism of complex spaces and & a
coherent analytic sheaf on X. Then the analytic sheaves R'f^) are coherent (q > 0).

100
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The proof will follow in three steps.
I. The construction of a free resolution.
1. The construction of an atlas. Consider a complex space X, an open subset
y of a numerical space <£.N and a proper analytic map/: X-* Y. Let x be a point
in X. By a relative cart in x we mean a closed immersion /': U -*■ D(r) x V, where
U is a neighbourhood of x, V is a neighbourhood of /(x) such that /(£/) cr V and
D(r) = {ze C"; l-z'jtl < r}, /' >0, a polydisc in some numerical space C" such that
the following diagram is commutative
U —'-^ D(r) x V
f\u\ i/pr,
V
Let y0 be an arbitrary point in Y. Since/ is a proper map, there exist a Stein
neighbourhood V% of y0 and a finite number of relative carts
jk:Uk^Dk{\) x F„ <)<*<*,
such that (J Uk =f~1(V*)- Suppose that the unit polydisc Dk(l) is taken in a
k = 0
numerical space C"(;c)-
We shall make the following notations: for r< 1 and for an open subset
V of V*, we set
X{V)=f~\V\ Uk(r, V) =/,TW) X V) and 1C(r, F) = (C/t(r, F)0<^.
Since / is a proper map, there exists /•„, < 1 such that X(V) = \^J Uk(r, V) for
k = 0
any r which satisfies >\ < r < 1 and for any open subset V cr V^. If V is a Stein
open set, then ^[(r, F) is a Stein covering of X(V), therefore by the Leray
theorem, H'{X(V), SQ ~ H'(C'{M(r, V), %=)), /■„</■< 1. In this formula denote
V
by C'(flt(r, V), &) the Cech complex of alternate cochains of the covering 1£(r, F)
with values in the sheaf 3\
2. Link systems of sheaves ("Verbundene Garbensysteme"). We use the above
notations. Let
A„ = {(k0, . ..,kn) j 0 < k0 < ... <kn^k*} and A = U A„.
If a = (k0, ..., k„)e A and p =(/„,..., /m) e A we connive to write
a cr p <^ {&0, . . , /c„} cr {/0, . . , /,„}.

111. PROPER MORPH1SMS OF COMPLEX SPACES
101
We further define
Ufa V) = C) Ukfa V) and DJr) = fi D^r)-
The fibred product of the maps jk determines a closed immersion
J„:UfaV)^Dfa)x V.
If a <= |3 we denote by
7cap:Z)p(r) X V^Da(r) x V
the canonical projection. The diagram
Ufa V) - C/„(r, F)
Dfa XV ^ Da(r) X V,
where the map £/p(r, V) -* UJj, V) is the canonical inclusion, is commutative.
By a link system of sheaves over (Da(r) X V, 7rap) we mean :
(i) a family (ifa)a6A of analytic sheaves <fa over Ba(r) x V;
(ii) a family Oppjacp of morphisms of analytic sheaves
such that <paa = â/ and <pYct = ((7rap)* <pYp) <ppa for i c p c y.
If (<fa, 9pa) and (cj^, cppa) are two such systems, then a morphism between
them
û:(<?<x> <Pp<x) -»• (<?'«» 9pa)
is by definition a family (8a)aeA of sheaf morphisms 6a: <$a-*§'a, which are
compatible with the morphisms <ppa, <ppa.
One can easily prove that one gets an abelian category, which is called
the category of the link systems of sheaves over (Da(r) X V, Tza^).
If & is an analytic sheaf on X, then it defines naturally a link system of
sheaves ]*(&), by (J*^)a =(ia)*(&)- A morphism <p: W -> W of analytic sheaves
on X induces in a natural way a morphism j*(<fi): j*(&) -*j*(&')■ One can easily
see that the correspondences are functorial.
V
3. TheCech complex associated to a link system of sheaves. Let <$=((|a, <ppa)
be a link system of sheaves over (DJj) x V, izap). For any n > 0 we define
C"(r; V; <J) = II r(Z)a(r) X F, ^).
aEA„

102 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
C"(r, V; §) is a T(V, c9y)-module. Define the differentials
S:C"(r, F; <?) - C+i (r, F; <?)
by the formulae: for \ = (Ça>,eAi> e C"(r, F; <£), (SÇ)P = "£ (- 1/ 9PPi(Çp,)> where
1=0
P = (/0, .. .,/„+1)eA„+1 and P,-= (/0, ...,/,-_!, /i+1, ...,/„+1). We denote by
C(r, <f) the sheaf V ^ C\r, V; <£), F open subset of V*. C"(r, <f) is an
<SV„-module. If S- is an analytic sheaf on X, one gets an isomorphism of complexes
CW, F), ff) =J C'(r, V; ]*(&)).
4. TTze construction of a free resolution. A link system of sheaves SI =
= (cka, 'ppct) is called a free system of finite rank if all 6D (r)x (/-modules Sla
are free of finite rank.
Lemma 2.2. Le? <$ = (<$a, <ppa) Z>e a //«& system of sheaves and, for any
a e A, /e? za : êa -> <|a Z>e a morphism of analytic sheaves §a Z>e/«g a /ree
®Da(r) x v-module of finite rank. Then there exist a free system of finite rank
Si = {Sia, (ppa) a«J a morphism 0 : ât -► <| iwcA /Aai' Im 6a => Im sa for all a e A.
Proof. For every y e A we define a link system Sly = (<$£, <pjj[a) as follows:
7t *oc (£Y) if y <= a.
If y c a c p, then (pjja : 7i*a (§Y) -»• (^ap)* tz*p(£Y) is the morphism associated by
adjunction to the identical map (7tYp)*(SY) ~ n^ 7r*a(fY) -> 7r*p (£Y). If y<£a, then
we put <p£a = 0. The morphism eY induces naturally a morphism 6Y: SV -> <|.
We denote 51 = © JlY and define 6: Si -> § as being the sum of the morphisms 6Y.
YE*
Obviously, Im 6a => Im ea for any a e A.
Lemma 2.3. Let & e Coh (X). For any relatively compact Stein open subset
V cr F* and for any r', '"*</■'< 1, ^ere exists a resolution
... -»• Jl^+i -»• Jl* -»• ... -»• Jl1 -»• Jt° -»• j^) -»• 0
over (Z)a(r')x F', 7rap), w/zere rôe systems Sik are free of finite rank.
(Such a resolution will be briefly called a free resolution).
The proof will derive from the following assertion which will be proved
by induction on k: "there exist a number r, r' < r < 1, a Stein open set F such
that V cr c F c cr F*, and a resolution by free systems of finite rank over
(£>«(/■) X F, 7tap), Sik -»• &*-1 -»• ... -»• <3t° -»• j\{^) -»• 0". For any r, ;•' < r < 1 and
whenever F is a Stein open set, V cr cr Fcr cr Vt, there are surjective morphisms
z*'- ^a-*j\(^)a, 3\x being free 6Da (r)x(/-modules of finite rank. By applying the
preceding lemma we find a free system Si0 over {DJj) x F, 7tap) and a surjec-

III. PROPER MORPHISMS OF COMPLEX SPACES
103
tive morphism 31° -+]*(&). For the general induction step ki-^k + 1 one
proceeds analogously, by considering the system of sheaves %k = Ker (Sik -> §ik~x)
instead of ]*(&).
5. The calculation oj the direct images. Let SF e Coh (X) and r** be a number
so that r* < r** < 1. By lemma 2.3, if we denote V by V* too, we can assume
that there is a free resolution over (AtO**) x V%, 7iap)
... -><&*->• S^-1 -»• ... -»• Jl1 -»• Jl° -^(SO -»• 0.
For any open set V cf, and for all r, r* < r < /■„..„, we consider the double
complex (C'(r, V; 3lk))ik. We shall denote by C'(r, V) the associated simple
complex, that is
CO, V)= II C'O, F; &*).
7-ft-a
The (9K,-module F m* C"(r, F), F open subset of Fs, will be denoted by C"(r). C'{r)
is a complex of <SV„-modules. Since C\r, V; Slk) = 0 for I > k%, it follows that
CO) =0 for « > &*. The morphisms Cn(r, V) -> C"(r, V; Si0) -> C"(r, V;j*(®))
define the morphisms of complexes
C'(r, V)^C\r, V,jJ?))> C\r)^C (r, jJSf)).
If r' < r, then we have canonical restriction maps
CO, V) - CO', F)
which induce morphisms of (^-modules C'(r) -> CO')-
Lemma 2.4. For any r, r% < r < r** a«J whenever V is a Stein open subset
oj V*, the canonical morphism oj complexes C'(r, V) -* C'(r, V; j^iS")) is a quasi-
isomorphism. In particular, the 6V*-morphism oj complexes C'(r) -> C'(r, j^))
is a quasi-isomorphism.
Prooj. Since every Da(r) X V is Stein, it results by theorem B that
H%C'(r, V; SI')) = 0 for q > 1 and H°(C'(r, V; 31')) ^ C'(r, V;j*(&)). Then the
assertion of the lemma follows from the properties of the spectral sequences
associated to the bicomplex (CO, V; Slk))i^-
Corollary 2.5. For any r,r* < r < r^, and jor any Stein open subset
V cr V*, the canonical morphisms H"(C'(r, V)) -*■ H"(X(V), Sr) are isomorphisms
jor all n. In particular, H"(C'(r)) ^ R" j*(&)\V* jor any n.
Corollary 2.6. Whatever the numbers r and r', r#<r'<r<r#:t, and
the Stein open subset V of V^, the restriction map C (r, V) -> CO', V) is a quasi-
isomorphism. In particular, the restriction èv ^-morphism C'(r) -*■ C(r') is a
quasi-isomorphism.
II. The induction scheme.
1. The statement oj the induction assertions. We will state two lemmas to be
proved by induction.

104 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Let C'(r) be the complex of fiy^-modules built above, r* < ;• < rt..,.r and
0% be a compact subset of V*.
Lemma A(n). There exist a Stein open set V„ such that Q* <= Vn <= V^,
a number rn so that r% < rn < r^, a complex £" oj free 6u -modules of finite rank
... _> o -► £" -A' £"+1 -»• . • • —> £k* -»• 0,
a
and a morphism of complexes £' —> C (rn) where, for any Stein open subset V
of Vn, the morphism £'(V)—> C'(rn, V) is an n-quasi-isomorphism. In particular,
a is an n-quasi-isomorphism.
In accordance with corollary 2.6, for any r satisfying /■#</■< /"„, the
o
composite map £* —> C'(rn) -* C'(r) denoted by a too, still verifies the assertion
of lemma A(n).
Again under the hypothesis of lemma A(n), we are going to formulate the
second lemma.
Denote by K'(r) the cone of the morphism £" -> C'(r), hence K"'(r) =
= C""(r) © £'"+1 for m 5s « - 1 and K'" (r) = C'"(r) for m < n - 1. The
differentials of K'(r) will be denoted by S.
For any open subset V c Vn, let K'"(r, V) =T{V, Km (r)). If V is a Stein
open set, then the hypothesis of quasi-isomorphism from A(n) implies the
exactness of the sequence
K'-1 (r, V) -Î-» K" (r, V) -% K"+i (r, V) -►.. .
Let Z"-1 (/■) = Ker (K^1 (r) -% A" (;•)) and
Z"~l (r, V) = Ker (A""1 (r, K) -% A" (/-, F)).
Lemma i?(« — 1) (the projection of cycles). For any Stein open set V <=<= V„
and for any pair of real numbers r, r' such that r^ < r' < r < rn, there exists
over V a continuous morphism of Qy-modules t : A"_1(r) -> Z"_1 (r') «/eft //m? rôe
following diagram is commutative
\ / r"tr-
Z"-!(r)
The significance of the word "continuous" in the above statement will be
made clear in the followings.

III. PROPER MORPHISMS OF COMPLEX SPACES
105
The above stated lemmas will be proved by induction in section III, following
the scheme:
0. For n > k^, A(n) and B(n) are clearly true.
1. A(n) and B(n) => B(n - 1).
II. A(n) and B(n - 1) =^> A(n - 1).
2. Preparation for induction. Let V be an open subset of Vt and D(r)
be the polydisc of radius r centred in origin which lies in the space (£'"• Let
t1,...,tm be the coordinates in <£."*. Every element / e T (D (r) x V, <3g'»xV) can
be expanded in convergent series /= X eu,? with ave V(V, 6V), v = (v1;..., vm)e [Nm
V
and f — ti1 ■ ■. t„". For a compact Q a V and for any positive real number
p < r define
II/IIpo= £IH[epw;,
where \\aw\\Q — sup \av(y)\ and [v[ =vx+ ... + v,„. [[ j|pG is a seminorm on
V(D(r) X F, (Îim^). The family of seminorms || \\pQ, Q compact in V and
p < r, defines the usual Fréchet topology of r(D(r) X V, <?<e"'xk)-
We will also consider [[ |jPG for p = r or Q = V, but in this case one
obtains only a pseudonorm. For p' < p and Q' cr Q, we have
II llp'o- < I! II po-
If fler(F,flK) and /er(fl(r) x V^^y), then ||a/|!pe < ||fl||0 ||/||p0.
Lemma 2.7. If 0 < r' < r" < r, then the family (t/r")\ v e IN'", has the
following properties :
(i) every element f eT(D{r) x V, Ga>»xy) can be uniquely written f —
= S"v (*/'"")v such that [[ av |[e < |[/ ||,-q for any compact Q cr V.
V
00 X II(t/r'T \Wv < oo.
V
The lemma follows from definitions.
We now consider a finite number of polydiscs Dk(r) cr (£mW and denote
*(r, V) = II HAfr) X P> VWxk)-
/c
For / = (fk) e K{r, V) we define
II/IIpq = max jj/JpQ.
From the previous lemma one easily deduces the following
Lemma 2.8. Let 0 < r' < r" < r. There is a countable family (<?/),-e/ of
elements of Kir, V) enjoying the following properties:
(i) For any open subset V cr V, every element f e Kir, V) can be uniquely
expanded in convergent series / = X afii with a-, e TiV, Qv) and jja,-[[g < ll/llr"<?
where Q is an arbitrary compact in V.
OO XI WeiWr'v < co.

106
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
restr.
The T{V, 6vymodules C'(r, V) and K1 (r, V) already defined in section I
are of the type K(r, V); they are even topological T(V, <9F)-modules. We also
remark that the morphisms a in lemma A(n) and the differentials S are
continuous.
Under the hypothesis of lemma A(n + 1) we give another form of lemma B(ri).
Lemma B*{n). For any Stein open set V cr cr Vn+1 and for any pair of
real numbers r, r' enjoying the property r* < /■' < r < rn+1, there exists a
continuous morphism of Qv-modules over V, x: Kn{r) -*■ Z"(r'), such that the diagram
K"(r) —U- Z"(r')
\ /
Z"(r)
is commutative and moreover, the following assertion is fulfilled:
There are a countable family (et)iej of elements of K" (r, V) and a number
f, r' <7 < r, such that
(i) For any open subset V" cr V', every element f e K" (r, V") can be uniquely
expanded in convergent series f = 2j aiei with aier(V", 6y>) and jjtf;jje < II/II~q
i
Q compact of V".
(ii) S jj xe; \\r>v < co-
i
Proposition 2.9. A(n + 1) and B(n) => B*(n).
Proof. Choose a Stein open set V such that V' <= cr V <= cr Vn + l and
real numbers r, p, p' enjoying the property r' < p' < p < r < r. By B(n), there
exists a projection of cycles x: K" (p) -> Z" (p') over V. Consider the diagram
K" (r) —i-> K" (p) —^- Z" (p') —L» Z" (V)
/
Zn(r) > Z"(p)
where (3 and (3' are the restriction maps. We will show that the morphism
x = p'x[3 has the properties required by B*(n) over V. Indeed, by making use
of lemma 2.8, we find a family (e;)ieI of elements of K"(r, V) so that 5j ||p<?,l
PK
< oo
such that the condition /) from B*(n) holds. Since x: K"(p, V) -► Z"(p', V) is
continuous, there is a constant M such that jjxg[[r'V' < M\\g\\pp for any ge
eK"(p, V). We get |jp'x>,j|,-r = Hx^^r < M||p<?,||pp<, hence S II«,11^ <.oo.
tel
III. 7V7e />roo/ o/ ?#<? induction assertions and of the theorem.
1. A(n) and B(n) => B(n — 1). Let r, r' be real numbers such that r* < /■' <
< r < r„ and F' be a relatively compact Stein open subset of Vn. Choose a real

111. PROPER MORPH1SMS OF COMPLEX SPACES
107
number r", r' < r" < r and a Stein open set V", V <= <= V" <= cr Vn. Let
-r: A!"(r)-»• Z"(r") and (<?,-)/£/ be the entities obtained by applying lemma £*(«),
t being a morphism of 6V"-modules and et elements of Kn(r, V"). We have
S l|fej[|r"K" < oo. According to lemma A(n), the linear continuous map
i
S
K"~\r", V")—>Z"(r", V") is surjective. By applying the Banach theorem we find
a constant M and some elements ££ e A^"_1(r", V") such that SÇ; = xef and [[!;;[[,'K' <
< A/||Te;||r"K". We get S ll^illi-'k' < oo. The correspondence !]«;<?,• m* $]fli?i
determines a continuous <SV' -morphism
h: Kn{r) ^ K"-l{r'),
which makes the diagram commutative
K"(r) <—Z"(r)
i t |
K"-1^')-% Zn(r').
Consider the diagram
K"-\r) —> K"(r)
3 */
A:"-1(r')-^A:',(r').
The morphism x: K"'1^) -> Z"-\r'), x = (3 - £S verifies £(« — 1).
2. ^(«) a«rf B{n - 1) => ^(« - 1).
(a) Let Fn_j be a Stein open set such that Q^ <= Vn_1 <= cr V„ and let
/■„_i be a real number, r* < rn_1 < rn. By lemma A(n), for any p, r„_i < p ^ r„.
we have the diagram
£» _> £»+i -, . . .
... - C"-2(p) - C-Kp)—» C(p) - C"+1(p) - ...,
which induces for any Stein open set V <= Vn an epimorphism T(F, Ker a") ->
-► H-(C'(p, V)). Over F„ -i, we have to find a free sheaf of finite rank £" 1 and
morphisms a"-1 and ci"-1
cc"-1
£n-l ^ £"

108
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
which enjoy the following properties:
/) 1V-1 = 0, aV'1 = ^-V-1,
ii) for any Stein open set Vc V„_x, the induced morphism r(F,Ker a'yima"-1)-»
-^H\C'(rn^, V)) is an isomorphism and T{V, Ker a""1) -»• H"-l(Cm{rn_u V)) is an
epimorphism.
Since Zn~\rn^ = {(•/), s) e C^Oa-i) © £>"■« = 0, a"s = - d"'1^}, it is
sufficient to construct £"_1 and a morphism
{ù:£--i-»Z"-KrB_1)
such that, for any Stein open set V cr Vn^1, the sum of the morphisms in the
diagram
S"_1(F)
is a surjective morphism.
(b) Let /■', r„_j < r' < rn. For any Stein open set V cr Vn the restriction
map C (r„, V)-* C'(r', V) is a quasi-isomorphism (corollary 2.6). Thereby, it results
that the sum of the maps in the diagram
Zn~\rn, V)
Ï
C-\r', V)^Z"-\r', V)
is surjective (1.10).
(c) Consider a Stein open set V, Vn-i <= cr V cr cr Vn and a real number
r, r' < r < r„. By means of lemma B*(n — 1) we get a projection of cycles
t: K"-\r) -»• Z-1^') over F', a family (e,),e/ of elements e.-eK"-1^, V) and
a real number r ,r' < f < r such that the property /) of B*(n — 1) holds and
T] llTC;lli-'K' < oo. Since
i
S t rcstr.
Im (KK-1(r^^K'-1(r) —> Z"^(r')) => Im (Z"-'(r„) > Z""1^')),
it results that the sum of the maps of the diagram
K"-\rn, V)
C"-\r', V) -► Z""1^'. F')
is surjective.

HI. PROPER MORPHISMS OF COMPLEX SPACES
109
According to the Banach theorem, there exist a constant M and elements
It e K.—l(r„ V), y,, e C-\r', V) (i e I) such that t^+9t),=«, and max {\\U\rVn „
Ik-lk-.K,-,) < M\\ret\\r'y: We then derive that ]£ || y,rB_, < oo and
Y\ II _/]i IL-jK,,.! = A/i < oo. Then there exists a finite set J cr I such that
Put £"_1 = <£^„_! and let w: £"_1 -> Z"1(rn_1)bc the morphism which maps
the canonical generators (g,)ieJ of S""1 onto the elements (p'TÇ,),e/. By p':
Z"-1^') -> Z"_1(;-„_i) we have denoted the restriction map.
(d) We are going to prove the following assertion: for any open set V <= Vn_1
and for any element/ e K"~l{r, V), there are elementsf, e K"-l{r, V), geT(V, S""1)
and 7) e C"-*(rB-i. *0 such that p'</) = w(g) + Srj + p'-rCA) and
ll/ilUe <S -y 11/11 re. llslle< ll/be, Nk_l0 < Afiil/llro,
Q being an arbitrary compact of V. In order to do this, let / be expanded in
convergent series / = £a;e;, where a^r^A), ||a,-||e < ||/||re f°r any
I
compact QcV. If we set/j = J] a£j, g = Jfljgj and r\ = J]a,7);,
''e/\/ /e/ ïe/
then we have
II/ilire < S llfl«llollÇ/llr0 < ll/be £ ||^||,0< ~ ll/llre,
'e/V iei\J 2
We = max ||fl,||0 < \\f\\rQ, hl|,„_l0 <£ ||fl,||0lhf||,n.l0 < MJI/flye
and the assertion is proved.
(e) We now check that w is the morphism we need. Let V be a Stein
open subset of V„_1 and/e K"~~1(r, V). By iterating the assertion proved in (d),
we find elements g e T(V, .t""1) and 7) e Cn-\rn_1, V) so that
(*) P'<0 = <*Gf) + S>).
Since the restriction map C*(r, F) -> C"(rn~n P) is a quasi-isomorphism, the sum
of the maps in the diagram
Z"-\r, V)
. i

110
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
is a surjective map (1.10). By using this fact, the equalities (*) and the equalities
P'x(f) = restr. (J) for / e Zn~\r, V), it follows that the sum of the maps in the
diagram
T(V, r-1)
1-
is surjective. In this way the lemma A(n — 1) is proved.
3. Proof of the theorem. We may assume that Y is an open subset of a
numerical space (£N. Let y0 e Y. By lemma A(—l) and corollary 2.5, there exists
a complex £' of free 6y-modules of finite rank in a neighbourhood V of y0
0 -»• £-J -»• £° -»• f1 -»• ... -»• S.k* -> 0
such that #«(£•) ~ ^/^(S?)! F for any ^eDN- Accordingly, every sheaf Rqf*(®)
is coherent and the theorem is completely proved.
We end this paragraph giving some applications of theorem 2.1. The first
one is the following Cartan-Serre finiteness theorem [17].
Theorem 2.10. Let X be a compact analytic space and gF a coherent
analytic sheaf on X. Then the complex vectorial spaces H"(X, gF) are
finite-dimensional for any n > 0.
Proof. Consider the complex space e = (*, <£) which is reduced to one
point and has as stalk the complex field. The unique morphism /: X -> e is
proper and since R'f*(&) = H'(X, &), the assertion follows obviously from
Grauert's theorem.
The next application is the projection theorem of Remmert [65] :
Theorem 2.11._ The image of a closed analytic set by a proper morphism
is a closed analytic set.
Proof. It is sufficient to show that f(X) is a closed analytic subset of Y,
for any proper morphism / : X -*■ Y. Since the support of any coherent analytic
sheaf is an analytic set and since we have f(X) = Supp (/*(<9^)), the conclusion
results from 2.1.
The last application deals with Stein decomposition.
Let / : X -> Y be a proper morphism of complex spaces. Any connected
component of a fiber /"^(y), y e Y, is called a level set off. We shall denote
their set by Y' and let p : Y' -> Y be the corresponding natural map.
If we assign to any point x e X the connected component of /_1(/(x)) which
contains x, then one gets a map /' : X -> Y' such that / = pf. The map /'
is surjective and we can endow Y' with the quotient topology. Then /' will be
a proper map and p is a finite continuous map. We further consider on Y' the
structure of ringed space, where f*(<9x) stands for the structural sheaf.
The topological decomposition X-> Y'-* Y of / naturally determines a
decomposition cf ringed spaces and the next theorem shows that it is even
C"-\rn^, V)-

111. PROPER MORPH1SMS OF COMPLEX SPACES
111
a decomposition of complex spaces, which is called the Stein decomposition of
the morphism f [14], [85].
Theorem 2.12. The ringed space (J",/.j.(<S;r)) " a complex space.
Proof. By 2.1 the ^-algebraf*(<S>x) is coherent. Denote Z = Specan (f*(<3x))
and let q : Z -> 7 be the structural morphism. The natural morphism f*f*(<9x) ->
-> 6X gives rise to a morphism g : X -* Z such that / = qg.
We are going to prove that for any y e Y, the connected components of f~*(y)
are in 1 — 1 correspondence with the points of q^iy) and that within this
correspondence, for a point xe X, g(x) is carried into the connected component of
f~Kf(x))> where x belongs. In this case a 1 — 1 correspondence 6 between Z and Y'
which agrees with g, /', p, q is obtained. The map g is proper and surjective,
hence Z has the quotient topology from X. We derive that 6 is a topological
isomorphism. Since £Y — <]*(®z)> it will result that the image of the morphism
®z ~* g*(®x) under the functor q% is an isomorphism; therefore 6Z -> g!f(®x)
is an isomorphism and 6 can be extended to an isomorphism of ringed spaces
(Y',f*(&x))^ Specan f^6x).
We now prove that the fibres of g are nonempty and connected; if so,
the assignments z e q'1(y) t-> g"1^), y e Y, will establish the above stated bijective
correspondences and the proof of the theorem will be concluded.
It remains to prove the assertion concerning the fibers of g. Suppose/ : X -> Y
is a proper morphism such that the morphism <SY -^f*(<S>x) is an isomorphism.
The map/ is then surjective. We now consider an arbitrary point y e Y and claim
that /_10>) is a connected set. For otherwise, let /_100 = T1 u T2 where 7\, T2
are disjoint closed proper subsets of /_100- Then there are neighbourhoods £/,•
of Th i = 1,2 such that U1 fl U2 = 0. By shrinking eventually this
neighbourhood, we can assume that U = U1 U U2 has the form f^iV), where V is a
neighbourhood of y. The section of 6X which equals the unity on V-^ and zero
on U2 determines a section <p of <9y on V; therefore <p(y) = 0, 9(7) =1, a
contradiction.
Corollary 2.13. Let f : X -* Y be a proper morphism of complex spaces
and X' be the set of all points of X isolated in fiber (it is an open subset). Then the
restriction of f to X' can be decomposed into an open immersion and a finite
f p
morphism; more precisely, if X -* Y' -* Y is the Stein decomposition of /, then
there exists an open set V c Y' such that X' = /'_1(F') and f'\X' :X'^V is
an isomorphism.
Proof. For a point xeX, /'_1/'W is tne connected component of /_1/(x)
to which x belongs, hence it is an open subset of /_1/W- Thus x is isolated in
/_1/(x) if and only if it is isolated in f'^f'ix). Let x be a point in X'. Since the
fibers of /' are connected, f'^f'ipc) = {x}. Hence the inverse image under /' of
a fundamental system of neighbourhoods of y' =f'(x) is a fundamental system
of neighbourhoods of x. This fact and the equality QY> = f*(<9x) imply that the
canonical morphism ®Y",y' ~* ®x. x ls an isomorphism. Therefore there exists

112 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
a neighbourhood V'y- of / so that/'-1^'') c x' ^Aj'\f'-\V'y-):j'-\V'y) -* V'y- is
an isomorphism. The set V = [J V'f>(K) satisfies the assertion of the corollary.
xex
§ 3. The comparison and the base change theorems
Let / : X -> Y be a morphism of complex spaces and y e Y. We denote by nij,
both the maximal ideal of 6Y,y and the natural ideal-sheaf given by this; by m^, we
mean the ideal-sheaf of 6X generated by the inverse image of m^,. Consider an
analytic sheaf Sr on X and q ^ 0 an integer. Define
(*«/*(*),)* = lim (R"f^ymky(R"f^)y)).
k
If Rqf.j.(&) is 6>y-coherent, then this module is the completion of the fi^-module
of finite type R9f^.(S;)y in the m^-adic topology.
We now define a natural morphism
9»: (tf«/„(S0,)A - lim W'OO, W/mp)
k
as follows. For any integer k > 0, the exact sequence
0 -»• mfë -»• & -»• BF/m*^ -»• 0
yields the exact sequence
... - 2F/*(m*ff) - *«/*(*) - B?U{$lmky9)
The relations mkyR"f^)y = Im(tfV*(m*S%^?/*(S%) and H%f^(y), ff/m**) =
= R"f*(!3;/my$;)y give rise to a natural morphism
ç£: R'MSr)ïlmy(K'f*(?),) - ""(/^(y), arfàn
This morphism can be deduced also from the morphisms
constructed in § 1, (a) by taking 3K, = 6Yjmy6Y.
The family (<pg)t is compatible with the projective systems and we thus
obtain the desired morphism <p«.
The following theorem, which is called the Grauert comparison theorem
(Vergleichssatz), is the main result of the paragraph.

111. PROPER MORPHISMS OF COMPLEX SPACES
113
Theorem 3.1. Let f : X -> Y be a proper morphism of complex spaces,
y a point in Y ,q > 0 an integer, and 3r a coherent analytic sheaf on X. Under
these assumptions:
(i) there is a function F: DN -> DN such that lim F(k) = oo and for any k >0,
k-*crj
Im (tf%(m*S% - &MSF),) c ,<<*>(*»/* W>.);
(ii) the natural morphism defined above
<p«: CR*/*(3%r - lim H"{j^{y), SF/m^)
k
is an isomorphism.
First of all, we recall the following fact of algbera. Consider a complex
space X, an Ê^-module W, a section f eY(X, &x) and a point xel We say &
is without f-torsion in x if the canonical morphism
is injective.
Lemma 3.2. If W e Coh (A"), f/je/? /or a«j' compact K of X there exists
an integer d ^ 0 iwc/j rôa? the sheaf /rfSF m without f-torsion in all points of K.
Proof. Let x e X. The ascending sequence of 6>v-submodules of ¥x
(0 :/,) cr (0 :/|) cr . . . cr 9X
is stationary and let d be such that (0 :fx) = (0 :f$+1) = ... Then /rf8F will
result without/-torsion in x. The morphism fdW ->/</Sr, <p >->/<p is injective in x,
hence it remains injective in a neighbourhood of x. In its points, f& will be
without /-torsion, etc...
The smallest d enjoing the property required in this lemma will be denoted
by d(j,&;K).
Proof of the first assertion of the theorem. We may assume that Y is an open
subset of a numerical space (£"' and that y is the origin of this space. Let
tu ..., t„, be the coordinate functions on (£'".
If a e DN'", then by ma we mean the ideal-sheaf V tf'6Y and by ni^ the ideal-
! =1
sheaf on X generated by the inverse image of nia. We will prove the following
assertion: for any a e DN'" there is p e DN'" such that
Im (R"ftXm^X - Rqf*(S=)0) = ma(tf«/*(300).
For an integer k > 0 we denote by F(k) the largest integer ^ 0 where
Im (tf«/,(m*ff), - R>f *{&),) c= m™ W,(8%)
(if the left term vanishes we agree to set F(k) = k).
S- c 2398

114 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
For any y e ON'",
Im (R"Um^\ - R>MFti = Im (tf«/*(mlYln> - *«/«,(*)*,),
where |y| = yi +... + y„„ because mYSF <= m[YlSr. Then it follows that lim F(k) =00
and part (/) of the theorem is so proved. It remains to prove only the above
stated assertion: let a. e ON"'. Denote
sfm = 0, ®m = 37$^ = s\
For 1 ^ r < m we define recursively the entities
i = l
# = rf(f,, ff[r.y; /-HO)), rfr = rf(fr, Ri+y^r-^; 0)
and (3, = ar + d'r + t/,. Obviously, SF[m] = S7mp3\
If F is a neighbourhood of the origin in C"', which is small enough, then
we get the exact sequences of f~\V)
a.r~dr _r_!
0 - ^[r-l] '""» *W] ^ *lrj ^0 (1 < r < m).
We also have the exact sequences on X
0 _► sF[r-1] -> sf ^V a^,^ -► 0
(by x£_1, t,_! we have denoted the canonical morphisms .. .). One then derives
the exact sequences
(*) H"U-\V), t<Srlr_1}) '—» H<KT\V), &lr^j) ±-> H«V-\V), ffrr]),
(**) **/*(*0o ^ ^/♦(fftr-l^ ^ ^+1/*(Sr[,_1])o •
It is enough to prove the following fact: if s e R"f%(Br)0 satisfies t,(j) = 0
(1 < r < m), then there exists 5' e Rqf*(&)0 such that x^O — t?s') = 0. We thus
suppose s e R"f*(SF)0 enjoys the property t,(j) =0. Since irr'1Tr^1 = xr, we derive
that Tf-iCs) e Ker x'-1. In virtue of (*), there exists ^ e ^?/*(^rr_i])0 so that
^+d'j1 = t,-!(j). From (**) it follows that fr^isj = 8(£r+S1) == «(t,-^*)) =0.
By the definition of dr we deduce S(?rr51)=0; hence there exists an element
s' e R"J^)0 such that x^O') = t^s^ Hence x^O - t?rs') = 0.

III. PROPER MORPH1SMS OF COMPLEX SPACES
115
Proof of the second assertion of the theorem. We first check the injectivity
of <p«. Let s = (ji)4>0elim(2F/,(ff),/m*(2î«/*(S%) be such that <P*<» = °-
k
Let .F be the function from (/); if we eventually replace F by the function
k i-> min {k, F(k)), we can assume F (k) < A\
For any k e ON, there exists / e [N such that F(l) 2s Ar. From the commutative
diagram
where s,, >;, are the natural morphisms...), one can find an element si e Rqf*(&)y
such that zt(si) = 5,. Since <p;%s,) = 0, we get t),(j/) = 0. According to (;'),
£fm(s'i) = °- Since sjr,;,(.s/) = ^(0 and P(/) 2s A:, it follows that sk = 0. As A: is
an arbitrary integer, we derive that 5 = 0.
We now check the surjectivity of <p?. Let s— {sk\enie\im{Rqf^l\hkyS:)y)
k
Let us fix an integer / 5= 0. For k ^ / we have the commutative diagram
**/*(S0 A 2P/*(S7m*80 A 2F+1/*(m*Sr)
h* J,",-k
tf«/*(S0 —» *«/*(§7m^) —^ 2P+1/*(w
7^ and 7/'" are the morphisms given by the natural maps SF/in*SF -►SF/m^oF
^ ;
We now apply the assertion (/) for the sheaf rit^SF. Then there exists a
function i7 : ON -»• IN such that lim F(k) = 00 and for k > I,
k-* oc
Im (,/■*), cr m;<*-" (^/*(W
From the above diagram we have §;(>;) errt^'*-'' {Rq+1f*(mlyS;)y) for A: 2= /. In
accordance with Krull's separation theorem we deduce that 8t(si) = 0. Therefore
there exists an element s't e Rqf^(S=)y such that yfei) = st. The image of si in
R"f*(s:)ylmy(R'lf*(^)y) has st as image by ç>/. The kernels of the maps cpq are
artinian (they are fi^/m^-modules of finite type) and, as in the proof of 1.11,
we can find s" such that q>"{s") = s. This completes the proof.

116 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Corollary 3.3. Under the assumptions of the theorem, the canonical morphism
tt*(8Q,r ^limr(/-M>), IF/m^)
k
is an isomorphism.
We will give now applications of the comparison theorem. Let/ : X-» Y be
a morphism of complex spaces and ye Y. The ringed space (Xy=f~Hy), ®xhny®x\Xy)
is an analytic space, which is isomorphic to XxY(y,&y/my); it is called the analytic
fiber over y. If & e Coh (X), then we denote frJ, = Sr/m>Sr. Wy is a coherent &xhny®x-
module and is called the analytic fiber of W over y.
Let M be an (S^-module of finite type. In a neighbourhood V of >■ we may
define a coherent analytic sheaf SM. such that Dllj, = M.
If fv:f~,(V) -> F is the restriction of/ then we will denote
B?h($®etM)y = **/£(* ®<9,^),
(for convenience we put Si®eYS\l instead of Sr ®exf*(S\V) and we will use this
notation in what follows). Rqf*($;®0yM)y has naturally a structure of an
<9^-module and it does not depend on I'll; moreover, if f is a proper map, then
it will be of finite type over Qy. The canonical morphism
*'/£(*) ®e^ - Rqfi (* ®e^Tc)
induces a canonical morphism
If A/ is of the form OyJm% then
*'/•(* ®<9,M), = B*U{SFImky&)y = H"(f-^y), S/m**),
and the map K>U(aF)y®e M -* B^f^Sr^^M) equals <$.
We also remark that all the above associations are functorial in A/.
Now we prove, by applying 3.1, the following theorem of base change:
Theorem 3.4. Let f : X -* Y be a proper morphism of complex spaces,
IF a coherent analytic sheaf on X which is flat with respect to f, y a point in Y,
and q an integer. The following assertions are equivalent:
(a) The functor M h> Rqf.^(S: ®&rM)y is right exact.
(a') The functor M h> jR*+1/*(^ ®6rM)y is lefi exact.
(a") The canonical morphisms Rqfif(&)y®eyM -* Rqf^.(S: ®$YM)V are
isomorphisms, M being an arbitrary 6y-module of finite type.

IIJ. PROPER MORPHISMS OF COMPLEX SPACES
117
(b) The canonical map A')/*(Sr)j, -»• Wf^/mffly is surjective.
(c) The canonical maps Rqf.:,{S;jm^+1S;)y -> Rqf^/my¥)y are surjective,
k^ 0 being an arbitrary integer.
(d) For any base change g : Y' -* Y and for any y' e Y' such that g(y') = y,
and taking into account the morphism f : A" = X X YY' -> Y' deduced from f
and the sheaf W on X' which is the inverse image of & on A", the canonical
morphism g*(Rqf^.(S;y) -> Rtf'^') is an isomorphism in a neighbourhood of y'.
Proof. Let 0 -> M' -> M -> M" -> 0 be an exact sequence of (^-modules
of finite type. In a neighbourhood V of y there exists an exact sequence of coherent
analytic sheaves 0 -> oTc' -> of, -> $11" -> 0, which induces the given sequence
in v. Since W is/-flat we get the exact sequence of/~\V)
o -► ® ®etmi' -> & ®&yski -► w®etsi\i" -> o.
Thus one obtains the exact sequence
■.. - &f*{& ®e^), - &U& ®6rM")y - R"^f^®&YM')y
Consequently, the equivalence (a) <=> (a') becomes obvious. The implication
(a") => (a)is also clear.
In order to prove that (a) => (a") we consider an exact sequence of the form
«9"1 -> 6"" -> M -> 0; a suitable commutative diagram and the fact that the
morphism from (a") is obviously an isomorphism for a free (S^-module of finite rank
lead immediately to the conclusion.
The implication (a") =* (b) follows by taking M = @y/my and the
implication (b) => (c) results from the factorization
We prove that (c) => (6). It is sufficient to show that we get a surjection if
we pass to TTtj,-adic completion. According to the comparison theorem,
(*«/*(S0; =* lim (RJ^/mky + ^)y).
k
Since ^(f/m,?), = H\Xy, Er y) is a vectorial space of finite dimension over
ôylmy ~ (C, it follows that
tf«/*(87nV); ^ R>f,(?/my8r)r
The conclusion results from 1.11 because the kernels of the maps from (c) are
modules of finite type over the artinian rings <Sym*+1, hence they are artinian
modules.

118
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
We prove the implication (b) => (a). We must show that for any epimorphism
M -> M" of 6^-modules of finite type, the map
Rqh(? ®8rM)y - R%(¥ ®&YM")
is surjective. By making use of the commutative diagram
Rqf*(®)y®eyM -► R"f^)y®6yM" -► 0
\ I
&/*(.& ®erM)y - #/,,(* ®@YM")y
the assertion (a) is proved as soon as the following assertion is verified:
(*) "For any module of finite type M, the map
Rqf^)y®0yM ^ Rqf^ ®eYM)y is surjective".
We prove this fact. It is enough to verify that the map from the assertion (*)
becomes surjective if we pass tom^-adic completion. We have (Rqf^(S;)y®eyM)'' =
= \im(RJ^)y®ôyM®ôyôy/mk+1)^lim (R%(W)y®0s,Mlmk+1M). By 3.1 we
k k
also have (Rqf*(& ®&rM)y)A
~ lim (*'/*(* ®ÔYM/mk+\ff ®erM))y) = lim (&/,(? ®6YMlmk + 1M)y).
k k
By applying 1.11 we have to prove that the maps
Rqf^)y®eyM/mk + 1M -► K>f*{W®eYMlmk^M)y
are surjective. In other words, it remains to check (*) in the particular case when M
is annihilated by some power of my. We will show by descending induction that
the maps
Rqf*my®ôy{mkM) - R'MSF ®&YmkM)
are surjective (k ^ 0 integer). For k = 0 we obtain our very assertion. For
sufficiently large k, mkM = 0 and the conclusion is patent. For the general
induction step we consider the exact sequence
0 -► mk+1M -► mkM -► mkM/mk + 1M -► 0.
We get the exact commutative diagram:
RqM®)y®Qy{mk + 1M) -► R*f^)y®es(mkM) -► tf'/*(30, ®<9,K M/m^M) -► 0

Ill PROPER MORPH1SMS OF COMPLEX SPACES
119
The module mkM/mk + 1M is a vectorial space of finite dimension over ôy/my ^ <£,
hence by using (b) and additivity the third vertical arrow is surjective. Thereby
the induction step k + 1 i-> k is achieved.
The proof of the implication (a") => (d). We will prove that the maps
which are deduced from the composition of natural maps
are isomorphisms. By 3.1, we get that the maps
g*(*v*(^)); - (*W'vr
are bijective.
The assertion (d) will then follow from the properties of the functor of
completion. It is convenient to set
Yik} = 0, <Vm* + 1), Y'{k) = (/, <S>,,/tti*,+1),
XM =(/-!(», 6xlmky + 13x\f-\y)), A"<*> = (f'^iy'), ^x'/mk^6x,\f'^(y'))-
We denote by f{k), g{k), f'ik), respectively, the morphisms which correspond to/
g, f and consider the diagram
x
î\
\
>lk)
A
i
V
A
\
\
X'
\
(k)
> Y
i
\\
\
- j
r
Y'
\
>Y
By means of (a") applied to modules M of type 6y/\\Xy+1, we can easily deduce
that the source module in (**) is canonically isomorphic to g{k)*(Rqf(£)($;ik))) and
the target module is canonically isomorphic to Rqf'^k)(Si'ik)), where Sr(fc) is the
restriction at X{k) of &jmk+1Sr and W'{k) is its inverse image under the morphism
X'{k) -> Xlk)-®{k) is flat over Y{k) and the hypothesis (a") can be easily checked
for (*<*>, y<*>, /<*>, #<*>, _>-, q).
To prove the bijectivity of the maps(##), it is then sufficient to prove the assertion
{d) in case when Y and Y' are reduced to a point and their structural sheaves are
artinian rings. In this case, (d) easily follows by applying (a") for the module
M = 6 . (which is an (S^-module of finite type!).

120 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
It remains to prove the implication (d) => (a"). Let M be an ^-module
of finite type. If we substitute eventually Y for a neighbourhood of y, we can
assume that there exists a coherent analytic sheaf Sic on Y such that SM, = M.
Consider the analytic space Y' = (Y, &Y ® SHI) where &Y © ofll is the Nagata
sum (1.4) and let g : Y' -*■ Y be the canonical morphism. In accordance with (d)
we have a canonical isomorphism in a neighbourhood of y
Rqf*(®)®e*(GY © m ^ RqfÀ® ® er{$Y ®SH))-
Then it will result that the morphism
**/*(*) ® a^ - *»/*(* ® <9r#n.)
is an isomorphism and passing to stalks in y, one gets the isomorphism stated
in (a").
The proof of the change base theorem is thus over.
We now consider a case when this theorem applies.
Corollary 3.5. To the conditions of the theorem we suppose in addition
H\Xy, ®y)=0. Then for any module M of finite type over Oy, RqfJ&®QYM)y= 0;
in particular, R^f^W) vanishes in a neighbourhood of y. As a consequence,
the equivalent conditions from 3.4 are fulfilled for the integer q — 1; in particular,
for any k ^ 0, the canonical maps
R^f^)y/mky(R^f^)y) -, m-HJ-Hy), ff/rit*»)
are bijective.
Proof. Let M be an <9 -module of finite type. By shrinking Y around y,
we may assume the existence of SU e Coh (Y) such that SHI = M. We must show
that Rgf*(&®exf*(Sf\l))y = 0. By 3.1 we have to prove that for any integer k^Q,
H\Xy, &®Qxf*(mivt&S® ax/*W)) = 0.
We shall prove this by induction on k. For k = 1,
*®Sir/*TOm,(ff®ôir/*(Sll)) ~ ®y®<LSnly/my$ly
(we identify Oyjmy with C) and the conclusion follows by the hypothesis already
made and by additivity.
In order to verify the general induction step, by using a suitable exact
sequence, we have to prove that
H\Xy, mky{® ®eJ*{SI\l))lm^\W®eJ*{S)\l))) = 0-

111. PROPER MORPH1SMS OF COMPLEX SPACES
121
Reasoning on stalks and using the flatness of 9 over Y, one can easily deduce
the isomorphism
™ky(® ®&xf*(^))lmky+1($ ®exf*(3\i)) =* ff,®Œ(Tît*§>H/m*+1Sn)
and again the conclusion follows by additivity from the hypothesis.
Remark. In the course of the proof the complex field has been identified
with the constant sheaf C which was defined on Xy and the vectorial spaces
oEy/m/^j,, m^ly/m*"1"1^ were identified with the constant sheaves (of C-modules)
defined on Xy by them. The structure of C-module of Sf/m^ is given by the
inclusion C -»• ®x/Wy®x and coincides with that deduced naturally from the
identification @y/my ^ (£.
A consequence of 3.5 is the following:
Corollary 3.6. Letf : X -> Y be aflat proper morphism of complex spaces
and let y e Y be such that H\Xy, Qxhny6x) = 0. // gF and § are two invertible
sheaves on X such that the sheaves §•y and §y are isomorphic, then there exists
a neigbourhood V of y so that aF\f~1(V) ~ §\f'\V) and this isomorphism induces
the given one &y ~ §y.
Proof. Consider the sheaf 5£ = Home{&, <£) ^ S^1®^. We have %y ~
~®yX®Q ime 1j> ~ a*/™A> hence H1(Xy,Wy)=0. According to 3.5, the
canonical map
/* Wm//*(xg - r(xy, %y)
is bijective.
Suppose \y : 8y-* §y is an isomorphism; it defines an element in F(Xy, %y).
Then there exists on some neighbourhood V of y a morphism Ç : SF|/_1(F) ->
->■ §\f~\V) which induces the isomorphism £r Analogously, by shrinking
eventually V, we deduce a morphism tj: <||/_1(F) -»• &\f~\V), which induces the
isomorphism i)y, which is the inverse of \r
Let x be an arbitrary point of the fiber f~\y). By identifying &x with 6X,
the morphism -r\x\x : §•* -► 8X corresponds to an element a of 6X. The image
of this in 0Jmy0x corresponds to the morphism {y\y)J$,y)x = identity, hence
it is just the unity. It will then result that a is invertible, hence f)£x is an
isomorphism. Similarly, we can deduce that ^xi]x is an isomorphism. In this
way, the composite maps Etj and rfc induce isomorphisms in all points of f~\y).
The same property holds on a neighbourhood of/-1^); since/ is proper, we
may assume this neighbourhood of the form/-1^).
Another consequence of the base theorem is the following exactness
criterion :
Corollary 3.7. Under the assumptions of theorem 3.4, the following
assertions are equivalent:
(a) The functor M >-+ Rqfij!(S:®eYM)y is exact.
(b) The maps

122 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
are surjective.
(c) For any integer k > 0 the maps
are surjective.
(d) The map
*«/♦(*), - R>ft(S/m,S),
is surjective and Rqf*(&)y is a free 6y-module.
The proof follows straightforwardly from 3.4 (recall that whenever TV is an
<9j,-module of finite type and the functor M h> N®eyM is exact, N will be free!).
If the equivalent conditions of this criterion are fulfilled, one says that /
is cohomologically flat in dimension q in the point y. Here is a case when this criterion
can be applied.
Corollary 3.8. Under the assumptions of theorem 3.4, we suppose in
addition that H\Xy, $y) = Hq-%Xy, gg = 0. Then &-%(&) is free in a neighbourhood
of y and the canonical maps
IP-ihWMW^M*)), - Rq-lf*{®lmhy$)y
are bijective, k ^ 0 being an arbitrary integer.
The proof is immediate. In particular one obtains:
Corollary 3.9. Let f : X -> Y be a proper morphism of complex spaces,
gF a coherent analytic sheaf on X flat with respect to f and y eY. IfH\Xy, $■ y) =0,
then /*(oF) is free in a neighbourhood of y and the maps
/*(*),K(/*(*),) - n%tKy), *7in*ff)
are bijective.
Corollary 3.10. Under the assumptions of theorem 3.4, the following
assertions are equivalent:
(a) gF is cohomologically flat in y, whatever dimension p,p ^ q.
(b) Rpfif{W)y is a free 6y-module for any p ^ q.
Proof. The implication (a) =*■ (b) follows obviously from the exactness
criterion. For the inverse implication one proceeds by descending induction on p,
and considers the fact that Rpf^)y = 0 for p sufficiently large.
Corollary 3.11. Under the assumptions of 3.4, the following assertions
are equivalent:
(a) Rf*(&), = 0 for any p > q.
(b) H»(Xy, »,) = 0 for any p > q.
The proof derives by means of 3.7 and 3.10.
Next we consider an example.

III. PROPER MORPHISMS OF COMPLEX SPACES
123
Proposition 3.12. Let f : A"-» 7 be a flat proper morphism of complex spaces
and y e Y. Suppose that the analytic fiber Xy is a reduced space. Then 6X
is cohomologically flat in dimension zero in y ; in particular f*(®x) is free
around y and
h(0x)ylm,f*(0x)y^ H\Xy, exlmy6x).
Proof. By 3.7 it is sufficient to show that the maps
f*(&x/mk,+1&x), = HX, &x/my+1Gx) _ T{X, Gx/my0x)
are surjective for any k ^ 0. By the change base Y' = (y, 6yjm.y+1) -, Y, it is
enough to prove the surjectivity of the map
r(x, &x) -, r(f-Xy), eximyex)
under the hypothesis of the statement, where Y is supposed in addition reduced
to a point. Since/_1(j) = X, we may assume X connected. In virtue of the
assumptions previously made, the structural morphism C -»• T~(Xy, ®xhny®x) is an
isomorphism and the conclusion follows from the composition of natural maps
C - r(7, eY) -, T{x, ex) -, r(x„ ox/myex).
Corollary 3.13. Under the assumptions of the proposition, if we require Xy
connected, then the canonical morphism <3Y -*f*(@x) is an isomorphism in a
neighbourhood of y.
§ 4. The semicontinuity and continuity theorems.
The invariance of Euler-Poincaré characteristic
Let / : X -> Y be a proper morphism of complex spaces, W e Coh(A") and
£HeCoh(y). We denote, as in the previous paragraph, & ®eJ*(§>]l) by &<g)QYffiL.
If V is an open subset of Y and 511 e Coh(F), then we set for convenience 8F ®QySRL
(iesp.R%(&®erSH)) instead of 9 \f'\V)®evmi (resp. R-fl{®\f-\V)®eYSXOj),
where fv:f \V) -, V is the restriction of/.
The sheaf SF is said to be transversal on 8ÏÏI [47] if for any x e/-1(F),
ToifHff,, ®rg =0, for i > 1, y =f(x).
If a? is flat with respect to Y, then it is transversal on any 8ÏÏL.
The first important result of the paragraph is the following theorem of
Kiehl-Verdier and Schneider ([47] and [72]):

124 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Theorem 4.1. Let X -> Y be a proper morphism of complex spaces, &• a
coherent analytic sheaf on X, and y0 e Y. Then there exist a neighbourhood V
of y0 and a complex £' of free sheaves of finite rank on V, which is bounded above,
and which satisfies the following assertion:
''''For any coherent analytic sheaf Sïi defined on an open subset of Y such
that §• is transversal on it, there is an isomorphism
which is functorial in SU and compatible with the restrictions and with the short
exact sequences in the argument STL (hence it is a functorial isomorphism of d-
functorsl)".
Moreover, if gF is flat with respect to Y, then the complex £' can be supposed
bounded.
The proof needs some preparations. We shall use the notations and the
facts from § 2.
(a) Let F*,^, r^^Jf.,... be the entities from the proof of thefiniteness theorem.
Let Si be a link system of sheaves. If S\i e Coh(y), then we denote by Si®eYS\i
the link system of sheaves, of components 3\ a® QyS\1(—Sia® e p*(8Rl),
where pa : DJr^) x V* -*■ V* are the canonical projections).
In fact we will consider coherent sheaves SIR, on open subsets on Y; in
this case we adopt the abuse of notation which has been recalled at the beginning
of the paragraph.
If Si -► SI' is a morphism of link systems of sheaves, then one naturally
obtains a morphism Si®ey3}l -> <&'(g^Dfl. On the other hand, if &ÏI -> SI is
a morphism of sheaves, then one naturally gets a morphism of link systems
Si®QYS!\i -> Si' ®qyS)1. One can easily check the functorial character of these
correspondences.
If 0 -> SU' -> S\l -> STi" -> 0 is an exact sequence of coherent analytic sheaves
(defined on an open subset of Y), then the sequence
0 -»• Si®er$R,' -»• Si®erSH -»• Si®ei.S\l" -»• 0
is exact for any free system SI; this fact follows easily from the flatness of the
projections Z)ax^-. K+.
One can also check without difficulty the existence of a canonical isomorphism
(b) Let Si' be the resolution of/*^) built in §2. Thus we have the exact
sequence of systems of sheaves
... _> &1! _> &1 _► &° -^(so -» o.

111. PROPER MORPH1SMS OF COMPLEX SPACES
125
Let Sic be a coherent analytic sheaf defined on an open subset of Y. By (a)
one gets a complex Si' ®@YS>\1 and an augmentation morphism
Lemma 4.2. // gF is transversal on Sit, then Si'®erSM, is a resolution for
M$®eYm.
Proof. We have to show that for any simplex a, the sequence
is exact. This fact follows easily by dividing the exact sequence
... -aj-aj^as-j^ao-o
into short exact sequences ; the reasoning goes from the right to the left and
by using the hypothesis of transversality and the flatness of the projection
(c) Recall that for any open subset V c V% and for any r, r* < r < r**
v
we denote by C'(r, V; Si') the Cech complex of components
C\r, V;W)= IlrWOx V,Si<$.
The simple associated complex was denoted by C'(r, V). The assignement V h>
i-> C'(r, V), together with the restriction maps that have been deduced canonically,
give rise to a complex of sheaves C'(r) on V9.
Let now Sïïi be a coherent analytic sheaf defined on an open subset of V%.
We denote by C'(r, II.) the complex of sheaves defined as above, replacing Si'
by Si' ®eYSf\l. Therefore, if V c V* is an open subset contained in the open set
where Sfk, is defined, then
T(V, C (r, mi)) = II C'{r, V; Si«®eySXl),
where C(r, V; &k®6rS)}l) = II T(Da(r)x V, Sika®eYS)VC).
Lemma 4.3. There exists a canonical isomorphism of complexes
C-(r)®erS>}i^C'(r,3\i),
functorial in S>]1.
Proof. The lemma easily results from the definitions if we use the following
fact: if y is a complex space, D an open polydisc, X = D x Y, p : X -> Y the
projection, SHleCoh(Y) and Si a free sheaf of finite rank on X, then the natural
morphism
/>*(&) ® ay5Tc -► pt(.A®ex p*(SJ\l))
is an isomorphism.

126
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
We prove this fact; we can assume that SI = Qx, hence we must prove
that the morphism
is an isomorphism. If SVt is a free <9y-module of finite rank this is obvious. For
the general case, the problem being local on Y, we may assume the existence of
an exact sequence of the form
&$ -► 6$ -> M -> 0.
Then the conclusion follows from a suitable diagram, by means of the fact that
functor p* is exact on coherent sheaves.
The augmentation morphism
SI°®&yS)}1 -^(SFCg^Sll)
induces naturally an augmentation morphism
C'(r, S>Vl) -+ CXr,U(S®eM)),
v
where the target complex is the Cech complex with respect to the system
If V is as above, then we have:
r(V,C(r,j^<S,ermi)))= Il T(Da(r)xV, h(Sr®erSHL)) =
a £A„
= II nuar\f-\v),9®eYm.).
Hence T(V, C'(r,j*(W ®eym,))) = C\r, V;j*(® ®ayS1l)) coincides with the Cech
complex associated to the covering 1f(r, V) and to the sheaf &<g)er8l\L.
Lemma 4.4, // gF is transversal on S>]i, then for any r, r!!t < r < r^, we
have natural isomorphisms
R"f*(®®6r®1) - H'\C\r, S)\l)).
In particular the restriction maps
C'(r, 0IL) -► C•(!■', ^0
are quasi-isomorphisms {even on Stein open subsets) for all real numbers r. /■„. <
< r' < r.
Proof. Let V c V.^ be a Stein open subset contained in the open set
where S\l is defined. Since the sheaves §\.^®eYS\l are coherent, exactly as in 2,4
one deduces that the augmentation maps
T{V. C'(r. SÏL)) - C'(r, V;j^<S,6rM))

III. PROPER MORPH1SMS OF COMPLEX SPACES
J 27
are quasi-isomorphisms; we thus get isomorphisms
H'\T{V, C'(r, ill))) =f H\t\V), ff®e^ll).
All these agree with the restrictions given by inclusions V c V; in this way, we
derive the isomorphisms stated in the lemma.
(d) We use the conditions and the notation from above and prove that for
any integer n the following holds.
Lemma C(n). Let V„, r,„ £*, £' -> C'(r„) be the entities which verify lemma
A(n), which was built in the proof of the finiteness theorem. Then for any
Stein neighbourhood V'n of y0, V'„ cc Vn and for any real number p„, r,f < p„ <
< r„, the following assertion is true:
"For any coherent analytic sheaf SM. defined on an open subset of V'n such
that gF is transversal on it, the induced morphism
s an n-quasi-isomorphism".
Proof, Recall that for any r, /„: < r < r*ff, we denoted by K'{r) the cone
of the map £• -> C'(r). Let V'n, p„ and $11 be as in the statement. Similarly, we
denote by K'(r, STl) the cone of the map £' ®ay^lt -> C'(r, #11). From lemma 4.3
one easily derives a natural isomorphism
We shall prove the exactness of the complex K'(p„, M.) in any dimension / ^ n
and the lemma will be concluded.
The problem is of local nature, so we can suppose that there is an epimor-
phism <9?-> Sit -> 0 on the domain where SU is defined. Let i be an integer,
i ^ n. Denote by d the boundary and by (3 the maps induced naturally by
restrictions. We agree to use also 6 for different maps naturally induced by 6.
By the very way of constructing the complex £.', there exists a continuous
<2Kn-morphism of sheaves
h : K!+\rn) - K!(Pn)
such that the diagram
Ki+1(rn) +-> Zi+\K\rn))
»l IB
KKe.) i. Z'-Wpj)
is commutative. Then we obtain canonically maps of 6Vi -modules, which are
also denoted by h, Ki+1(r„, 6Y) -> Kl(9„, 0Y), Kl+1(r„, eTi) -+"K'(p„, S>}1), such that

128
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Ki+\rn
'i
K'(Pn, <
, e\)-^zu
d«Y)!U
Z'
\K'(r„,
Ie
r'(K'(Pn.
0Y))
&>y)).
the diagram
K+\rn,e«Y)\Ki+\rK,8m)
/<<(?„- fy) - *'(?„, ®IC)
is commutative. Moreover, the maps 6 result to be epimorphisms at the level
of sections over Stein open sets.
The commutativity of the diagram (*) implies by additivity the commuta-
tivity of the diagram
(**)
By applying lemmas 4.4, 1.9, it results that the map K'(r„, $\i) -> K'(pn,S)]i) is
a quasi-isomorphism. We now prove the exactness of the sequence
K'-\Pn, SKi) - K!(Pn, All) - K^\P„, S\l).
Let <p e Z'(K'(pn, S]l )) (we omit to indicate the domain of definition which is always
supposed to be a Stein open set). Choose <\i e Z'(K'(r„, 8Vj) such that <p = p(i|/) +
+ coboundary. Let ty e K!(r„, 6qY) such that 6((|/) = <J>. As follows from the
diagram (**), the element (3(<I/) — h(d^i') of K'(pn, £Y) is just a cocycle.
Since the sequence
K'-](p„, &>Y) - K\9n, eY) -, k'+Kp„ &V)
is exact, there exists <p' e K!-\pn, 6qY) such that d<p' = (3(<J/) — h(d^'). We
then get d(09') = 0(3<p') = 6(p(<j/)) - 0(A(3<J/)) = P(6(+')) - W+')) = P(<|0 -
— /z(d(6t|/)) = p(<p) — h(d<\i) = p(^). Therefore 9 is a coboundary and the lemma
C(«) is completely proved.
(e) The proof of the first assertion of the theorem follows from lemmas C(— 1)
and 4.4. The functoriality and the compatibility with the short exact sequences
follow easily from the functoriality of the isomorphisms given by 4.4 and from
the exact commutative diagrams of the type
£• ® &l —> £• ®3l 0 -»• £• ® Ml' -»• £' ® PI -»• £• ® 5)11" -»• 0
C*(P„, P"0 - C'(p„, ^) 0 - C'(p„, 5Tl') - C'(p„, Pc) -C-(pB, Sit") - 0
In order to prove the second assertion we need some other preparations. We
use the preceding notations.

III. PROPER MORPH1SMS OF COMPLEX SPACES
129
(f) Lemma 4.5. Let A be a noetherian local ring and M an A-modide of
finite type which has a resolution
0 -»• L"« -»• ... -»• L1 -> L° -> M -> 0
by free modules. Under these assumptions, if
. .. -► A1 -► #° -► Af -► 0
is a resolution of M by free modules of finite rank, then N = Ker(R"°-L -> R"°~2)
is a free module.
The proof is an elementary fact of homological algebra [16]: from the first
resolution we deduce that Ext'A(M, *) = 0 for / > n0 and from the second one,
Ext'A(N, *) =i Ext"j+i(M, *), hence N is a projective module, therefore it is free.
Proposition 4.6 [20]. Let Y be a complex space, let U be an open subset
of a numerical space, and & e Coh (U X 7), flat with respect to Y. Under these
assumptions, for any point (x0, y0) e U X Y there is on a neighbourhood of (x0, y0)
a finite resolution of aF by free sheaves of finite rank.
Proof. Let (x0, y0) be an arbitrary point of U X Y. Consider the closed
immersion U -> U x Y, x h> (x, y0). Denote by &(y0) the sheaf j*(&). If
p : U X Y -> Y is the projection, then jj^iy^)) is the sheaf SF/m^ S\ By syszggies
theorem there exists on a neighbourhood of x0 a finite resolution £° for ^(jo)
by free sheaves of finite rank. We set £1i = &(y0). If £? is of the form £>£/,
a"
then we set £; = <9rt/xy, i>0. Denote also £_1=a? and 3C?= Ker(£? —^^i).
We construct in a neighbourhood of (x0, y0), morphisms d;: S, -> S.;_1 such that
i*'(dd = df and show that the sheaves Slt = Ker dt are 7-flat and moreover
^iCVa) (=J*(^d) = °>>'/- To do this we proceed by induction on /. Suppose dt
already constructed and the properties relative to 3C, verified. One can construct
di+i '■ £j+i -»• £,- such that the diagram
*"i + l ^ ^i
J*(«?+i) -$ J*(3C?)
is commutative. By Nakayama lemma we shall then deduce that lmdi + 1 = 3it
in the point (x0, y0), hence also in a neighbourhood of this point. The exact
sequence
0 ->• 3Ci+1-> £;+!-»• 3ï;->-0
where 3C,- and £i+1 are 7-flat shows that the sheaf 3CÎ + 1 is 7-flat and that
^i + iOv) = °"/ + i- The beginning of the recurrence is similar when the hypothesis
of flatness concerning & is used. One thus obtains the resolution required in the
proposition (remark, by construction, that Im di+1 = 3ï;)-
9_c. 2338

130 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(g) The proof oj the last assertion of the theorem. Since A is finite, then by
lemma 4.5 and proposition 4.6 we may suppose that the resolution oil* of j^ÇS")
is bounded. Accordingly, the complexes C'(r) are bounded and let n0 be such that
C\r) = 0 for n < n0.
By the proof of theorem 2.1, there exist a Stein neighbourhood V of y0,
a number r, r* < r < r**, a bounded complex
£•:... -> 0 ->■ l""-1 -> £"° -»• £"°+1 -> ...
of free (9K'-modules of finite rank and an (n0 — l)-quasi-isomorphism a : £* -> C*(r).
We set #1 = Ker (S"»-1 -»• £"«) and let £*(>) be the cone of a. We have A>-2(r) =
= S'1»-1 and K"'-\r) = £"°, hence Z"°-2(K'(r)) = ^.
By lemma 5(«0 — 2), there is a morphism t : fi"»-1 -> ôt such that tv = id
where v : HI -> £"«-J is the inclusion. Thus the sheaf 31 results to be locally
free and by shrinking eventually V we may suppose it free. Consider the complex
l' : ... -> 0 -»• oK. -> i"»-1 -»• £"» -»• ...
A A A
The morphism a extends to a quasi-isomorphism a : £* -> C'(r). The complex £"
and the morphism a satisfy lemmas A(ri) and B(ri). The proof of the theorem is
concluded by applying again lemma C(n0 — 3) and lemma 4.4.
Remark. Theorem 4.1 allows us to give a new proof for the comparison
theorem for sheaves of the type Sr®(9>ro'lc, Sf flat with respect to Y (EGA III,
7.4.8). In this way the change base theorem is also a consequence of 4.1.
We confine ourselves to prove the following complement to the change base
theorem.
Theorem 4.7. The equivalent conditions of 3.4 are also equivalent to the
following ones:
(e) If £* is a complex of sheaves defined in a neighbourhood of y and
satisfying the assertions from 4.1, then the sheaf Z'q+1(£') = Coker (£« -> £«+1)
is free around y.
(f) One can find a complex £" in a neighbourhood of y, which verifies the
assertions from 4.1 and such that the differential £'? -> £'«+1 is null.
(g) There exists an 6y-module of finite type SS' {which is unique, up to
isomorphism) such that the functorial isomorphism
Rq+1h(® ®eYM)y =* Homay (ïïy, M)
holds, M being a module of finite type over Oy.
Proof. (a')=>(e). It is sufficient to prove that Z">+1(£')y = Z'?+1(£;)
is a flat (9j,-module. Let 0 -> M' -> M be a monomorphism of <9>rmodules of

Ill PROPER MORPH1SMS OF COMPLEX SPACES
131
finite type. From the right exactness of the tensor product one can easily deduce
the exact commutative diagram
whence the conclusion.
(e) =>(/). We have the exact sequences
0 -»• Z«(£*) -»• £« -»• £?+1(£') -» 0, 0 -»• fi»+1(f) -► £«+1 -»• Z'«+1(£*) -»• 0
where £?+1(£*) = Im (£« -»• £«+1) and Z«(£#) = Ker (£" -»• £«+1). Then one gets
that Zq(£.')y and 5«+1(£j.) are free (S^-modules. By shrinking eventually the
neighbourhood V from the statement of theorem 4.1, we may assume that Z'?+1(£*),
Z?(£') and Bq+1(&') are free sheaves.
Consider the complex £" of components £"' = £' for i *£ q, i *£ q + 1
and £'« = Z«(£'), £'«+1 = Z'«+1(£'). The differentials d" are equal to rf'' for
i*q-\,q,q+\ and J'»"1 : £«-* -► Z«, rf'« : Z« -► Z'«+1, d"?+1 : Z'"+1 -> S."+2
are deduced naturally from the differentials of £*. Obviously d'q = 0. It remains
to check that £" satisfies the assertions of the theorem. Let Sli be a coherent
analytic sheaf defined on an open subset of V. One can deduce without difficulty
the equalities
Ker (S."-1 ® SHI -► £* ® Sll) = Ker (&-1 ® Sll -+Zq® §>Z),
Im (£«+1® Sll -»• £«+2® SU) = Im(Z'"+1® SÎI -»• £«+2® PI)
and isomorphisms
#«(£• ® Sit) = Z"(£*)® SIt/Im (£?-J® Sit -► Z«(£*)® SU),
H" + 1(&' ® Sit) ~ Ker (Z'«+1® 31c -► £«+2® Sit).
One thus obtains an isomorphism
H\S,' ® Sit) ~ H'(l" ® STi).
One verifies canonically that this isomorphism is functorial in Sit and agrees
with the short exact sequences. Thereby the assertion of 4.1 is verified.
(/) =*■ Or)- Let F be a neighbourhood where £" is defined. Denote by " v "
the passing to dual. Let' ÏÏV be the cokernel of the map £'?+2 -»• £'«+1 and 8\.
be its stalk in y. 3 is an <9j,-module of finite type. Let M be an (S^-module of

132 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
finite type. We have the exact sequence
0 -► Horn^S-,, M) -► Ho^+i, M) -► Homay(£;«+2, A/).
On the other hand
R9+1f*(® ® M)y ~ (W«+1(£" ® Sit)), =* //«+1(S;* ®6yM)
= Ker (£;«+1® A/ ->• £;«+2(g) A/),
where ©11 is a coherent analytic sheaf defined in some neighbourhood of y and
SS\ly = A/. The isomorphism from (g) follows by the natural identification
£;• ®&yM =i Homav(£;, M).
Its functoriality can be proved in a canonical way. The uniqueness from (g) follows
from functoriality in a well-known way.
Since the implication (g) => (a') is clear, the proof of the theorem is complete.
/
Corollary 4.8. Let X -* Y be a proper morphism of complex spaces,
8r a coherent analytic sheaf on X which is flat with respect to Y, and q an integer.
If the change base theorem is true in a point y e Y, it remains true in all points
of a neighbourhood of y.
The proof arises straightforwardly from (e).
In particular, we get
Corollary 4.9. Under the assumptions of the previous corollary, if Sr is
cohomologically flat in dimension q in a point y eY, then the same property holds
in a neighbourhood of y.
We actually consider the change base theorem in the global form.
We shall write §\l for a coherent analytic sheaf defined on an open
subset of Y.
Theorem 4.10. Let f : X -* Y be a proper morphism of complex spaces,
& a coherent analytic sheaf on X which is flat with respect to Y, and q an integer.
Then the following assertions are equivalent:
(a) The functor SÏÏL h> R'f^ ®<9^) is right exact.
(a') The functor St h> R?+if*(& ®ôïS\l) is left exact.
(a") The canonical morphisms
Rqfd®)®er^ - R"f*(® ®6yS>TL)
are isomorphisms.
(a'") The functor STi h> Rqf^ ®eY^~i) is isomorphic to a functor of the
form Sïï. *->8Jl®0,8fH, SZeCoh(Y) (SK. will result isomosphic to R*f*(&)).

111. PROPER MORPH1SMS OF COMPLEX SPACES
133
(b) The canonical morphisms
R>MS) - KM*,)
are epimorphisms for any y e Y.
(c) The canonical morphisms
*7*(S7m*+1S0 - R%($/my$)
are epimorphisms for any ye Y and for any k ^ 0.
(d) For any change base g : Y' -*■ Y, if one considers the morphism
f : X' = X xYY' -*■ Y' deduced from f and the sheaf SP on X', the inverse image
of Sr, then the canonical morphism
is an isomorphism.
(e) There exists ST e Coh(7) (unique up to isomorphism) such that the following
functorial (with respect to ofll) isomorphism
R'^MSr ®6lSl) ^ Horned, ®l)
holds.
Proof. All the implications, but for the construction of oT, follow
straightforwardly from the previous considerations. In order to construct ST, we take
again the proof of the implication (J) => (g) from 4.7. The sheaf ïïv (defined on V)
enjoies the following property: for any coherent analytic sheaf ok defined on an
open subset of V there exists an isomorphism Rq+1ft.(& ®<3rSf\i) ~ HomeY(S'v, S\l)
and moreover, such isomorphisms are functorial in $11 and agree with the
restrictions. If (V±, SVx) and (V2, S~y2) are two pairs which verify the above properties,
then there is an isomorphism
?v1--S>v2\v1nv2^srVi\v1nv2.
One easily checks the conditions of gluing together the ïïv's and in this way one
gets the required sheaf ST.
Corollary 4.11. Let f : X-*■ Y be a proper morphism of complex spaces
and SF a coherent Qx-module which is flat with respect to Y. Then there exists
a coherent Qy-module ST (unique up to isomorphism) and an isomorphism
functorial in S\l
j*(® ®&YS>]1) =î HomeY(®, ®1)-
The proof follows by means of (a') •** (e) when q = — 1.
We shall say that & is cohomologically flat in dimension q over Y if the functor
$11 h> Rqfif(& ®fiyoJlt) is exact. This means that 3r is cohomologically flat in dimen-

134 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
sion q in any point of Y. By 3.4, 3.7, 4.7 and 4.10 we obtain criteria of cohomo-
logical flatness. In particular, it will result that W is cohomologically flat in
dimension q over Y if and only if the maps
Rlf*(8F)y/myRf^)y -, Rf^jmyW)y = H\Xy, 9 y)
are bijective for any y e Y and i = q, q — 1, if and only if R^^S?) is locally free
and the maps
wuarymyR'uar), - H%xy,wy)
are bijective for any y e Y.
The main result of the paragraph is the following theorem of Grauert :
Theorem 4.12. Let f : X -* Y be a proper morphism of complex spaces,
$• a coherent analytic sheaf on X which is flat with respect to Y. Then:
(i) (Semicontinuity theorem). For any fixed integer q, the function
y^dimH\Xy,9y)
is upper semicontinuous.
(ii) (Continuity theorem). Consider an integer q. If W is cohomologically flat
in dimension q over Y, then the function
y h> dim H\Xy, 9 y)
is locally constant. Conversely, if this function is locally constant and Y is a
reduced space, then Sr is cohomologically flat in dimension q oyer Y ; in particular,
the sheaf Rqf*(&) is locally free.
(Hi) (Invariance of Euler-Poincaré characteristic). The function
y^5](-i)« dim #«(*,, ag
is locally constant.
Proof. We apply theorem 4.1 and consider sheaves S)]i of the form 6Yjmy6Y.
The assertions of the theorem being of local nature on Y, we may assume that
there exists a bounded complex £* of free (9y-modules of finite rank such that
//•(£•) ~ R-U(?) and //'(i'/m/') =* H\Xy,¥y)
for any point y e Y.

III. PROPER MORPHISMS OF COMPLEX SPACES
135
The assertions (/) and (Hi) follow straightforwardly from 1.7.
(ii) If 8F is cohomologically flat in dimension q over Y, then R9/^) is locally
free and
R"f^ymy{R"f^)y) * H%Xy,S?y).
Accordingly, the function
y *-+dim H%X„aFy)
is locally constant.
We now prove the converse. By 1.7 (ii), Coker (I9'1—>£.") and
d"
Coker (£«—>£?+1) are locally free. From 4.10 it will then result that ¥ is
cohomologically flat in dimension q over Y.
Remark. The continuity theorem is also called the exactness criterion of
Grauert (EGA, III). The "clasical" statement (Y reduced, y h> dim Hq(Xy, Wy)
locally constant => R"fi.(S:) locally free) can be directly obtained from the exact
sequence
0 ->• H\Z') ^ R9/^) -» Coker d9'1 -»• £?+1 -»• Coker d« -► 0.
Bibliographical indications
The proof of Grauert's finiteness theorem is taken from Forster and Knorr's paper [23] and
it is reproduced here without any modification! Another proof besides that "classical" one
([29], [48], [58]) can be found in the paper of Kiehl and Verdier [47]. For a more general context
one can consult the work [42] of Houzel and [21],
The original proof of Cartan-Serre finiteness theorem, of the Remmert projection theorem,
respectively, can be found in [17] and [65]. The Stein factorization theorem is presented following
the algebraic model [(37], Ch. Ill, 4.3). The original proof (for manifolds) is given in [85]. The
paper [14] of Cartan and [95] present a general theorem of quotient spaces (in whose proof
the main argument is again the finiteness theorem).
The proof of the comparison 3.1 is that of the paper [48] of Knorr. In Chapter VI
(according to [6]) a more general and more precise statement will be proved; in particular some
A.
supplementary informations about the projective systems (H9(Xy,®im.y:§y)k^,0 will be obatined.
The base change theorem as well as the other facts from § 3, are either taken from [15]
or they are transpositions from the algebraic case ([36], [37]).
Theorem 4.1 is proved for the case of sheaves of the form SK = &Ylm 6Y (sufficient for
theorem 4.12) in Schneider's paper [72] and it was stated in the general form in ([47],
Bemerkung 4.4.1). The proof given here follows [72]. We would also like to mention that this
theorem is analogous to theorem 6.10.5 from EGA, Ch. Ill, [37].
As soon as theorem 4.1 is proved, the proof of the other facts from § 4 can be made without
any difficulty following the algebraic model (EGA III, SGA 6). For another proof of theorem 4.12
one can consult [66].
The case of the analytic families of compact complex manifolds is treated in the paper
of Kodaira and Spencer [49].

Chapter IV
Projective morphisms
of complex spaces
Introduction
In his paper Faisceaux algébriques cohérents [76], Serre proved the following
theorem :
"Let \Pr be the projective space of dimension r over an algebraically closed
field and & be a coherent algebraic sheaf on \Pr. Then there exists an integer m0
such that:
(A) the sheaves &(m) = gF (x) &(m) are generated by their global sections
for any m ^ m0.
(B) Hi (\Pr, & (m)) = 0 for any q > 1 and m ^ m0."
The case of coherent analytic sheaves over the complex projective space
is analysed in [77]. The authors' aim in this chapter is to present the extension,
due to Grauert and Remmert [32], of this result to projective morphisms of
complex spaces.
The main result is theorem 2.1, which shows that the projective morphisms
"verify at + oo theorems A and B". As a particular case we find again the
absolute case of the projective spaces. As an application, we obtain by means
of the above stated theorem of Serre, the comparison theorem 2.6 between coherent
algebraic sheaves and coherent analytic sheaves on a projective manifold [77].
Following Grothendieck [36], the behaviour at — oo of the projective
morphisms is analysed in § 3. The result so obtained (theorem 3.1) corresponds to
the characterization of the dimension and the depth of a coherent analytic sheaf
on a Stein space and shows that the projective morphisms (under suitable
hypothesis of flatness) "behave even at — oo like the affine morphisms". Some results
from the next chapter are used in this paragraph.
Two criteria for ampleness are proved in § 4. The first one is a converse of
theorem 2.1. The second one gives conditions for a morphism X -> Y to be
projective around a point y e Y as soon as the fiber Xy is a projective manifold.
The first paragraph contains as usual preliminaries: here we recall the notion
of projective fiber bundle, ample sheaf, projective morphism...

138 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
§ 1. Preliminaries
(a) We consider the projective space \Pr over the complex field as a complex
manifold. Let (9 be its structural sheaf, that is the sheaf of germs of holomorphic
functions.
Denote by t0, ... , tr a system of homogeneous coordinates in \P'. By 'IC =
= (U0, ..., Ur) we mean the natural covering of \Pr with affine spaces given by
these coordinates, hence £/; = the set of the points such that /; # 0, 0 < i < r.
For an (9-module & we shall denote by & ; the restriction of W to Uh 0 < i < /•.
Let m be an integer. The multiplication by tf/t1? defines an isomorphism on
Ut fl Uj between SF{ and Wj. The sheaf obtained by gluing together the sheaves &■\
via these isomorphisms is denoted by SF(m). If cF is coherent, then the sheaves SF(m)
are clearly coherent.
We have canonical isomorphisms
6(m + m') ^ 6(m)®e6{m'),
® (0) ^ &, W (m) m ^ ®e 0(m), 3F (m + m') =± & (m) (m')
and also,
#oma (3% <J (m)) =; #owa (^ (- w), (?) ^ Home (&, §)(m\
§ being another (9-module. In particular, €(— m) is isomorphic to the dual
of 6(m).
Recall that if (X, (9) is a ringed space and £ an invertible <S-module, then
one sets
m times
£ ®e ... ®e £ if m > 0
Home(£.®-m, 0) if m<0.
From the above said one derives that (9(m) ~ 6>(1)®'", hence <? (m) =*
~ SF®^^)®'".
Let ? be an homogeneous polynomial of degree k in the coordinates t0,...
. .., tr. If s — (.?;); is a section of SF(m), then the system ((t/tf)sj)i is a section of
<F(m + &:). We thus obtain natural morphisms ^(m) -> SF(m + k); these will be
called the morphisms obtained by multiplication with t; the image of a section s
will be denoted by ts.
One can easily verify the existence of an isomorphism
<9(- r- 1) ~ Q
where Q is the sheaf of germs of differential forms of the type (/•, 0) with analytic
coefficients.

IV. PROJECTIVE MORPHISMS OF COMPLEX SPACES
139
We shall use the following
Lemma 1.1. H%\Pr, 6) = 0 for any q > 1.
Proof [24]. We use the isomorphisms H'(\Pr, 6) ~ H'(%!, &). If q > r then
Hi(\Pr, &) = 0. Let now -k: Cr+1\ 0 -> \Pr be the natural morphism which
corresponds to the homogeneous coordinates t0,..., tr. Denote Llf" = tc-1^') and let 6'
be the structural sheaf of Cr+1- Let q be an integer, 1 < q < r — 1. By lemma II.
3.4, i/<?('l£', 6>') = 0. Consider a cocycle \ = (E,c...lV) e C^lf, 6). By composition
with 7r we get a cocycle £' = (tz \h..A ) e CÇW', 6'), hence there exists tj'e
e C*-1^!!', 6"'), so that Svj' = £'. Every holomorphic function ti- ,■ can be uniquely
0-* * g — 1
written as a sum of the homogeneous components of given degree by expansion
in Laurent series. Denote by vjj .. ,• the component of zero degree of vj,' /
0 q -1 q - J*
Since the holomorphic functions on an open subset U of \Pr correspond to the
homogeneous holomorphic functions of zero degree on 7t-1 (£/), vj = (■/);„...; )
defines an element of O?-1^, <S>) such that Sv) = Ç.
It remains to check that Hr(\Pr, <S>) = 0. We first remark that any cocycle
£' e C(f[l', £>') = <9'(^o n ... fl Ur) can be expanded in a Laurent series
S ffcfgo ... f«r, Oae<£..
«o ar=-co
Moreover, £' e iT(^C', 6') if and only if aa = 0 as soon as oc0 < — 1,..., ocr < — 1.
Consequently, £' is cohomologic with a holomorphic function that admits a Laurent
expansion of the form
Y! a t*» ... t"r
ZJ "• 'o • • • r •
^0< - 1. . . . . «r< - 1
The homogeneous component of zero degree is null within such an expansion so
that the desired conclusion follows easily as above.
(b) We now consider the projective space \Pr as an algebraic manifold and
let 6 be the structural sheaf, that is the sheaf of germs of rational functions. In
order to prove the comparison theorem 2.6 we shall use the following result of
Serre [76]:
"Let & be a coherent algebraic sheaf on \P'. Then there exists an integer
m0 = m0(Sr) such that
(A) The sheaves SF(m) are generated by the global sections for m > m0.
(B) H\\Pr, ®(m)) = 0 for q 5* 1 and m > m0."
The following fact will be useful too:
"The cohomology groups Hq(\Pr, 6) vanish for any q > 1."
/
(c) Let us fix a complex space Y and a sheaf Se Cob.(7). Let X—>7be
a complex space over Y. Consider the set of all pairs (£, <p), where £ is an inver-
tible ^-module and <p : /*(©) -> £ is a surjective fi^-morphism. Two such pairs,

140 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(£, <p) and (£', <p'), will be called equivalent if there is an ^-isomorphism x : £ -► £'
/'
such that <p' = T<p. If X' —> 7 is another complex space over Y and g : X -* X
a 7-morphism (/' =/g), then any pair (£, <p) on A' defines a pair (#*(£), g*(<p))
on X' (for convenience, we identify g*f*($) with /'*(£>)).
As can be easily seen, one obtains a functor from the category of complex
spaces over Y with values into the category of sets. In [15] the author proves
the existence of a complex space denoted P($), which is called the projective fiber
bundle associated to £>, together with a morphism p : P(B) -> Y, which is called
the structural morphism and a pair (6(1), s) on P(&), which represents this functor.
So for any complex space X —> Y over Y, there exists a 1 — 1 correspondence
between the 7-morphisms X -*■ P(&) and the equivalence classes of pairs (£, <p)
on X defined above; if r : X -* P(&) is a 7-morphism, then the associated pair
is (£ = r* (6(1)), <p = r*(e)). The system (P(B), p, 6(1), e) is uniquely determined
(up to isomorphism) by this property.
If g = 6rY+l, then one can actually conclude that P(B) coincides (modulo
an isomorphism) with 7 x \Pr, the structural morphism p coincides with the
projection, 6(1) with the inverse image of 6pr (1) by the projection 7x \Pr^\Pr and e
is deduced via this projection from the morphism 6j.t -+6pr(\) which is defined
r
by means of formula (s0,..., sr) i-»- J] ttSi. In this case one denotes by 6(m) the
i = l
inverse image of <%>■ (m) by the projection 7x [Pr -> \Pr; for an «Syxip^ -module W,
we denote ^(m) = & ®0 6(m) (these sheaves can also be easily obtained by
patching just as in (a)). Properties similar to those of the absolute case hold here.
We also recall the following two properties of the projective fiber bundles.
If & -> S is a surjective morphism in Coh (7), then one obtains a closed
immersion P(i) -> P(&), which agrees with the structural morphisms. In particular,
we derive that, locally on 7, any projective fiber bundle P(B) can be embedded
in a space of the form 7x \Pr, and the embedding is compatible with the structural
morphisms.
The other property is the following: if (7, £>) is as above and <p: Y' -*■ Y
is a morphism of complex spaces, then there exists a natural isomorphism
p(ê)xYY'^p (<{,*($)).
In particular, for any point y e 7, the analytic fiber P(B)y (which is isomorphic
to the fiber product of X and (y, 6y/my) over 7) can be identified naturally with
the projective space associated to the vectorial space %yjmyS>y.
f
(d) Let X —> 7 be a proper morphism of complex spaces and £ an inver-
tible ^-module. We assume that the canonical morphism /*/*(£) -> £ is surjective.
By (c), it defines a morphism
X^P(U(£)),
which agrees with / and with the structural morphism P(f%(£)) -> 7.

IV. PROJECTIVE MORPH1SMS OF COMPLEX SPACES
141
£ is said to be very ample with respect to Y {or j) if the morphism X -*
-* P(J*(£)) is a closed immersion. The sheaf £ is called ample with respect to Y
if for any point y e Y there exists a neighbourhood V and an integer n > 1 such
that the sheaf £®"|/-1(F) is very ample with respect to V.
A morphism/ : X -* 7 of complex spaces is called projective if there exists
an ample sheaf with respect to Y on X.
In particular we consider the case when Y is the final object (*, (£)■ An
invertible sheaf £ on a compact analytic space X is called very ample if the
global sections T(X, £) generate the stalks £x, x e A", and the canonical morphism
X -*■ P(T(X, £)) is a closed immersion. £ is said to be ample if some of its power
£®" (« > 1) is very ample. These definitions agree with those given above.
(e) Recall some facts in local algebra.
Lemma 1.2. Let A -> B be a local morphism oj noetherian local rings and
M a B-module oj finite type, which is flat over A and such that M/mAM is a jree
B/mAB-module. Under these assumptions M is a jree B-module.
Prooj. Choose x±,..., xre M such that their classes in MjmAM form a
basis over B/mAB. By Nakayama lemma, the morphism Br -* M defined by these
elements is surjective and let N be its kernel. Hence we have the exact sequence
of 5-modules §-*N-*Br-*M->0 and by tensoring ®A A/mA we obtain an
exact sequence
0 ->• N/mA N -► (B/mA E)r -► M/mA M ->• 0.
It results that NjmAN = 0. Since m^TV = (mAB)N, we deduce again by Nakayama
lemma N = 0, therefore M is 5-free.
Proposition 1.3. Let A -* B be a local morphism oj noetherian local rings
such that B is a flat A-module and B/mAB is a regular ring oj dimension r. Let M
be an A-flat B-module oj finite type and N an arbitrary B-module. Under these
assumptions
(a) Exti (M, N)=0 jor i > r.
(b) Ij moreover the modules N and Extg(M, N) are A-flat, then
ExtB {M, N)/mA ExtB (M, N) ^ Ext; ,mAB (M/mA M, N/mA N).
Prooj. (a) Let us consider an exact sequence of 5-modules
Lr_x -> ... -> L0 -> M -> 0,
where Lt are free modules of finite type and let P be the kernel of the map
Lr_! -> Lr_%. P is a .B-module of finite type which is flat over A (one can see this
by splitting the above exact sequence into short exact sequences). By tensoring
®A A/mA one then obtains the exact sequence
0 -> P/mAP -> Lr_1jmALr_1 ->...-► L0/mAL0 -> MjmAM -> 0.

142 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
By the hypothesis on B/mAB one derives that P/mAP is free as a B/mAB-modu\e.
In virtue of the previous lemma, P is 5-free. The conclusion follows by calculating
ExtB(M, N), taking into account the exact sequence
(*) 0 -> Lr -*■ Lr-l -*■ . .. -*■ L0 -> M -> 0,
where we have put Lr instead of P.
(b) Denote by L. the resolution of M given by (*). By tensoring ®AA/mA,
one gets a resolution L./mAL. of M/mAM. Then Ext'BimAB(M/mAM, NjmAN) are
the cohomology groups of the complex HomB!mAB(LJmAL., N/mAN). Consider
the complex K' = HomB(L„ N). Its components are flat ^4-modules and the
associated homology groups enjoy the same property since they are isomorphic to
Exti (M, N). Whence (by taking into account the short exact sequences given by
cocycles; coboundaries and cohomology...), some isomorphisms
H'(K')/mAH'(K') * H'(K'/mAK').
Since K'jmAK' ^ HomBlmAB(LJmAL„ N/mAN), the conclusion follows.
We shall also use the following
Lemma 1.4. Let A -> B be a local morphism of noetherian local rings such
A A
that the morphism induced between completions A -* B is flat. Then A -> B is flat.
A A
Proof. The morphisms A -*■ A and B -*■ B are flat. Let now 0 -> M -> N
be a monomorphism of ^-modules of finite type and P = Ker (M ®AB -* N<g>AB).
Then one obtains the exact sequence
0 -> P®BB -*■ {M®AB) ®BB -> (N®AB)®BB.
But (M ®AB)®BB ~(M®AA)® ,B, (N®AB)®BB ~ (N®AA)®,B.By hypothe-
A A
A
sis we have P®BB =0, therefore P = 0 according to Krull's theorem.
§ 2. The behaviour at +oo of the sheaves ^(w)
The main result in this paragraph is the following theorem of Grauert and Rem-
mert [32]:
Theorem 2.1. Let / : X -* Y be a projective morphism of complex spaces
and £ be an invertible sheaf on X, which is ample with respect to f. For any Qx-
module Er, denote ^(m) = &■ ®@xS.®m, m arbitrary integer.
Then for any coherent analytic sheaf & on X and for any compact Kof Y there
exists an integer m0 = m0(K, fr) such that:
(A) The canonical morphism
is surjective in all points of K for any m > m0;
(B) Rqf\.(&(m)) vanishes on K for all q > 1, m > m0.

IV. PROJECTIVE MORPHISMS OF COMPLEX SPACES
143
Proof. We first remark that if the theorem is still true when £ is replaced by
$,®d, d > 1, then it is true in the stated form. In order to prove this, let & e Coh X.
For any integer m > 1 we can write &(m) = (SF (g) £®r) ® £®M where h, r are
integers, h > 0, 0 ^ r < d; the conclusion follows for the coherent sheaves & ® £®%
0 < r < d, which is nothing else than our hypothesis.
Since the theorem is local on Y we may assume that £ is very ample with
respect to/. Hence we have a natural commutative diagram
X >Y
PU* («))
where X -> P(Jm (£)) is a closed immersion and P (/* (£)) -> Y is the structural
morphism. Again, if we take into account the fact that the assertions of the theorem
are local on Y, we can assume that there exists a surjective morphism <3y+1 -►/.,.(£) ->
-> 0. We thus obtain a closed immersion P(fJ&)) -> P(GrY+1) = Y x \Pr, which
is compatible with the structural morphisms. Thus we find a closed immersion /:
X -*■ Y x \Pr such that the diagram
X—'-+ Y
M-
Y x\Pr
is comutative; moreover, we have an isomorphism z'*(<?y x[Pr(l)) ~ £. For any
^-module SF, we get i*{&{m)) ~ (i*(&)) (m).
Taking all these into account we can easily reduce the proof of the theorem
to the case when X is of the form Y x \Pr,f is the canonical projection and £ is
the sheaf <5x(l) (the inverse image under the projection Y x \Pr -*■ \Pr of the sheaf
<9Pr(l)). We shall use in this case induction on r. If r = 0 then the assertions from
the statement are obviously fulfilled for any integer m. We suppose now that the
theorem is true for r — 1 and prove it for r. Let & e Coh (X).
The existence of some m0 which verifies (A). Denote by t0, ..., tr the
homogeneous coordinates in \Pr. Let x be a point in X and m an integer such that the
morphism f*f*(^(m))x -> &(m)x is surjective. We show that this fact holds for any
integer m', m' > m too. Let x = (y, z), y e Y, z e \Pr and let Uk be one of the
canonical affine carts of |pr which contains z. Consider the morphism
6: 6pr(m) -> 6\pr{m)
defined by multiplication by ï£'-m(for any cart £/; this means the multiplication
by (?t/?i)m'"m). This morphism is an isomorphism on Uk. Thus one derives canon-

144 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
ically a morphism of ^-modules oF(w) -> ^(m'), which is an isomorphism on
Y x Uk, particularly in the point x. The conclusion follows immediately. We
also remark that as soon as for an integer m the morphism f*J^(^(m))x -*& (m)x
is surjective in x, the same property holds in virtue of the coherence in a
neighbourhood of x.
These two remarks reduce the proof of the existence of the integer m0 to the
proof of the following assertion:
(*) For every xeX there exists an integer m which depends on x and EF,
such that the morphism
/*/*(*(«)) - *(m)
is surjective in x.
Let x= (y, z),ye Y and z e \Pr. Consider a hyperplane E which passes through
z, whose equation is t = 0. Denote by 6E the structural sheaf of E, which is extended
trivially in \Pr \ E. Y x E is a closed subspace of Y x \Pr and denote by Oyxe its
structural sheaf trivially extended out of Y x E. We have the exact sequence
0 -»• <V(— 1) -»• <V -»• ge -»• °
where the morphism <2ff»-(—1) -> 6ffr is given by multiplying by t. Taking
into account the projection Y x [Pr -> [Pr one obtains an exact sequence
By tensoring ®<9X SF one gets the exact sequence
0 _► Tor?<§S <S>yxE) - SF(- 1) - ff - SF®,sx<S>yx£ - 0.
For convenience we denote
® = ^Oex^xE and (2 = Tor<?*(S% 6ïxE).
Si and S are coherent ^-modules and for any integer m we have an exact sequence
0 -»• £(m) -»• 9{m — 1) -»• SF(m) -»• <&(m) -»• 0.
Denote 3m = Ker (SF(m) -> $(m)). We thus obtain the exact sequences
0 -»• e(m) -> ®{m - 1) -»• 2m -»• 0, 0 -»• 2m -»• ^(m) -»• $(m) -»• 0.
By applying the functor/* we derive the exact sequences
R%(SF(m - 1)) - ^/*(2m) - *2/*(<S(m)),
/P/^J - RlU{&{m)) - &S*mn%

IV. PROJECTIVE MORPHISMS OF COMPLEX SPACES 145
If 3(7 x E) is the maximal ideal-sheaf associated to the subspace 7 x E <= 7 X \Pr,
then
3(7 x E)3b = 0 and 3(7 x E)& = 0.
We realize that & and (2 can be regarded as 6,yxE-modules, even as coherent <9yx£-
modules. By the induction hypothesis applied to the morphism 7 x E -> 7 and
to these sheaves, there exists an integer m0 such that R2f*(£(m)) and Ry^aèÇm))
vanish in a neighbourhood of y whenever m > m0. Then the maps
tf1/*^™ - 1)) - ^y*(SJ, ^/*(2J -, VMHrn))
are surjective in >' for any m > m0- We deduce that for these integers m the
composite maps
R%(Hm - l))y - VhiSrim)),
are surjective. The sheaf R1/*(&(»% — 1)) is <?y-coherent by the finiteness theorem.
By the noetherianity of the «S^-module ^(^(/«d — 1))^, we deduce the existence
of an integer m1 ^ m0 such that the maps R^-f^im — l))r -> Rlf^(m))y are
bijective for all m > «?!■ Then it follows that the maps R}j'%(%,„) -»• Ry*(!f(m)) are
bijective in >? for m > m^ From the exact sequence
A(SF(«)) -+MHm)) - ^(SJ - IPMHm))
we deduce that the maps/^(^(m))^ -*JJJ&{m))y are surjective for any m~^mx. As a
consequence, the maps/*/#(S"(m))x-»/*/^(^(m))x are surjective for any m~$>mx.
By induction hypothesis there is an integer m2 such that the map /*/*(<&(>«)).,.->
-> Sb{m)x is surjective for any m > m2. Therefore, for m sufficiently large, the maps
f*UHm))x ^f*M®(m))x and f*MWm))x - «-(«)x
are surjective. We have
&(m)x = ff(/n)x/3x(r x £) ff(m)x.
The assertion (*) will be concluded from these facts by means of the commutative
diagram
Ï I
&(m)x —> &(m)x
and by Nakayama lemma.
10 - c 2398

146 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The existence of some m0 which verifies (B). We proceed by descending
induction on q. For q sufficiently large the functor Rqf* is null. We suppose now
that for an integer q > 2 and for any triplet (Y, K, &) there is an integer m0 which
verifies (B) and we are going to prove the corresponding assertion for the integer
q — 1. Let (Y, K, 8F) be as in the statement of the theorem (recall that we are
considering the case X = Y x \Pr, £ = <S> (1) and / projection). Let y e Y be an
arbitrary point. It is enough to find a neighbourhood V of y such that Ri-y^tFÇm))
vanishes on V for m sufficiently large. Let V be a relatively compact Stein
neighbourhood of y. Consider an integer m0 such that the morphism
f*MS(mJ) - ffK)
is surjective on/_1(F). There exists a surjective morphism QPY ->/*(aF(m0)) on V.
We then deduce via/* a surjective morphism &"x ->/*/*(<? (m0)) on f~\V) and
by composition, a surjective morphism S^ -» ff(m0) on/-1(F). Let <95-(— m0) -> 8F
be the morphism obtained from this and let § be its kernel, (J e Coh (/_1(F)).
For any integer m we have an exact sequence on/_1(F).
0 -»• §(m) -»• <95.(m — m0) -»• SF(/ti) -»• 0.
Denote by/ too the morphism/_1(F) -> F deduced from/. One obtains the exact
sequence on V
Rr%(.e^im - m0)) - Ri^MHm)) -» /^*(<?(«)).
By the induction hypothesis and by shrinking eventually F around j>, we get
Rqf*(§(m)) = 0 for m sufficiently large.
The proof of the theorem will be over by applying the following
Lemma 2.2. Let Y be a complex space, X = Yx [Pr andj: X -* Y the projection.
Then
R>h(Pxim)) = 0
/or any m > 0, q~$>\.
Proof. If F is a Stein open subset sufficiently small in 7, then
H%Vx1Pr, <9A:)=0for 9>1 ; this fact can be proved either exactly as in lemma 1.1
or by using this lemma and a simple Kiinneth formula. Thus we get Rqf%(6x) = 0.
We shall prove the assertion of the lemma by induction on m. The case
m = 0 is already proved. For the general induction step we use again induction
on r. If E is an arbitrary hyperplane of IP'" then we have already pointed out the
exact sequence
0-<S>*(-l)-<2>*-<2>rx*-0.
The conclusion easily follows by means of the exact sequences
0 - Ox (m - 1) - Ox{m) - eYxE(m) - 0.

IV. PROJECTIVE MORPHISMS OF COMPLEX SPACES
147
Remark. The theorem can be applied in the particular case when X is the
/-dimensional projective space over Y, £ = 6^(1) and/ is the projection. However,
the theorem follows easily from this case, as one could realize from the proof.
Corollary 2.3. Under the assumptions of the theorem, let &■ -*■ § -*■ % be an
exact sequence of coherent Qx-modules. Then, for any compact K of Y, there is an
integer m0 such that the sequence
is exact on K whatever m > m0.
Proof. Let &', §', <$" be the kernel, image and cokernel, respectively, of the
morphism § -> <$. Then <$' is the kernel and <$" is the image of the morphism
i| -> %\ let 3C" be the cokernel of the latter morphism. All these are coherent
é^-modules. Since the functor & i-»- aF(m) is exact, it is sufficient to prove that for m
sufficiently large, each of the sequences
0 -/.(*' (m)) -+M*(m)) -/,(<?' («)) - 0,
o -/,(<?' («» -+uqQn)) -»/*(<?"(«)) - o,
0 -/,(<?"(«)) -/»(3C(/n)) -/,(3C"(«)) - 0,
is exact on K. In other words, we may assume that the sequence
0-> g; -► <J-> ft -► 0
is exact. To conclude with, we apply the theorem and the exact sequences:
0 -/,(*(«)) -/*(<?(«)) -/*(3£(«)) - tf1/*^)) - ... .
Consider y as a complex space reduced to a point, whose stalk is the
complex field and X = \Pr ; then one gets the theorems A and B on the projective space
of Serre [77]:
Theorem 2.4. Let \Pr be the complex projective space of dimension r and <3
be the sheaf of germs of holomorphic functions on it. For any coherent Q-module W
there exists an integer m0 = m0(oF) such that
(A) The sheaves 8r(m) = Sr®& <3(m) are generated by the global sections
for m > m0;
(B) Hi{\Pr, ^(m)) = 0 for q > 1 and m > m0.
Corollary 2.5. Any coherent analytic sheaf on \Pr admits a finite resolution
by locally free sheaves of finite rank.
Proof. Considering the regularity of the local rings of \Pr, it is enough to
show that any sheaf & e Coh (\Pr) is the quotient of a locally free sheaf of finite

148 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
rank. For m sufficiently large we deduce from the theorem an epimorphism of the form
6'g.r -> W(m) -> 0, hence an epimorphism
<S>S.r(—m) -»• & -> 0
and the conclusion follows.
We conclude this paragraph with the proof of the comparison theorem of
Serre [77].
Denote by X the projective space \Pr considered as algebraic manifold and
by Xh the same space considered as complex manifold. By 6 we mean the
structural sheaf of X and by &h the structural sheaf of Xh. Since any Zariski open set
is open in the usual topology and since any rational function is holomorphic, one
naturally obtains a morphism of ringed space Xh -► X.
For any point x e X the morphism 6X -+ Qhx induces an isomorphism by
passing to completions (the C-algebras 6X and Q% are in fact isomorphic to the
algebra C[[^i, ■ • -, zr]] of formal series in r indeterminates). By applying lemma 1.4
it follows that the morphism Xh -> X is flat.
If 8F is a coherent algebraic sheaf on X, then its inverse image under the
morphism X1' -> X is a coherent analytic sheaf on Xh, denoted by oF*. In this way
one defines an exact functor from the category Coh(X) to the category Col^A^).
We have 6(m)h = <Sh(m), m being an arbitrary integer.
Let gF e Coh(X), ^f a finite affine covering of X and "If'' its inverse image in
X1'. Since any Zariski affine open set is a Stein open set, 1£* turns out to be a Stein
V
covering. We naturally deduce a morphism between the associated Cech complexes
C*(lf, &) -> C'(ellh, &h), hence a morphism between the cohomology groups
H'(X, §9 -► H'{Xh, &>')
(the existence of this morphism is in fact, a general property of the ringed spaces).
It is easy to check the functoriality of these morphisms and their
compatibility with the short exact sequences.
Theorem 2.6. The functor W i-»- Wh from the category of coherent algebraic
sheaves on the projective space X = \Pr to the category of coherent analytic sheaves
is an equivalence of categories.
Moreover, the canonical morphisms
H\X, SF) -► Hi{X\ S=h)
are isomorphisms, q being an arbitrary integer.
Proof. We first prove the last assertion. If HF = <9, then H°(X, &)=H°{Xh, 6h) =
= C and H"{X, 6), H\Xh, ô*1) vanish for q > 1 (lemma 1,1 and § \.b).
Next, consider the case of the sheaves & (m). We proceed by induction on r.
As the case r = 0 is obvious, we prove the general induction step. Let E be a hy-
perplane of the homogeneous equation t = 0. We have the exact sequence
0-> <s>(—1) -»• 6 -> <3E -> 0

IV. PROJECTIVE MORPH1SMS OF COMPLEX SPACES
149
where the morphism <S>(—1) -> & is obtained by multiplication by t and 6E is
the structural sheaf of E = IP'"1 which is trivially extended on \Pr \ E. For any
integer m we get the exact sequence
0 -»• 6{m — ])-»• <3(m) -> eE{m) -> 0.
Similarly we get exact sequences
0 -»• e\m — 1) -»• e\m) -»• 6E*(m) -> 0.
We have a commutative diagram
... -> H"(X, 6{m - 1)) -► #«(*, 6(m)) -► #*(£, <9£(w)) -► H«+l(X, <S>(m-l)) -► ...
4- 4-4-4-
.. .^>H«(Xh, e{m-\)h)^H"{Xh, <9(m);r)->-#«(£'', 6E(pif)^>H'>+\Xh, <9(/m-1 )*)->•. ...
According to the induction hypothesis the morphisms Hq(E, <5E(m)) -+Hq(Eh, &E(m)h)
are isomorphisms. By the lemma of the five the morphisms in the
statement are isomorphisms for âF = 6(m) if and only if'they are isomorphisms for
âF = &{m — 1). Since m = 0, this fact has been already verified; the required
conclusion follows.
We now consider the case of arbitrary coherent algebraic sheaves and proceed
by descending induction on q. For q sufficiently large H"{X, âF) and H^X'', ®h)
V
are null (by using Cech cohomology !), hence we have to prove the general induction
step only. By § 1 (b) there exists an exact sequence 0-><SI->£->gF->0 where
£ is a direct sum of sheaves of the form 6 (m). We have the commutative diagram
...-»• H"(X, Si) -► H"(X, £) -► H"(X, âF) -► Hi+1 (X, SI) -► H"+\X, £)->...
\y \y \y \^> \y
'...-> H"(Xh, Sit) -► W(Xh, £*) ->• H»(Xh, Wh) -► H*+\Xh, Slh) -► Hi+l(Xh, lh) -► ...
The morphism e2 is an isomorphism; by the induction hypothesis the morphisms
e4 and e5 are also isomorphisms. The lemma of the five shows that e3 is surjective.
This fact holds for any âF e Coh (X), hence for SI too, therefore e^s surjective.
By using again the lemma of the five we deduce that e3 is bijective and the proof of
the second assertion of the theorem is completed.
We now prove the first assertion of the statement. Let SF, 6j! e Coh (X). There
exists a natural morphism
Home($, §)h -> Home»{Wh, qh).
We will show that it is an isomorphism. The problem is local on X, hence we may
assume that there exists an exact sequence &"x -> &qx -> âF -> 0. The conclusion

150 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
follows without difficulty from the exactness of the functor "h". By applying the
last assertion of the theorem, the morphism
Horned, q) = H°(X, Homei®, §)) -»• H°{Xh, Home(®, §)h)
~ H°(Xh, Home*($h, <f)) = Hom^a^, <f)
is bijective.
Thus, in order to complete the proof of the theorem, it remains to show
that for any <F e Coh (Xh) there exists (and hence is unique) a sheaf <F e Coh (X)
such that gF' ~ gF*. By 2.4, for m sufficiently large, âF' (m) is isomorphic to a
quotient of a sheaf by the form (<3h)n and hence âF' is isomorphic to a quotient of a
sheaf of the form £\— m)'\ If we denote by £0 the sheaf £>(— m)n, then we
obtain an exact sequence 0 -> §' -> £§ -> âF' -> 0. By iterating the above
reasoning we find an exact sequence
£* X £§ -► âF' -► 0
where the sheaves £0 and S.x are isomorphic to some finite direct sums of sheaves
of the type 6 (m). By the antecedence, there is <p e Home (£1; £0) such that <p;' = <p'.
If we denote by & the cokernel of <p, then we deduce that IF' ~ Wh and the theorem
is proved.
Remark. The results from 2.4, 2.5, 2.6 could be indeed stated for an
arbitrary projective manifold X (in 2.5 X is assumed nonsingular). The proofs easily
follow from the above considered case by embeddings X -* \P'.
§ 3. The behaviour at — qo of the sheaves £F(m)
Let/: X -*■ 7 be an arbitrary morphism of complex spaces, âF an ^-module and
y a point of 7. Denote as usual by Xy the fiber/-1 (y) endowed with the restriction
of the sheaf 6Xy = <3x/my<3x to ^l by &y we mean the sheaf &/my& (my is
the maximal ideal of 6 and "A" stands for the sheaf generated by it in 6X).
Wy is an ^-module and denote by prof &y its depth,
profff„ = inf profo, (ff,)x = inf prof^/m^^/m,*,).
Analogously,
dim gFj, = sup dimax (âF^ = sup dim(9x/my(9;e(âF;c/mJ,âF;c).
x./U)=.>' y''v *./(*)-.y

IV. PROJECTIVE MORPH1SMS OF COMPLEX SPACES
151
For convenience we introduce the notation
prof y âF = inf prof âF the depth of âF with respect to Y, dimy âF
yeY
the dimension of §• with respect to Y.
We have the inequality
profy âF < diiriy âF(if âF # 0).
The main result in the paragraph is the following theorem of Grothendieck :
Theorem 3.1. Letf: X -> Y be a projective morphism, £ be an invertible <3X-
module very ample with respect to Y,W a coherent Qx-module flat over Y, and q > 0
an integer.
(a) If prof y âF > q + 1, then for any compact K of Y there exists an integer
m0 = m0(K, âF) such that R'f %(&•(— m)) \ K = 0 for any i < q and m > m0.
(b) diiriy âF ^ q if and only if for any compact K of Y, there exists an integer
m0 = m0 (K, âF) so that R'f^ (âF (—m)) \ K = 0 for any i > q and m > m0.
Proof. The assertions of the theorem are of local nature on Y, hence we may
assume the existence of a surjective morphism of sheaves ô^1 ->/# (£). This induces
a closed immersion X ^ X' = Y x \P' such that £ is induced by Qx. (1) and, in
addition, the diagram
X » Y
\/r-r
X'
is commutative.
If âF' is the image of âF by the immersion X -* X', then âF' is a coherent
^'-module which is flat over Y. We have
R'fU®') - R'f*(®)> Prof y âF = prof y âF', diiriy W = diiriy W.
Therefore, in order to prove the theorem we may assume X of the form Y x \Pr,
S. — 6X (\) and f the projection.
For an analytic subset Z of Y (locally closed) and for an integer j > 0 we
denote by Xz the product Zx \Pr (= Zx y X), by âFz the inverse image of âF under
the morphism Xz -► X and by &{Z) the sheaf ExtiQx (âFz, QXz{— r — 1)). âFz is
flat over Z. £>7 (Z) is a coherent fi^-module which is equal to zero if j does not
belong to the interval [0, r], as it can be seen by passing to stalks and applying
proposition 1.3. In particular, for the sets Z = {y}, y e Y we get a family of
coherent analytic sheaves BJ'(y) = ExtJ@x {&y, 6Xy (— r — 1)) on the fibers Xy, yeY. We
are going to prove the following assertion :
(*) For any compact Kof Y there exists an integer m0 such that H'(Xy, &{y)(m)) =
= Ofor any i > 1, j ^ 0, m ^ m0, y e K.
= sup dim âFj,,
yeY

152 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
It is enough to confine the problem to semianalytic Stein compacts ([25] or
Chapter V, § 1). Let K be such a compact. The ring <3(K) of the germs of holo-
morphic functions on K is noetherian ([25] or Chapter V, § 3).
Let ae Spec <5(K); we denote by Za the germ of analytic set defined by a
around K, and we designate so any representative of this germ. Let Xa = XZa,
and &'(</.) = ê'(Za). Apply V. 4.3 for the morphism Xa -+ Zx and the sheaf £>'(oc):
there exists a dense open subset Va of Za such that Za \ Va is an analytic set and
&i(ot.) is flat over Va. Since the sheaves S-'(a) are null, except a finite number of
indices j, one may assume the open set Va "good" for all indexes j. The analytic
set Za \ Va yields a closed set in {a} e Spec &{K) and let Da be its complement in
{a}: it is a nonempty open subset of {a} (otherwise, by restricting eventually the
representative Za around K, one finds Va = 0, a contradiction), so a e Da.
We are going to prove, by noetherian induction, that there exists a finite
partition of Spec <S(K) by locally closed sets of the form Da. Consider the set of
all closed subsets of Spec <3(K) which do not enjoy this property. We shall prove
that it is empty and the desired conclusion will follow. On the contrary, suppose
that this set is nonempty : since £(K) is noetherian, this set has a minimal element
T. The minimality implies that T is irreducible, hence T = {a}, a e Spec <9(K).
Let Da be the open subset of {a} = 7 constructed above; the minimality of T again
and the equality T = Da U (T \ DJ lead us to a contradiction. Thus let
(.DJa be such a finite partition of Spec <2>(K). By the above constructions
we derive that the corresponding sets Va cover the compact K (if x e K and mx e Da,
then x e VJ). Apply theorem 2.1 to the morphisms /a: Xa -> Za deduced from /
and to the sheaves &(a) : there exists (eventually by restricting the sets Za around K)
an integer m0 such that R'fa (&J'(a)) (m) = 0 for all i > 1, j > 0, m > m0 and for
any index a of the above partition. Since the sheaves &(<x) (m) are flat over Va then
by III. 3.11,
H'(Xy, §'(«) («),) = 0 for i > 1, j > 0, m > m0, yeVa.
By virtue of proposition 1.3 we derive isomorphisms
S/(a) («), ^ fi'' C) (m),
for any y ^ 0, m arbitrary integer, yeVx and the assertion (*) is thus proved.
From (*) we now deduce the following assertion:
(**) for any compact K of Y there exists an integer m0 such that
H' (Xy, $y (- m))' ~ T (Xy, Ext'-'(S„ QXy) (m))
for any m ^ m0, y e K and i > 0 (the accent means the algebraic dual over C and
Q, as usual means the sheaf of germs of holomorphic forms in the maximal degree).
Let K be an arbitrary compact of Y and let m0 be an integer which verifies (*). For

IV. PROJECTIVE MORPH1SMS OF COMPLEX SPACES
153
any integer m and for any point ye Y there is a spectral sequence which converges
to Ext: (X; &y, fiy (m)), such that
EV(y, m) = W{Xy, Ext'ex{SFy, QXy(m))).
We have
Exf@x(S„ nXy(m)) ~ Exfex(Wy, aXy) (m).
For the projective space \Pr, 6^ (— r — 1) ~ Qy. The assertion (*) shows that
the spectral sequences Eu'{y, m) degenerate, hence
Ext^OT,; ff„ aXy) (m) ~ T(Xy, Ext'@x(Sf y, ttXy) (m))
for any y e K and m ^ m0.
The assertion (**) will follow then from the duality theorem on the projective
space (VII, § 4) and from the isomorphisms
ExfQx{Xy; Sy(- m), QXy) x Ext'@x{Xy; Sy, nXy(mj).
We prove now the assertions of the theorem.
(a) Let A! be a compact of Y and m0 an integer which verifies (**). Let y be a
point of K. We have prof &y $s q + 1. By passing to stalks and applying I. 1.15,
it follows that
Ext'e-ysy,nXy)(.m)=0
for any i < q and whatever m. From (**) we derive that
H'\Xy, Wy (— m)) = 0 for any i < q, m > m0 and jel
By applying III. 3.5 it follows that
^'/* (^ (~ w)) | A! = 0 for any i -^ q and »i > m0.
(6) The direct implication can be proved in the same manner. We prove the
converse. Let ye K. By hypothesis, the sheaves /?!/*(<? (— m)) are null in a neighbourhood
of y, for / > q and m large enough. According to III. 3.11, H'(Xy, ^y (— m)) =
= 0 for i > q and m large enough. For m sufficiently large the sheaves
Ext' (SFy, Q.Xy) (m) are generated by the global sections. By applying the assertion
(**) (to the compact K = {y}, a case when the use of (*) is superfluous !) we deduce

154 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
that Ext' (&y,ClXy)=0 for i < r — q. By 1.1.17, dim f^, < ?. Accordingly,
dimycF < q and the theorem is proved.
Remarks, (i) The proof of (b) can be made without the use of the assertion (*).
For the converse implication, we have already remarked this fact. For the direct
implication it is enough to check that R!f*(8r) — 0 for i > q, for any gF e Coh (X)
which is flat over Y and such that diirijJF < q. For any yeY, H'(Xy, § y) = 0,
whenever i > q ^ dim W y (apply [64]); the conclusion will follow by III. 3.11.
(ii) In the course of the proof we have applied the duality theorem on the
projective space \Pr for coherent analytic sheaves. By using 2.5 this duality can be
deduced from the classical case [75] of the locally free sheaves of finite rank.
Corollary 3.2. Let f: X -> Y be a flat proper morphism such that the fibres
Xy are perfect spaces of dimension r, let £ be an invertible <3x-module very ample
with respect to Y and W e Coh (X) locally free.
Under these assumptions, for any compact K of Y, there exists an integer m0 =
= m0 (K, §0 such that
R'f* (gF (— m)) | K = 0 for any i # r and m $s m0.
Proof. For any point yeY the sheaf Wy is locally free of finite rank over Xy,
hence prof Wy = dim Sfy=r. As a consequence profy-oF = dimyoF == r and apply
the theorem.
The corollary can be applied in the following situation : Y an arbitrary complex
space, X = Y x \Pr,f the projection and £ = &x (1).
We now restate theorem 3.1 in the absolute case.
Corollary 3.3. Let X be a projective manifold, £ a very ample sheaf on it,
S. e Coh (X), and q be an integer. Denote ^(m) = Sr®(gx£®m, m arbitrary integer.
(a) prof S7 ^ q + 1 if and only only if H'(X, Sr(— m)) = 0 for i < q and m
large enough;
(b) dim & < q if and only if H'(X, W{— m)) = Ofor i > q and m large enough.
Proof. We have to prove the inverse implication from (a) only. By a suitable
embedding we reduce the problem to the case X = \Pr and £ = <%»• (1). The
conclusion follows from the assertion (**) (which can be easily established in the
absolute case), exactly as in the case of assertion (b) from 3.1.
Corollary 3.4. Let X be a projective manifold, £ a very ample sheaf on it
and W e Coh (X). Then &• is Cohen-Macauley {prof S:= dim 8r) if and only if there
exists an integer q such that
H'(X, oF(— m)) = 0 for i # q and m large enough.
To conclude this paragraph, we give the following result due to Grauert.
Theorem 3.5. Let f: X -> Y be a projective morphism of complex spaces, £
an Qx-module very ample with respect to Y, t a global section of £, 3 the maximal
ideal-sheaf associated to the subspace of all zeros of t, & e Coh(A'), and q an integer.
Suppose in addition that & is flat over Y, the section t is &-regular and profY W ^ q + 1.

IV. PROJECTIVE MORPH1SMS OF COMPLEX SPACES
155
Under these assumptions, the canonical morphism
R!f^) - lim Rf^SF/T'Sr)
in
is an isomorphism for i < q and a monomorphism for i = q.
(Recall that a point x is said to be a zero for t if txemxS.x).
Proof. The problem is local on Y and so we may replace 3 by any coherent
sheaf of ideals having the same zeros. If we replace £ by some of its tensor power
and section t by some of its power, we may assume that £ is very ample with respect
to Y. Moreover, by 3.1 we can suppose that there exists an integer m0 such that
R!f*(&(— tn)) =0 for i ^ q and m ^ m0.
In order to prove the theorem it is sufficient to check that if moreover Y is Stein,
the canonical morphism
H\X, &) -> lim H'(X, S:/3mS:)
m
is an isomorphism for i < q and a monomorphism for i = q.
We have isomorphisms H'(X, H~ tri)) =± T (X, R'f^® (- m))) = 0 for
i < q and m ^ m0. On the other hand, the multiplication by tm considered as
morphism from qF(— m) to SF yields an exact sequence
0 -»• ^(— m) -> W -»• S^/a'"^ ^-0 (m > 0).
From the associated cohomology exact sequence, one derives that the canonical
morphism H'{X, &) -»• H'(X, &/3m&) is bijective for i < q and injective for i = q.
Thereby the proof is completed.
§ 4. Two criteria for ampleness
The first criterion stands for a converse of theorem 2.1.
Theorem 4.1. Let f: X -> Y be a proper morphism of complex spaces and
let £ be an invertible sheaf on X which enjoys the following property : for any compact
K of Y and for any W e Coh (X), there is an integer n0 = n0 (K, SF) such that
*"/*<?®£®n)l K = 0 for q >l, n>n0.
Under these assumptions, £ is very ample with respect to Y.

156
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. The problem is local on Y. Let us fix a point y0 e Y. Take x e Xyo =
=j~1 0'0). We first prove the following assertion:
(*) For any invertible sheaf £ on X, as soon as the canonical morphism
/*/*(£) ^ £ is surjective in x, the same property holds for the morphisms /*/<,(£®") -»•
-y £®", n > 1 arbitrary integer.
Indeed, the hypothesis shows that any element of £x is a linear combination
of germs in x of elements from/*(£);,„ = T (Xyo, £) with coefficients in &x; then
any element of (£®")x ^ (£J®n is a linear combination of elements of the form
si s", steF (Xya, £) with coefficients in 6X; the conclusion follows.
By mx we denote for convenience the maximal ideal of 6X as well as the
coherent ideal-sheaf on X defined by it. From the exact sequence
o -► mx -► ex -> exjmxex -► o
we get the exact sequences
0-tiiij® £®" -»• £®" -»• £®n/mx £®" -> 0.
For n sufficiently large, R1fif{mx ® £")„ = 0, hence the morphisms
/*(£®"k -/.(««"/m, £®"),,
are surjective. The sheaves £®"/mx £®" are concentrated in the point x where their
stalks are equal to £®7'«* £f "■ In this way the morphisms r (Xyo, £®") ->•
~^S,f/mxS,f" are surjective for 72 sufficiently large. By Nakayamma lemma, the
morphisms /*/*(£®")x -»• £?" result surjective.
So there exist a neighbourhood [4 of x and an integer nx such that the
morphism /*/!K(£®"J!) -> £0"* is surjective on Ux. Since the fiber A^0 is compact, we find
a finite covering (t/"i)i<K* of it and some integers «; such that the morphisms
/*/*(£®n0 -»• £®"' are surjective on £/,-. If we set n = % ... «t, then the morphism
/*/*(£®") -»• £®" will be surjective on (J Uh hence it will be surjective on an open
subset of the form f~\V), V neighbourhood of y0. If we replace £ by £®" and Y
by V, then we may assume that the canonical morphism/*/*(£) -> £ is surjective.
By (*) the same property holds for the morphisms
/*/*(£®") -" £®",
n ^ 1 being an arbitrary integer.
Denote by
6 : X - />(/*(£))
the morphism over Y which is deduced from the epimorphism /*(/*(£)) -> £.
This morphism becomes explicit by passing to fibers. The fiber in y0 of the structural

IV. PROJECTIVE MORPHISMS OF COMPLEX SPACES
157
morphism P(/*(£)) -»• Y can be identified with %(%>.,,/*(%.)■ We have a
natural commutative diagram
^(/*(*k/"V„/*(Sk)-> A/*(£))
Let s°,..., sr e/#(£)j,0 = r(A^0, £) be such that their images in the vectorial
^0/ittyo-space/*(£)yo/my0/*(£)y0 form a basis. If x is a point of Xyo, then 6(x)
regarded as point in P(f*(£)yJmyJ*(£)ya) has the homogeneous coordinates
■j°(x),. .., s'(x) (as usual, for a section s of £, ^(x) is its image in £Jmx£x ^ C)-
If ' " -—- is a rational function defined around 6(x), P and g being homoge-
Q(s°,..-,sr)
neous polynomials of the same degree such that Q(s°,..., sr) (6(x)) # 0, then the
germ of holomorphic function defined by it around the point xe Xy through 6 is
just —, where a is an isomorphism £x ~ <S> (we remark that the
e(o(53,...,fffe0)
element 0(<t(s°), ■ ■ • > <*(■*£)) is invertible).
We will denote by 6®" the morphism X -»• P(f*(£.&")) and show that for «
large enough, by shrinking eventually Y around y0, 6®" is a closed immersion.
In this way, £®" will be very ample with respect to 7 and the theorem will be proved.
We first prove that for n sufficiently large 6®" is a local immersion in every
point. In order to do this we will use the characterization of local immersions by
means of Zariski tangent spaces. Consider a point x e Xyo. We prove the following
assertion:
(**) For any invertible sheaf £ such that the morphism /*/*(£) -> £ is sur-
jective, if moreover the induced morphism X -»• P(f*(£)) is a local immersion in x,
then the morphisms X -*■ P(f*(£®")) enjoy the same property, n ^ 1 being an
arbitrary integer.
Again, consider sections s0,..., srer(Xyo, £) such that their images in
/*(£)/„M.vo/*(£k form a basis- The morphism Xyo -»• i,(/*(%Mj'*(^W is an
immersion in x. Consider a section s e T{Xy, L) such that s(x)^0 and let s"~* = s® ... ®s;
the elements s""1® s0,. .., s"-1® s' lie in r(Xyo, £®") and the morphism Xn -► IP'
induced by their images in r(Xyo, £®"/myo£®n) coincides, in a neighbourhood
of x in Xyo, with the morphism induced by 6. One then derives that 6®" is
immersion in the point x and the assertion (**) is verified.
From the exact sequence 0 -> m* -> &x -> <3xlmx -> 0 we deduce the exact
sequences
0 -»• m;® £®" -»• £®n -»• £®"/m|£®n -»• 0.

158 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
If n is large enough, then the morphism
f*&*%. -/«(«•7m=*«'0J,„
is an epimorphism. The sheaf £®"/ml£®" is concentrated in x and its stalk
in this point coincides with £®"/m*e®"- We thus realize that the morphisms
r(Xy<), £®") -> £®"/m;£f " are surjective. Then it will result that the morphisms
induced by 6®" between Zariski tangent spaces in the point x and 6®"(x) are
surjective. Accordingly, for an integer n as above, the morphism 6®" is a local
immersion in the point x ([15]).
We actually find open sets Ult..., Uk which cover Xy„ and integers n[,. ..,n'k
such that the morphisms
6®"< : X -► />(/*(£*"'))
are immersions on U;. Let n' = n[... n'k . By (**) the morphism 6®"' : X -*
-> />(/*(£*"')) is a local immersion in every point of [_) U{. By shrinking eventually
i
Y around y0, we may assume that X = [_) £/,-. We will show that for n large enough,
i
6®" is an injective map. Let x, x' be points in X such that f(x) = /(x'). We are
going to prove the following assertion:
(***) For any invertible sheaf £ onl such that the morphism /*/*(£) -> £
is surjective, provided that the map 6 : X -> /"(/*(£)) verifies the condition
6(x)#6(x'), then the same property holds also for the maps 6®" for any n $s 1.
Let y =f(x) —f(x') and s0,..., sr e T{Xy, £) be as in the proof of (**). By
hypothesis the points (s°(x),..., sr(x)), (s°(x'),..., sr(x')) differ from each other in
the projective space \Pr. Let s e T(Xy, £) be such that s(x) <£ 0, s(x') <£ 0. The
elements s""1® s0,..., s"'1® sr are in T(Xy, £®") and separate the points x and x'.
One then derives that 6®"(x) # 6®"(x') and (***) is proved.
k
Let A! be a compact neighbourhood of y0. Denote U = U Ut X yC/;.
i = i
Consider a point (x, x') e (X x y^)\ [/ such that >> =f(x) =f(x') e A!. We have
an exact sequence
O-fin.nm.^fl^ 6x/mx ® <3x/mx- -» 0.
As in the previous reasonings, one derives that for n large enough the canonical
morphism
T(Xy, £®«) - £«7mx£*" © &$n/mx>&}n
is surjective. Then one can easily check that 6®n(x) ^ 6®"(x'). Since 6®" is a
proper map, it follows that there exists a neighbourhood W in X x jJf for (x, .x')
such that 6®"(x) # 6®n(x') for any (x, x') e W.

IV. PROJECTIVE MORPHISMS OF COMPLEX SPACES
159
Thus there is a finite number of open sets W; which cover ((X x YX)\ U) n
n (/ x yfy1^) and some integers n'[ such that the maps 6 separate the points
of Wt. Let n" be the product of these integers. The map 6®"" separates the points
of {{X xYX)\ U) n (/ xYfTl(K). By shrinking eventually Y around y0, we may
assume that the map 6®"" separates the points of (X xYX)\ U. Let n = n'n".
We show that the map 6®" is injective. Let x ^ x' such that f(x) = f(x')
(6®" being a morphism over Y, it is sufficient to confine to such pairs of points).
If x and x' belong to an open set Ui; then the conclusion follows because n\
divides n. If (x, x') e (X xrX)\ U, then 6®nO0 <£ 6®n(x') as n" divides n.
Therefore the morphism 6®" is injective, proper and local immersion in
any point of X. Consequently, 6®'1 is a closed immersion and this completes
the proof.
The second amplitude criterion is a consequence of the results established
in Chapter III.
Theorem 4.2. Let f : X -> Y be a flat proper morphism of complex spaces,
let £ be an invertible sheaf on X, and y a point in Y. Assume that S.y is a very
ample sheaf on Xy and H\Xy, &y) = 0.
Then there exists a neighbourhood V of y such that £|/~1(f/) is very ample
with respect to V; moreover, the sheaf f *(£) is free in a neighbourhood of y.
Proof. We set & =/*(£). By III. 3.9, & is free in a neighbourhood of y
and êj/mA = r(xy< £j>)- since the sheaf £j> = £/™/ is verv ample, hence the
elements of r(Xy, S.y) generate the stalks (S.y)x = £Jmy£x, x ef"\y). Hereof, by
Nakayama lemma one easily derives that the natural morphism of sheaves
/*/*(£) =/*(&) ^ £ is surjective in the points of the fiber f~l{y). Then it will be
surjective in a neighbourhood oî f~1(y).
By shrinking eventually Y around y we can assume that the morphism
/*(§) -> £ is surjective and let i : X -> P($) = P be the morphism over Y given
by it. The morphism induced by i on the analytic fiber over y is just the morphism
Xy^Py = P(§(y)), $(y) = BylmyBy = H\Xy, £,),
defined by the very ample sheaf S.y on Xy, whence it is a closed immersion.
The conclusion of the theorem will result from the following:
Proposition 4.3. Let Z be a complex space, X and Y two complex spaces
over Z such that the structural morphisms g: X -* Z, h: Y' -»• Z are proper and let
f : X -> Y be a Z-morphism. Let zbZ and f, : Xz -> Yz be the map induced by f.
(0 V fz is a finite morphism (viz., a closed immersion), then there exists a
neighbourhood U of z such that the morphism g~1(U) -> hr1(U'), which is the restriction
of f, is a finite morphism (viz., a closed immersion).
(ii) Suppose in addition g flat. If fz is an isomorphism, then there exists a
neighbourhood U of z such that the morphism g~1(U) -> hr^U), the restriction
of f, is an isomorphism.
Proof. In both cases it is sufficient to prove that for any y e h^1(z) there exists
a neighbourhood Vy of y such that the restriction of /, f'KVy) -»• Vy is a finite

160 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
morphism (respectively closed immersion, isomorphism). Indeed the restriction
of J, f~\V)-* V, results clearly to be a finite morphism (closed immersion,
isomorphism, respectively), where V is the union of the neighbourhoods Vy\ since h
is proper, there exists a neighbourhood U of z such that h~*(U) <= V and the
proposition is proved.
We remark that/ is also proper.
(z) Let y be an arbitrary point in the fiber /j-1(z). We first analyse the case
of the finite morphisms. Consider a point x ej~\y). The morphism 6y/mz6y -*■
-> 6Jmz6x is a finite morphism of rings. Then 6Jmz6x, hence 6Jmy6x is an
fi^-module of finite type. According to Weierstrass preparation theorem ([15],
[33]), 6X will be an ^-module of finite type, hence/ is finite in x.
Then one can deduce easily the existence of a neighbourhood V of y
such that the morphism j~\Vy) -*■ Vy is finite in every point. Since this restriction
of/ is even proper, it is finite.
We now deal with the case of closed immersions. By the previous
considerations, we may assume that / is a finite morphism. Thus one gets X = Specan (&)
where <3L is a coherent <9y-algebra. Let u : 6Y -> <& be the structural morphism.
We are going to prove that there exists a neighbourhood Vy of y such that the
restriction u\Vy is surjective; in virtue of the properties of the functor Specan it
follows that the morphism induced by j,j~\Vy)^> Vy is a closed immersion.
It is enough to check that the morphism uy : &y -> dy is surjective. By hypothesis,
the morphism <3yjm2<3y -*■ GLjmzGLy deduced from uy is surjective and the conclusion
follows from Nakayama lemma.
(ii) By the same reasoning as above, all we need is to prove that uy is bijec-
tive, as soon as the morphism 6y/mz6y -> £lylmzSLy, induced by it is bijective and
that GLy is a flat ^-module (GLy is isomorphic to the direct sum of the ^-modules
®x> f(x) — y)- The morphism uy is surjective and let K = Ker uy The exact sequence
o^K->ey->ay-+o
yields the exact sequence
0 -+ K/mzK -» 6ylmzey -» ajmAy -»• 0.
Then K/mzK = 0, so much the more K/myK = 0. In virtue of Nakayama lemma,
K = 0 and the proof of the proposition is over.
Recall that a morphism of complex spaces / : X -* Y is said smooth (or
simple) if, for any point x e X, there exist some neighbourhoods Ux(Vy
respectively) of x (y =/(x), respectively) and an open set Dx in a numerical space such
that f(Ux) <= Vy, Ux cz Vy x Dx andf\Ux : Ux -* Vy is obtained by means of the
projection Vy X Dx -> VT In particular, the morphism/ results to be flat. Moreover,
if / is proper and the fibers are Riemann surfaces of genus g, then X is called
curve of genus g over Y. In this case the sheaf fl^/y of relative differentials is an
invertible ^-module [15].

IV. PROJECTIVE MORPHISMS OF COMPLEX SPACES
161
Corollary 4.4. Let X be a "curve of genus g over Y" and let O = ilXIY
the sheaf of relative differentials.
(a) If g = 0 then Q®-'1 is very ample with respect to Y for any n^-\.
(b) If g = 1 and S is the marked section of X over Y which corresponds
to an &x-ideal 3 on X, then 3®~" is very ample with respect to Y for any n ^ 3.
(c) If g = 2, then Q®" is very ample with respect to Y for n $s 3.
(d) If g S= 3, then Q®" is very ample with respect to Y for n^.2.
Proof. Apply the theorem, the conditions to be checked being consequences
of the Riemann-Roch theorem [78].
Corollary 3.2. can be applied if A' is a "curve of genus g over Y": by the
preceding corollary we can obtain very ample sheaves £ by means of Q.x/Y. For
instance we have:
Corollary 4.5: Let f : X -> Y be a "curve of genus 0 over Y", O = QXIY
the sheaf of relative holomorphic differential forms and & e Coh (X), locally free.
Then, locally on Y,f*(oF® fi®"') =0 for m large enough.
Indeed, £ = Q®"1 is very ample with respect to 7 and the conclusion follows
from 3.2.
Bibliographical indications
The former proof of the Grauert-Remmert theorem in the case X = Y x \Pr, £ = &x(l),
f = projection, can be found at [32] ; at the same time, the authors proved the coherence of the
sheaves R9/^). The proof given here, which uses the finiteness theorem for proper morphisms,
is the transposition to the relative case of the proof given by Serre [77] in the absolute case.
Theorem 2.4 is essentially equivalent {modulo its algebraic analogous [76]) to the comparison
theorem 2.6; for this reason, its extension 2.1 is also called the comparison theorem.
A systematical treating of the comparison of algebraic and analytic properties can be found
in ([36], SGA 1, Exp. XII).
Theorem 3.1 was proved by Grothendieck in the algebraic case ([36], SGA 2, Exp. XII);
the analytic case was considered in [7],
Theorems 4.1 and 4.2 are taken from [30], [57] and [15], respectively.

Chapter V
Flat morphisms of complex spaces
Introduction
Let / : X -* 7 be a morphism of complex spaces and W a coherent analytic sheaf
on X. For any point y e 7, the fiber Xy = (J~1(y), <3x/^y^xU"1(y)) is a complex
space. The sheaf Ëf y = SF/m^SF defines a coherent analytic sheaf on X . One thus
obtains an "analytic" family (Xy)yer of complex spaces and a family (&y)yeY
of coherent sheaves.
The key point of the flatness consists in the fact that under the hypothesis
that / is flat a "continuous" behaviour of the properties of the above families
is provided, as well as some connections between the properties of the source X,
the target space Y and the fibers Xv. Assertions 2.4—2.6, 2.8—2.11 stand for
examples in this respect; a systematical treating of these problems in the algebraic
case can be found in ([37], Ch. IV).
The hypothesis of the flatness of a morphkm also implies the remarkable
fact that it is open (theorem 2.12); theorem 2.13 stands as a converse result.
The main result in the chapter asserts that the set of the points where & is
not flat over 7 is a closed analytic subset (theorem 4.5). In its proof one makes
essential use of the following theorem: the ring of germs of analytic functions
defined on a Stein semianalytic compact is noetherian (theorem 3.1).
The flat morphisms can be interpreted as the extension of the notion of
analytic family of complex manifolds to the singular case.
All the results of this chapter were first proved in algebraic geometry and
were conjectured for the analytic case by A. Grothendieck [15].
§ 1. Preliminaries
(a) (cf. [82]) Let (X, dx) be a metric space. For xe X and A, C subsets of X,
denote dx(x, A) = inf dx(x, y) and dx(C, A) = inf dx(y, A). If e > 0 and xe X
yeA yec
we set Bx(x; z) = {y e X\dx(x, y) < z). If e > 0 and x, ye X, then a finite sequence
of points x0,..., xm of X is called an z-chain linking x with y whenever x0 = x,
xm = y and dx{x{; x;+1) < z for 0 < / < m.
Proposition 1.1. If x and y are points of a compact subset K of a metric
space X, then x and y lie in the same connected component of K if and only if,
for any z > 0, x can be linked with y by an z-chain in K.

164 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. We may assume K = X. Let C be the connected component of X
which contains x. Denote by Tz the set of the points z e X which can be linked
with x by an e-chain. Let T = f) Tz. We will prove the equality C = T and the
conclusion will follow. The sets Tz are simultaneously open and closed, hence
C c T. For the other inclusion, it is sufficient to prove that the closed set T is
connected. Suppose, on the contrary, that T = MU N, where M and N are closed
disjoint nonempty subsets which are distinct of T. Let d = d(M, N), d > 0 and
P = \zeX\d(z, T) ^ — [ • Suppose x0 e M. For any natural number n there exists
a chain linking x0 with a point of N. For any integer n > 3/d there exists a
n
point xn of this chain which lies in P. Any accumulation point of the sequence xn
lies in P n T and we thus obtain a contradiction.
Lemma 1.2. Let f : X -* Y be an open proper map between metric spaces.
Then for any compact connected subset K of Y, f~\K) has on/y a finite number of
connected components.
Proof. Let K <= Y be a compact connected set. We may assume K = Y.
Since / is open and proper,/(X) is simultaneously open and closed in Y; therefore
f(X) = Y. We first prove the following assertion:
(*) For any e > 0, there is S > 0 such that for any x e X,
f(Bx(x; e)) = BY(f(x); 8).
Suppose, on the contrary, that this is not true. Then there exist e > 0, a descending
sequence of positive numbers {§„} converging to 0 and a sequence {x„} in X such
that f(Bx(xn; e))J>BY(f(xn); S„). Let yn e BY(f(xn); 8„)\f(Bx(xn; e)). Since X and Y
are compact spaces, we can assume that there are x* e X, y* e Y such that x„ -* x*,
yn-+ y* (passing eventually to subsequences). We have/(x*) =y*, as dY(f(xn), yn)<8n
and S„-> 0. Since Bx(x*; e/2) and f(Bx(x*; e/2)) are neighbourhoods of x*
and y*, there exists m0 so that xm e Bx(x* ; e/2) and y„, e f(Bx(x* ; s/2)) for any
m ^ m0. Therefore, for m ^ m0, ym ef(Bx(x*; e/2)) <=f(Bx(xm; e)), a contradiction.
In this way the assertion (*) is proved.
Let us fix a point ye Y and let J~Ky) = {*i>- ■ -, xk}. Consider an
arbitrary point xe X. We shall prove now the assertion :
(**) For every natural number n there exists j„, 1 < jn < k such that x can
be linked with xJn by a 1 /«-chain in X.
Fix a natural number n. By (*) there exists S > 0 such that f(Bx(z; l/«)) =>
=> BY(f(z); 8) for any zeX. Since Y is compact and connected, there exists a
S-chain y0,..., y„ in Y which links f(x) with y (1.1). By induction on i, 0 < / < m,
we can find points z{eX such that z0 = x, /(z,-) = yt and dx{zi^.1, zt) < l/n,
1 < / < m. Since /(zm) = >>m = y, there exists /„, 1 < jn < £, such that zm = x7n.
So the assertion (**) is proved too.

V. FLAT MORPHISMS OF COMPLEX SPACES
165
There exists _/*, 1 <_/* < k, such that j* =jn for infinitely many natural
numbers n. Consequently, for any e > 0, x can be linked with Xj* by an e-chain
in X. According to proposition 1.1, x lies in the same connected component of X
as Xj*.
In this way, we have proved that X has only a finite number of connected
components.
Lemma 1.3. Let X be a compact metric space and A a compact connected
component of an open subset U of X. Then A is a connected component of X.
Proof. Let A be the connected component of X containing A. It is enough
to check that A <= U. Suppose the contrary. Let us fix y0eA\ U and x0e A.
We have z = d(A, X\ U) > 0 and let C = {xe X\ e/3 < d(x, A) < 2e/3},
D = {xe X\d(x, A) < 2e/3}. We need the following assertion:
(*) If» is a natural number > 3/e, then x0 can be linked with an element z„
of C by a 1/«-chain in D.
Let n > 3/e. The point x0 can be linked with y0 by a l/«-chain x0,..., x„,
in X. Let i be the smallest integer such that xt $ D. We have 0 < i < m. Since
c/(x;_i, Xj) < \/n < e/3, it follows that x;_x e C. Let z„ = x,_x. Then x0,... ,x,__i
is a 1/«-chair, in D linking x0 with z„ and (*) is thus proved.
Since C is a compact set, by passing eventually to a subsequence, we may
assume that the sequence z„ tends to a point z0. For any S > 0, x0 can be linked
with z0 by a S-chain in D. According to 1.1, z0 belongs to the connected component
of D which contains x0. As D <= U and A is the connected component of U
containing x0, z0 e A. This fact contradicts the inequality d(z0, A) ^ e/3 and the lemma
follows.
(b) We would like to call attention to several facts concerning flat modules.
"Let A -> B be a morphism of noetherian rings, / an ideal of A such that IB
is contained in the Jacobson radical of B (the intersection of all maximal ideals)
and M a 5-module of finite type. Then M is a flat ^-module if and only if
MjlM is a flat ^//-module and Tor^(Af, A/1) = 0" ([10], Ch. Ill, § 5, th. 1 and
prop. 2).
In particular, the following result holds:
"Let A and B be noetherian local rings, k the residual field of A, A -*■ B
a local morphism and M a 5-module of finite type. Then M is ,4-flat if and only
if Tor?(M, k) = 0".
Corollary 1.4. Let A -*■ B be a local morphism of noetherian local rings,
t a nonzerodivisor of A belonging to the maximal ideal, and M a B-module of
finite type. If t is M-regular and MjtM is A/tA-flat, then M is A-flat.
Proof. We have to verify that Torf(M, A/tA) = 0. This fact can be easily
checked by tensoring ®AM the exact sequence
0 -► A -U A -► A/tA -► 0.
Corollary 1.5. ([37], Ch. O,,,, 10.2.4). Let A ^ B be a local morphism
of noetherian local rings, let k be the residual field of A and let M and N be two

166 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
B-modules oj finite type, where N is flat over A. Let n : M -* N be a B-morphism.
The following conditions are equivalent;
(a) n is injective and Coker u is Aflat.
(b) u® 1 : M®Ak -* N®Ak is injective.
Proof. The implication (a) =* (b) follows by tensoring ®Ak the exact sequence
0 -»• M -^ N -»• Coker u -»• 0.
(b) =*■ (a). Let P = Im u, Q = Coker w and R = Ker w. The composition
M®Ak -> P®Ak -> N®Ak is injective and the map M®Ak -*■ P®Ak is surjective.
Therefore the map M®Ak -*■ P®Ak is bijective and the map P®Ak -* N®Ak is
injective. The exact sequence 0-*P-*N-*Q-*0 yields the exact sequence
0 = Tor^(iV, k) ->• Torftg, A:) -► Ptg^fc ->• 2V®^fc -► 0 ®^ ->• 0.
It will then result that Tor^(g, fc) = 0, hence by the result recalled above,
Q = Coker u is ,4-flat. From the exact sequence 0->P->iV-><2->0 it follows
that P is ,4-flat. The exact sequence 0->/?->A/->P->0 yields the exact sequence
0 -»• R®Ak -»• M®Ak -»• P®Ak -»• 0.
Since the map M®Ak -> P®Ak is bijective, R®Ak =0. By Nakayama lemma we
get 7? = 0, hence « is injective. The corollary is proved.
Let A be a ring (commutative and unitary), / an ideal of it and M an
A
^-module. Denote by M the completion of M in the 7-adic topology,
AAA A
M — lim (M/IkM). M is an ^-module and the correspondence M \-> M is
k
functorial. Recall the following result:
"If A is a noetherian local ring, M an ^-module of finite type and / an
A
ideal included in the maximal ideal of A, then M is canonically isomorphic to
A A A
M®AA and the natural map M -> M is injective. Moreover, the functor M >-*• M
A
from the category of ^-modules of finite type to the category of ^-modules of finite
type is exact and faithfull (M = 0 => M = 0)" ([10], Ch. Ill, § 3).
Corollary 1.6. ([10], Ch. Ill, § 5, n = 4, prop. 4). Let A -+ B be a local
morphism of noetherian local rings, mA and mB the maximal ideals of A and B
A
and M a B-module of finite type. Then M is Aflat if and only if M (the completion
A
in the mB-adic topology) is A (the completion in the mA-adic topology)flat.
A A A
Proof. By our above remark, it is sufficient to prove that Tori (A/m*, M) ot
A
A „
c^ TorA(A/mA, M)A (ntA = mAA is the maximal ideal of A). Let L. be a resolution

V. FLAT MORPHISMS OF COMPLEX SPACES
167
for A/mA by free ^-modules of finite type. TorA(A/mA, M) are the homology groups
of the complex L, ®AM. Since the completion is an exact functor on the modules
of finite type, Tor^/m^, A/)A can be canonically identified with the homology
A
groups of the complex (L.®AM)<g>BB. This complex is canonically isomorphic
AAA A A
to the complex L.®*M. Since L, is a resolution for A/m* by free ^-modules,
A A
the desired conclusion follows.
Proposition 1.7. ([37], Ch. OnI, 10.2.6). Let A and B be two nowtherian
local rings, A -> B a local morphism, 1 a proper ideal oj B and M a B-module
oj finite type. If moreover M„ = M/In+1 M is a flat A-module for any n^O,
then M is Aflat.
Proof. We must show that for any injective morphism u : N' -*■ N of ^4-modules
of finite type, the morphism v = 1 ® u : M®AN' -* M®AN is injective. M®ÀN'
and M®AN are 5-modules of finite type, hence the morphisms M®AN' -*•
-»• {M®AN'Y, M<g>AN-> (M<g>AN)* are injective, where the completions are
considered with respect to the 7-adic topology.
Consequently, it is enough to prove that the morphism v : (M®AN')* ->
-> (M<g)AN)* is injective. We have z; = limz;„, where v„ is the morphism
1 (g) u : Mn®AN' -> Mn®AN. By hypothesis M„ is ,4-flat, hence v„ is injective
for any n; the functor lim being left exact, the conclusion follows.
Corollary 1.8. ([37], Ch. O,,,, 10.2.7). Let A -> B be a local morphism
of noetherian local rings, f an element of the maximal ideal of B and M a B-module
of finite type. Suppose f is M-regular and M/fM is a flat A-module. Under these
assumptions, M is a flat A-module.
Proof. Denote M"=/"+1M, n^O. Since /is M-regular, the modules M"/M"+1
are isomorphic to M/fM, hence ,4-flat. By using an exact sequence of the form
0 -► M"/Mn+1 -► M/Mn+1 -► M/M" -» 0
and by induction on n, one derives that the modules M„ = M/M" = M/f"+,M
are ,4-flat. The conclusion then follows from 1.7.
Proposition 1.9. ([37], Ch. Oiv, 15.1.16). Let A ^ B be a local morphism
of noetherian local rings, k the residual field of A and M a B-module of finite
type. Let (Jdi<Kr be a sequence of elements of the maximal ideal of B and
(g/)i«i«r its image in B®Ak. The following conditions are equivalent:
(a) The sequence (g;)i<i<r is M ®Ak-regular and M is A-flat.
(b) The sequence (JW^^r is M-regular and M \\fiM is Aflat.
Proof. We use induction on r. For r = 1 the equivalence is true in virtue
of 1.5 and 1.8. We now assume the assertion already proved for r — 1 and prove
the equivalence of the statement for r (r ^ 2).

168
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(a) =* (b). By induction hypothesis, the sequence (J^a^r-i is A/-regular
and Ml £/;A/ is ,4-flat. We have a canonical isomorphism
' 1=1
{M®Ak)jr^g;{M®Ak) * (mI r%fiM\ ®Ak.
Consequently, the element gr is l M J]/jAf Jtg^fc-regular. According
jr-1
to the implication (a) =* (b) for r = \, fr is A// V /; A/-regular and
/ ;=i
(A/7 J] /;A/| //J A/ / ^ /,A/} ~ M I % f,M is ,4-flat. The conclusion follows.
The other implication can be proved similarly.
(c) Lemma 1.10. Let A be an integral ring, f a non-null element of A and
M an A-module svch that the module of quotients Mf is Affree. Then for any
g ç£ 0 in A, Mfg is Afg-free.
Proof. If g is the image of g in A/, then M/g =- (Mf)j. The conclusion
follows then from the fact that passing to quotients commutes with the direct sums.
Lemma 1.11. Let A be an integral ring, f and g non-null elements of A
and 0-*M-*N-+P-+0 an exact sequence of A-modules. If Mf is Af-free and
Pg is Ag-free, then N/g is Afg-free.
Proof. By the previous lemma it results that M/g and Pfg are free ^4/g-modules.
The sequence 0 ->■ M/g -*■ N/g -> Pfg -*■ 0 splits and the conclusion follows.
Recall a fundamental result on the algebras of finite type over a field ([10],
Ch. V, § 3, n = l,Th. 1).
"(E. Noether's lemma of normalization). Let A be an algebra of finite type
over a field k. Then there exists a finite number of elements fl5..., tm in A, which
are algebraically independent over k and such that A is integral over fc[?1;..., tm]".
Consequently one obtains the following result:
"Let A be an integral domain and B an ^4-algebra of finite type such that
the structural morphism A -*■ B is injective. Then there exist an element g^O
in A and elements t±,..., tm in B, which are algebraically independent over A and
such that the ring of quotients Bg is integral over the ring A^tx,..., tm]" ([10],
Ch. V, § 3, n = l, cor. 1, th. 1).
We also recall the following fact:
"If B is an ^4-algebra of finite type, integral over A, then B is an ^4-module
of finite type. If moreover the structural morphism A -> B is injective, then dim A =
= dim fi" ([79], Ch. III).
Lemma 1.12. Let A be a noetherian integral domain, B an A-algebra of
finite type and M a B-module of finite type. Then there exists f ^ 0 in A such that
Mf is a free Af-module.

V. FLAT M0RPH1SMS OF COMPLEX SPACES
169
Proof. Denote by K the field of the quotients of A; then B®AK is an algebra
of finite type over K and M®AK is a B®AK-moâu\e of finite type. We use
induction on the integer n = dim^® iK(M®AK). Consider first of all the case
when M®AK = 0. Let (m;)i<;<r be a system of generators of M over B.
There exists an element/ <£ 0 in A such that/m; = 0 (i = 1,..., r); accordingly,
Mf = 0 and the lemma is concluded in this case.
So we may assume n > 0 and the lemma proved for integers < n. Let
M = Mx =3 M2 =>... => Mg = 0 be a composition series such that the fi-modules
Nt = MjlMl+1 are isomorphic to fi-modules of the form Bjph pt being prime
ideals of B. If the lemma is true for every Nh then it will be true for M too
(1.10, 1.11).
In this way, if we replace B by a ring of the form Bjp (p being a prime ideal
of B), we may suppose M = B and B an integral domain. Let A' ~ Ajp the
image of A in B, p being the kernel of A -> 5. Consider the open set
Spec ^4\ V(p). If this set is empty, then the morphism A -> B is injective. Otherwise,
there exists/ ^ 0 such that D(J) <= Spec ,4 \ F(p); accordingly, (A-/p)f = 0, hence
the desired assertion is fulfilled for the morphism A -* B as soon as it holds for
the morphism A' -* B. We may thus assume that the structural morphism A -* B
is injective. Then there exist an element g ^0 in A and elements tt in 5, 1 < / < m,
which are algebraically independent over A and such that Bg is integral over
,4J?!,..., tm]. We may further replace A by ^ and 5 by Bg and in this way
we can assume the ring B integral over the polynomial ring C = A[tx,..., tm].
The extension C®AK = K[tx,..., t„] -> B®AK is integral (and obviously injective),
therefore m = w.
5 is a C-module of finite type without torsion, hence there exists an exact
sequence of C-modules of the form
0 ->• C" -> 5 -+ A/' -+ 0,
where M' is a torsion C-module (one considers elements y1;---.yr in B, whose
images in B®CL form a basis over thefieldL of the quotients of C; the morphism
C ~+ B defined by them is injective and its cokernel is a torsion C-module).
M'®AK is C®^-module of finite type, hence
dimc®AK(M'®AK) < dim (C®AK) = n.
By the induction hypothesis, there exists / °£ 0 such that M'r is ^4y~free. Since
(Cr)/^ Af\tx,..., f„,]r is ^/-free, it follows that Bf is a free ^4y-module and the lemma
is proved.
Lemma 1.13. ([37], Ch. 0m, 9.2.6). Let X be a noetherian topological
space {any nonempty family of closed subsets contains a minimal element) and E
a subset of X. Then E is open if and only if, for any irreducible closed subset Y
of X which intersects E, E fi Y contains a nonempty open subset of Y.
Proof. The condition is obviously necessary. We shall prove now it is also
sufficient. Consider the set of all closed subspaces (hence noetherian) of X which
do not verify the lemma. We will show that this is void and the proof will be

170 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
over. Suppose, on the contrary, that it is non-void, hence it has a minimal element.
Then we obviously come to a contradiction as soon as we are able to prove
the lemma under the supplementary hypothesis that it holds for any closed subspace
X' of X, X' # X. If the space X is of the form X = X' U X" where X' and X"
are closed proper subsets of X, then the lemma holds for X, since it is true for
X',X".
Thus we may assume that X is an irreducible space. Let E be like in the
statement. There exists a nonempty open subset U of X, U cr E. In virtue of
the supplementary hypothesis, Ef\ (X\ U) is open in A'X U. Then it will easily
result that E = U U (X\ (X\ E)) and this concludes the proof.
Let A -> B be a ring morphism, qeSpecfi and M an ^-module. We say
that M is A-flat (or (p-flat) in the ideal q if Mq is a flat ^cp-i(q)-module. By the
next lemma this is equivalent to the fact that Mq is ,4-flat.
Lemma 1.14. Let A be a ring and let S cr A be a multiplicative system.
If M is a flat A-module, then Ms is flat over the rings A and As. If M is an
As-module, then M is Aflat if and only if it is Asflat.
The proof immediately follows from the properties of the modules of quotients.
Theorem 1.15. ([37], Ch. IV, 11.1.1). Le? A^Bbe a morphism of noetherian
rings such that B is an A-algebra of finite type and M a B-module of finite
type. Then the set of the prime ideals of B in which M is A-flat is an open subset
of Spec B.
Proof. By applying lemma 1.13 (since the set from the statement cuts an
irreducible closed subset V{c\) of Spec B if and only if M is ,4-flat in q), it is
sufficient to prove the assertion:
(*) "Let q be a prime ideal of B in which M is ,4-flat and p is its inverse
image in A. Then there exists an element ge B\q such that for any prime ideal
q' rj q of B enjoying the property that g$ q', M is ,4-flat in q'."
Let us fix q such that Mq is ,4-flat and let p be its inverse image in A.
Let q' d q be a prime ideal in B. Consider i?q< as an ^4-algebra; we obviously
have pBq' cr q'i?q<. We now apply the criterion of flatness recalled at the beginning
of section (b): My is ,4-flat if and only if My/pMy is a flat ^/p-module and
Tor^(A/q, A/p) = 0. There are natural identifications
A/q-/pMi- cz (M/pAf)q< and Torj*(A/r, A/p) ~ (Tor^M, A/p)^
(for the second one, the functor Tor could be calculated by means of a projective
resolution of A/p). Let now geB\ q be such that g$ q'. There are also natural
isomorphisms
(M/pM)q<~ {{M/pM)g\,B^ and OV(M, A/p))q. ~ (Tor^(M, A/p)g\.Bq
where M/pM and Tor^(A/, A/p) are considered as 5-modules.
Since the passing to quotients is an exact functor, in order to prove (*) it
is enough to verify the assertion:
(**) "Under the conditions from (*), there exists an element ge B\c\ such
that (M/pM)g is a flat ^/p-module and Tor^(M, A/p)g = 0."

V. FLAT MORPHISMS OF COMPLEX SPACES
171
We prove this assertion. Apply lemma 1.12 to the integral domain A/p,
/4/p-algebra of finite type B/pB and the i?/pi?-module of finite type M/pM: there
exists an element he A\p such that if h is its image in A/p, then (M/pM)% is
a free (A/p)}, -module, hence a flat ^/p-module. On the other hand, since Mq is
a flat ^p-modu!e (consequently ,4-flat), one gets
(Torf(M, A/p))q ~ Tor^Mq, A/p) = 0.
Since A and B are noetherian, Tot^(M, A/p) is a 5-module of finite type, hence
there exists g' e B\ q such that
g'(TorA(M, A/p)) = 0, therefore Torf(M,A/p)g- = 0.
The element g = hg' satisfies (**). Indeed, (M/pM)g = (A//pA/)j = ((M/pM)j;)j
will be A/p-ûat (g and A are the images of g and h in 5/p5) and Tor^(A/, ^/p)^ =
= (Tor^(M, A/p)h)g, = 0. The theorem is proved.
The following variant, due to Kiehl [46], will be useful:
Theorem 1.16. Let A -> B be a morphism oj noetherian rings, let I be an
ideal in B such that Bjl is an A-algebra of finite type and M a B-module oj
finite type. Then the set oj the prime ideals oj V(I) in which M is A-flat is open
in V(l).
The proof can be made similarly by using the followings instead of 1.12:
Lemma 1.17. Under the conditions oj the statement oj the theorem, suppose
in addition that A is an integral domain. Then there exists an element j <£ 0 in A
such that Mf is A-jlat in all prime ideals oj V(If).
Prooj. The ring
gr7(i?) = B/I ®I/P® ...
is naturally a 5-algebra. By the hypothesis on /, gr2(B) is an ^-algebra of finite type
gr7(Af) = M/IM © 1M/PM ® ...
is gr7(.fi)-module of finite type. There exists an element / ^ 0 in A such that
(gTj(M))f =* grff(Mf) is ^-free, hence ^-flat (1.12).
For any p e V(If) we obtain that the gr/p(5p)-module grfp(A/p) ~ (grr/(Mf))p
is A-ûat (we identify the ideals of Spec Bf withi the corresponding ideals of Spec B).
In this way, all modules MV/1VMV, lvMv/lyMv,... are ,4-flat; it then results that for
any n, Mp/IpMp is ,4-flat, hence AvnA-fiat. By (1.7), the module A/p(~ (Mf)v)
is ApnA-fiat.
Remark. Under the assumptions of the theorem, there exists however an
ideal a in B such that
V(a) n V(I) = {pe V(I)\MP is not ^-flat}.

172 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
We will use 1.16 just in this form.
(d) Let / : X -> Y be a morphism of complex spaces and let âF be an
<9x-module. Consider a point x of X. By means of the natural morphism
£>Y.f(X) -* ®x.x, &x has a structure of <9y./w-module. The sheaf âF is called
j-flat (or Y-flat) in the point x if SFX is a flat 6Y. /(^-module. We shall say that
oF is j-flat (or Y-flat) if it is /-flat in any point x of X. In this case, if
o ->• ç -► q ->• <ij" -»• o
is an exact sequence of <9y-modules, then the sequence
o -/*(<s')®<9,* -/*(<?)®fx* -/*(<?" )®s** - o
is exact.
The morphism/ itself is called flat if the structural sheaf 0X is /-flat. Again,
if 0 -> <$' -> § -+ éj!" -> 0 is an exact sequence as above and if / is flat, then one
obtains the exact sequence
o _/*($') _♦/*(<!) -/*($") -, o.
Recall the following properties:
"Let/ : J -» F be a morphism of complex spaces, âF e Coh (X), Y' ->■ Y
another morphism of complex spaces, /' : X' = X ~XY Y' -* Y' the morphism
deduced from / by base change, âF' the inverse image of âF on X', x' a point of X'
and x, y, y' its images in X, Y, Y'. Under these conditions, if SF is /-flat in x, then <F
is/'-flat in x'. Conversely, if the morphism Y' -> y is flat in / and âF' is/'-flat
in a:', then âF is/-flat in x ([15], Exp. 13, prop. 2.4 and Cor. 2.5)."
As a consequence, one obtains:
"If JT, Y,X', Y'JJ' are as above and/ is flat, then/' is flat. Conversely,
if y ->• y and /' are flat, then / is flat".
"If Y and Z are complex spaces, then the projection morphism X = Fx
x Z-» y is flat" (indeed, if e is the final complex space, then the unique
morphism Z -> e is flat and yxZ = yxeZ; another proof of this fact can be
found in [20], th. 2, p. 60).
(e) Proposition 1.18. Let A -* B be a flat local morphism oj local rings.
Ij M is an A-module and M®AB = 0, then M = 0.
Prooj. If M' is a submodule of M, then the map M'®AB -> M®AB is
injective. It will then result that we can suppose M of the form A/a, where a is
an ideal of A. Since A/a®AB ~ B/aB = 0, it follows that B = aB. Thus a = A
(hence M = 0), otherwise, a <= mA and ct.8 <= mB.
Proposition 1.19. Le? A -> B be a flat local morphism oj local rings. Then
the jollowing properties hold:
i) The morphism <p is injective.

V. FLAT M0RPH1SMS OF COMPLEX SPACES
173
ii) // a is an ideal of A, then aB n A = a (one identifies A with its image
under 9).
Hi) If moreover B is noetherian, then the natural map Spec B -* Spec A is
surjective; more precisely, if p e Spec A and q e Ass (B/pB), then q n A = p.
Proof. (i) We have Ker<p®AB ~ (Ker<p)i? =0 and the conclusion follows
from proposition 1.18.
(ii) The morphism A/a -> B/aB ~ A/a ®AB is a flat local morphism of local
rings. It results to be injective, therefore a — aB n A.
(Hi) Let p, q be as in the statement. If we replace A -* B by A/p -* B/q we
may suppose p = 0 (hence A is an integral domain). Suppose, on the contrary,
that there exists a non-null element xeq n A. The homotety defined by it in A
is injective. By the flatness hypothesis, the homotety defined by x in B is
also injective. So x is nonzerodivisor in B, and this contradicts the hypothesis
x e q e Ass B.
Lemma 1.20. Let A -*■ B be a local morphism of noetherian local rings
and let M be a B-module of finite type, which is flat over A. Suppose in addition
that Supp M = Spec B. If p e Spec A, then any prime ideal q of B which contains
pB and is minimal with respect to this property verifies the relation <p-1(q) = p.
Proof. We proceed as in 1.19 (Hi). If we replace 9 by A/p -> B/pB and M
by M/pM ~ M®AA/p, we may assume p = 0, hence q is a minimal prime ideal
of B. Suppose, on the contrary, that <p-1(q) # 0. Let x # 0 be an element of A
such that <p(x) e q. The multiplication by x in A being injective, it results that the
multiplication by <p(x) in M is also injective. The hypothesis Supp M = Spec B
implies q e Ass M, hence <p(x) e q is a zerodivizor of M, a contradiction.
Proposition 1.21. ([37], Ch. IV, 6.1.2, 6.3.1). Let A ^ B be a local morphism
of noetherian local rings, m the maximal ideal of A, k = A/m its residual field
and M # 0 a B-module of finite type which is flat over A. Under these conditions
(i) dimBM = dim A + dimB9Ak(M ®Ak).
In partiadar, if 9 is flat, then dim B = dim A + dim (B ®Ak).
(H) profflAf = prof A + profB8j4ft(A/(g)^).
In particular, if 9 is flat, then prof B = prof ,4 + prof (B®Ak).
Proof, (i) We may replace B by 5/Ann M and we can thus suppose that
SuppA/ = SpecB. By Nakayama lemma, Supp(M®Ak) = Spec(B®Ak). So we
have to prove the equality dim B = dim A + dim (B®Ak) under the assumption
of the existence of a 5-module of finite type M, which is flat over A and such that
Supp M = Spec B. We proceed by induction on n = dim A.
If n = 0, then m (hence mB too) is nilpotent and the conclusion is clear.
We now suppose « ^ 1 and the assertion proved for « — 1. Let q;, 1 < / < j,
be the minimal prime ideals of B and pj, 1 < j < t, the minimal prime ideals of A.
For any /, the ideal p; = 9-1(q;) is different from m. Indeed, if we would have

174 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
P; = m for an index i, then there would be prime ideal p in A, p <= p. and p t£ pt;
we easily obtain a contradiction by using 1.20. Consider an element xem which
does not lie in any ideal p; or p'j ([10], Ch. II, § 1, n = 1, prop. 2).
Denote A' = A/xA, B' = BjxB and M' = M/xM. Then
dim A' = dimA — 1, dimB' = dimB — 1, SuppA/' = Spec£'
and M' is ^'-flat. Thus we may apply the induction hypothesis and get the equality
dim B' = dim A' + dim(B'®A,k).
On the other hand, B'®A,k = B®Ak = BjmB.
(ii) By Nakayama lemma the module M®Ak is non-null, hence the integer
n = prof A + prof B®Ak(M®Ak) is finite. We proceed by induction on n. We first
consider the case n = 0. It follows that prof ,4 = prof^^M®^) =0, hence
n\A e Ass A, mB/mAB e Ass (M/mAM). There exists an injective ^-morphism
AjmA -> A. By tensoring ®AM we get an injective fi-morphism M/mAM -> M,
therefore AssB(7k//mAM) <= Ass M. Accordingly, mB e AssBA/, hence profflA/ = 0
and the case n = 0 is thus treated.
We suppose now n > 0 and distinguish two cases:
a) Assume prof A > 0. Let x emA be an ^-regular element. Denoted' = A/xA,
B' = B/xB and M' = M/xM. Since'xemA, B'®A'k = B®Ak and M'®A'k =
= M® Ak. By the flatness hypothesis on M one derives that M' is ^'-flat and x
is A/-regular. Then
prof A' = prof A — 1, proffl- M' = profg M — 1 and
profs-® ^(AT ®A'k) = proîB0Ak(M®Ak).
Thereby, the general induction step is proved in the case a).
b) Assume profB@Aic(M®Ak) > 0. LetyemB be a regular element with
respect to M/mAM = M®Ak. By 1.9, y is Af-regular and M' = MjyM is ,4-flat.
We have
M'®A>k~(M/mBM)/y(M/mBM), hence profB@^(Af ®^fc) =profB®^,t(Af ®^)— 1.
In this case the proof of the general induction step is obvious too.
Corollary 1.22. Let A -* B be a flat local morphism of noetherian local
rings. For any ideal a of A(a t£ A) the following equality
ht a = ht (aB)
holds.

V. FLAT MORPHISMS OF COMPLEX SPACES
175
Proof. By definition, ht a = inf ht p and ht (aB) = inf ht q, p and q being
prime ideals in A, B, respectively. The conclusion follows straightforwardly from
the following two assertions:
(*) q e Spec B => ht (q n A) < ht q;
(**) p e Spec,4, q prime ideal in B containing pB and minimal with respect
to this property => ht p = ht q.
In order to prove (*), consider the flat local morphism of noetherian
local rings AqnA -> Bq. We have ht (q n A) = dim AqnA and ht q = dim Bq and
the conclusion follows from the proposition.
As for the assertion (**), we remark moreover that p = q n A (1.19) and
that dim Bq/pBq = 0, since Spec (Bq/pBq) consists of the prime ideal determined
by q only.
Corollary 1.23. Under the assumptions of the proposition,
coprofBM = coprof A + coprofB®^(M® Ak).
In particular, if <p is flat, then
coprof B = coprof ,4 + coprof {B® kA).
The proof follows from the equality coprof = dim—prof.
Corollary 1.24. Under the assumptions of the proposition, M is a Cohen-
Macauley B-module if and only if A is a Cohen-Macauley ring and M®Ak is a
Cohen-Macauley B ®A k-module. In particular, if <p is flat, then B is Cohen-Macau-
ley if and only if the rings A and B®Ak are so.
We will also use the following results.
Proposition 1.25. Let A -* B be a local morphism of noetherian local rings
and M a B-module of finite type. If B and M are A-flat, then
àhBM = dhB/mAB(M/mAM).
Proof. If L' -* M is a resolution of M by free 5-modules, then the flatness
hypothesis shows that L'/mAL' is a resolution of MjmAM by free B/mA 5-modules.
Thus dhBimAB (MjmAM) < dhBM.
We now prove the other inequality. If n — dhB/mAB (M/mAM) = oo, this is
clear. Suppose n < oo and let
L"-1 -»• .. . -»• L° -»• M -> 0
be an exact sequence of 5-modules, L! being free fi-modules of finite rank. Denote
N = Ker (L"_1 -> L"~2). By splitting the exact sequence into short exact sequences
from right to left, we derive that N is ,4-flat. By tensoring®A^/m^, we get an exact
sequence
0 ->• N/mAN -» Ln-llmAL"-1 -» ... L°/mAL0 -» M/mAM -» 0.

176 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
It will then result that NjmAN is B/mAB-free, hence N will be 5-free (IV. 1.2) and
the proof is completed.
Proposition 1.26 ([31], 2.4). Let A -*■ B be aflat local morphism of noetherian
local rings. If A is regular and B/mAB reduced, then B is reduced.
Proof. Proceed by induction on n = dim A. First, consider the case n=\
and let ? be a regular parameter for A. Let xe B be such that xr = 0 for an integer
r ^ 0. We get x e mAB, hence x = txx, xx e B. The element t, as well as any power
of it, is a nonzerodivisor in A, hence in B by the flatness hypothesis. Then we get
x\ = 0.
Analogously, we get xx = tx2, x2 e B. By the same reasoning one derives
that x e P) tkB, hence x = 0 (by Krull's theorem).
Assume now n ^ 2 and the proposition proved for dimensions < n. Let
/„...,(, bea regular system of parameters for A. The morphism A\txA -> Bj^B
is flat, A\txA is a regular ring of dimension n — 1 and (Bjt1B)jmA (B/t^) ~ B/mAB
is reduced. By the induction hypothesis, the ring B\txB is reduced. Since the
multiplication by any power of tx is an injective map B -> B, it follows as above that B
is reduced.
Let A be an analytic algebra, X a complex space and x a point of X such that
A = &x,x- Denote by A{T^ ..., Tr], T{ being indeterminates, the analytic algebra
®x -<Lr (x o)- The projection Xx C' -»• X yields a canonical inclusion ,4 e-»-
^{j,;...,^}. f
Proposition 1.27. Let A -* B be a morphism of analytic algebras such that
dim B = dim A + dim (B/mAB). Then there exist an integer r and a finite morphism
of analytic algebras of the same dimension A {Tx, .. ., Tr} -* B such that f is the
composition of this with the inclusion A <->■ A {T1; .. ., Tr}.
Proof. Let r = dim BjmAB. Choose elements x±, ..., xr of B such that their
classes form an ideal of definition in B/mAB. The morphism A {Tx, ..., Tr] -> B
given by / and by the substitutions Tt >-*■ xt is quasi-finite, since a suitable power of
mB is contained in the ideal generated in B by mA and the elements xt. By Weierstrass
preparation theorem, this morphism is finite. Moreover, dim A{T1}.. . ,Tr) = dim^ +
+ r = dim A + dim B/mAB = dim B.
(f) Let U be an open subset of some space R" and let x be one of its points.
Denote by Sx the smallest family of germs of subsets of U in x, which is stable
to finite unions, to finite intersections and to passing to complementary, and which
contains in addition the germs of the form {x' e U \f (x') < 0}x where/ is a real
analytic function defined in a neighbourhood of x. A part of U is called semiana-
lytic if for any point x of U its germ in x is an element of Sx.
We consider now an analytic subset X of an open set U of some space
C" oi [R2". A part of X is called semianalytic if it is semianalytic in U in the
previously defined meaning. By defining locally one easily obtains the notion of the
semianalytic subset of a complex space. For instance, the analytic sets are semi-
analytic.
By the very definitions, every point of a complex space admits a fundamental
system of neighbourhoods which are semianalytic and Stein compacts. Moreover

V, FLAT MORPHISMS OF COMPLEX SPACES
177
semianalytic compacts are stable to finite intersections and products, to images
under isomorphisms and to inverse images under finite morphisms.
One of the remarkable properties of semianalytic sets is that they are locally
connected [52]; in particular the compact semianalytic sets have a finite number of
connected components.
§ 2. Algebraic and topological
properties of the flat morphisms
We first give the transcription, in terms of flat morphisms of complex spaces, of
some results of local algebra from § 1.
Proposition 2.1. Let f: X -> Y be a flat surjective morphism of complex
spaces. The correspondence Y' t-*-/-1 (}") = X X Y Y', between the set of the siib-
spaces of Y and the set of the subspaces of X, is infective.
Proof. Let Y' be a subspace of Y. We have the equality of sets Y' =/ (f'KY')).
If V is an open set such that y is a closed subspace of it, then/_1(F) is an open
subset of X, J~\Y') is a closed subspace in/_1(F) and the morphism f~\V) -* V
induced by / is flat and surjective.
In this way we can confine to closed subspaces. If Y' is determined by
an ideal-sheaf 3 <= 6Y, then/"1^') is the subspace determined by/* (I) 6X. The
morphism/ being flat,/*(3) 0X is identified with/*(3).
If 3 and 3' are two coherent ideal-sheaves of 0Y such that/*(3)=/* (§'), then
we will show that 3=3' and the proposition will be proved. By the flatness
hypothesis,
/* (3 + 3'/3) ~/*(3 + 3')//*(3) ^/*(3 + 3') <2W/*(3)<9X
* (/* (3) Ox +/*(S') 6x)lf*{3)6x = 0.
By passing to stalks and applying 1.18, one gets 3 + 3'/3 = 0; hence 3' <= 3.
Similarly, 3 <= 3'.
Corollary 2.2. Let X and Y be two complex spaces over a complex space
S and let S' -* S be aflat surjective morphism of complex spaces. Then the map
Homs(A-, y) -► Homs- (XxsS', Y Xs 5'), f^fxsS'
is infective.
Proof. Since (X X s S') X s- (Y x s 5") œ (X X s Y) X s 5", the projection
morphism (X xs 5") xs- (Y X s 5") -> X xs Y is flat and surjective. The elements
of Homs (I, y) are in 1 — 1 correspondence with their graphs, which are subspaces
of X X s y. Analogously, one can characterize the elements of Homs- (X X s 5",
y XSS'). Moreover, if feHoms(X, Y), then r/xsS- = tc-1 (Tf), where T means
the graph. The corollary then follows from the proposition.
12 - c. 2398

178 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Here is some further addition to 2.1:
/
Proposition 2.3. Let X -> Y be a flat surjective morphism of complex spaces
and Y' cz Y a closed analytic subset. Then
codim (J-1 (Y'), X) = codim (Y', Y).
Proof. If x is an arbitrary point of X, then we will prove the equality coding
(/_1 (Y'), X) = codim/(jc) (Y', Y) and the proposition follows. Let 3 be the coherent
ideal-sheaf associated to Y'. We have coding (/_1 (Y'), A') = the height of the (9^-ideal
3f(x) 0X and codimy^Y', Y) = the height of the <9/(;c)-ideal 3f(xy The conclusion
follows from 1.22.
Proposition 2.4. Le? /: X —► Y Z>e a morphism of complex spaces and
& e Coh (X), which is flat with respect tof. For any point x e Supp S7,
dim &x = dim 0f(x) + dim (Srx/m/(x) &x) and
prof (ffj = prof &fM + prof (SJmf(x) 9X).
In particular, iff is flat, then for any x e X, the following equalities hold
dim 6X = dim &f(x) + dim (<3Jmf(x) 6X) and
prof Ox = prof <9/(x) + prof (@Jmf(x) ©J.
The assertions follow from 1.21.
Corollary 2.5. Under the conditions of the proposition,
coprof S^ = coprof 6f(x) + coprof (SVc/in^S^).
In particular, iff is flat, then for any x e X,
coprof 6X = coprof <9/(l) + coprof (&x/mf(x) Gx).
Corollary 2.6. Let f: X-* Y be a flat morphism of complex spaces and
let x be a point of X. Then 6X is a Cohen-Macauley local ring if and only if the rings
®f(X) and 6xlviif(X) 6X enjoy the same property. In particular, if f is moreover
surjective, then X is perfect if and only if Y and the fibers Xy, y e Y, are perfect spaces.
Further we use the next lemma.
Lemma 2.7. Suppose (X, 6) is a complex space, S'eCoh(A') and k an
integer. Then {x e X | dh 8X ^ k} is a closed analytic subset of X.
Proof. First, consider the case k = 1 (for k < 1 the assertion is clear). The
problem is local on X, hence one may assume the existence of an exact sequence of
(^-modules of the form
O-Kl-tfiP^J-tO.

V. FLAT M0RPH1SMS OF COMPLEX SPACES
179
If x is a point of X, gFv is (9^-free if and only if this sequence splits in x. Then
it follows that the set {x e X \ &x is not (S^-free) coincides with the support of the
cokernel of the map HomQ (&, 6P) -> Home (SF, SF), hence it is a closed analytic
subset.
Now we consider the case k ^ 2. The problem being local in nature, one can
assume that an exact sequence as above exists. For a point xe X, the inequality
dh 8X ^ k is equivalent to dh §x > k — 1, etc.
Theorem 2.8. Let f: X -* Y be a morphism of complex spaces.
For any coherent analytic sheaf §• on X,flat with respect tof, and for any integer
k, the sets {xe X\ dim (®xlmfM ®x) ^ k], {xeX\ prof (&Jrt\f(x) &x) < k}, {x e
e X | coprof (oF^/nty^) SFx) > A:} are closed analytic subsets of X.
Proof. Denote these sets respectively by Mk{W;f), Sk{& ;/) and Dk(&;f). Since
coprof = dim — prof, it follows that
Dk («S/) = U (Mt {S;f) n S,_t (ff ;/)).
Moreover, this union is locally finite. Thus, it is sufficient to show the analyticity
of the first two sets of the statement.
The problem is local on X and Y, hence we may assume that/ can be decom-
posed into a closed immersion X -> Y x U and the projection Y x U -> Y, U
being an open of some numerical space C". By considering the direct image IF of âF
under i, we have
'" (Mk (? ;/)) = Mk(W; tc), i (5, (ff ;/)) = 5, (#"; tc).
We may thus reduce the problem to the particular case X = Y x U,f:X^> Y the
projection, U being an open subset of C". Let x = (y, z) be a point of X. One has
&x. MySx,, * Ou.» Prof (?Jm, *,)=#!- dha (^/in, sy.
By 1.25, dh e^x/ntj, F*) = dh^^. Therefore, the set
Sk(SF;f) = {xeX\dh¥x^n-k}
is a closed analytic subset of X (2.7).
Denote by (X,-), the family of the irreducible components of yred. Let Xt =
=f'1(Y;),f;: Xt -> Yj be the morphism induced by/ and SF. the inverse image of Sf
on X;. The following equality
holds and the union is locally finite. In this way, in order to prove that Mk(&;f)
is a closed analytic subset, we may assume Y irreducible. Let / = dim Y. By 2.4,
for any xe X,
dim (§x/mf(x) SFJ = dim 8X — dim 6f(x) = dim 8X — I.

180 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
To conclude, it is enough to check that the sets M; (âF) = {x e X | dim %fx > /}
are closed analytic sets. But Mt (§•) coincides with the union of the irreducible
components of dimension $s i of Supp âF.
Remark. The fact that the sets {xeX \ dim fj/m/, > k} are analytic can
be proved without the hypothesis ^/-flat, as a consequence of the following theorem
of Remmert [65]:
"For any morphism/: X-* Y of complex spaces and for any integer k,
the set {xeX\ dimxf1f(x) > k] is closed analytic in X".
(One applies this result to the composition Supp âF -> X -> Y and realizes
that Supp gF n /_1/ (x) = Supp ($/mf(x) ^)> by Nakayama lemma).
The next corollaries show that the flatness hypothesis allows to convey some
properties of a fiber to the neighbouring fibers.
Corollary 2.9. Let f: X -* Y be a proper morphism of complex spaces-
âF e Coh (X) flat over Y and k an integer. Then the sets {ye Y\ dim Wy^k],
{ye Y |prof gFj, < k] and {ye Y \ coprof &y $s k} are closed analytic subsets of
Yiiff^iy) = 0, we agree to put dim Wy = — oo, prof âFy = + oo, coprof Wy = 0).
Proof. By the very definitions,
dim gFj, = sup dim (âF^/m^), prof ®y = inf prof (®Jmy$x) and
xef-Hy) ' xef-Ky)
coprof âFv = dim Wy — prof Wy,
whatever y ef (X), Wy # 0. The conclusion follows by the theorem and by
Remmert projection theorem (III. 2.11).
Corollary 2.10. Let f: X -* Y be a proper morphism of complex spaces,
§• e Coh (X)flat over Y and y a point of Y. If S'y is a Cohen-Macauley sheaf (coprof
oFj, = 0), then for any point y' of some neighbourhood of y, the sheaf oFy, is
Cohen-Macauley.
Corollary 2.11. Let f: X -> Y be a flat proper morphism of complex spaces
and y a point of Y such that the fiber Xy is a perfect space. Then, there exists a
neighbourhood of y such that Xy, is perfect for all points y' of that neighbourhood.
Theorem 2.12. Let f: X -> Y be a morphism of complex spaces and let âF
be a coherent analytic sheaf on X, which is flat with respect to f. Then the restriction
of f to Supp âF is an open map.
In particular, any flat morphism is open.
Proof. If we replace X by (Supp âF, (9^/Ann âF | Supp âF), we can assume that
Supp âF = X. For any y e Y,it follows that Supp Wy = Xy by Nakayama lemma.
If x is an arbitrary point of X, we derive the equalities
dim âFt = dim 0X, dim ($Jmy ®x) = dim (6x/mf(x) Ox).
By applying 2.4 we get
(*) dim 6X = dim Gf(x) + dim (6x/mf(x) 0X), x e X.

V. FLAT M0RPH1SMS OF COMPLEX SPACES
181
Let y be the normalization of YKd, g: Y' -* Y the normalization morphism
/' g
composed with Yted =-> Y, X' = Xx Y Y' and X' -> Y', X' -> X the natural morphisms
which appear.
If x' is a point of X' and x, y, y' are its images in X, Y, Y', then dim 6X- =
= dim 6X, dim 6y- = dim Gy, dim (&x'/my- 6X>) = dim (@x/my <3X). Consequently,
for the morphism X' -*■ Y' too,
dim 6X- = dim Qy^ + dim (<9x'/nVV) 0x') f°r aU x' eX'.
If U is an open subset of X, then g_1(/(C/)) =/'fe'_1 (C7))- Since y has the
quotient topology of Y', if/' is an open map, then/ itself enjoyes the same property.
In this way, it is enough to prove the assertion: any map of complex spaces/: X^ Y
which satisfies the equalities (*) and such that Y is integer in every point is an
open map.
Let x be an arbitrary point of X. We have to show that the image under /
of a neighbourhood V ofx contains a neighbourhood of y =/ (x). Apply 1.27
in connection with the correspondence between the analytic algebras and the
germs of analytic spaces. There exist a neighbourhood U of x, U <= V, a
neighbourhood V of y, f (U) <= V, an open subset W of some numerical space and a
finite morphism g: U -> W x V such that the morphism U -> V which is deduced
from / by restriction coincides with the composition of g with the projection W X
x V -»• V and such that dim U = dimxU = dim (W X V) = dim9(;c) (W X V).
The map W x V -* V being open, it is enough to prove that g (U) contains a
neighbourhood of g (x). The morphism g being finite, g (U) is a closed analytic subset
of W X V. One has dimv. U < dimB(T) g(U) < dim^ W XV, hence dimB(;c)g ([/) =
= dimB(;c) W X V. Since the analytic space H7 x F is integer in g(x), it follows that
the germs of g(U) and W X V in g (x) coincide and thus the proof is over.
A converse result would be the following:
Theorem 2.13. Let J: X -> Y be an open morphism of complex spaces. Suppose
in addition that Y is a manifold and the fibers Xy, y e Y, are reduced. Then f is flat.
Proof. We may assume that Y is an open subset of some numerical space.
Proceed by induction on dim Y. For an arbitrary point xelwe must prove that
6X is <9y(;c)-fiat. One can also assume that Y contains the origin and that f(x) = 0.
The case dim Y = /. Let t be the coordinate function on Y which is null in
the origin. In order to show that (S^is <9y.0 ~ C {?}"nat i*is sufficient to prove that
it is torsionfree. We have to check that the multiplication by t, 6X -* 6X, is injective.
Let 9 e 6X be such that 19 = 0. Let U be a neighbourhood of x where 9 extends
and where ttp is the null section. Consider an arbitrary point x' of U n X0. There
exists an irreducible component T of U which passes through x' and which is not
contained in X0 (otherwise, X0 would contain a neighbourhood of x', hence / (X0)
would contain a neighbourhood of f(x') = 0, a contradiction).
For any x" e T \ X0, 9 (x") = 0. So the function associated to 9 vanishes
on T, hence 9 (x') = 0. In this way, the function associated to 9 vanishes on X0 n U.
Since the analytic space X0 =(/_1(0), <3x/t6x |/_1(0)) is reduced, it follows that
9 = t<pl5 91 e 6X. Analogously, from the equality t\^ = 0 we can derive 9! = ftp2,
92 e 6X, hence 9 = ?292. By iterating the reasoning we get 9 e P) t'6x, hence 9=0
by Krull's theorem. r>0

1 82 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The case n = dim Y > 2. Suppose the theorem proved for dimensions < n.
Let tx, ..., tn be the coordinate functions of Y (null in the origin), and let Y' be the
submanifold of y given by the equation t1=0and X' = f-\Y'). The morphism X'-*Y'
will be open and its fibers will be reduced (if y' e Y', then X . = X'v), hence by the
induction hypothesis it will be flat. Moreover, X' is reduced (1.26).
We first prove that the multiplication by fl5 <SX -* 6X, is injective. Let <peôx
be such that t±(p = 0. Consider a neighbourhood U of x on which <p extends and
where t±cp = 0. Let x' be a point of U n X'. There exists an irreducible component
T of U passing through x' and which is not contained in X' (otherwise, X' would
contain a neighbourhood of x', hence / (X') would contain a neighbourhood of
/ (x'), a contradiction).
Since t^ = 0, it follows that <p(x") = 0 for any x" e T \ X'. Therefore the
function associated to <p is null on T, particularly <p (x') = 0. So we have proved
that the function associated to <p vanishes on U fl X'. Since X' is reduced and its
structural sheaf is Qxltflx, it follows that <p = fj<pl5 cpi e 6X. Similarly, one shows
that <p! = t-^2, <p.2 e Qx. The reasoning goes on and one finds that <p e Ç\ t[ 6X,
therefore <p = 0. h r>0
In this way we have checked that the morphism «9, -+ Qx is injective and since
0x/t1 6X is a flat ®Y.olh <9y,o-module (the morphism X' -* Y' is flat!), &x will result
to be a flat 6Y 0-module (1.4).
§ 3. A noetherianity theorem with respect
to Stein compacts
The main result in the paragraph is the following theorem of Frisch [25], Grothen-
dieck [36] and Siu [82].
Theorem 3.1. Let K be a Stein compact oj the complex space (X, 6). Then
T (K, 6) is a noetherian ring if and only if, for any analytic set Y defined in a
neighbourhood of K, Y n K has a finite number of connected components.
First we will prove the following two lemmas.
Lemma 3.2. Letf: X -* Y be a finite morphism of complex spaces which are
normal and of the same dimension. Thenf is an open map.
Proof. Let x be a point of X and let D be one of its neighbourhoods. We
must show that f (D) is a neighbourhood of/(x). We may assume that the closure
D is compact and that D fl/_1/(x) = {x}. Then f (x) £f(dD), where dD is the
boundary of D. Let G be a connected neighbourhood of/ (x) such that G [) f (dD) =
= 0. We will consider it as a complex space, namely, an open subspace of Y. Since Y is
normal, G is irreducible. Let H = D n /_1(G) and g: H -* G be the morphism
induced by /. The morphism g is finite, hence / (H) is an analytic subset of G, of
dimension n = dim X = dim Y. Since G is irreducible and dim G = n, it follows
that / (H) = G, therefore / (D) 3 G.
Lemma 3.3. Let X be a complex space and let K be one of its compacts such
that for any analytic set Y defined in a neighbourhood of K, Y{\K has finite
number of connected components. If f: Z-> X is a finite morphism of complex
spaces, then /_1 (K) has a finite number of connected components.

V. FLAT M0RPH1SMS OF COMPLEX SPACES
183
Proof. We may assume X and Z reduced and finite-dimensional. We will
use induction on dim Z. The case dim Z = 0 is trivial.
Let n ^ 1 and suppose the lemma proved whenever dim Z < n. We are going
to verify the assertion of the lemma in the hypothesis dim Z = n. Consider the
normalization morphism
7t : Z —► Z.
It is enough to show that (/7t)_1(A!) has a finite number of connected components;
so we can assume that Z is normal. Since f~\K) does intersect a finite number of
connected components of Z, we reduce the problem to the case when Z is moreover
connected (that is irreducible, being normal). Y =/(Z) is an analytic set of
dimension n in X, which is irreducible. Let
a: Y -»• Y
be the normalization morphism. There exists a unique finite morphism /: Z -> Y
such that/ — fa. By the previous lemma,/ is an open map. From 1.2, it is enough
to show that a~\K) has a finite number of connected components only. Let Y' be
the set of the singular points of Y, Y" = a'\Y') and -r = a\Y'. Since dim Y'< n,
by the induction hypothesis t_1(A!) has a finite number of connected components
only, Dj, ..., Dk. Let a~\K) — U Et be the decomposition into connected
corniez
ponents. Since Dj is a connected subset of çj_1(A!), hence Dj <= E: for some index
IjE 1. Let J = I \ {iv ..., ik}. For le J, Ex n ?' = 0, hence £; is a connected
component of tr_1(.K) \ Y'. Since Y \ Y' is isomorphic to ï\ î", cjCEj) are
distinct connected components of K Ç] (Y \ Y'), le J. By lemma 1.3, a(Et) {le J)
are distinct connected components of K fl Y. Accordingly, J and thereby / are
finite sets.
The proof of the theorem, (a) Let A! be a Stein compact such that r(K, 6) is a
noetherian ring. Suppose on the contrary that there exists an analytic set Y defined
in a neighbourhood U of K such that Y Ç] K has infinitely many connected
components.
We will construct by means of induction on n subsets Y„ (n ^ 1) in Y n K
such that:
(i) Y, = Y n K;
(ii) Yn+i is a ProPer open and closed subset of Yn;
(iii) Yn has infinitely many connected components.
Let Yx = Y f\ K. Suppose Yl7 ...,Yn already constructed. Since Yn is not
connected, Yn is the disjoint union of two proper subsets B and C simultaneously
open and closed. Since Y„ has an infinity of connected components, at least one of
these subsets, say B, inherits the same property. We will set Y„+1 = B and the
construction is thus completed.

184 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Define an = {feT(K, 0) \f(x) = 0 for any x e Y„}. an is an ideal of T(K, 6)
and an c an + 1. We will show that for any n, a„ ¥= an+1 and thus we will come to a
contradiction. Fix an integer n. Yn+1 and (Y n K) \ 7„+1 are disjoint compact
subsets of U. Let G and H be two disjoint open subsets of U, which contain Yn+1
and (Y n K) \ Yn+1 respectively. (U \ Y) U G U H is a neighbourhood of K,
hence it contains a Stein neighbourhood D of K. Clearly, D fl F c 6 U //. Let 3
be the ideal-sheaf of Y in U. Let j e T(D n y, 6/3) be the section induced by the
section s e T(G U H, &) which is 0 on G and 1 on H. As Z> is Stein, s is induced
by an element teT{D, 6). Let t = t\K. It follows immediately that tea„ + 1 \ a„.
(b) Let K be a Stein compact satisfying the topological condition in the
statement of the theorem. We will show that the ring T(K, 6) is noetherian. We may
assume that X is of bounded dimension and let n = dim X.
Consider an arbitrary ideal a in r(K, 6). We will prove by descending
induction on k, 0 < k ^ n + 1, the following assertion:
(*)t: "There exist an open neighbourhood Uk of K and elements j^\ .. .,/^t) e
e F(Uk, 6) such that
(i) f!k)\Kea, 1 < i <?(*);
(u) provided that j e T(C/, Ô), U being a neighbourhood of K
included in Uk, and f\Kea, then in some neighbourhood U' of K in U, the analytic set
«(ft) , 1
xeU'\fxt£if?xeA
is of dimension < k".
The assertion (*)„ + i is utterly obvious. Suppose (*)t verified for m < k ^
< n + 1 and we are going to verify it for k = m. We consider the ideal-sheaf
î(m + l)
3 = £ //"+i)<9 on Um+1. The subsheaf 3cS,(<9/3) = (6/3) [m] of (9/3 defined by the
;' = 1
sections whose supports are of dimension < m is coherent and its support
has the dimension < m (II. 5.3 and II. 5.5). Let "J be its inverse image under the
morphism 6 -> 6/3. "J is a coherent ideal-sheaf containing 3. If x e C/m+1, then one
can see that
}j = {je <3X\ there exist a neighbourhood D of x, an analytic subset V of Z) of
dimension < m and ? e T(D, 6) such that tx = s and ^ e 9^ for any ye Um+1 \ V}.
From the exact sequence
0 _► a _► °J _► (<s/3) [m] -»• 0
there results that Y = {xe Um+1\% ¥> 3X) coincides with Supp ((<3/9) [m]).
Therefore Y is an analytic subset of Um+1 of dimension < m. Consider Y endowed
with the structure of a reduced space and let
7t: y-> y

V. FLAT M0RFH1SMS OF COMPLEX SPACES
185
be the normalization morhism. By 3.3, 7i_1(A! n Y) has a finite number of connected
components Cl5 ..., Cp. Define zj e C} and Xj = tz(zj), 1 < j < p. Since the fibers
Gx are noetherian rings, there exist a neighbourhood Um of K in £/m+1 and sections
g[, ...,greT(Um, 6) such that gl\K e a, 1 < i < r, and {feOxjl 1 ^ ' < r]
generate the same ideal in <9XJ as the set {fXJ\f e a], 1 <j </>.
Let <7(m) = ?(/« + 1) + r. Define /,""> =flm+1)\Um for 1 < / < ?(tm + 1)
and /j'+^m + D = gy for 1 <j < r. We will prove that these entities verifiy (*),„.
Let / e T(U, 6), U being a neighbourhood of K in Um such that/|A!ea.
By (*)m+i, f\U"eT(U", }) on a neighbourhood £/" of A! in U. The analytic set
Z = JxeC/"|/.^ £(//"") A J
is contained in y, hence it is of dimension < m. Let Z' be an irreducible
component of Z of dimension m. We have to prove that Z' n K = 0; if so, by
considering Zj = the union of all irreducible components of Z of dimension m,
the open set U' = U" \ Zi contains A!, dim (Z n £/') < m and (*),„ then follows.
Suppose, on the contrary, that Z' n K^0. Since dimZ' =m and dim Y^m,
Z' coincides with an irreducible component of Y n U", hence it is the image under
tz of a connected component T of 7t_1(y n £/"). Obviously, Tn7t_1(£n Y)>£&.
Choose j, 1 < j < />, such that C, n T ^ 0- Since T n ^(A! n y) is a union
of connected components of -k~\K fl y), then Cj c T. Hence z} e T and
consequently Xj e Z'. This contradicts the very selection of sections glt ...,gr. Thus
the assertion (*)k is proved for any integer k, 0 < k ^ n + 1.
We now prove the elements fj0)\K, 1 < i =$ ^(0), generate the ideal a and
thereby the proof of the theorem will be concluded. Define gea. There exist a
neighbourhood U of K in U0 and a section/ e T(C/, 6) such that g =f\K. By (*)0
we can choose a neighbourhood V of Kin U so that/.,, e V (jf\Qx for any xeV.
Then one easily derives by choosing a Stein neighbourhod H7 of A" in £/' and by
means of theorem B that f\We'£(Jl0)\W)r(W, &). Therefore g=f\Ke
i= 1
e ^<JHK)T(K,6).
t-i
Corollary 3.4. Let K be a Stein compact of complex space (X, &), which is
contained in an analytic set of dimension 1. Then the ring Y(K, 0) is noetherian if
and only if the compact K has a finite number of connected components only.
Proof. The inverse implication is an immediate consequence of the theorem.
We prove the direct implication. If Y is an analytic set defined in a
neighbourhood U of K, we should prove that Y n K has a finite number of connected
components. Let Z c I be an analytic subset of dimension 1, containing A!. If Z n U is
irreducible, then Y n K either coincides with A" or is a finite set. In the general
case the conclusion easily follows since only a finite number of irreducible
components of Z n U cut K.

186 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The theorem can be applied to a Stein semianalytic compact K. Indeed, if
Y is an analytic set defined in a neighbourhood of K, then Y n K is a semianalytic
subset, therefore it possesses only a finite number of connected components (§ 1, f).
In this way, each point of a complex space admits of a fundamental system
of compact neighbourhoods K, which is stable to finite intersections and such that
the rings £(K) are noetherian.
We are going to close this paragraph with another proof of the noetherianity
of the ring 0(K), where K is a Stein semianalytic compact subset of a complex
space (X, 6).
We have to prove that any ascending sequence of ideals of finite type
Û! Œ Ct2 <= Û3 <=
from T(K, 6) is stationary. Every ideal a„ is of the form a„ = T(K, S„) where 3„
is a coherent ideal-sheaf, defined in a neighbourhood of K. By theorem A one has
the sequence of inclusions
3,\K cz 32\K <= 33\K cz ...
of coherent ideal-sheaves in 6K = Q\K. The conclusion will follow from the
compactness, provided that one verifies the assertion:
(*) "Let âF be a coherent (S^-module and let (gF„) be an ascending sequence
of coherent <9x-submodules of W. Then for any xe K there exists a neighbourhood
U of x such that the sequence (âFJt/ n K) is stationary".
In this assertion we agree to call coherent 6K-module the restriction to A! of a
coherent (9^-module, U being a neighbourhood of K.
Let xe K. The ring 6X is noetherian, hence there is an integer »0 such that
(^»)x = (®Jx for n>n0. If we replace SF and the sequence (SF„)„>„„ by SF/aFno and
(^J^nX>na respectively, we may assume that (âF^ = 0 for any /;.
Define F = &x. F is an (9^-module of finite type and consider a filtration
F = F° ^ F1 ^ ... 3 Fq = 0
such that quotients Fl/Fi+1 are isomorphic to (9T-modu!es of the form OJp',
p! e Spec &x. We "localize in x" (that is, replace X by a suitable neighbourhood U of
x and the compact K by K n A!', where A!' is a Stein semianalytic compact
neighbourhood of x in U). We can thus suppose that there exist a filtration of âF by
coherent sheaves
s= = âF° => âF1 3 ... 3 âF« = o
and coherent ideal-sheaves 3' such that

V. FLAT M0RPH1SMS OF COMPLEX SPACES
187
To prove the assertion (*) with respect to the stationarity of the sequence (<?„)„
around x is to prove (*) in the point x for the sequences induced by this in the sheaves
§•/&'+! ~ &/$. In other words, we may assume the sheaf gF of the form 0/3 with
3X a prime ideal.
If X is replaced by the subspace given by 3 and K by its intersection with this
subspace, we may assume gF = 0 and 0X an integral domain. Further, localizing
in x, X can be supposed reduced. We will show that there exists a neighbourhood ZT
of x such that gF„|[/ n K = 0 for any n and the proof will be concluded.
Consider the normalization morphism
■k:X ^ X.
tz~\K) is a Stein semianalytic compact in X. For every integer n one obtains, by
taking the inverse image, a coherent ideal-sheaf 8Ïn of 0%, defined in a
neighbourhood of tz~\K). Clearly, gF„ <= iz#(&„). More precisely, tz*(&„) coincides with the
ideal-sheaf generated by Wn in the integral closure of 0X (which coincides with
7t *(<%)). If xe7t-1(x), then (&n)~ =0. From these facts it follows that we can
suppose X just normal in the assertion we need; in particular, X is integer in every
point.
Let us fix an integer n. The sheaf & n is the restriction to A! of a coherent ideal-
sheaf defined on a neighbourhood Un, which is denoted also by SF„.
Consider A = {x' e U„ | (S^)*' = 0}. A is an open subset. Define x' e Un
such that (^J*- ^ 0. Denote by Y the analytic set given by qF„, Y = Supp (<9X/Sr„).
If p is a prime ideal of &x- which contains (&n)x', then its height is ^ 1 (6X- is an integral
domain!). It then results that codim^y, U„) > 1. The same property holds in a
neighbourhood of x' ([15]). The stalks of gF„ will be nonnull in the points of this
neighbourhood. Accordingly, A is also a closed subset. Let T be the connected
component of K containing x. Since xeAiïK, T cz A. In this way ¥„\T = 0
for any n. Since K is locally connected (§ 1, f), T is of the form U n K, U open
subset of X containing x.
§4. The flatness locus of a morphism
Let A' be a complex space, K <= X a Stein semianalytic compact and SF e Coh(A').
The ring T(K, 0X) is noetherian (theorem 3.1) and T(K, &) is a T(K, (5x)-module
of finite type. For a point xe K, denote by m(x) the maximal ideal of T{K, 0X)
given by x (the kernel of the composite map T(K, 0X) -> 0X -> «S^/m^ =; C)-
Lemma 4.1. 77?<? completion of r(K,&) with respect to the m{x)-adic topology
is canonically isomorphic to the completion of 5sx with respect to the mx-adic topology.
Proof. Let 3 be the coherent sheaf of ideals given by x. Since K is a Stein
compact, we have natural isomorphisms
T(K, ar)/m(xy T(K, 9) = T(K, S)/T(K, 3)" T(K, ¥) ^
T(K, ff/3'SF) ~ (y/S'ff), * &J3&, = Sr,/m^x,

188 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
p > 0 being an arbitrary integer (generalities about Stein compacts can be found
in Ch. VI, § 1, b). Passing to projective limits one obtains the required isomorphism.
Let now /:!-» y be a morphism of complex spaces and let K <= X and
L<= Y be Stein semianalytic compacts such that f(K)cL and <F e Coh (X). T(L, &Y)
and T(K, <3X) are noetherian rings and denote by T(J) the map induced by /
between them. Consider a point xof K and let y =f(x). Obviously, rC/)"1^) = mx.
Lemma 4.2. 8 is f-flat in the point x if and only if T(K, 8) is T(f)-flat in
m(x) (that is, r(K, ®)m(x) is T(L, 6)mMflat).
Proof. 8X is fi^-flat if and only if 8X (the completion with respect to the mv-adic
topology) is &y (the completion in the rryadic topology)-flat (1.6). Similarly,
T(K, ®)m(x) is T(L, 6y)m(y)-ûa.t if and only if the same property holds when
passing to completions. The conclusion follows from the previous lemma.
Theorem 4.3. Let Y be a reduced complex space, D an open of some
numerical space, X = Y X D, f : X -* Y the projection, x = (y, z) a point of X and 8 a
coherent analytic sheaf on X. Then there exist an open neighbourhood V of y and an
analytic subset T of V for which dim T < dim V, such that SF is f-flat in the point
(/, z) for any y' e V \ T.
Proof. Let L be a Stein semianalytic compact of Y, neighbourhood of y. It
is enough to find an analytic germ T around L, which is distinct of the germ
defined by the whole space and such that & is /-flat in (y', z) for any y' e L \ T.
Let yls ..., Yr be the analytic sets denned in a neighbourhood of L which
correspond to the minimal prime ideals of the noetherian ring T(L, 6r). Since the
germ denned around L by [J (Yt n Y J) has the dimension < dim V, it is sufficient
to find T in the case we do the base changes 7; -> Y. We can thus suppose that the
ring A = T(L, 6Y) is an integral domain. Let 3 c 6X be the ideal associated to
Y x z. For a point x' = (/, z), 3X> = mz6x-. Denote K = L x z. K is a Stein
semianalytic compact of X. Let / = r(K, 3), B = T(K, 6X) and M = r(K, SF).
Since the composite morphism
r(L, er) - r(K, ex) - r(*, gx)/t(k, 3)
is an isomorphism, we may apply lemma 1.17. Then there exists a nonnull element
<p e A such that Mv is ,4-flat in all prime ideals of V(IV).
Consider the analytic germ T of analytic set around L defined by <p. Let
y' e L \ T. We will show that gF is /-flat in the point x' = (j', z) and the theorem
will be proved. By 4.2 we have to show that the module r(K, 8)^-) is
T(L, <9j-)tn(3,')-flat. But this fact easily results from the inclusion m^'),,, => Sv.
Let Y be a complex space and Y' the set of all regular points of YTea- Y' is a
dense open subset in Y and its complementary is analytic. A part A of Y is called
negligible if A n Y' is an at most countable union of the submanifolds of Y', which
are locally closed with empty interior. From the definition it follows that an at most
countable union of negligible sets is negligible. Since the complex spaces are Baire
spaces, the interiors of negligible parts are empty and their complementaries are
dense subsets.

V. FLAT M0RPH1SMS OF COMPLEX SPACES
189
Lemma 4.4. Let f : X -* Y be a morphism of complex spaces. Suppose Y
reduced and X with countable topology. Then f(X) is a negligible set in Y if and only
iff admits no sections a : V -* X (fa = inclusion) on any nonempty open subset V of Y.
Proof. One can suppose Y a complex manifold and X a reduced space. A" is a
locally finite (hence countable) union of locally closed complex manifolds Xt such
that the rank of / | Xt is constant (one can reason by induction, making use of thx.
singular locus of A"!).
Accordingly, we may assume that A" is a complex manifold and/ is a holo-
morphic map of constant rank. Locally, / is then a composition of a submersion
and an immersion. Thus we have to prove the lemma under the assumption that X
and Y are complex manifolds, that f(X) is a submanifold of Y and/ a submersion
of X on /(A"). In this case, the set f(X) is negligible if and only if its interior is empty,
that is if and only if there is no point x e X where/ is a submersion. The required
conclusion follows.
The main result in the paragraph is the following theorem of Frisch:
Theorem 4.5. Let f : X -> Y be a morphism of complex spaces and SF a
coherent analytic sheaf on X. Then the set of the points of X where SF is not f-flat is a
closed analytic subset.
If moreover Y is reduced and X has a countable topology, then its image in Y
is negligible.
Proof. The first assertion is local in nature on X and Y. Therefore we may
. . ' p . .
assume that / is the composition X -* <£,"x Y -* Y, where / is a closed immersion
and p the projection. By considering the sheaf i#(&) we can easily reduce the
problem to the particular case X = (E"xY and/ the projection.
Let x = (z, y) be an arbitrary point of X. Consider some Stein semianalytic
o o
compacts L c Y and Q <= <£," such that y eL and z e Q. K = QxL is a Stein
semianalytic compact in X and x is one of its interior points. Denote
Z = A'XyA' and let Z z£ X be the natural morphisms. A/=KxtA!isaSteinsemiana-
?!
lytic compact in Z. Denote by A: X -* Zthe diagonal map. A is a closed immersion
and let 9 be the coherent ideal-sheaf of Z which defines it. We set / = T(M, S)
and & = q*(&). The factor-ring T(M, <3z)jl is canonically identified (by the
morphisms r(çj) and r(q2)) with F(K, <5X). Apply theorem 1.16 to the morphism
r(çj) : r(K, 6X) -»• r(M, 6>z), to the ideal / and to the module T(M, SF). Then there
exists an ideal a of T(M, <£>z) such that
V(a) n V(I) ={pe V(1)\T(M, SF) is not T^-flat in p}.
By 4.2, the set
A = {z e A(K)\(f>(z) = 0 for any <p e a}
coincides with the set of points of A(K) where SF is not ^j-flat. Since / is a flat
morphism, S is /-flat in a point x e X if and only if SF is t^-flat in the point A(x)

190 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(§ 1, d). It then follows that A~JG4) is the set of the points of K where SF is not
o
/-flat. The restriction of A_1(^) to the interior of A! is a closed analytic subset of K
and in this way the first assertion is proved.
Suppose now Y reduced and X with countable topology verifying the last
assertion. The problem being local on X and Y, we may suppose that X = Y x D
where D is an open subset of some numerical space (£"' and / is the projection. We
can also assume Y a complex manifold.
Let Z be the set of points of X where SF is not /-flat. Apply lemma 4.4 to the
composite morphism
Z^X = Y x D^ Y.
In order to achieve the proof we show that if V is a nonempty open subset of Y
and <p : V -> D a morphism, then the graph F of <p can not be contained in Z.
Let D' be the image of V X D under the morphism
VxD -»• C", (y,z)\-+ z — cpO).
The morphism
a : VxD-»- VxD', (y, z) i-* (y, z — <p(»)
is a F-isomorphism and transforms F into V x {0}. The conclusion results from
theorem 4.3 applied to the morphism V X D' -> V and to the sheaf y.^{®\VxD).
We now give some consequences of the theorem.
Corollary 4.6. Let / : X -* Y be a morphism of complex spaces, & and 8'
two coherent Qx-modules and u :&' -* & an <3x-morphism. Suppose SF is j-flat.
Then the set of the points xe X where the induced morphism
(fr/w), = *>V(x,»i - (?M)X = 9Jmflxfx
is injective is open in X and its complementary is analytic.
Proof. Let SZ and 3C be the kernel and the cokernel of u, respectively. By 1.5,
the morphism
Klmnx)K - *Jmf(x?x
is injective if and only if Slx = 0 and 3C is/-flat in x. The conclusion then follows from
the theorem and from the fact that the support of a coherent sheaf is a closed
analytic set.
Corollary 4.7. Let f : X -> Y be a flat morphism of complex spaces and
<p e r(A", &x). Then the set of the points xe X such that the image of <px in
<S>Jmf(x)<3x — <3Xj. is a nonzerodivisor is open and its complementary is analytic.
The proof follows from the previous corollary applied to the endomorphism
of 6X given by <p.

V. FLAT M0RPH1SMS OF COMPLEX SPACES
191
Corollary 4.8. Let f : X -* Y be a morphism of complex spaces, 8 a
coherent &x~m°dule, which is flat with respect to f, and (<Pi)lS;j<r a sequence of
sections of T(X, 6X). Then the set of the points xe X such that the image of the
sequence ((pi,x)i^i<r m ^/%W^ ls a regular 8Jmf(x)8x-sequence is open in X
and Its complementary is analytic.
r
Proof. Let U be the set from the statement. Denote § = 8 V <f>jSF. Since 8
is /-flat, it follows from 1.9 that U is just the set of the points xeX enjoying
the property that the sequence (<p(jX) is a regular cFx-sequence and (J is /-flat in x.
The conclusion follows from the theorem and from
Lemma 4.9. Let X be a complex space, 8 e Coh(A") and (<pi)i<i<r a sequence
of global sections of 6X. Then the set of the points xe X such that the sequence
(<?i x)i<i<r Is 8x-regular is open and its complementary is analytic.
Proof. Consider the sheaves <$i = 8 £ <pj8 and the endomorphisms ut :
<?; ~* §i given by <p!s 1 < / < r (we take 8 as é^). If U stands for the set from the
statement, then
X\U = {j Supp (Ker «,)•
We now reformulate the main result of the paragraph for proper morphisms.
If X -> 7 is a morphism of ringed spaces, y a point of Y and 8 an (9x-module,
then we say 8 is f-flat in y if 8 is/-flat in all points of/-1^).
Theorem 4.10. Let f : X -> Y be a proper morphism of complex spaces
and 8 a coherent analytic sheaf on X. Then the set of the points of Y where 8 is
not f-flat is a closed analytic subset.
If moreover Y Is reduced, then its dimension is greater than the dimension of
the previous set.
Proof. Let Z = {x e X18 is not /-flat in x} and T = {y e Y\8 is not/-flat in >>}.
By theorem 4.5, Z is a closed analytic subset of X and hence /(Z) is an analytic
closed in Y in virtue of Remmert's projection theorem.
The first assertion follows from the equality /(Z) = T. In order to prove the
second assertion, we may suppose Y with countable base. Since / is proper, X
follows with countable base. By 4.5 the open Y\T is dense and the conclusion
results.
Corollary 4.11. Letf : X -* Y be a proper morphism of complex spaces and
8 e Coh(A"). Suppose dim Y < oo. Then there exists a finite partition (Yi)l<iiir of Y
with locally closed analytic subsets such that, for any I, the sheaf 8t = 8 XYYi is
flat over Y\ where Yt is considered with its reduced structure of complex subspace of Y.
Proof. We may suppose Y reduced and proceed by induction on n = dim Y.
The case n = 0 is obvious.
Suppose the corollary proved in the case when the base is of dimension < n.
By the theorem the set T of the points of Y where gF is not /-flat is analytic and

192 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
closed. Moreover, dim T < n. We perform the base change T -* Y and apply the
induction hypothesis. The partition so obtained for T, to which we add the open
subspace Y\T, verifies the corollary.
Bibliographical indications
Theorem 2.12 is due to Douady and Kiehl ([19] and [45]); the proof given here is that from [45].
Theorem 2.13 and the proof presented here belong to Kerner [441. Theorem 2.8 is due to
the authors. The other results from § 2, as well as the corollaries from § 4, are from ([37], Ch. IV).
Theorem of noetherianity 3.1 (for Stein semianalytic compacts) belongs to Frisch [25]
and Groethendieck [36]. The more general statement given here and the proof are from the
paper of Siu [82]. The proof given at the end of the third paragraph in the semianalytic case
is that from ([36]. SGA 2).
The results of § 4 were obtained by Frisch [25]. The proof of the main result (the first
assertion of theorem 4.5) is given in Kiehl's paper [46]. Theorem 4.10 was already pointed out
by Grauert in [29].
A presentation of the flatness in the case of the complex spaces can be found in Douady's
paper [20].

Chapter VI
The formal completion of a complex space
with respect to a subspace
Introduction
Let (A", <S>) be a complex space, let A" be a closed analytic subset of X and let 3 <= <S>
be an ideal of definition of A". Many problems suggest the consideration of the
complex spaces {X',ôj3k+ï\X') (the "infinitesimal neighbourhoods of A" in A"').
The ringed space X = (A", (lim 6>/3*+1)| A") is called the formal completion of X
with respect to A". For any <?-module Er, one may consider the 6>A-module W =
= lim (SF/S^+W), called the completion of & with respect to A". If / : X -»• 7 is a
morphism of complex spaces and A", 7' are closed analytic subsets of X, 7 so
A A A
that /(A") <= 7', then one obtains a morphism f : X -> Y.
Our aim in this chapter is to study the behaviour of the direct images R'f%
with respect to completion.
Theorem 4.1 connects the invariants R%(&)*, R'f*(&) and lim RJ^Sr^+W).
The coherence of some graded sheaves is assumed; thereby one may derive conditions
of Artin-Rees and Mittag-Leffler type and the assertions follow from the reasonings
of projective systems. Suppose, for instance,/ proper and A" = /~1(7'). In order to
apply the theorem here, the difficulty in the case of complex spaces is to check the
hypothesis of coherence of the © f--modules ®Rqj^(pk^)Q\% an ideal of definition
of 7' and 3 =/*CJ)<S)a:)- Theorem 4.4 ensures just this; its proof makes essential use
of a finiteness theorem for ringed spaces of polynomials over complex spaces
(theorem 3.1).
As an application of the above facts, one obtains the following result (4.6
and 4.7):
"Let / : X -> 7 be a proper morphism of complex spaces, y a point in 7
and & e Coh(A"). Then there exists an integer k0 such that
Ker (H\Xy, SF) -► #'(*„, Sr/m*+rar)) = rrf, Ker (H\Xy, $)^Hi(Xy, SF/m*SF)),
Im(H\Xy, S7m*+r59 ^H\Xy, ff/in;^)) = ïm(H%X„ Sfjm^TSf)^H\Xy, ff/in^)),
for 9 > 0, r > 0 and A: > k0.
13-c. 2398

194
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The first paragraph contains preliminaries and in § 2 one proves the elementary
properties of the completion functor.
§ 1. Preliminaries
(a) Let A be a noetherian commutative ring, let a be an ideal in A and M an
A-modu\e. Denote by M the completion of M with respect to the ct-adic topology,
A A A
M = lim {MjakM). M is a module over the completion A = lim Afak of A. Recall
k k
the following results ([10], Ch. Ill, § 3, n = 4 and § 5, n = 5):
A A
— the ring A is noetherian and separated in the ct/l-adic topology;
A
— if M is an ^4-module of finite type, then the canonical morphism A ®A M ->
-> M is an isomorphism;
— if M -* N -* P is an exact sequence of ^4-modules of finite type, then the
V\ A A
sequence M -* N -* P is exact;
A
— the canonical morphism A^ A is flat;
A A
— an ^4-module, which is separated in the a^-adic topology, is flat over A
A
if and only if it is ,4-flat.
Lemma 1.1. Let A^> B be a flat morphism of noetherian rings, let a be
A A
an ideal of A and let b be an ideal of B such that aB cr b. Denote by A and B
the completions of A and B in the a-adic and b-adic topology respectively. Under
A A
these assumptions, the morphism A -*■ B induced to completion is flat.
A A
Proof. B is separated in the bi?-adic topology, hence it its separated in the
A
û/4-adic topology. Since the morphisms A -*■ B and B -> B are flat, the conclusion
follows.
(b) Consider a complex space (X, 6). Let g be a Stein compact, hence a
compact which admits of a fundamental system of Stein neighbourhoods. If gF e
e Coh(A"), then from theorems A and B there results that the cohomology groups
Hq(Q, §•) vanish if q > 1 and in addition, SF admits an exact sequence of the form
Q" -> <3« -> S -> 0 on Q. In particular, if &' -> 8 -> gF" is an exact sequence in
CohCA-), then the sequence T{Q, 8') -»• T(Q, 8) -»• T(Q, 8") is exact.
If ôi) Ô2 are Stein compacts, then Qx n Q2 is also a Stein compact, as follows
easily by means of diagonal morphism X -> X x X. Consequently, the Stein compacts
are stable to finite intersections.
Lemma 1.2. Let Q be a Stein compact in X. Then:
(i) For any 8, (| e Coh(A"), the canonical morphism
r(e,ff)®r(o.<9)r(e,<?)-r(e>ff®(9<?)
is an isomorphism.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
195
(//) For any SF e Coh(A") and whenever S is a coherent ideal-sheaf,
r(e,3ff) = r(e,3).r(e>ff).
Proof, (i) We may assume that there is an exact sequence of the form &" ->
_> Qi _,.§;->. 0, by an eventual substitution of X for a neighbourhood of Q.
The conclusion follows from the exact commutative diagram
r\e, ep)®viQ,6)rm <?) -H6, <s*)®iw.<9>ne,§) - ne, s^r^ne, <D-o
ne, ^®ai?) -► rde.e)»®^) -► ne, ^®a(|)^o
07) 9^ is the image of the morphism 9 ®(9Sr -> S^ and apply (/)•
Lemma 1.3. Let Q' <= Q be Stein compacts. Then
0") -For awj cF e Coh(A"), ?fte canonical morphism
TXQ, §r)®r(e.e)r(e', <S>) - T(Q', 8r)
is an isomorphism.
(ii) The restriction map
ne, o) - ne', <s>
« flat.
Proof, (i) We may assume that there exists an exact sequence of the form <5" ->
-» 6" -> S^ -> 0. The conclusion will result from the exact commutative diagram
T(e, &)®ria.e)T{Q', ^)^ne, <S?)®r(e,a)r(e', ©)-r(0, ^)®r(e,e)r(e', <9) - 0
ne;, «s*) -»• ne;, «s*) -»• ne', so -* o.
(ii) If a <= T(e, <5) is an ideal of finite type, we have to show that the
canonical morphism a ®r(o, 6)^(Q\ <S) -»• T(Q', &) is injective. There exists a coherent
ideal-sheaf 3 on a neighbourhood of Q such that T(Q, 9) = a and apply (/).
We shall sometimes use the notation
«see) = ne, <s>), hq) = ne, §o-
(c) Let A" be a locally compact topological space and (Br;);6/ a family of
sheaves of abelian groups on X. If Q is a compact subset of X, then the canonical
morphism
© ne, ^) - ne, © *«)

196 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
is bijective. By passing to cohomology (for instance, by means of resolutions with
flabby sheaves) one gets isomorphisms
© H\Q, SF,-) -+ H\Q, © Srd-
i i
From this fact, by lemma III. 1.3 one obtains
Lemma 1.4. Let f : X -* Y be a proper map of locally compact topological
spaces and (S^X-e/ a family of sheaves of abelian groups on X. Then the canonical
inorphisms
© R'Marà - R"U © ard
i i
are isomorphisms.
Let a? e Ab(A"). Recall the construction of Godement's flabby resolution [26].
Denote by <B° = (2°(Sr) the sheaf U \-+ JJ &x, U open subset of X (the restriction maps
x£U
are obvious). £° is a flabby sheaf and there exists an injective morphism W -> £°,
s h_> (sx)x- Denote £x = £°(Coker (&-* &>)), etc. One thus obtains a flabby resolution
o _► w -»• £° -»• e1 -> £2 -»•...
If oF -> § is a morphism in Ab(A"), then one obtains a morphism of complexes £*(SF) ->
-> <£*(<5). These correspondences give rise to an exact functor.
(d) ([37], Ch.Om, §13.2.3). We say that a projective system (A„)n>0 of
abelian groups satisfies the ML condition (Mittag-Leffler) if for any n ^ 0 there
exists an integer m > n such that, for any r ^ 0
Im(/4m+r ->• A„) = Im (/4m -► An).
Lemma 1.5. Let 0 -> /4„—>B„—>C„ -> 0 beanexact sequence of projective
systems of abelian groups. If the systems (A^) and (C„) satisfy the ML condition, then
the same property holds for C6„).
Proof. Let n > 0. Choose m > n such that Im (Am+r -> An) = Im (A,,, -> An)
for any r > 0. We then choose p ^ m such that Im (Cp+r -> Cm) =Im(C/I -> Cm)
for all r > 0. A simple reasoning with exact sequences shows that
Im (Bp - £„) = Im (Bp+r - £„), for all r > 0.
Proposition 1.6. Let
0 -»• A„ -»• £„ -»• C„ -»• 0, « ^ 0,

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE 197
be an exact sequence of projective systems of abelian groups. If (A„)„ satisfies the ML
condition, then the sequence
0 -» lim A„ -*■' lim B„ -+ lim C„ -> 0
is exact.
The proof can proceed exactly as for proposition III. 1.11.
Proposition 1.7. Let {K'n)n>0 a projective system of complexes of abelian
groups having the differentials of degree + 1. For every m there exists a canonical
morphism
hm: «"'(lim K'„) -+ lim H'"(K'n).
If, for any degree m, the projective system {K"à)n>o satisfies the ML condition,
then all the morphisms hm are surjective. If moreover, for a degree m, the projective
system (//'"""1 (K„)) satisfies the ML condition, then the morphism hm is bijective.
Proof. Denote for an integer m, Km = lim K™. The morphisms hm are to be
n
deduced from the commutative diagrams
...-»• K"'-1 -> Km -> Km+1 ->■...
4" 4 4
. . . -► K™-1 -» K™ -» K?+1 -»...,
the differentials of K' being the projective limits of the differentials of the
complexes KÛ ■
Consider the exact sequences
(*) 0 ^ B"\K-n) ^ Z"'(K-„) ^ Hm(K-„) ^ 0
(**) 0 -► Zm-\K'„~) -> K?-1 -> Bm(K'„) -> 0
(as usual, B dessignates the group of coboundaries and Z the group of cocycles).
By hypothesis one derives easily that the projective systems (B'"(K'„))„>0 satisfy
the ML condition for any m. By 1.6 we get the exact sequence
0 -»• lim B'"(K'„) -»• lim Z'"(K'„) -> lim Hm(K'„) -> 0.
From the left exactness of the projective limit one obtains the isomorphisms
Zm{K')^\imZm{K'n).

Ï98
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACE
Then the fact that hm are surjective results from the exact commutative diagram:
0 -► lim Bn\K'n) -»• lim Z"'(K'n) ->• lim H'"(K'n) - 0
«>0 n>0 «SsO
î lî Î'-
0 —> B"\K') —> Zm(K') —> Hm(K') -»• 0.
A ssume now that the projective system (Hm~\K'n)) enjoys the ML condition. From
lemma 1.5 and the exact sequences (*) one derives that the system (Zm~1(K^l))n^0
also satisfies the ML condition. From (**) one then obtains the exact sequence
0 -► lim Z'"-\K '„) -► lim R™"1 -► lim Bm{K'„) -»• 0.
Then it results that the morphism B'"(K') -> lim Bm(K'„) is bijective, etc.
Further we will give an application of proposition 1.7.
Lemma 1.8. Let X be a topological space and (a^„)„^0 a projective system
of flabby sheavessuch that the maps 8r„+i -> &„ are epimorphisms and their kernels
are flabby. Under these assumptions, the sheaf & = lim Wn is flabby.
Proof. T(X, SO =? lim T(X, &„). Let U be an open subset of X. r(U, S7)^
^lim T(U, SF„). The hypothesis implies that the maps T(X, SF„+1) -► T(X, &n),
F(U, &„+1) -> T(U, &„), n > 0 are surjective. Let s =(s„) bean element of T(U, &).
Consider an element t0 e r(A", S^) such that t0 \ U = ,j0. Let S; be an element of
r(A", SFj) so that its image in r(A\ SF0) is just t0. The element ^ — Ç [ U lies in the
kernel of the map r(U,&J ^>r(U,&0), hence there exists 7) e Ker (IX*, S^)-►
-»• T(A-, S^)) such that 7) | (7 = sx - \\ U. We set tx = I + 7) e r(A\ S^). The image of ^
in r(.y, JF0) is t0 and tx\ U = ^. The reasoning goes on and one can find an element t
of IX*, SO such that t\U = s.
Proposition 1.9. Let X be a topological space and let (SF „)„>() be a
projective system of sheaves of abelian groups. Suppose the following conditions fulfilled:
(i) There is a base Si for the topology of X such that for any U e Si, //*(£/,§•„) =
= 0 for all q > 1, n ^ 0.
{it) For any U e Si, the morphisms F(U, §"„+1) -> T(U, §•„), n > 0, are surjective.
Under these assumptions, for any m > 0, the canonical morphism
hm : Hm(X, lim ^„) -► lim Hm(X, &„)
is surjective ; if moreover for some m the projective system (Hm~\X, &„))n>0 satisfies
the ML condition, then hm is bijective.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
199
Proof. For every sheaf SF„ we consider Godement's flabby resolution <B'(8„).
Denote 8 = lim 8„ and <2" = lim (S'^J. Let ^„+1 = Ker (S^ -»• 8„). From the
exact sequence 0 -> q£b+i -»• 8n+1 -*■ 8n -»• 0 one derives the exact sequence of
complexes of sheaves
o - e-(^„+1) - e-(SF„+1) -, e-(ff„) - o.
Consequently, the kernels of theepimorphismsS'^n+i) -»• Ê'^^) are flabby sheaves.
By lemma 1.8 it follows that the sheaves £m = lim &"(8n) (m > 0) are flabby.
For any open set U e o&, the condition (/) shows that the sequences
o - r(u, 8n) - ne/, e<W) - r(c/, ew)
are exact. By using (//) and the surjectivity of the maps F(U, &m(8n+1)) -»•
-> r(t/, &"(8n)) (m > 0, « > 0), we derive from 1.7 the exactness of the sequence
o -► r(u, 8) -> r(u, e°) -► r(t/, e1) -►...
So £* is a flabby resolution of 8 and therefore the cohomology groups H'(X, 8)
are isomorphic to the cohomology groups of the complex r(A", S") =
= lim (T(X,&'(&„)). The proof will be completed by applying again propositio n 1.7
§ 2. Definition and elementary properties
Let (X, 6) be a complex space and let A" <= X be a closed analytic subset.
Lemma 2.1. (/) If S <= & is a coherent ideal-sheaf such that Supp ((9/9) = A",
7/zctz Supp (lim (9/3*+1) = A".
(//) // "J cr (9 /j another coherent ideal-sheaf such that Supp ((9/'}) = A", ?/ze«
?/zere exists a natural isomorphism of 6-algebras
lim (e/3k+1) ^ lim ((9/f+1).
fc A:
Proo/. (/) Obviously, Supp (lim (&/3k+1)) c A". Let xe A"; we have to show
that the canonical morphism & -* lim (&/3k+1) is injective in x. Let / be a section
k ■
defined around x so that the image of fx in the projective limit is null. There exists
a neighbourhood U of x such that the image off\U under the morphism
T(U, (9) -* T(U, lim (c9/3*+1)) = lim r(U, <9/3*+1)
k k
s null. Therefore, fx e Pi 3k+1 c PI ™*+1 = 0.

200
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(//) Replacing °\ by 3 + °\, we may assume moreover that 3 <= 'J. Then there
exists a natural morphism of (9-algebras
lim (6l$k+1) -+ lim (0lf+1)
k k
and we need to show that this is an isomorphism. By Nullstellensatz it follows
that, locally, some power of *J is contained in 3. The assertion we have to prove is
local in nature, so its proof becomes clear.
Let us fix an ideal 3 as in the lemma and let us call it an ideal of definition
of A". The ringed space
(A", lim (6/5k+1) | A")
k
is called the forma! completion of X with respect to A" and it is denoted by (X, (Sa).
In virtue of the above lemma this does not depend on the sheaf 3 and Supp &* = X.
The inclusion A" -> X and the canonical morphism 6—> lim (Qj3k+1) yield a morphism
k
of ringed spaces X -> X, which will be denoted by i.
Let SF be an (9-module. Denote SF = lim Srw, where 8(k) = SF/S*+1SF. It is
k
easy to check that SF is independent of the ideal of the definition 3 of A". Clearly,
A A
Supp SFc A" and by restriction one gets an 6 * -module which is also denoted by SF.
In particular, 6 = <9 a. There exists a canonical morphism SF -► SF. If C/ is an open
subset of X, then r(C7, SF) = lim r(U, SF<«). Remark that if A is a subset of X,
then lim T(A, SF**') does not coincide generally with T{A, SF) (even when A is reduced
k
to a point!). If A a B are two subsets of X, then the restriction map defines a
natural morphism lim Y(B, SF<*') -> lim T(^, SF(i)). These morphisms are functorial
k k
with respect to inclusions.
Lemma 2.2. (/) .For a«>> Ste/« semianalytic compact K, the F(K, 6)-module
lim r(A!,SF(*i) ij canonically isomorphic to the completion T(K, SF) of the r(K, 6)-module
k
T{K, SF) with respect to the T(K, 3)-adic topology.
(ii) For any xe K there is a canonical isomorphism
(SF), ~ lim HaCsf),
K
K being a Stein semianalytic compact which contains x as an interior point.

Vf. THE FORMAL COMPLETION OF A COMPLEX SPACE
201
Proof, (/) From theorems A and B we easily derive the isomorphisms
lim r(K, »<*>) = lim T(K, ff/3*+1ff) ~ lim (T(K, 9)IT(K, 3*+1ff)) ^
k k k
st \im{T{K, S)IT(K, »k+v) T{K, 9)) ^ lim (T(K, *W(K, S)k+1r(K, 9)).
k k
(ii) 8x = lim T(U, lim 8<k>) = lim lim T (U, ®{k)) = lim lim T (K, 8<k>) ^
U k U k K k
~ lim F(K, 3F), where U is a neighbourhood of x and K is as in the statement.
K AAA
An (9-morphism SF -> éj induces naturally an <9-morphism SF -> (J; one can
easily check that a functor is so obtained. For any (9-module SF the canonical
morphisms
8®e (lim c9/3*+1) -► 8®e (6j3k+1) ~ ffW
induce a morphism
/*(#) -► SF,
which is functorial in S*.
A
Proposition 2.3. (i) 77ze functor §•(->-§• w exac? o« ?Ae coherent sheaves.
A
(//) If 9 e Coh(X), ?/ze« ?/ze morphism z'*(oF) -> <? w a« isomorphism.
Proof, (i) Let 0 -> SF' -> SF -> SF" -> 0 be an exact sequence in Coh(X). For
any Stein compact K one obtains an exact sequence
0 -► T(K, SF') -► r(K, SF) -► r(£, ff") -► 0
of <9(A!)-modules of finite type. If moreover K is semianalytic, then <S(K) is a noether-
ian ring (V. 3.1) and by completion with respect to T(K, 9)-adic topology one gets
an exact sequence
o -► r(k?s') -► mC*) -► r(£V') -► o.
The conclusion follows by the previous lemma.
(//) The problem being local, we may assume that there is an exact sequence
of the form 6" -* Qq ->§•->• 0. The morphism from the statement is obviously
an isomorphism if SF = <S>, hence if SF is free of finite rank. The conclusion follows
by means of the exact commutative diagram
/*(£/>) _> /*((9?) _> ;*(SF) _> o
4' 4 4*
(eey ->(<s>«)A -► sf ->o.

202
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Corollary 2.4. // W and § are coherent 0-modides, then there are canonical,
isomorphisms which are functorial in §• and §,
SF (x) a (3 ^ (8 ®e <f) A , (#oma (SF, 6Q) A ^ ftra. (£, <f).
Proof. There are functorial canonical isomorphisms
i*(&) ® a i*(<2) ^ /*(# ®a <£), i*(Home (8, <?)) ^ flora* (**<», /*(<§)).
fi S
The first one holds for any morphism i of ringed spaces, without supplementary
hypothesis on SF and ($; as for the second, one applies the exactness of
functor i* on the coherent sheaves.
A
Proposition 2.5. The sheaf of rings 0 is coherent.
A
Proof. We have to show that the kernel of any <9-morphism of the form
A A A
0P -> 0, defined on an open subset U, is an (9-module of finite type.
o
Let xeU and K <= U be a Stein semianalytic compact so that xeK. As
above, denote by " A " the completion of the T(K, <9)-modules with respect to the
T(K, 9)-adic topology. Since T(K, 0} is a noetherian ring, there is a morphism
•s /"s
of the form r(A!, <9)« -»• r(X, 0)' such that the sequence
(*) mC®)9 -► r(£"V -► r(jC<s)
A A A O
is exact. This morphism induces an <9-morphism &q -> 0" on K. We are going to
AAA
prove that the sequence 0" -> 0" -* 0 is exact and so the proof will be over. It is
o
sufficient to show for any Stein semianalytic compact K' <= K, the exactness of
the sequence
T{K'^0)"^ T(K'?0y^ TiK^ô).
This can be obtained by tensoring ® /s, T(K', 0) the sequence (*) and by means
of lemmas 1.1 and 1.3.
Corollary 2.6. If 8 e Coh(^), then ê e Coh(f).
Proof. The problem is local, so we may assume the existence of an exact
sequence of the form 6" -> 0" -* & -> 0. One then applies 2.3 and 2.5.
Proposition 2.7. For any 8 e Coh(T), the kernel of the canonical map
T(X, 9) - T(X, SF)
'■ s formed by all the sections which vanish on a neighbourhood of X'.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
203
Proof. By the very definition of & the image of such a section is null. We
A A
now prove the converse. Let s e T(X, §•) be such that its image in T{X, §•) =
r= lim T(X, WI3k+1&) is null. For every point x e X', sx e PI 3*+1§V Since 3*+ 1 cm*+\
k
one derives by Krull's separation theorem that sx = 0, etc.
A
Corollary 2.8. For any S e Coh(Af), Supp S = (Supp &) n A", In parti-
A
cw/ar, £F /5 «w// //" a«d o/ï/y // //zere *5 a neighbourhood U of X' such that $ \ U = 0.
Corollary 2.9. Le/ w ; £F ->■ 6j! fo a morphism of coherent 6-modules. The
A A A
morphism u : & ^> § is null if and only if u is null in a neighbourhood of Xr,
A
Proof. By 2.3, (Im w)A =; Im (w) and apply the previous corollary.
Corollary 2.10. Let u : & -* § be a morphism in Coh(A'). The morphism
A A A
u : of -> 6j! /j a monomorphism {epimorphism, respectively) if and only if u is a
monomorphism {epimorphism, respectively) in a neighbourhood of X'.
Proof. Let SU = Ker u and 2 = Coker u. By 2.3 it will follow that êl = Ker(w)
A A
and 2 = Coker(w). One then applies 2.8.
Let / : X -* Y be a morphism of complex spaces and let X' and 7' be two
analytic subsets of X and Y such that f{X') c 7'. If "J c c\ is a coherent sheaf of
ideals such that Supp {&yIT) = Y', and 3 is an ideal-sheaf of 6X which is maximal
with respect to the property Supp (<9j-/9) = A", then/*('J)£'A- c 3. For any integer £,
one gets an inclusion f*Qk+1)®x c Sk+1 and so, morphisms of ringed spaces
{X', 6xj3k+1) -»• (7', <SV/f+1). By passing to the projective limit these yield a
morphism
A A A
f-X^Y.
One can check easily the functorial dependence on this with respect to / and the
commutativity of the diagram
A
A f A
X—> Y
X—> Y
Proposition 2.11. Let f \ X -* 7 be a morphism of complex spaces and let
X' and Y' be closed analytic subsets of X and Y respectively, such that f{X') c Y'.
For any S;6Coh(7) there exists an isomorphism {functorial in §•) of G ^-modules
f*{®y ^f*(9).
Proof. If/*(§0A is identified with i$(j*(&)) and W with i\\W) (by 2.3), then
A
the proposition follows from the equality iYf=fix-

204 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proposition 2.12. Suppose X and Y are two complex spaces over a complex
space S and let X' and Y' be two closed analytic subsets of X and Y, respectively
and f, g two S-morphisms from X to Y such that f(X') <= Y' and g(X') <= Y'.
A A
Then f = g if and only if f and g coincide in a neighbourhood of X'.
Proof. The direct implication follows straightforwardly from definitions.
A A
We now prove the converse. Suppose/ = g. Let us fix a point x' of X'. We should
prove that / and g coincide in a neighbourhood of x'. Obviously, f(x') — g(x').
The problem is local, hence we may assume moreover that 7 is a Stein space.
We may also assume that there exists a finite number of sections s1}. . .,sre
e T(Y, èy) so that the set of polynomials in these sections is dense in r(Y, 6Y).
Apply theorem 1.4.11: the morphisms / and g correspond to two morphisms <p
and <jj of topological C-algebras from T(Y, 6Y) to T(X, 6X).
The morphisms T(f), r(g) : T(7, <9a) -► T(X, 0*) induced by/and g coincide.
A
Then it results that the images of <p(^-) and <\>(s;), 1 < / < r, in F(X, 0^) coincide.
By 2.7, there exists a neighbourhood U of X' such that <p(st)\U — <\)(s;)\U for any /.
t
The composite maps T(Y, 0Y) ^ r(X ®x) ~* F(U, &x) coincide on the elemens sh
hence they are equal. By applying again I. 4.11, one derives that <p = <\> on U.
Proposition 2.13. Let X and Y be two complex spaces over a complex
s It
space S such that the structural morphisms X -> S and Y -> S are proper. Consider
a closed analytic subset S' of S, X' = g'\S') and Y' = h'\S') the inverse images
A A A
of S', X, Y and S the corresponding formal completions, f : X -> Y an S-morphism
A A A
and let finally f : X -*■ Y be the morphism induced by f.
A
In order to assure that f is an isomorphism (a closed immersion, respectively)
it is necessary and sufficient to have a neighbourhood U of S' such that the morphism
g""1(C/) -> hr\U), induced by f, is an isomorphism {closed immersion, respectively).
Proof. The assertion concerning "the sufficient" follows immediately. For
the converse assertion, it is enough to prove that every point y e Y' has a
neighbourhood Vy such that the restriction/-^^) -> Vy is an isomorphism (closed immersion,
respectively). By hypothesis the fiber/-1(j?) is reduced to a point. Since / is proper,
then there is a neighbourhood V of y so that the restriction f'\V) -* V of / is
a finite morphism.
Thus we confine the question to proving the inverse assertion of the proposition
A
under the supplementary hypothesis that / is finite. It results easily that /*(£>*) can
be identified with /*(<9a:)a(/* is exact and commutes with the projective limits;
much more, if 3 is a coherent ideal-sheaf defining S', then g*(S)&x and h*(3)6Y
A A
define X' and Y', respectively) and the morphism (9a -►/^((S'a) given by / becomes,
after this identification, equal to the completion of the morphism 6r -^/^C^) given
by/. The conclusion required follows then from 2.10, by the fact that any
neighbourhood of Y' contains a neighbourhood of the form h~HU) for some
neighbourhood U of S'.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
205
§ 3. A finiteness theorem
Let (X, &) be a complex space and let N > 0 be an integer. Consider a system
J = (Tj,..., r„) of indeterminates and denote by <2[7] the associated sheaf of
polynomials: if Q c X is a compact, then
HQ, £[TJ) = r(g, <S)[J] - 6(<2)[7'].
<2[T] has a natural structure of sheaf of graded rings. An ©[rj-module UK, is called
graded if it is isomorphic (over 0) to a direct sum © $11, such that <9[7']„-Jïïc <=
<= STip+q,p and 9 integers > 0 (<9[r]p is the component of degree^ in «SIT]).
Now we consider a morphism X -* Y of complex spaces. We canonically get
a morphism of ringed spaces (X, Ox[T]) -»• (7,<9 y[T]). tf ©11 is an a^[T]-module,
then the sheaves R9fx(al\i) display a structure of <9y[T']-modules.
Theorem 3.1 . Let f : X -* Y be a proper morphism of complex spaces,
T = (7\,..., TN) a system of indeterminates and otl a graded coherent 6x[T\module.
Then the sheaves Rqf'^(SHV) are coherent QY\T\modules for any q > 0.
For the proof we need some preparations.
I. Use again the notation from the beginning of the paragraph.
Lemma 3.2. 6[T] is a coherent sheaf of rings.
Proof. Let 0[T]P -* 6[T] be a morphism on an open set U and let x be a
point of U. Consider a Stein semianalytic compact Q cr £/, neighbourhood of x.
We have an exact sequence
0 - T(Q, Ker q>) - 3(0[7T ^6 (Q)[T].
Since <9(0[7j is a noetherian ring, there exists a surjection 0(Q)[T]g ->T(0 Ker <p)->0.
We get an exact sequence
0(Q)[T]" - 3(0[7T - O(0[T]
and accordingly, a sequence of (S^TJ-modules on Q
G[T]* -»• a[Tp -»• <S[71.
We will show that this is exact and the lemma will follow. Let Q' cr Q be a Stein
compact. The required conclusion then results from 1.3 by the isomorphisms
<9(0[7T(g)a(e) (9(0) ~ (9(2')[7T, r > 0 being an arbitrary integer.
Lemma 3.3. Let S\l be a coherent &[T]-module, For any point x e X there
exists a neighbourhood U enjoying the following property: for any Stein compact
Q <= U, H"{Q, £11) = 0 for q > 1.
Proof. If Q c X is a Stein compact, then H"{Q, 6[T]) = 0 for q > 1.
Indeed, <S[T] is isomorphic to a direct sum of sheaves, each of them being isomorphic
to 6, and apply theorem B to Stein compacts.

206
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES '
Let x 6 X. In virtue of the coherence we get a resolution
(*) S,d -> ... -► SP -> §>}l -► 0,
on a neighbourhood U of x, where S,1 (0 < / < J) is a free (SITJ-module of finite
rank and d ^ 2dim £/. For every Stein compact Q <= U, Hq{Q, #11) = 0 for q >1.
Remark. The lemma holds for any Stein compact Q on which S\l admits
a resolution of the type (*); we shall say that such a compact is sufficiently small
with respect to SU.
Corollary 3.4. Let #11 Z><? a coherent 6[T]-module, Q <= X a compact and
{Q;}; a finite covering of Q by Stein compacts sufficiently small with respect to SYl.
Then the canonical map H'({Q^h #11) -> H'(Q, §>]l) is an isomorphism (the first term
V
represents the Cech cohomology).
Proof. This follows from the lemma by using Leray's theorem (a finite
intersection of Stein compacts is a Stein compact too; the condition of being sufficiently
small with respect to #11 is obviously fulfilled).
Lemma 3.5. Let X' —> X be a closed immersion of complex spaces and #11
an 6x.[T]-module.
(Ï) #11 is a coherent 6x,[T\module if and only if i^SW) is a coherent ^[jH]-
module.
(ii) If #li admits a resolution on X'
fd _> £<*-! _> . , , _> £0 _> g,Ti _> 0)
where d = 2dim X' + 2 and if £'"(0 < / ^d) are free 6x.[T]-modules of finite
rank, then:
(a) there exists an epimorphism 31° -*■ i*($ftl) -> 0 on any Stein semianalytic
compact of X, where 31° is a free 6x[T]-module of finite rank;
(b) for any Qx\T\morphism 31° -> /*(#ll), where 31° is a free 6x[T]-module of
finite rank, there exists an exact sequence of the form Si1 -* 31° -* i^(olli) on any
Stein semianalytic compact, Si1 being a free 0x[T]-module of finite rank.
Proof, (i) It is sufficient to show that i*(&x> [T]) is d^lTJ-coherent. Since
i*(6x. [T]) ^ ij&x> )[T], the conclusion results if we notice that &®ex®x[T] ls
<9^[r]-coherent for any W e Coh(^).
(ii) (a). Let Q <= X be a Stein semianalytic compact, Q = i~\Q) is a Stein
semianalytic compact. We get the exact sequence
(*) T(Q', £i) -► T(Q', Sf) -► T(Q', #11) -► 0
(we use here the hypothesis d = 2dim X' + 2). In particular, it follows that
r(g,z*(#Tl)) =r(Q',Sf\L) is of finite type over 0(Q')[T~\ hence of finite type over
&(Q)[T]. We thus obtain a surjection &(Q)[T]n<> -► f(Q, i*(SH)). Denote 31° = &X[T]">,
hence we get a morphism 31° -* i*(SÏÏl) on Q. We will show that this is an
epimorphism.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
207
Let gj c Q be a Stein compact. Obviously, TiQ^ $1°) ^ T{Q, 31°) ®e(Q) &{Q^).
There are also the isomorphisms T^, i„{S)\l))=T{Q'1, SU) ~ T(Q', SH.)®eiQ,)e(Q3 -
~ T(Q,i#($11)) (g)eiQ)®(Qù, Q'i—i(Qù (the former isomorphism derives from
the sequence (#) and from the similar sequence for Q[ ; for the latter we use the
equality Ha, 3) = T{Q, 3)-&(Qi), where 3 is the ideal of the immersion /). We deduce
that the map r(Qi, <&°) -»• T(QU i*(SHl)) is surjective, etc.
(//) (b). Let 31° -*■ i^SUl) be such a morphism and Q a Stein semianalytic
compact. T(Q, Ker <p) is a submodule of T(Q, Si0), hence it is of finite type over
0(Q)[T]. Then there exist a free 6x[T]-modu\c of finite rank SI1 and a morphism
SI1 -> 31° on Q such that the sequence
ne, si1) - ne, &°) - ne, asn»
is exact. One can prove the exactness of the sequence SI1 -> 31° -> ^($11), as in the
proof of (a).
II. Let Sit and ÔK be two graded <9[T]-modules. A morphism <p: SIR, -* SK. is
called homogeneous if <p(#Hp) <= SKp for any ^. The graded (9^ |T]-modules, together
with the homogeneous morphisms constitute an abelian category. Let d be an integer
(we will consider only integers d ^ 0). As usual 6[T] [d] stands for the graded
©[rj-module of components 6[T] [d]„ = &[T]n+i, A finite direct sum of modules
of this type is called free graded (9|T]-moduie of finite rank. In lemma 3.5, if SM
is moreover graded, then the sheaves Si0 and SI1 may be supposed free graded of
finite rank and the morphisms which appear may be supposed homogeneous.
If X -*■ Y is a morphism of analytic spaces, SIR, is a graded (9x[T]-module
and SK. a graded <9y|T]-module, then an (SVITJ-morphism SK ->f*(a)K.) is called
homogeneous if for any integer n one has the factorization
SK.n^U(SKn)
n n
III. Let r > 0 be a real number and D(r) <= (£'" the open polydisc of radius r;
denote by £>(< r) the associated closed polydisc. Let Y be an open subset of some
numerical space and Q <= Y a compact. Consider on T(D( < r)xQ, <9<E™xr) the
topology of the uniform convergence of germs: one obtains a DFS structure (I, § 1, c).
Denote by t1}..., tm a system of coordinates in <£"'• Every f 6 T(D(^r) x Q, ôç,™xr)
can be expanded in series
/ = Yia^ /v e ne, 0Y), v = (Vl,..., vj, f = #... fa
(the expansion holds in a neighbourhood). For r' ^ r and Q <= Q compact, denote
ll/llr'.o'= S Klle-''^

208 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
where ||aJQ. = sup {|avO)|, y e Q'} and |vl = vx + .. . + v,„. One thus obtains
a continuous seminorm on r(Z>(< r) x Q, ^œ^xf).
Lemma 3.6. Let 0 < r' < r" < r. The family (t/r"y, v 6 Nm, enjoys the
following properties on any compact Q:
(i) every f e F(D( < r) X Q, i?i»xi') can be uniquely written {on
a'neighbourhood of D(^r) X Q) f = I av(f/r")v, such that ||avllG' < ||/||r" & for any compact
Q' ^Q;
(«) £ 110/01,'. q < co
The proof is immediate.
Let us now consider a finite number of polydiscs Dk(r) <= (Em<*>. Denote
'_#« r, Q) = U r(Dt(< r) x g, <9Œ...<Wxr) (r > 0, g c 7 compact).
Endowed with the product topology, 'K(^ r, Q) is a DFS space. This topology
coincides in fact with the topology inductive limit of Frechet spaces 'K(p, U)
(p > r, U => Q open)
'K(P, U): = H IWp) X U, <SW>xr).
K
For / = (/,) 6 'K«r,0 we set
11/11,.. c = max ||/J,.. ^(r' < r, g' e g compact).
k
From the previous lemma we derive
Lemma 3.7. Let 0 < r' < r" < r and Q <= Y be a compact. Then there exists
a countable family (e^iei of elements of 'K(^ r, Q) such that:
(i) for every Q <= Q, any element f e'K(^r,Q') can be uniquely written
/ = 5j afii (the convergence is in the above defined topology), where a{ e T{Q', QY)
i
and ||a,-||L ^ 11/11 r",Lfor any compact L <= Q' ;
(il) Y> \\ei\\r'.a< ».
Remark. Suppose p chosen such that r" < p < r. From the construction of
the family (e;), it follows that its image under the restriction map 'K(^ r, Q) ->
-> 'K(^ p, Q) also satisfies the properties (/) and (ii).
It is useful to consider spaces of the following form:
*« r,Q) = H T(Dk(^ r) x Q, 0^WxY[T] [~-dk]),
k

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
209
where T = (7^,.. ., TN) is a system of indeterminates and dk > 0 integers. These
are graded <3(g)|T]-modules (positive). By grouping conveniently the terms with
respect to the integers dk, it follows that there exists an integer q > 1, integers 0 <
^ dx < d2 < . .. < dit and <9(0-submodules 'K^ r,Q),..., 'KH( < r, Q) of
K(^ r, Q) of the above considered type, 'A"t(< r, Q) submodule of the component
of degree dk ('Ki( < r, Q) coincides with this component) such that : any element
fe A'(< r, Q) can be written / = £ Tafa (finite sum), fa e %(< r, Q) © . . . ©
a
©'À",(< r, Q). If / is homogeneous of degree d, then every /„ has the component
in 'Kk (< r, Q) null whenever d — ja| # dk; in particular, fa = 0 for |a| > d.
For r' < r and g' c g compact, we set
||/lk 0, = max ||/J|r,. 0,.
a
For every a one has \\Taf\\r',Q> = \\f\\r>q'. The homogeneous components of
K(^ r, Q) are spaces of the type 'A"(< r, g) (hence they have a topological DFS
structure) and the seminorms || ||r/? q> induced from È (< r, Q) are the seminorms
defined previously. From lemma 3.7 one derives:
Lemma 3.8. Let 0 < r' < r" < r and Q <= Y be a compact. Under the
above conditions, there exist an integer q, integers 0 < dy < d-z < ... < dq, and
for every k,l < k < q, there is a countable family (e£),-6/ of homogeneous elements
of degree dkfrom À"(< r, Q) which enjoy the following properties:
(i) for every compact Q' c Q, any element f e À"(< r, Q') can be uniquely
written
f= £ T«ataelataeG(Q')
a. i. 7c
(the sum with respect to a is finite) such that \\af-a\\L < \\f\\r",Lfor any compact
L <= Q'. Morevoer, if f is homogeneous of degree d, then a£a = 0 for (i, a, k)
verifying |a| + dk # d;
(«) Elk?llr'.0<00.
t.k
Remark. If p satisfies r" < p < r, then the image of the family (ef)iit in
À"(< p, Q) also verifies properties (i) and (ii).
The proof of the theorem, (a) By means of lemma 3.5 the problem is confined
to the case when Y is an open subset of some numerical space. We may assume
dim X < oo.
Let Uk, V*, k*, r^Jq. be the entities from the proof of Grauert's theorem of
finiteness (III, § 2). F* can be supposed to be a polydisc. We may assume in addition
that §\1 admits a resolution of the type
14-c. 2398

210 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
on any open subset Uk, where d = 2dim X + 2 and £'(0 < i < d) are free sheaves
of finite rank (over &X[T])-
For a compact Q <= V* and for a number r, r.% < r < 1, denote £/t( < r, 0 =
= Jk1(Df.(^ r)xQ)), 1 < k < A:*. If g is Stein, then the compacts C/fc(< r, 0are
also Stein. We have
\jUk(^r,Q)=X(Q)=f-\Q)
k-l
and denote
1I(< r, 0 is a finite covering of/_1(0 by Stein compacts which is sufficiently small
with respect to $11. By 3.4,
H-(C\n< r,Q), Sll)) ~ H\X(Q),$\l).
Let A„, A, Ua(r, V),Da(rJ,j„ 7cap be as in (III, §2). Denote C/„« r, 0 =
= H C/*v(< >", 0 where a = (fc0,. . ., kn).
v=0
We shall use link systems of sheaves SK. = (ôta, <pap) of a particular type,
namely graded systems:
(i) the sheaves Sfla are graded 6Da{nxV[T]-mod\i\es;
(ii) the morphisms <pap are homogeneous.
One can easily see that j*(aftl) is such a system. If every <Ka is a free graded
®Da(n x K[^]-module of finite rank, then SK. is called a /ree graded system of
finite rank. By a morphism between two graded system we mean a family of
homogeneous morphisms verifying the corresponding conditions of compatibility.
One thus obtains an abelian category.
For a compact Q <= V% and for a real number r, r.A, < r < 1, define
C"U r, 0 0t) = II r(Ac« r) x 0 #U.
a g A„
C"(< r, Ô; OK) is a graded (9(0[r]-module and the differential
S : C"« r, 0 £1) -► C"+1(< r, 0 £1)
is homogeneous. Similarly, for an open subset V <= V*, one can define
C'« r, F; 0t) = n r(Ac« r) X V, Sfta)
ccêA„

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
211
(sections in the sense of the sheaf theory). There exist isomorphisms
C'Clf« r, Q), 8X1) * C-« r, Q;j*(SH))-
(b) Lemma 3.9. For any Stein open set V' <= <= V* and for every r,rH <
< r' < 1, there exists a resolution
...->• 3Lk+1 -»• Sik -»• ... -»• Jl1 -»• <ft° -»- 7*(®lt) -»• 0
on (DJr') X V, izap), w/jere (<&*)& are free graded systems of finite rank and the maps
of the resolution are homogeneous.
Proof. We construct such a resolution for any Stein semianalytic compact Q,
V c Q c V*: proceed by induction on k, as in (III, §2). For k = 0 and k = 1,
apply lemma 3.5 and for k > 2 use a reasoning as in the proof of 3.5 (ii).
So we may assume that there exists a polydisc, denoted also by V%, a number
r**,r* < r** < 1 and a resolution like in the lemma on (DJ/m) X V*,Tzap). For
a Stein compact gcf, and r, r^ ^ r < r^^., consider the double complex (C'( < r,
Ô; °kk))i,k- All the components are graded 6(Q)[T]-modu\cs and the differentials
are homogeneous. Denote by
C"«r,Q)= n CX<r,Q;SLk)
l~k = n
the associated simple complex. Its components are also graded (9(0[T]-modules
and the differentials
C"« r, 0 - C+1« r, 0
are homogeneous. For every open V <= V^ one can define
C"(^r,V)= U C\^r, V;Sik).
l-k=n
The correspondence V i-»- C"(< r, F) gives rise to a sheaf on V^, which is denoted by
C"(< r). One can easily check the equality
1X6, C"« r)) = C"(< r, Q).
We have thus obtained a complex C"(< /•) whose components are graded @y,[T]-
modules and the differentials are homogeneous.
Lemma 3.10. Let r, r^ < r < r**, and Q <= V* be a Stein compact. The
canonical morphism
C'(^r,Q)^C\^r,Q;M®l))
is a quasi-isomorphism.

212 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. The lemma follows by the properties of the spectral sequences, by
lemma 3.3 and the remark made after this lemma.
Corollary 3.11. For every r,r% < r < r^^, and for any Stein compact
Q c v*,
H-(C\^r,Q))^H\X(Q),3]L).
Corollary 3.12. For every pair (r, /■'), r* < r' < r < r^, and for any Stein
compact Q c V#, the restriction
C'(^r,Q)^C-(^r',Q)
is a quasi-isomorphism.
(c) Consider a compact Q% in V%.
Lemma A(ri). There exist a Stein open set V„, Q% <= V„ <= V%, a real number
rn, r* < rn < r*^, a complex £" on V„ of the form
... _► o -»• £" X £"+1 -»• .. . A £*♦ -»• 0
o/ /re<? graded &Y[T]-tnodules of finite rank and a homogeneous QY\T\-morphism
of complexes
£• A C*« r„)
5wc/z that for any Stein compact Q of Vn the morphism £"(6) —>£"(< /■„, g) is
an n-quasi-isomorphism. In particular, a is an n-quasi-isomorphism.
a
By corollary 3.12, for any r, r* < r < r„, the composite map £" -> C"(< /"„) -»•
-> C"(< r), which will be denoted by a too, satisfies also the assertion from
lemma A(n).
We use the hypothesis of lemma A(n). Denote by K'(^ r) the cone of the
composite map £" -> C"(< r) (r^^ r < r„)andfor a compact g of V„, K'(^r, Q) =
= Ho, A:'(</•))■ K'(<r, Q) is a complex of graded <S(0[r]-modules. Its
components Km(^ r, Q) are of the same type as the modules studied in section III,
hence their homogeneous components have a DFS structure. All morphisms
defined till now are continuous with respect to these topologies.
Lemma B(n — 1). For every Stein compact Q' <= Vn and for any pair (r, /■'),
r% < r' < r < rn, there exists a homogeneous and continuous morphism on Q'
t : Kn-\^ r) -> Z^iK'i^ r'))
such that the diagram
K"~\^ r) ^ Z'-W« r'))
\ / restr.
Z"-\K-(^r))
is commutative.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
213
We also consider under the hypothesiss of lemma A(n + 1) the following
Lemma B*(ri). For any Stein compact Q cr V„+1 and for every pair (r, r'),
r_ < /•' < r < >'„-n, there is a homogeneous and continuous morphism on Q'
t: K"(^ r) -► Z"(K\^ r'))
such that the diagram
K-ter)2* Z"(JC(^r'))
Zn(K\^r))
is commutative.
Moreover, there exist an integer q > 0, integers 0 < d1 < ... < dcl, countable
families (e*)(- 6/, 1 < / < q, of homogeneous elements from K"( < r, Q') of degree
dk and a number r, r' < r < r, such that:
(i) for every compact Q" cr Q', any element f e K"(^ r, Q") can be uniquely
written
/= £ T*alÂ, 4,ae0(Q")
a, i. k
such that ||a£jL ^ WfWr. l for any compact L cr Q" (the sum with respect to a
is finite). In addition, if f is homogeneous of degree d, then dfM — 0 for systems
(i, a, k) such that | a j + dk # d;
(H) S l|-re?|ko.<oo.
i.k
Proposition 3.13. A(n + 1) and B(n) => B*(n).
Proof. Let Q cr V„+1 be a Stein compact and (r',r) such that r„, < r' <
< r < rn+1. Let g be a Stein compact in V„+1 such that Q' cr c\Q (that is
o
Q' c 0 and p', p, ? so that r' < p' < p < r < r. Apply lemma B(n) for Q and
(p', r') : let Ï : K"(^ p') -»• Zn(K'(^ r')) be the morphism so obtained on O. The
morphism
restr, t
K"(^ r) —>K"(^ p') -> Z"(K\^ r'))
has the required properties on Q'.
In order to prove this, apply lemma 3.8 for 0 < p < ? <r and K(^ r, Q).
Hence there exist q,dk(l < k < q), (ef)ieI such that the expansions from (/) take
place and S l|e/l|p,2 < °°- F°r every k, the map
T:K%<P',Q')dk^Z%^r',Q')dk
is continuous.

214
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
There exist M'k > 0 and a seminorm || || on K"(^ p', 0k such that
ll?(/)llr'.0, <M'k\\f\\, feK"{<9',Q\k.
There exists M'^ > 0 such that
|| restr. (g) \\ ^ M'k'-\\ g \\p>q
for any g e X"(< p, 0^ (restr. means the restriction, we will often omit to write
it). Therefore for some M > 0,
l|™,*||r,,0, <M-||e?||p>g
for all z and &:. One then obtains (//).
Remark. From the proof there results that under the assumptions of lemma
B*(n) one can assume that there exist r", r < r" < r and a morphism z: K"(^ r") ->
-> Z"(K' (< r')) such that the morphism X"(< r) -> Z"(K' (< r')) from fi*(«)
coincides with the composition of x with restriction X"(< r) -> /£"(< r").
The remark made on lemma 3.8 shows that for every /■"', r" < r'" < r, the
morphism
K"{^ r'") -»• #"(< r") -► Z"(/T « /")),
the integers 9, rf1} ...,dq, the number ? and the restriction of the family (ef)ik
to •£"(< r'", 0) also verify B*(n).
(d) Proposition 3.14. A(n) and B(n) => B(n — 1).
Proof. One may suppose that Vn <= cr FB+1 and that r„ < rn+1. Let 0 <= F„
be a Stein compact and (r, r') such that r* < r' < r < r„. Choose Q and g Stein
compacts such that g'ccgccgcF, and p', p real numbers, r^ < r' <
a' < p < r < r„. Apply the lemma fi*(«) for Q and for the pair (p, r); let
t, q, dk, ek;, r(p < r < r) bs the entities so obtained. We have ^ || xef || g < 00.
'.ft
By lemma A(n), the map
K'-'K p', 0-^ ZW< p', 0)
is surjective, hence for all k, 1 < k < q, the map
tf'-'K p', 0**-^ ZW< p',0k
is also surjective. Z"(K' (< p', 0k is the kernel of the map Kn(^ p', 0^ -»•
Kn+\^ p', 0dk, hence it is a closed subspace of K"( < p', 0^, therefore a
DFS space.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
215
The continuous and surjective map A["_1(< ?', Q)dk ~* Z"(K' (< <?', Q))ik is
open. Then there exists a seminorm || || in K(^p',Q)dk, a constant M'k > 0
and a family of elements (£*),, E* e X""1 (< p', 0^, such that SE? = xef(we omit
to write the restriction #"« p, 0 -► #"« p', 0) and || E.? ||p,,e < M£ ■ ||xef ||,
for all /. There exists also a constant M'k' > 0 so that || restr. (f)\\ < A/^' • ii/llp g
for any /ef« p, Q)ik. For some A/ > 0, we get || Ef ||p-,e < M • || ref II*'Q
for any iel and A:, 1 < k < 9. Consequently, J] || E,f ||p,e < 00. Define
h:K"{^r) -+Kn-\^ r')
by the formula
The morphism A is continuous and makes the folowing diagram commutative
£"« r) <-Z-CK'K r))
The morphism t: K1'1 (< /■) -»• Z"-1^'^ r')), t = restr. - AS, verifies fi (« — 1).
Proposition 3.15. A («) awrf £(«) => ,4 (« — 1).
Proof. Let 0 0" be Stein compacts such that 0 <= <= g cr <= 0" <= F„, and
r„_j such that r% < r„_i < r„. Consider p', p enjoying the property r„_! < p' <
<p<r„. Apply lemma fi*(« — l)for 0 and(p', p); define x, 9, <4,0?f);< k, p(p' < p<
< p) the entities so obtained. We have £ || x<?f ||p'e' < 00. Define /■', r„_i < /■' < p'.
i,k
The map C"(< p, 0 -> C"(< J"', 0 is a quasi-isomorphism, hence the sum of
the maps of the diagram
;
C-'Ur^-Z-Htf'Kr'.e))
is a surjective map. The same fact will result also for the diagram
J'

2!6 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(we have denoted again by t the composition of x with the restriction
Z"-\K'(^ p')) ^ Z^iK'(^ r'))). We now use the surjectivity for the
homogeneous components : as in the proof of 3.14, it results that there are elements
tf e K"-\^ p, 2k, 7)f e C"-2 « r', Q)dk
and a constant M > 0 such that
<lf) + 8(y*) = ™? and max(|| tf \\ftQ, || rj? ||r,,e) < M-1| «? ||p-,0.
for / e /, 1 < k < q. Then it will result that
(*) S II If lip, 0 < co, J] || tf || t.iQ =: Mj < co.
i,k i,k
Let J <= / be a finite part such that £ || £? || < , 7V0 being an integer
_ tei\J,k 2N0
greater than the number of multi-indices a such that | a \^dk. Consider the free
graded £(Q) [T]-module of finite rank, whose basis is indexed over the set
{(;', k)\iej, 1 < k < q}. Denote this sheaf by S""1 and by
a: S.""1 -► Z"-!(A:\'< rB_]))
the homogeneous morphism gf f-> xEf, where gf are the canonical generators. The
morphism w defines two morphisms on Q
a„-i. £n_i _, £Bj ffB-i. £«-i _> c"'-^^ r„_!).
We show that lemma A (n — 1) is fulfilled in the interior of Q. Just as in III, § 2,
it is sufficient to prove that for any Stein compact Q <= <= Q, the sum of the maps
T(Q, £-i)
C-1 « r„-i, 0 - Z""1 (**« r„-i, 0)
is surjective. By the remark made at the end of proposition 3.13, one may suppose
the existence of some p", p < p" < p, such that the morphism x: A!"_1(< p) ->
-> Z"~\K'(^ p')) is the composition of a morphism (denoted x too) X"_1(< p") -»•
-► Z"-1(^,(< P')) with the restriction X""1(< p) -»• K"-1 (^ p").
Just as in III, § 2, in order to complete the proof of the proposition, it
is enough to prove the following:

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
217
Lemma 3.16. For every f eK"~l (< p"', g) there exist geT(Q, £"-1) and
7] e C"-1 (< rn-!, g) such that
<f) = co(g) + 8(f/).
Proof. By using the expansions given by (ef)t k, we may suppose / to be a
homogeneous element of degree d, d < dq. Let r, p* be numbers such that r„_j <~r
< r', p" < p* < p and let g* be a Stein compact, g <= <= g* <= <= g, such that
/ e X"_1(< p*, Q*)d- We have the following expansion in convergent series
a, i, fc
where there is no term such that \&\> d. Define
gl= £ ^,i«fer(fi'.n
a, k, IeJ
and
ra= Y> T«aka>iy!;eC»-\^~r,Q*)d
a, k,iel
(the convergence is assured by (*)). Then it will result (in K"~\^ ~r, Q*)d) that
t(/) = cofei) + 8(y))i + t(/i). We get the evaluations
II/iIIp-.o'<-N0- max II £ ûS,«Çf IIp'.o«<
<* l,i£l\J
<7V0-max( S ||<,-||e-||Ei||p.,e.)<
cc k,i£l\J
<N0>( £ ll/llp.c-- ||Çf ||p.0)< -^II/IIp.q-
and similarly
llsille < 11/IIp. o«, Millie* < -JV^r11/ Up, e*-
In order to conclude, we make iterations; the above inequalities imply the
convergence in A»-i(< p", Q)d, T(g, £»-i)B. and C-^ r„_1; g).
Thereby lemmas ^ («) and 5 («) are completely proved (for n sufficiently
large they are obviously satisfied).
The proof of the finiteness theorem can be concluded exactly as in III, § 2.

218
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
§ 4. The comparison theorem
Let X -> Y be a morphism of complex spaces and let A", Y' be closed analytic
subsets of AT and Y, respectively, such that/(A") <= Y'. Choose 3 <= 6X, *} <= 6Y as
ideals of definition of A" and Y' such that/* Q) 6X c 3. Denote by X, Y the res-
A A A
pective formal completions and by/: Af-> 7 the extension of/. Recall that there
A A A
are natural morphisms ix: X -*■ X and iY: Y -*■ Y which agree with/ and/.
In the following formulae we will sometimes identify a sheaf defined on a
topological subspace with its trivial extension.
Fix an (9^-module §\ For any integer k > 0 the morphism gF -> SF<* >(§;(* > =
= gF/3*+1 §•) induces morphisms
jt/*(sf) _, *•/,(*<*>).
Since SFW is an (9x/3/t+1-module, R'ft($(k)) is an <9y/f ""-module. One thus obtains
the morphisms
tf'/*(^)®fi,(3r/?+1) - R'MSlk>)
and by passing to limit, the morphisms
y.R'MSy - lim *•/*(*<*>).
k
We now define morphisms
* : tf-/»(ff) - lim ^-/*(SFW).
There are isomorphisms 77'(A", SF) ~ #•(-*"> %) = H\X, lim âF<*>) and these yield
k
natural morphisms
H\X, ê) -► «'(A-, SF(*>), jfc > 0.
For every open set F of 7 we thus obtain morphisms
<\>v: H\X n /-1 (F), gF) -► lim W(/_1 (*0> S(ky) -> «m F (F, fl"/^*))).
The family (^k)k determines the required morphisms <p.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
219
Next we are going to construct a morphism
Let V be an open set of Y; by means of ix one gets morphisms H'(f~1 (V), SF) ->
-JÏ-O^MZ-W), '*(*)) = H'{f^{iyW)), i*(ff)). By composition with
H'(tKij\V)),i*{$))^T{iYXV), R'UiiK*))) we obtain morphisms H'(f-\V),&)-*
A
-> T (/f \V), R'f* (/*(oF))) and since V is arbitrary we obtain morphisms R'j-J^) ->
-KiVMWIOjIW)). fiy adjunction we derive morphisms if(R'M*)) ^R'MW*)).
A A A
The composition of these with the morphisms ^7*0"*(^)) ->R*f*(&) given by /*(^)->-
A
-> &■ determines p.
We finally denote by r: iY(.R'f^)) -> R'fJ^)A the natural morphism defined
in § 2 (here we have identified again the completion of an <3rmodule with respect
to Y' with its restriction to Y').
Thus we have obtained a diagram
**/*(8QA-^Um *«/*(*<*>)
k
for any integer 9^0. One can easily check the commutativity of this.
Consider the graded <9y-algebra
* = <2>y © Ï © ? © • • •
In virtue of lemma 4.3, $> is a coherent sheaf of rings. By I [J] we mean the
graded S-module of components S [d]n = $>n+d. For an integer q^O consider
the (9y-module
"% = © R%(3k^).
Let F be an open set of Y and let s be an element of F (V, }'"). Denote by s' the
image of s in T(T1 (P), /* Q'") <2>*) = r (J'1 (V), 3m)-
For any k the homotety given by j' defines an (9^-morphism on/-1 (F)
hence a morphism Rqj^{Sk^) -»• R'f^(3k+m^) on F. Thus on each « one obtains
a structure of graded S-module.
Suppose in the following that & e Coh (X).

220 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Theorem 4.1. Let n be an integer. If the ^-module q% is S-coherent for q =
= « — 1 and q = n, then :
(0 The morphisms r„-1 and (ph_1 are isomorphisms, p„_1 is a monomorphism
and fyn-i is an epimorphism.
{it) The morphisms p„ and <p„ are monomorphisms and r„ is an isomorphism.
(Hi) For any compact Q of Y, there exists an integer k0 such that the following
equalities hold on Q
Ker CR?/*(S9 -»• R"f^ (§^+r>)) = J Ker (R"f^) -»• R«f*(®<-k% whenever r ^ 0,
k^k0 and q = n - 1, n and Im CR«/„c(§r(ft+r>) -»• R«f*(&(r))) =Im(JR*/=!:(§;(*o+r>) -►
-»• RgU(.^(r))), whenever r ^ 0, k ^ k0 and q = n — I.
Proof. We first establish some facts required in this section.
(a) Suppose q = n — 1 or q = n. Locally, q% is generated as an S-module
by a finite number of sections, which can be considered homogeneous. Thus, locally
on Y, there are homogeneous S-epimorphisms of the form £° -> q% -> 0, where £°
is isomorphic to a direct sum of sheaves of type S [d]. The kernels of these
morphisms are ^-modules of finite type; thus we can use the same reasoning.
Consequently, locally on Y, there are exact sequences of the form S,1 -» £° -> q% -> 0,
where S1 is of the same type as £° and the morphisms are S-homogeneous.
Passing to homogeneous components one derives that the QY-modules
R"f^kW) are coherent for all k ^ 0. In particular, R^fJ^) and R"f*(&) are
coherent (9y-modules, hence rn-x and rn are isomorphisms (2.3). We deduce that the
sheaves R"-1/^^), k $s 0, are coherent too.
(b) Fix a Stein semianalytic compact Q of Y. Denote A =T (Q, 6r), J =
= T (Q, J), S = ® Jk and "H = ® T{Q, Rif^®)). A is a noetherian ring, S is
an ^-algebra of finite type hence also a noetherian ring and qH is a graded S-module.
The relation T (Q, »3f) = "H holds. The components of q<K are (9y-coherent
for q — n — 1 and q = n, hence by theorem ^ there exists an epimorphism £° ->
-> «3C -> 0 as above even on a neighbourhood of g. The kernel of this epimorphism
is a direct sum of coherent (9y-modules, therefore it is T (Q, *)-acyclic. The mor-
phism T (Q, £°) -> T (Q, qX) is thus surjective, hence "H is an S-module of finite
type for q = n — 1 and q = n.
There exists an integer kx = kx (Q) such that Rl+r = J'R^for k ^ kx, r ^ 0,
q = n — 1, q — n, where for an arbitrary integer k, one has denoted
Rl = Ker (T (Q, R"U(W)) - T (g, *«/,(*<*>))).
In order to prove this, consider the exact sequence
tf*/*(3*+lr> - R<fJ?) - R"U(ar^, k>0.
Since the first two (9y-modules are coherent (q = n — 1, n) one derives easily the
exact sequence
r (Q, r% (3*«ao) - r (q, R'uar)) - r (g, *«/„,(*<*>)).

Vl. THE FORMAL COMPLETION OF A COMPLEX SPACE
221
We can consider on the direct sum © R9- the structure of graded S-module, since
J'"Rqk cr Rk + m (this follows easily from some suitable commutative diagrams given
by the homoteties of the elements of Jm). By the above exact sequence, it follows that
thisS-module is isomorphic to a quotient of the S-submodule © Y(Q,Rqj'%(3k+1&))
of "H, hence it is of finite type since S is noetherian. The assertion follows easily.
(c) Denote M" = T(Q, RqU(®)), HI = T(Q, Rqf^k))). Consider the
completion of Mq with respect to the J-adic topology, hence (Mq)A = lim (Mq/JkMq~).
k
The canonical morphism
{MqY ^-> lim (Mq/R%)
k
is an isomorphism (q = n — 1, n).
This follows immediately in virute of (b).
The canonical morphism
lim (Mq/Rqk) -^-> lim Hqk
k k
is a monomorphism (q = n — 1, n), as follows easily from the left exactness of the
projective limit.
(d) Let A^ = Coker ÇT (Q, Rqf^)) -+ T (Q, *y*(ff<*>))); so we get a
projective system of exact sequences
(*) 0 -► Rqk -> Mq -> HI -> Nqk -> 0.
The homoteties given by the elements of Jm induce naturally commutative diagrams
of the type
T(Q, R%(8r)) - T{Q, *«/*(*<»))
1 I
T(Q, R%(SF)) -+ T{Q, RqU®^)),
hence morphisms N% -*■ Ng+m. One thus obtains a structure of graded S-module
on Nq = © N%. It easily follows that a(Jk+1)-Nl = 0 (k^ 0, q arbitrary), where
a". Jk+1 -* r (Q, 6y) <= S is the inclusion; the notation a points out that in this
formula the elements of Jq+1 are regarded as elements of zero degree.
Suppose now q = n — 1. The exact sequence of coherent <9y-modules
R'^Uar) - Rn-'f^k)) - R1^k+^)

222 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
yields the exact sequence
r (q, R"-1/^)) - r (Q, R'-1/^)) - r (q, jpy, (3*+iff».
Accordingly, A?"-1 is an S-submodule of "H, hence an S-module of finite type.
Then there exists an integer k2 = k2 (Q) such that N£+\ = Jh N^1 for k ^ k2 and
h^O. There results a(J*2+1) AT1 = 0 for k ^ k2. The integer k2(Q) depends
only on the degrees of a finite system of homogeneous generators of TV"-1. If Q' <= Q
is an arbitrary Stein semianalytic compact, then we can similarly construct integers
k-y (Q') and k2 (Q)- We notice that, by the construction of these integers, one can
assume that ky (Q') = ky (Q) and k<, (Q') = k2 (Q) (we make use of the relations
T(Q',R"U(3^^)) ~ T(Q,R«f^Mn®6iQP(Q') and AT Ho') =* ATKÔ) ® SioPiQ'),
which are consequences of lemma 1.3).
Let r ^ 0 and k ^ 2k2 + 1. Obviously, N?~l = J*2+1 Ngr1 where k' = r +
+ k — k2 — 1 ^ k2 + r. Then one can easily see that ?&e image of the morphism
N"rl -> N"'1 coincides with the image of a(Jkl+1) Ngr1 under the morphism A^r1 ->
-> N"_1, hence it /j ««//. From the exact sequences (*), one obtains the assertion:
for any r^O and k ^ 2k.z + 1, Im (T (g, R-J*(&«+"'>)) -»• T (g, JR"-1/*(Sr(')))) =
= im (r (g, 2i»-i/„(ff«fc+!+'>» - r (g, *"-y*(s^>))).
(e) By using again (*), we get the exact sequences
o -► M'-yRr1 -»• «r1 -» at1 -»• o,
hence the exact sequence
0 -► lim(Mn-1/Rnk-1) ^> lim Ht1 -»• lim AT1-
a ft , ft
Since lim Nl"1 = 0, it follows that the morphism tn_x is an isomorphism.
k
Let us prove now the assertions of the theorem.
(/) The morphism r„_j is an isomorphism as we have seen in (a). By (c) and (e),
the canonical morphism
r (Q, *"-y*wr - lim r (Q, tf"-y*(3^>)),
ft
which coincides with the composition ?„_! ^„_i, is an isomorphism. Since Q is an
arbitrary Stein semianalytic compact of Y, it results (inserting sufficiently small
open sets between such compacts and using suitable canonical commutative
diagrams) that <pll_1 is an isomorphism.
The assertions with respect to p„_: and $„-! follow from the equality
9»-i'"ii-i = <K-i P»-'-

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
223
(//) We have already seen that /•„ is an isomorphism. Since s„ and t„ are mo-
nomorphisms, the morphism cp„ is a monomorphism. Therefore p„ is also monomor-
phism.
(///) It is enough to consider Stein semianalytic compacts. Let Q be such a
compact and k0 = k0 (Q) an integer such that k0 ^ k^Q) + 2k2(Q) + 1. The
first assertion follows from (b) and the second one from (d).
Thereby the theorem is proved.
Complement to Theorem. Under the conditions of the theorem, suppose in
addition the sheaves Rf^Er^) coherent for i < n — 1 and k ^ 0. Then the morphism
(ji„ is an isomorphism.
Proof. We have to prove that the canonical morphism
R% (lim §^fc>) -► lim R%(®<-k))
k k
is an isomorphism. The problem is local on Y, so we may assume the existence of
an integer k0 such that
lm(K-*MSW) - tf-y„(S^>)) = ImiR-1/^*'™) - R^f^^))
for all r ^ 0 and k ^ k0. For each Stein open subset V of Y there are isomorphisms
H'V-1 (V), *<*>) ^T(v, *'/«(*<*>))
for all A: ^0 and i < n. Then we derive that
Im (H"-1 (/-1 (V), ff<*+'>) -► H"-1 {f-1 (V), S*'))) =
= ImC^"-1^-1 <T), gF(fc°+r)) -► H'-ifJ-1 (V), S*>))
for all A: ^ fc0, /■ ^ 0. In virtue of proposition 1.9, the canonical morphism
H*(j-i (p)s iim §;(«) _> i;m H'-if'1 (V), &<•»)
k k
is an isomorphism. Since
lim H'Xf-1 (V), ff<*>) ~ lim T (F, 2î"/»(3f(*))) = T (V, lim ^/^(ff'*))),
A k k
the conclusion follows.
Now, we consider two cases when the theorem applies.

224 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proposition 4.2. Suppuse "J is generated by a regular section t' of <3Y and
3 =/*CJ) 6X = t&x, where t is the image of t' in Gx. Suppose the following conditions
are fulfilled:
(/) t is S-regular;
(U) R"f!f (&) is coherent for q = n — 1 and q = n.
Then the hypothesis of theorem 4.1 is fulfilled.
Proof. There exist some isomorphisms 3kW ^ SF. One then deduces the
isomorphisms q% ~ R"fJ(&) ®0rOY [J], where T is an indeterminate. Since 8 ~ 6Y [J],
the conclusion follows from lemma 3.2.
If SF is /-flat, then condition (/) is satisfied. If/ is an open immersion
and Y \f (X) is an analytic set, then the finiteness theorem for local analytic coho-
mology from Chapter II gives conditions when the assertion (ii) holds.
The second case when the theorem can be applied is when/ is proper.
Let (X, 6) be a complex space, let 3 <= 6 be a coherent ideal and oF e Coh(A').
Suppose there are sections /J; ...,/Nof T(X, 6) which generate 3. These yield
naturally a structure of graded 6 [T]-module on © 3k&(T = (7\, ..., TN)). We also
remark that there exists a canonical epimorphism &[T]-> © 3k -* 0.
Lemma 4.3. © 3k& is a coherent 6 [T]-module; in particular, © 3k is a
coherent sheaf of rings.
Proof. Let Dll = © S*SF. Consider a Stein semianalytic compact Q of X.
k^>0
Obviously,
ne, su) = © ne, s*») = © ne, sy ne, *>.
T(e, STc) is a module of finite type over the noetherian ring 6(Q) [T], hence there
exists an exact sequence of the form
& (Q) [TY - e (Q) [T]< - r (e, cTl) - o.
We get a sequence of 6 [T]-modules on Q and if we prove that this is exact, then
the lemma will follow from 3.2. Consider a Stein compact Q' <= Q; by lemma 1.3
we get exact sequence
e (eo [ty - (9 (eo m» - r (e', ^ - o, etc.
Theorem 4.4. Letf: X -*■ Y be a proper morphism of complex spaces, "} <= &r
a coherent ideal, 3 =/*Q) 6X and SF a coherent sheaf on X. Under these assumptions,
© Rqfil.(3kS:) is a coherent © "}k -module for any integer q.
// e is a Stein compact of Y, then there exists an integer k0 = k0(Q) such that
H"(J-\Q\ £*+'») = T(e, 3)r • H"(J-\Q), 3kW)
for all integers q $s 0, r $s 0, k ^ k0.

VI. THE FORMAL COMPLETION OF A COMPLEX SPACE
225
Proof. The problem is local on Y, hence we may assume that there exist
sections flt... ,fN e T(Y, X> such that (/i, ; • • ,/*) <2>r = I Then (/1? ... ,/*) 6Y = 3
(we have denoted also by fu .. . ,/N the images under the morphism T(Y, 6Y) -*
- T(X, 6X)).
Let afti = © 3fc§\ By lemma 4.3, $11 has a natural structure of coherent <9^ [T]-
/c>0
module. By the finiteness theorem 3.1, the sheaves R9f* (Dll) are <9j- [T]-coherent.
The first assertion of the theorem follows from 4.3, since Rmf*(SH) ^ © R-f*(3k&).
Let g be a Stein compact of Y. There exist canonical isomorphisms
hv-\q), s**) « ne, *7*(sfcâF)), *; > o.
Then it easily follows that, in order to prove the second assertion of the theorem,
we may suppose Q sufficiently small. Then by the coherence of the &Y [T]-modules
R'fJ^i), the 6{Q) [T]-modules
ne, *v*w) =* © ne, *•/*(»)) ~ © h-(j-\q>, &*)
are of finite type and the required conclusion is obtained easily.
From theorem 4.1 and its complement we get the following
Corollary 4.5. Let f: X -*■ Y be a proper morphism of complex spaces, Y' a
closed analytic subset of Y, X' =f~\Y'), "J cr QY an ideal of definition of Y', 3 =
=f*(Dex and® e Coh(^). For an integer k ^ 0, denote §^fc> = Ïï/S^1®. Then:
(/) The morphisms from the diagram
k
are isomorphisms (q ^ 0).
(n) For each compact Q of Y there exists an integer k0=k0(Q) such that one
has on Q
Ker (*«/*(*) - R«f*(®ik+r))) = Y Ker (*«/*(*) - tf«/*(*(fc))),
Im(tf«/*(*(fc+,)) - R"j;(»('>)) = Im (tf«/*(*lfc" + ,)) -► tf«/*(*(,)))
for r ^ 0, 9^0 ow^ A: ^ A:0.
p.r-l A A
Remark. The isomorphism R'f^S)" >• R"f*{&) shows that the direct image
functors .R"/* commute with the completion; for these reasons theorem 4.1 is
called a comparison theorem (between the analytic theory and the formal theory).
In particular, from theorem 4.4 we get the following
15 - C.239S

226 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Corollary 4.6. Let f: X -* Y be a proper morphism of complex spaces and
let y be a point of Y and SF e Coh (X). Then there exists an integer k0 such that
H"{f-\y), m*+rSF) = m'yH«(f-\y), m*ff)
for all integers q > 0, r ^ 0 and k > k0.
From 4.5 we find again, in a little stronger form (the explanation of the
function F), theorem II 3.1.
Corollary 4.7. Let f: X -> Y be a proper morphism of complex spaces, y a
point of Y and S e Coh(T). Denote Xy =f~1(y) and ®m = §7m*+1SF. Then:
(f) The canonical morphism
R"f*(^)y -> lim H"(Xy, 8<k>)
k
is an isomorphism for any q ^ 0 (the completion is taken with respect to the my-adic
topology).
(if) There exists an integer k0 such that
Ker (H«(Xy, 9) -► H"(Xy, §^+r>)) = m^Ker (H"(Xy, 9) -► H"(Xy, »<*>)),
Im (H"(Xy, ®{k+r)) -► H"(Xy, S*'))) = Im (H"(Xy, §^o + <>) _► 77?^, S^))
/or all r ^ 0, q ^ 0 and k ^ fc0.
Proof. We have to check only the isomorphisms
(R«f*(®y)y =* (*v*TOA,
where the first completion is with respect to the ideal-sheaf defined by y and the
second one with respect to the m^-adic topology. This fact is achieved in following
way. if V is a Stein neighbourhood of y and m (y) is the maximal ideal of T(V, &r)
which corresponds to y, then
T(V, R'U*))lm <»*« T(V, R'U8)) « R'f*($)ylmky»R-f*(®)v,
the sheaves R'f*(oF) being coherent.
Bibliographical indications
The results from § 2 are taken from [37] and the comparison theorem 4.1 from ([36], SGA 2,
Exp. IX). The other results in this chapter are due to the first author [6].

Chapter VII
Duality on complex spaces
Introduction
Let A'be a Riemann compact surface, Q the sheaf of germs of holomorphic forms of
type (1, 0) and £ an invertible sheaf on X. A classical result (Roch's half of the
Riemann-Roch theorem) shows that the complex vectorial spaces H°(X, £) and
V
H^X, £ (g) Q) are in duality. The generalization of this result for arbitrary
dimension is the following duality theorem of Serre [75]:
"Let A' be a compact complex manifold of dimension n, let Q. be the sheaf of
germs of holomophic forms of maximal degree and 8 e Coh (X) locally free. Then
V
the complex (finite-dimensional) vectorial spaces H"(X, &) and H"~P(X, & (g) Q)
are in algebraic duality".
V
If & is an arbitrary coherent sheaf, then H"~P(X, & (g) Q) should be replaced
by Ext"-^; 8, Q). If the manifold X is not compact, then the invariants must be
topologized and the algebraic duality becomes a topological one (moreover, Hn~"
must be replaced by H"~").
The aim of this chapter is to extend the duality to the singular case. When
compared with the case of manifolds, the difficulties are considerable. Grothen-
dieck overcame them in algebraic geometry [39] and Ramis and Ruget for complex
spaces.
Another way of studying the nonsingular case can be found in Andreotti
and Kas' paper [4].
Let A' be a complex space. In § 2 we construct a complex Kx of (9x-modules
which is called the dualizing complex and which plays the part of the sheaf Q. The
Ext's, whose arguments are complexes of sheaves defined in § 1, B, allow us to
lend a meaning to the invariants Ext* (X; 8, K'x), 8 e Coh (X). In the case of complex
manifolds, the couple which achieves the duality can be obtained from the exterior
product of the forms and from the trace map H"(X, Q.) -*■ C, which is deduced from
the integration of the forms of maximal degree. In the singular case one defines
similarly a trace map H°(X, Kx) ~* <E- The last mentioned map and the Yoneda
bilinear maps define bilinear maps
W(X, 8) x Ext-'(A-; S, Kx) -► C,
Hpc{X, 8) x Exrp(X; ff, K'x) -»• C-

228 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
In § 3 one defines naturally some topologies on these invariants (generally
nonseparated) and proves that these pairs achieve topological dualities between
the associated separated spaces (theorems 3.7 and 3.10). The separation theorems
concerning the above invariants, which are proved in [63], stand for a complement
to these results and they are necessary for applications.
In § 4 we consider the case of manifolds. The proofs given in § 3 are resumed
in this case (now the simplifications are essential since the dualizing complex and
the formalism of the trace map are not in the least necessary). Moreover, we
provide a minute presentation of Serre and Malgrange's proof by means of theDolbeault
resolutions (which, however, makes use of the results from [54]).
The chapter is concluded by the presentation of the dualizing sheaves
introduced by Andreotti and Kas [4].
In § 1 we give the construction of the Cousin complex of a ring and the
definition of the derived functors of functors whose arguments are complexes of objects
and we also present some topological facts required in the study of duality.
§ 1. Preliminaries
(A) In this section we prove some facts concerning the local cohomology of the
rings and make the construction of the Cousin complex ([36] SGA 2, [39]).
(1) Fix a topological space X. Let Z <= X be an arbitrary subset. For any
$■ e Ab (X), denote
TZ(X, 9) = {s e T(X, 9) | Supp s c Z} and
rz($0 = the sheaf associated to the presheaf U h> Tunz(U, 9). Thus, we get two
functors and the associated derived functors are denoted by H'Z{X, 9) and °il'z(9}
respectively. Hz coincides with Tz and 3C^ with Hz- In Chapter II, § 1, we have
considered the particular case when Z is a locally closed subset (moreover, Tz
and Fz were defined in a little different way; if Z is closed set both definitions
coincide).
Let now Z' <= Z be two subsets of Z. For any sheaf of abelian groups gF
on X, denote
Tz,ziX, 9) = TZ(X, 9)[TZ.(X, 9).
A morphism 9 -* § induces naturally a homorphism of groups
TziziX, *) - TziziX, <?).
It is obvious that these correspondences yield a functor
Tz/z{X, *) : Ab(*) -► Ab

VII. DUALITY ON COMPLEX SPACES
229
and its (right) derived functors are denoted by HZlz'(X, *). Generally, the functor
Yziz' is not left exact and hence HZ/z' can be different from TZjzf-
Now, define invariants of local nature which are associated to the inclusion
Z' <= Z. If 8 e Ab(X), by T_ziz-(&) we mean the sheaf associated to the presheaf
U h> ^unziuPiZ'W, ^)- One obtains a functor
rz/z,: AbCY) - AbW
and its derived functors are denoted by 'Kz/z-- For any gF e Ab(A'), one get?
exact sequences
0 -► HZ,{X, 8) -► HZ{X, 8) -► tfj|/z-(A , S=) -► HZ.{X, 8)^ ...,
0 -► %z.(8) ->• ^(ff) -► ^/z-C^) ->• SK^) -»•...
Finally, we will define invariants of punctual nature, which will allow us to compute
the invariants %'Ziz-- Let x be a point of X. If 8 e Ab(Af), we may consider the
subgroup r^SF) of §• x formed by the element 0 and by those elements sx which admit
a representative s in some neighbourhood U of x such that Supp s = {x} fi C/.
The derived functors of the functor thus obtained are denoted by 3t^(oF). One can
easily see that
h;qs) * (x-z(S))x, ifz = {x}.
Recall some topological facts ([10], Ch. II, § 4 and [37], 07), which are
straightforward consequences of the definitions. In fact, we have already used some of them
(IV.3.1, V.1.13).
A topological space is called noetherian if any descending sequence of
closed subsets is stationary. Any open subset of such a space is also noetherian.
A topological space is called locally noetherian if any point owns a neighbourhood
which is a noetherian space.
The topological space X is called irreducible if it cannot be written as a union
of two of its proper closed subsets. A part of X is called irreducible if it is an
irreducible space with respect to the induced topology. A part of X is irreducible if
and only if its closure enjoies this property. If X is noetherian, then any closed
subset can be written as a finite union of irreducible closed subsets and whenever
this representation is reduced, it is unique (modulo order).
If Y <= X is a closed set, then a point y e Y is called generic if Y = {y}. A
topological space is called sober if any irreducible closed subset has a generic point and
only one. Any open set of such a space is also sober. The space X is said to be
locally sober if each of its points owns a neighbourhood which is sober with respect
to the induced topology.
A specialization of a point x is a point x' such that x' e {x}. We express this
symbolically by x -* x'. A subset Z <= X is called stable to specialization if, whenever
x e Z and x -> x', x' e Z.

230 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proposition 1.1. Let X be a locally noetherian topological space, which is
also locally sober. Consider two subsets Z' cr Z stable to specialization and so that
any point of Z\Z' is maximal in Z (xe Z\ Z', x -> x' => either x' = x or x' e Z').
Under these assumptions, there exist functorial isomorphisms
X'z/z-p) =* © /, (#*'(»)), 9 e Ab(A"),
xez\z'
where for a group G, ix(G) means the sheaf on X which is equal to G on {x}, and
is null on X\ {x}.
Proof. It will be enough to prove that for any SF e Ab(A") there exists a functorial
isomorphism
rz,z-(*)* © ar»(ff)).
xEZ\Z'
For any open set U which is sufficiently small (hence noetherian and sober), define
the map
rzni/tt'.*)- © r,(ff),
xe{Z\zr)nu
which associates to a section the system of its germs in the points of (Z\ Z') n £/•
The definition is meaningfull: indeed, if se Tzc\v(U> 9), then by the hipothesis
Supp s = {z-^} U ... U {z„} and one can easily see that no element z' e (Z\Z') fl U,
which is distinct of z1;..., z„, belongs to Supp s.
A section s has null image if and only if it belongs to Tz-ç]v{U, 9). Thus,
we obtain an injective map
rz(\u(u,swz.nu(.u,9)^ © r,(»).
xe[z\zF)C\u
One then derives a monomorphism of sheaves
Ez,z<*)- © f»(T,(ff)),
xez\z'
which is functorial in 8\ It remains only to prove that this is an epimorphism. It
is sufficient to check that any germ sx of T^SF), xeZ\ Z', has a preimage; but by
the definition of rx(SF) there exists a representative s on a neighbourhood U such
that Supp s = {x} fl U, etc.... The proposition is proved.
Consider a filtration
X = Z° => Z1 => Z2 => ...
of A" by arbitrary subsets. Associate to any & e Ab(A^) a complex of sheaves on X,
which is called the Cousin complex with respect to this filtration. Let 3' be an
injective resolution of 9. Deduce a filtration of complexes of sheaves
s- = r>(3') => rZi(s*) => !>(£•) ^ ...

Vil. DUALITY ON COMPLEX SPACES
231
For any p > 0 we have an exact sequence
0 -► Fzp+ •(£*) -► rZp(3-) -► rz,/z,»,(S') -► 0.
hence an exact sequence
... -► 3fZpW -► t'ZP/zp + i («0 -»• 3Czii W -»•••.
For the index p + 1 we obtain similarly the exact sequence
... -► %pztU(&) -► W'ziU/ZP+iiS) -> Wztl'iS) -»•...
From these two exact sequences one obtains by composition a morphism
3CZP/ZP+i(§0 -► 3CzJii/z, + ,(ff).
From the above exact sequences written for p =0, one obtains a morphism
» -► 9ez./zi(*).
One can easilly prove that these morphisms do not depend on the resolution o■• and
they have a functorial character in SF.
Proposition 1.2. Let X be a topological space, X = Z° => Z1 => Z2 => ...
a filtration and SF e Ab(A^). Under these assumptions, the above constructions yield a
complex of sheaves on X
0 -► a'zo/zi(§0 -► 3fzi/z,(ff) -► 3cz,/z.(ff) -►...,
together with an augmentation morphism SF -> 2CZo/Zi(S") ; moreover, these associations
are functorial in SF.
// ?/î<? filtration is finite and if 3CZp/zp+i(^) = 0 /or q ¥" p, then the above
complex is a resolution of SF.
.Proo/. Only the last assertion presents some difficulty. We first show that
3CZP(SF) = 0 for q < p. Proceed by descending induction on p. For p sufficiently
large, Z" = 0 and the assertion is obvious. The induction step follows in virtue of
the exact sequence
... -► 3CZp + i(») -► 3Cz,(ff) -► 3tz,/ZP + i(ff) -► ...
Now we prove that 2£zp(§0 = 0 for q > p. Proceed by ascending induction on p.
The case p = 0 is clear since Z° = X. The induction step will result from the exact
sequence
•••-•• ^-'/z^) -► 3CZp(») -► 3fZp-W -p • • ■

232 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Consequently, %qZp(&) = 0 for q ^ p and the proof of the proposition can be
completed by use of the definition merely of the morphisms SF -*■ 3Czo/zi(SF),
(2) Concerning the properties of the Zariski topology associated to a ring
one can consult ([10], Ch. II, § 4 or [37], Ch. I, §1.1). Recall the following facts.
Let A be a commutative and unitary ring. Spec A is a sober space. The irreducible
closed subsets are of the form V(p), p being a prime ideal of A. Here p stands
for the unique generic point of V(p). For an element / e A one may consider the
main open set D(J) = {pe Spec A\f $ p}, which is identified with Spec Af. The family
^(f)> / e A, constitutes a basis of open sets in Spec A. If / and g are elements of A,
then D(f) n D(g) = D(fg). Any open covering of Spec A can be refined to a finite
covering with main open sets. Moreover, if/1;...,/r are elements of A such that
r
Spec A = [_) D(ft), then the ideal generated by these elements equals A.
If A is noetherian, then Spec A is a noetherian space and consequently, any
open set of Spec A is a finite union of main open sets.
We denote for convenience X = Spec ,4 and Xf = D(f), f e A. Let M be an
^-module. For an element/ of A, Mf means the module of quotients with respect
to the multiplicative system of the powers/", n ^ 0. My is an ^-module. If D(f) <=
<= D(g), then a suitable power of / lies in the ideal generated by g. One then derives
a canonical morphism Mg -*■ Mf. In particular, if D(J) = D(g), then Me ~ Mf.
Let us denote by M the sheaf associated to the presheaf on X
Xf h> Mf, Xf c Xg h> Mg -► Mf.
If M=A one obtains a sheaf of rings A. M has a natural structure of ,4-module. One
can easily check that the stalk of M in the point which corresponds to the prime
ideal p is canonically isomorphic to the module of quotients Mv.
Theorem 1.3. ([37], Ch. I, 1.3.7). For any A-module M and for any f e 4,
the morphism
df-.Mf^ V(Xf, M)
is bijective. In particular, M can be identified with T(X, M) by 6l4
Proof. We first show that Qf is injective. Let s e Mf be so that Qf(s) = 0.
For each ideal p e Xf = D(J) the image of s under the morphism Mf -* Mv is null.
Consequently, there exists h <£ p such that hs = 0. Thus, the annihilator of s (consider
Mf as an ^-module) is not contained in any ideal of Speedy-; therefore, it equals Ar.
Then s = 0.
Now we prove the surjectivity. Notice the following: if g is an element of A
such that D(g) <= D(f), then Mg is canonically isomorphic to the module of quotients
Mfy, where g is the image of g in Af ([10], Ch. II, § 2, n° 3, prop. 7).

Vil. DUALITY ON COMPLEX SPACES
233
If we identify Xf by Spec Af, then by this remark there results that the sheaf
M\Xf can be canonically identified with (Mf)~. As a consequence, we reduce the
problem to proving the surjectivity of the map 6 = Q±. Let s e F(X, M). There exist
a finite covering (£)(/,-));e7 of X and elements st —t'i/f"' of Mfl such that 6y.(j;) =
= s \Xft. We may assume that all integers n\ are equal to an integer n'. The injectivity
of the maps Qf assures the existence of an integer mV3 such that (Jifjynu{fj't'i^fft'])=Çi
for each pair (i,j). Since / is finite, we can assume all mrj equal to an integer m.
Let n =n' + m and t: = fft\. Obviously, sx = t-Jfl, f"t; =fftj. We also have
£>(/") = £>(/), / e /. Since D(/;) cover X, the ideal generated by the elements ff
coincides with A. Therefore, there exist elements gte A so that £#,/? = 1. The
element t = 1.gitt of M has the property 6(0 = s (both sections coincide on
any Xf) and the theorem is proved.
(3) Let A be a ring and/ = (/;)i^;<r a system of r elements of A. We will
denote by K.(f) the Koszul complex associated to / ([37], Ch. Ill, § 1 and [79],
Ch. IV, A). Recall its definition. For every element /,-, consider the complex £.(/,)
of components
Koifd = Ki(fd = A, Kn(fd - 0 for n * 0,1,
whose differentials are defined by the multiplication by /;. Then K.(f) is the tensor
product of complexes K.ij^® K.(fù® ■ • • ® K.(fr)> and it is endowed with the
total degree ([26], 1.2.7).
Denote by e{ the unit element of A, regarded in A^(/,•).
Then Kp(f) is a free ^4-module of base en® ... ® e;p, ix < i2 < ... < ip;
in particular, it is isomorphic to the exterior product f\"{Ar). ^(f) is isomorphic
to Ar and by this isomorphism the elements ex,.. ., er are identified with the canonical
basis of Ar. Via this identification we shall write sometimes e-h A ... A e,-p instead
of e-h (x) ... (g) e;p. The differential d of K.{f) is determined by the formula
d{eh® ...® eip) = J] (- l)k+1fikeh® ... ® âfc® ... ® eip.
k
In fact, one could construct the complex KXf) taking into account the formula
Kp(J)= Ap(Ar) and the above differentials.
For any ,4-module M define the Koszul complex of chains
K,<J,M)=K.(j)®AM
and the Koszul complex of cochains
K'(f, M) = UomA(K.(f), M).

234 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The module Kp(f, M) is the direct sum of the modules eti® ... ® e!p® M, i\ <
< i2 < ... < ip and the differential dp : Kp(J, M) -> Kp-X(f, M) is given by the
formula
d{etl®...(g) eip®m) = £ (- lf+1e,k® ... (x> e-,k ® ... (x> elp <g? (fikm).
k
The elements of Kp(f, M) are identified with the alternating maps from [1, r]p to M
by the correspondence
g e K>(f, M) h* g{iu. .., ip) = g(eu A ... A etp).
By this identification, the differential d" : K"(f, M) -► Kp+1(f, M) becomes
(d"g) (A, i2, ..., ip+1) = "£ (- If-'f^gih,. ..,£,.-•, ip+1).
k = l
Define
//.(/, M) = //.(£.(/, A/)),
//•(/, M) = H%K-(f,M)).
There exist A -isomorphisms K,(f, M) ^ K'(f, M) which associate to any chain
z = S(e?1A... Aeip)® zfl...tp the cochain gz given by gz(j\,. ■ -,jr-p) = ezi,...,>
where (Jk)i<k^r-P is the strictly ascending sequence which is complementary to the
sequence (ik)ie:k^p in [1, r] and e is the signature of the permutation [1,/■]-»•
-> (z1;..., ip,jlt. ■ .,jr-p). One can verify the compatibility with the differentials,
and thus obtain isomorphisms
//'(/, M)^ Hr_p(f,M).
One can easily deduce the equalities
//«(/, M) = (0 : (A,... ,fr))M and //'(/, M) = M[(fu... ,/r)M.
We shall use the following result ([37], Ch. Ill, Prop. 1.1.4 or [79], Ch. IV, A,
Prop. 2): // (/1;.. .,/r) ij M-regular, then H"(f, M) = 0 for p <£ r.
Proposition 1.4. // (/1;.. . ,/r) generate the whole A, then the complex K.(J)
is homotopically trivial. Moreover, for any A-module M, the complexes K.(J, M)and
K'(J, M) are also homotopically trivial.
Proof. The assertions with respect to the module M follow from the first
assertion by additivity. Let gu.. .,gr be so that 1 =figi + ... + frgr. The morphisms
Kp(f) -»• Kp+1(f), given by the formulae
z = E(e(l A ... A e!p)zh ..,-p h> S(e,x A ... A e!p+1) th .Jp+l,

VII. DUALITY ON COMPLEX SPACES
235
p + i
where tllm..tp + l = J] (— l)*^ z. * ; t define a homotopy between the identity
k-l
map and the null map of K.(f).
For each integer n > 0 we denote by/" the system (/",...,/"). If m > « > 0,
then the "multiplication" by /"'"" defines morphisms of complexes
e„m : K.{f") - *.(/»).
If £.(/'") and £.(/") are identified with the tensor products K.(J'?)® ... (g) £.(/;")
and K.(f0® ■ ■ ■ ® K.(f") respectively, then 6„„, is the tensor product of the
morphisms %\,m : K.(ff) -> K.(fy) which are reduced to the identity in degree 0 and
to the multiplication by jf~" in degree 1. Asa matter of fact, if we denote by e'l
the unit element of K^fl) = A, then we may directly set
QUe'tl A ... A e%) = {el A ... A <)/r" •• •/,"-"■
The morphisms 6„m yield morphisms of complexes
6'"" : £•(/", A/) ->• £-(/m, A/).
o we obtain an inductive system and denote
C-((f), M) = lim K-(f, M).
n
//•((f), M) = H-(C-((fl M))« lim //'(/", M).
n
By the previous proposition, we deduce
Corollary 1.5. // (Jlt.. .,/r) generate the whole A, then //'(/, M) = 0 a«d
H'((f),M) = 0 for any A-module M.
Indeed, for any n > 0, the ideal (/f,...,/?) coincides with A.
(4) We will use the above results to prove the triviality of the cohomology of
the affine schémas. Use the previous notation. Let in addition Ut = /)(/,-), U =
r
= U Ui and "^ be the covering (C/,)i<,-^r of U. For any (/„,..., ip),
1 = 1
£>'..../„ = fi Ulk = Xrh . flp, hence T(Uio ..,„, M) = M/v../lp.
Fix a system (/„,..., ip). For any n > 0, denote A/,'"1..,^ = M. For m > « consider
the morphism

236 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
which is given by the multiplication by (/,-0 ... ftp)",~n. One thus obtains an inductive
system and it can be easily seen that Mr. r, is identified with lim M[n) . (one can
consult [37], 0ls 1.6).
If Cfn) (A/) means the set of the alternating maps from[l, r]p+1 to M, then one
can consider maps
<?'»" : Cf„(A/) - Cfm)(M), m>n>0,
as above. If C^i, M) stands for the group of the alternate />-cochains of IE with
respect to the sheaf M, then by the previous ones we get isomorphisms
C(nf, M) ^ lim C{n){M).
n
But each C{n){M) is identified with Kp+1(f", M) and by these identifications the maps
6'™ and <pm" correspond to each other.
One thus obtains for any p > 0, an isomorphism
cp(ne, m) ~ cp+1((f), M),
which is functorial in M. Moreover these isomorphisms are compatible with the
coboundary operators.
Proposition 1.6. By the above notations, there exist isomorphisms
functorial in M
H"{M, M) ~ H"+1((f), M), p>\
and an exact sequence, which is functorial in M,
0 -► H°((f), M)^> M-> H°(% M) -► //>((/), M) -» 0.
Proof. The isomorphisms from the statement are deduced from the isomorphisms
already established. On the other hand, C°(% M) ~ CJ((/), A/), and by this
isomorphism H°Ç\l, M) is identified with the subgroup of cocycles of dimension
1 of C"((/), M). Also M — C°((f), M) and the sequence from the proposition
follows by the definitions.
Theorem 1.7. ([37], Ch. Ill, 1.3.1). Let A be a commutative and unitary
ring, let M be an A-module and X = Spec ,4. Then H"{X, M) = 0 for any p > 0.
Proof. Let 'U be a finite covering of X by main open sets Xf, = D(ft), 1 <
</</■. The ideal generated by the elements/i,. ..,/r coincides with A. From 1.5
and 1.6 we then deduce that Hp(^\i, M) = 0 for p > 0. Since any open covering can
V
be refined to a covering as above, it follows that the Cech cohomology groups
H"(X,M) vanish for p > 1. In particular, H"(Xf n ... n *),.fc, A/)=0,for p>\
and for any (/„,..., ik). The conclusion follows from ([26], Ch. II, 4.9.2).

VII. DUALITY ON COMPLEX SPACES
237
Corollary 1.8. For any open subset U of X, the cohomology groups H'{U,M)
can be calculated by means of a covering with main open sets.
(5) We now connect the Koszul complexes to the local cohomology.
Theorem 1.9. Let A be a commutative unitary ring, f = (f1}.. .,fr) a finite
family of elements of A, X = Spec ,4, Y = the closed subset of X defined by f,
and let M be an A-module. Under these assumptions there are functorial isomorphisms
Hy(X, M) ~ //•((/), M) (* lim H'(f, M)).
Proof. By theorem 1.7 we get the exact sequence
0 -► H%(X, M) -»• H°(X, M) -> H°(X\ Y, M) -> H\(X, M) -> 0
and isomorphisms
HpY(X, M) ~ H"~1(X\ Y, M), p>2.
If 1.6 and 1.8 are applied to the open set U = X\ Y and to the covering
'•U = (Dififii^i^r of U we get the exact sequence
0 -► H°((f), M)^M^ H°(X\ Y, M) -» H\(j), M) ^ 0
and isomorphisms
H>-\X\ Y, M) ~ H>((J), M), p>2.
By making use of these facts (and of the isomorphism M ~ H°(X, M)), the proof
can be easily concluded.
Corollary 1.10. The invariants H'((f), M) depend only on the closed set
determined by the ideal generated by f1}.. .,fr.
Consider in what follows a special case. Suppose A is a local noetherian ring
and let m be its maximal ideal. We will denote Hm(M) = H(m}(SpecA, M).
Every Hm(M) has a natural structure of ,4-module.
Corollary 1.11. // A is a Cohen-Macauley local ring of dimension n and
m is its maximal ideal, then H^(A) = 0 for p # n.
Proof. Let/i,.. .,/„ e m be a regular sequence. The ideal generated by these
elements is an ideal of definition, hence the associated closed set F((/i,.. .,/„))
s reduced to m. In virtue of theorem 1.9, H&(A) ~ H"((j\A) ~ lim Hp(jk, A).
k
For an integer k > 0,/A=(/i,.. .,/*) is a regular A -sequence; therefore, H"(fk, A)=Q
whenever p # n. ~ —

238 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Let A be a commutative and unitary ring and M an ^-module. Recall that
an injective ^-module / together with a monomorphism 0 -> M -* J is called an
injective envelope of M ([12], [59]) if the following condition is fulfilled: for any
submodule P of /, P fl M = 0 => P = 0. One proves that any module admits
an injective envelope. A resolution 0 -> M -* I0 -* Ix -* I2 -* ... of M, where I0
is the injective envelope of M and any Jk + 1 is the injective envelope of Im(4->
-* Jk + i), k > 0, is called a minimal injective resolution. Its length coincides with
the injective dimension of M. The next theorem requires the following assertion:
"Let A be a noetherian local ring, M an ,4-module and / an element of
the maximal ideal of A, ^-regular and A/-regular. Then diA!fA(M/fM) < diA(M) — 1
(di = the injective dimension)".
The proof will proceed as follows ([60], vol. II, p. 233) : Let 0 -> M -*■ I0 -*■
-* /j -> I2 -* ■ ■ ■ be a minimal injective resolution of M. Denote by N the image
of the morphism I0 -> Ix. Consider the exact commutative diagram
0->M^I0->N->0
|" |B I"'
0->Af->-/0-►//->■ 0,
where the vertical arrows are just the multiplications by /. By hypothesis, the first
vertical arrow is injective. Since Ker v fl M = 0, it follows that Ker v = 0, hence
the map v is also injective. We show now its surjectivity. Let y e J0. Since fA ~ A,
there exists a morphism <p : xA -* I0 such that <p(x) = y. From the injectivity of
I0, <p extends to a morphism <\i : A -* I0. If y' = ^(1) then fy' = y.
Thereby w is also surjective. From the serpent lemma we deduce the
isomorphism Coker u ~ Ker w, hence M If M ~ HomA{AlfA, N). If / is an injective
^-module, then HomA(A/fA, I) is an injective A/fA-mod\i\e; for there exists a
canonical identification HomA/fA(P, HomA(A/fA, /)) =* Hom^^, /) for any A/fA-
module P.
Thereby the proof of the assertion will be completed as soon as we prove
the exactness of the sequence
0 -► HomA(A/fA, N) -► HomA(A/fA, h) -» HornA(A/fA, I2) -+ ...
f
From the exact sequence 0 -> A -> A -* A/fA -*■ 0, it follows that ExtkA(A/fA, •) = 0
for k > 2. By splitting the exact sequence 0 -»• M -> /0 -> JY -* I2 -> ... from left
to right into short exact sequences, the proof is concluded by applying the functor
UomA(A/fA, •), etc.
Theorem 1.12. // A is a regular local ring of dimension n and m is its
maximal ideal, then Hm{A) is an injective A-module.
Proof. Let / =(/1;.. .,/„) be a regular system of parameters. H^(A) ~
~ lim H"(fk, A) ~ lim A/(fk,... ,fk)A. The maps of the inductive system are
k k
A/(fk)A ->• A/(fk+1)A, the class of a h> the class of f-,.. . f„a, aeA.
Since(/1;.. .,/„) is a regular ^-sequence, it easily follows by a recursive reasoning

VII. DUALITY ON COMPLEX SPACES
239
that these maps are injective. Since A is regular, hence its injective dimension is n,
by applying the above assertion it follows that the rings A/(jk)A are injective (in fact
A/(Jk)A is a Gorenstein ring of dimension zero, hence it is injective !).
We now prove the assertion of the theorem. Let <\> : M -* N be a monomorphism
of ^-modules, supposed of finite type, and let <p : M -> lim A(Jk)A be an arbitrary
k
,4-morphism. Since M is of finite type, there exists an integer k such that <pfactorizes
by a morphism <px : M -* A/(Jk)A. From the Artin-Ress theorem there is an integer s
which can be supposed larger than k so that
<K(f )M) => (/) jv n <KM).
Hence the map ^ : M/(fk) M -> N/(JS) N + <\>((jk)M) induced by <\> is injective. The
map <p! factorizes by a map Mj(fk) M -*■ A/(fk) A, which, composed with the
map A/(fk)A -> A/(js)A, gives rise to a map 6 : M/(Jk)M -> A/(J") A. The morphisms
(J^ and 6 are ,4-linear and the corresponding modules are annihilated by (J")A-
Since A/(f)A is an injective A/(fs)A-modu\e, there exists an ^-linear map 6j : N/(fs)N+
+ <\>(fkM) -► A/if^A such that 0^ = 6. It will then result that the composed
morphism
N -► N/(f) N + <K/") M \ A/(f) A -► lim ^/(/') ^
t
extends <p.
(6) Let A be a commutative unitary ring and A' = Spec ,4. For an integer p
write
Z" = {a e Spec A\ht a = dim ^a > />}•
Obviously, X = Z°, Zp => Z'+1. The sets Zp are stable to specialization and any
point of Z"\ Zp+1 is maximal with respect to specialization.
By (1), there exists a complex of ^-modules which is called the Cousin complex
of A associated to the filtration given by codimension
0 -► %h,ztÀ) -► %z>,z*{A) -► ..-,
together with an augmentation map A -> %z°iziA~). By applying the functor
r(A^, •), one obtains a complex of ^-modules which is called the Cousin complex
associated to A and to the filtration given by codimension.
Lemma 1.13. // A is noetherian, then the sheaves of the Cousin complex
are flabby.
Proof. The conditions of 1.1 are fulfilled, hence for any q > 0 we have
isomorphisms
(*) X'zP/zp+tA) =* © UH&Â)).

240 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Each sheaf ia(H^(A)) is flabby, since it is constant on the irreducible set {a} and
null on its complementary. The conclusion follows from the fact that a direct sum
of flabby sheaves on a noetherian space is flabby.
We recall that a noetherian ring A (not necessarily local) is called Cohen-
Macauley (respectively regular) if the localizations Av are Cohen-Macauley (regular,
respectively) rings, p e Spec A.
Theorem 1.14. // A is a Cohen-Macauley ring, then the Cousin complex
is a resolution of A; moveover, if A is regular, then the stalks of sheaves of
this resolution are infective.
Proof. The filtration is finite. We will show that Tz'/zr + ^A) = 0 for any
q # p and the first assertion will result from proposition 1.2. In virtue of the
isomorphisms (*) and by the proof of the previous lemma, it is enough to show that
H&Â) = 0 for a e Z" \ Zp+1. It is easy to prove the isomorphisms H «(A) ^ H^Aa(Ax)
and the conclusion follows from 1.11.
The second assertion will follow from theorem 1.12, again by the
isomorphisms (*).
Corollary 1.15. Let A be a Cohen-Macauley ring. Then the Cousin complex
of A is a resolution of A. Moreover, if A is a regular ring, then the components
of this resolution are A-injective.
The proof follows from 1.7, 1.13 and 1.14 if we notice that
r(x,xZPIZP+>(A))~ © //*(i)~ © mAa(Aa).
a€Zp\2p + 1 <x£Zp\Zp + 1
Remark. Let A be a regular local ring of dimension 1.
Z° = Spec A = {0, m}, Z1 = {m}. We have TZVZi{A) zl © ia(H°(A)), hence
<x6.Z°\.Z'1
it is isomorphic to the constant sheaf given by the field M of the quotients of A.
The term °X.zi(A) is isomorphic to 'Xzvzi(A)/A, hence the Cousin complex is
in this case
0 ->• M -»• M/A -► 0,
which explains the terminology!
(B) In this section we will define derived functors of the functors whose
arguments are complexes of objects [39].
"(1) Let A be an abelian category. For convenience one may suppose that A
is the category of modules over a ring or the category of sheaves of modules over
a sheaf of rings, the only cases considered below.
Denote by C(A) the category of complexes over A, where the objects are
complexes of objects of A and the morphisms are classes, modulo the homotopy
relation, of morphisms of complexes (of zero degree). We have a canonical functor
A -> C(A), which assigns to any object X of A the complex X' whose components
are all null but X° = X.

VII. DUALITY ON COMPLEX SPACES
241
Denote by T : C(A) -> C(A) the functor (automorphism) which translates
the degrees one step to the left and changes the sign of the differentials. By T"
(« arbitrary integer) one denotes the corresponding iteration of T. Hence for an
object A" of C(A), T%X') is the complex of components T"(X')P = X" + p and
of differentials t/r,.,^., =(— \)"dx.. We sometimes write X'[n] instead of T"(X').
By H" we mean the functor C(A) -* A which associates the cohomology
object in the dimension n.
Recall some definitions (although repeating some facts stated in Ill.l.e!).
An object of C(A) is called acyclic if its objects of cohomology are null. Let
u : X' -* Y' be a morphism in C(A); u is said to be a quasi-isomorphism if
it induces isomorphisms between the cohomology objects; hence the morphisms
//"(«) : H"(X') -* H"(Y') are isomorphisms for each integer n. In this case we
may say that -> Y is a resolution of X'. Let X be an object of A. A complex
0 -> Y° -> Y1 -> Y2 -> ... of objects of A, together with an augmentation mor-
phism X -> Y°, is a resolution of X in the usual sense (i.e. the sequence 0 ->
-+X~-+ Y° -*■ 71-»... is exact) if and only if the canonical morphism X'-* Y' given
by s is a quasi-isomorphism (that is if and only if -> Y' is a resolution of X'
in the above sense). The notion of quasi-isomorphism can be reduced to the simpler
notion of acyclicity. Recall first a definition.
Let u : X' -» 7" be a morphism in C(A). The cone of u is the complex
C\u) of objects T(X') © Y' whose differential is given by the matrix
T(dx) 0 \
T(u) dY)
(one can easily see that a complex is so obtained).
There exists an exact sequence of complexes
0 -► y* -► C\u) -► T(X') -► 0.
From the exact cohomological sequence associated to this, one obtains (III. 1.8):
Lemma 1.16. A morphism u is a quasi-isomorphism if and only if its cone
C'(u) is acyclic.
We have assumed, as we will do in the following, that the differentials have
the degree + 1 ; similar considerations can be made in the case when the
differentials have the degree — 1.
(2) Suppose now A is an abelian category with enough injective objects.
Denote by C+(A) the full subcategory of C(A) whose objects are the complexes
bounded below. So a complex A" is in C+(A) if X" = 0 for any n sufficiently small.
A complex /' is called injective if all of its components /" are injective objects.
f f
A morphism X' -> /" is called an injective resolution of X' if -> /" is a resolution
(hence/ is a quasi-isomorphism) and /" is injective.
16-c. 2398

242
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
f
Lemma 1.17. Let A" -> 7" be a morphism of complexes of C+(A), where A"
is acyclic and 7" injective. Then f is homotopic to zero {hence a null morphism
in C+(A)).
Proof. We may assume that X" = J" = 0 for n < 0. By hypothesis, we have
a commutative diagram with the first line exact and the second one formed by
injective objects
0 -> X° -► A"1 -> X2 -► ...
/o| fi fi
0 -► 7° -► 71 -► I2 -► ...
Since 7° is injective, there is a morphism g1 : X1 -* 7° such that gx d°,. =f°. The
morphism f1 — df.g1 vanishes on \m{d%-), hence it factorizes through A^/Im t/£.=
= A'1/Ker d\- ^ Im (d^.). Since 71 is an injective object, this factorization can be
extended to a morphism g2 : A'2->71, etc. ... One thus obtains a family of morphisms
(g"),i^i, g" '■ X" -> 7""1 which define the stated homotopy.
Proposition 1.18. Each complex X' of C+(A) admits an injective resolution
and two such resolutions are homotopically equivalent.
Proof. Suppose, for convenience, that X" = 0 whenever n < 0. Choose a
monomorphism 0 -> A"0 -> 7°, where 7° is an injective object. We set 7" = 0 for
n < 0. Consider the fibred sum of A"1 and 7° over A"0, Y = A"1 J] a-» 1° -
A'1©70/Im {}id%-—/2w°), where z'i and 4 are the canonical morphisms A'1->A'1©70,
7° -> X1 © 7°. Let 0 -> Y -* 71 be a monomorphism such that 71 is injective. Denote
by d°. (respectively w1) the composition of the maps 7° -> Y -> 71 (A"1 -> Y -> I1,
respectively).
The reasoning can be repeated and one thus obtains an injective complex /'
and a morphism of complexes u : A" -> 7". One can easily check that u induces an
isomorphism between the cohomology objects. The uniqueness from the statement
follows from:
u i
Proposition 1.19. Let A" -> Y' be a morphism in C+(A) and let X' -> J',
Y' -* J' be injective resolutions. Then there exists a morphism v : I' -> J' such that
vi = gu in C+(A). Moreover, two such morphisms are homotopic.
Proof. By 1.16 the cone C'(i) is acyclic. By lemma 1.17 the composition
projection T(u) T(j)
C'(0 -^ T(Xm) -4 T(Y') -A TOT)
is homotopic to zero. Consider the maps C"(i) = A""+1 © 7" -»• r(J')"_1 = J" which
define this homotopy. By composition with the canonical injections we obtain
morphisms x" : X" -> J"-1, if : 7" -> J". One can easily see that the family (if)„
defines a morphism of complexes and that the maps vi and ju are homotopic (by
means of the maps t„).

VII. DUALITY ON COMPLEX SPACES
243
The last assertion of the proposition is a consequence of the following:
Lemma 1.20. Suppose X' -* Y', Y' -> /" are two morphisms in C+(A) such
that i is a quasi-isomorphism, ji is homotopic to zero and I' injective. Then j is
homotopic to zero.
Proof. Let (s")n be a family of morphisms which achieves the homotopy
between 0 and;/, s" : X" -► J"'1. One has j"i" = dpr1s" + sn+1d"x.. Let Z' = C'(i).
Consider, for any integer n, the map w : Z" = X"+1 © Y" -> /", the sum of the maps
5"+1 and;". One immediately verifies that the maps w" commute with the differentials,
hence they define a morphism w : Z" -* J'. Since Z' is acyclic and /" is injective,
iv is homotopic to zero by lemma 1.17. One has the equality j = wk, where k is
the inclusion morphism Y' -> Z* and the conclusion follows.
(3) Let A and B be two abelian categories, A with enough injective objects
and F : A -* B an additive functor which is left exact. F extends naturally to a
functor C+(A) -*■ C+(B) which is also denoted by F.
By 1.18, for any object X' of C+(A) there exists an injective resolution X' -*
-y J'. The cohomology objects of the complex F(I') do not depend on this
resolution: indeed, in accordance with proposition 1.18, two such resolutions /" and J'
are homotopic and hence the complexes F(I') and F(J') are homotopic (F is
supposed additive). We shall denote by R'F(X') these cohomology objects:
R>F(X') =Hi(F(Im))eB.
If X' -» y is a morphism in C+(A), then by means of 1.19 we get morphisms
R'F(X-) ->• R'F(Y-).
In this way one obtains a family (R'F)t of functors defined on the category
C+(A) with values in B, and which are called the right derived functors of F. The
compositions of functors A -> C+(A) —> B coincide with the derived functors of F
in the classical acceptance, as follows easily by the definitions. If X' is an object
of C+G4)and n is an integer, then R!F(X'[n]) a RK+'F(X'). If A" -^ Y' is a quasi-
isomorphism in C+(A), then the morphisms R'F(u) : R'F(X') -► R!F(Y-) are
isomorphisms.
An object Z of A is called F-acyclic if R'F{Z) = 0 for any i > 0. Any
injective object is F-acyclic.
Lemma 1.21. // X' is an acyclic complex of C+(A) formed by F-acyclic
objects, then F(X') is acyclic; if u is a quasi-isomorphism between two complexes
of C+(A) formed by F-acyclic objects, then F(u) is a quasi-isomorphism.
Proof. Let X' be as in the statement. We may assume X" = 0 for n < 0.
Since Coker d°x. = X1/^ d°x. = A^/Ker dx. ^ Im dx. and since H°(X') = 0, we
get the exact sequence 0 -> X° -> X1 -> Im(dx.) -»• 0. Then it will result that lmdx.
is F-acyclic and that the sequence
0 -► F{X°) -► FiX1) -> F(Im d\.) -► 0

244 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
is exact. Analogously, by using the exact sequence 0 -> ImJj. -> X- -*■ \md\. -> 0
one derives that Im d%. is /"-acyclic and that the sequence
0 -► /(Im d\?) -> F(X2) -► F (Im d\.) -> 0
is exact. By iterating the reasoning, we get a sequence
0 -» F(X°) -► /(A"1) -► /(A"2) -► ...
which is exact; hence the complex F(X') is acyclic.
Let now u : X' -*■ Y' be as in the statement of the lemma. If Z' is the
cone of w, then by the additivity of F it follows that F(Z') is the cone of F(u).
The required conclusion is obtained by the first part and by 1.16.
From this lemma we deduce that the derived functors of F can be calculated
by means of resolutions with /"-acyclic objects (the analogous of de Rham's abstract
theorem).
Lemma 1.22. Let X' be an object of C+(A) and X' -* Z' a resolution
such that the components of Z' are F-acycIic. Then for any integer i, H'(F(Z')) ~
~ R'F(X-).
Proof. Let X' -*■ I' be an injective resolution. By 1.19 there exists a morphism
Z' -> /' which makes (modulo homotopy) the diagram commutative
x- >z-
\/
I-
It follows easily that this morphism is a quasi-isomorphism and the assertion is
a consequence of the previous lemma.
(4) We explain the previous facts in the case of functors Horn. Let A be
an abelian category with enough injective objects and let X be an object. Consider
the functor Horn (X, •) : A -»• Ab. Its derived functors R'Hom(X, •) : C+(A) -»• Ab
are denoted by Ext'^A", ■ ). If Y' e C+(A) and Y' -* J' is an injective resolution, then
Ext"(A", Y') are the cohomology groups of the complex
... -► Horn (X, Y'-1) -► Hom(A", Y'~) -»• Hom(A", 7,+]) ->• ... .
For any integer n, Ext\X, Yw[n]) ~ Ext"+'(A\ Y'). If Y' is the natural complex
associated to an object Y of A, then the usual Ext's are obtained.
Recall Yoneda's method of calculating the functors Ext. Let X, Y be two
objects of A and /• (J', respectively) an injective resolution of X (respectively Y).
Define a complex of abelian groups Hom'(/', J') by formulae
Hom«(/-, J') = U Hom(P, J^*),
p
df = (d'J^f + (- 1)"+1 /'+1 df.)p>0, f = (f% e UomV, J').

VII. DUALITY ON COMPLEX SPACES
245
One can easily see that in this complex the g-cocycles are just the morphisms of
degree q from /• to J' (equivalently, the morphisms of complexes /• -> J'[q]) and
the ^-boundaries are exactly the morphisms of degree q which are homotopic to
zero. The augmentation morphism X -* J° determines a morphism of complexes
Hom'(/', J') -> Hom(A", J') (in the degree q, f = (fp)p *->f°i), hence by passing to
cohomology, morphisms
H"(Hom-(J-, J')) -► H"(Uom(X, J')) = Ext«(T, Y).
Proposition 1.23. Let X, Y be objects of A and I' (J', respectively) an
injective resolution of X (Y, respectively). Under these assumptions, the morphisms
fl«(Hom-(/-, J')) ->• Ext'CA", Y)
are isomorphisms.
Proof. We first prove the injectivity. Let / = (J")p e Hom«(/', J') be a q-
cocycle such that the image of the associated cohomology class is null. This image
is the cohomology class of f°i e Hom(A", J"), hence there exists a morphism
v° : X -* J"^1 such that the diagram
00 \ / dl
is commutative.
Consider the acyclic complex
0 -► X -» 7° -» P ->• J2 -» . ..,
where the object X has the degree (— 1).
The family of morphisms (f)p^0, to which we add the morphism (— l)9v0
in the degree (— 1), yields a morphism from the above acyclic complex to the
injective complex J'[q], According to lemma 1.20, this morphism is homotopic
to zero. Then / is homotopic to zero, hence a ^-boundary.
We now prove the surjectivity of the maps from the statement. Let g : X -> J"
be such that dj. g = 0. Choose a morphism/0 such that g =f°i. We set/'' = 0 for
any / > 1 and it is easy to verify that the family/ = (fp)p e Hom'(/*, J') is a cocycle
and the image of its cohomology class under the morphism H"(l{om'(r, J')) ->
-> //<?(Hom(A', J")) equals the cohomology class of g.
Now we are going to define the operations of composition for the functors Ext.
Let X, Y, Z be three objects of A and let /', J' and K' be some injective
resolutions of theirs. The composition of maps defines a morphism
Hom-(/% J') <g) Hom"(J-, K') -► Hom'(/-, Km)
which agrees with the differentials.

246 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Passing to cohomology one gets the maps
Ext'CAT, Y) X Ext«(7, Z) -► Extp+"{X, Z)
which are called the Yoneda bilinear maps. One can show that these maps are
well defined, functorial and agree with the short exact sequences.
Remark. One can see that an analogous result with respect to proposition
1.23 also holds if Y e C+(A). Accordingly, the above bilinear maps are also
defined when Z is in C+(A).
(5) Let A" be a topological space and $ a family of supports. The functor
gF h> r®(X, âF), from the category Ab(A") to the category Ab, extends to a functor
C+(Ab(A")) ->■ C+(Ab). The associated derived functors will be denoted by H®(X, •).
Let âF* be a complex bounded below of sheaves of abelian groups and let
§' -> 3' be an injective resolution. By definition fl®(X, âF') are the cohomology
groups of the complex
... - r9(X, 3"-1) - U(X, 3") - U(X, 3"+1) - ... .
If gF" is the complex associated to a sheaf âF, one obtains the cohomology groups
of âF with supports in $.
As in I. 4.16 we get the Mayer-Vietoris exact sequence:
Lemma 1.24. Let X be a paracompact space, which is a union of two open
subsets U and V. For any âF" e C+(Ab(A")), the following exact sequence of natural
maps
...-> Hqc(V n V, SF*) -► H%U, âF*) © WC(V, 9') -* H§(X, ff')-* Hf\U r\V,9')-+...
holds.
Suppose now (X, 6) is a ringed space and âF an (9-module. The functor
(| h> Home(âF, 6£), from the category Mod((9) to the category Ab, extends to a
functor C+( Mod(<9)) -► C+(Ab). The associated derived functors will be denoted
by Ext^A"; âF, •). If q' e C+(Mod((9)) and §' -> 3' is an injective resolution, then
Exta(A"; SF, iÇ") stands for the cohomology of the complex
...-»• Uom0(9, 3"'1) -► Hom^SS 3") -> Hom^f, 3*+1) -►....
Let <b be a family of supports. Starting with the functor § h> Homo.e^, ($) =
= Tq,(X, Home{^,(9^)), one can define the invariants Extô.a(^"; &, §'), §' e
C+(Mod ((9)). We will consider only the particular cases: $=the family of all closed
sets and O = the family of all compact sets (in the latter case we denote, as usual,
by Extc,g the obtained invariants).
If gF — Q, then there exist remarkable isomorphisms
Exti,a(A-; 6, <f ) ~ h'9(X, <?')» §' e C+(Mod((9)).

VII. DUALITY ON COMPLEX SPACES
247
We define now composite maps for Ext*. &. Fix two (9-modules SF and <$ (§ can
be in fact an object of C+(Mod((9))) and let 3', y be injective resolutions of theirs.
Consider the complex of abelian groups Homô.a(3". D given by the formulae
Uoml,e(r, D = II Horn».,9(3', Yu\
p
df = {d^"f + (- \)«^f^d%, f = (f*)e Homl,e(r, T).
The augmentation 8F -> 3° defines a morphism of complexes
Hom3>,a(3\ y) -► Homi.aC^, D>
hence morphisms
H%Homi.e(3', D) -► H«(Hom<»,6(®, y)) = Ext£.e(*; SF, §).
Lemma 1.25. For any q, the map
H«(Homo.e(3m, D) -► Exû,a(^; ^. <?)
is bijective.
The proof is analogous to the proof of 1.23, by making use of the
following
Lemma 1.26. Let 0 -> SU -> SK. be a monomorphism in Mod (<9), let 3 be an
injective Q-module and f : $11 -> 3 an Q-morphism. Under these assumptions, there
exists an Q-morphism g : SK. -* 3 such that f = gi and Supp g = Supp f.
Proof. Let U = X\Y, where Y = Supp/. Consider the sheaf S!\lu(SfLu,
respectively), equal to M{oZ, respectively) on U and with stalks null on Y. <KV, gflv
are subsheaves of Sic, SI and f{Sfdv) = 0 (one can reason on stalks). As 31 v n /(Sic) =
= i(Sf\lv), f can be extended to a morphism /' : Slv U i(STi) -> 3, by putting /' = 0
on iJly. Any extension g : HI -* 3 of f satisfies the assertions of the lemma.
The following result will be useful.
Corollary 1.27. // Y is a closed subset of X and 3 is an injective Q-module,
then X°3 is an injective Q-module.
Proof. For any (9-module Sic, Home(Stl, 2iy3) = Homy,â(a)lc, 3) and apply
the previous lemma.
We now consider a third (9-module X, and a new family of supports Y.
Let 3C be an injective resolution of %. The composition of maps defines a morphism
Homes', y)® Hom^Q-, 3Q -, Hom^^O', ST),
which agrees with the differentials.

248 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Passing to cohomology we get the Yoneda bilinear maps (in fact F(X, £>)-
bilinear)
ExtS,. &(X\ Sr, §) X Ex$r,e(X; §, 3C) -► E^V ô(X; Sr, 3C).
In the case when X is paracompact, O = the family of closed sets of X, W = the
family of compact sets of X (or conversely) one obtains the bilinear maps
Ext&CT; Sr, §) X Ext« e(X; §, T) -> ExO*; ^, ^*),
Ext£ô(*; 3\ <?) X Ext9ô(X; §, 3T) -► Ext^'C*; ff, 3T>
Remark. In the above considerations we may replace the sheaf % by a
complex %' of sheaves, bounded below. If SF = <S> then one obtains the bilinear maps
#'(*, §)x ExtteiX; §, 3f) - H?+* (*, 3C),
/#(*, <£) X Exfe(X; §, 3f) - //r9(*, 3t).
We conclude with some words on the local Ext's. Fix again an (9-module 8r~.
Starting with the functor q h> HomQ(&, §), we can define for any ($" e C+(Mod £>)),
the (S-modules Ext'e(8r, §'). In this case one can also check easily that each Ext^fë, §')
is the sheaf associated to the presheaf U h> Exte;u(t/; S^ | U, §' | £/). If SF=<9, one
obtains isomorphisms
Extqô{6, q-) ~ H"(q-y
(C) In this section we recall some facts concerning topological vector spaces
([61], [75]). We have denoted by FS (DFS, respectively) the spaces of Fréchet-
Schwartz type (strong duals of Fréchet-Schwartz spaces, respectively). By QFS
and QDFS we mean quotients of such spaces (according to I, § 1, C).
(1) Let M, N be two topological vector spaces defined over the complex
field and u : M -* N a linear and continuous map. Recall that u is called strict (or
topological homomorphism) if the quotient topology on u{M) coincides with the
induced topology from N. If M and N are FS or DFS spaces, then u is strict
if and only if u(M) is a closed subspace of N. This will be used in the following
U V
form: if M -* N -* P is a complex of FS (or DFS) spaces, then the cohomology
space Ker vjlm u is separated (in the natural topology induced on it) if and only
if u is strict.
Lemma 1.28. Let u : M -> N be a linear and continuous map between FS
spaces. Then u is strict if and only if the transposed map u' : N' -* M' is strict.

VII. DUALITY ON COMPLEX SPACES
249
Proof. We easily reduce the problem to the case when u is injective or sur-
jective and then the assertion follows by properties of reflexivity.
Lemma 1.29. Let u : M -* N be a linear and continuous map between FS
spaces. If u(M) is of finite codimension in N, then u is strict.
Proof. Let P be an algebraic complement of u(M) in N. Consider on it the
induced topology from N; by hypothesis there results that P is also FS. The map
v : M X P -> N, (m, p) h> u(m) + p
is continuous and surjective. By the Banach theorem it is strict. If W is an open
subset of M, then u{W) = v(W x P) n u(M); therefore u is strict.
(2) Two topological vector spaces M and N are by definition in topological
duality if there exists a bilinear and separately continuous map MxN-»C, such
that the induced maps M -* N', N -* M' are topological isomorphisms (the accent
means the topological strong dual).
Lemma 1.30. Let M -*■ N -*■ P be a sequence of FS spaces and linear
u' v'
continuous maps such that vu = 0; let M' <- N' <- P be the transposed sequence.
Then the duality between N and N' induces a topological duality between Kem/Imw
and Keru'/Imv', where these spaces are endowed with their natural topologies.
Proof. For convenience we denote by < > the canonical bilinear maps
M X M' -> <£,, N x N' ->■ (£,, P x P' ^ <£,. By restriction one obtains a bilinear
map Ker v X Ker u' -* C- 0Qe can see easily that this vanishes on Im u x Ker u'
and Ker v X Imw'; by continuity we get a bilinear map
Kery/Imw X Kerw'/Im v' -> C
and we will prove that the duality stated in the lemma is fulfilled by this map.
Let Kerw'/Im v' -> (Kery/Im «)' be the induced map. We check its injectivity
Let n' e Kerw' be such that the image of the associated class is null, hence <«, «'> = 0
for all n e Ken;. In order to show that n' e Im v' it is sufficient by the Hahn-Banach
theorem to prove that whenever L is a linear continuous functional which is null
on Im v', one has L(n') = 0. The functional L has the form <«, • > for some
ne N. We have to prove that <«, n'y = 0 and this follows as soon as n e Kera
But for any p' e P', (v{n),p'y = <#i, v'(p')y = L(v'(p')) = 0.
The surjectivity of the map Kerw'/Im v' -> (Ker v/lmu)' is immediate in virtue
of the Hahn-Banach theorem. Its continuity is merely a consequence of the fact
that any bounded subset of Kery/Im u is the image of a bounded subset of Kerv
(Kerf; is an FS space); then the map will result a topological isomorphism, by
open map theorem for DFS spaces.
Since the space Kerv/lm u is FS, hence reflexive, one derives also the
topological isomorphism Kerf/Imw ^> (Kerw'/Im v')'. The lemma is proved.
We now prove a result which ensures the uniqueness of the topologies of the
spaces in duality.

250 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Lemma 1.31. Let M, N be two QFS spaces and P, Q two QDFS spaces.
Let a : M x P -*■ C and v : N X O -* C be bilinear maps which put in
topological duality the separated spaces associated to M and P, to N and Q, respectively.
Under these assumptions, any linear maps u : M -* N and v : Q -*■ P which
agree with \ and v are continuous.
Proof. By hypothesis the map M -*■ P' given by X is surjective and its kernel
is {0m}- A similar assertion holds for the map N -* Q' given by v. Then «({0M}) <=
c= {0N}; analogously, one deduces that v({0Q}) <= {Op}. Hence the maps u and v
induce maps u and % between the associated separated spaces. We will show that
these are continuous, whence one derives the continuity of u and v. Let F be the
A A A A A
graph of u ; we show that it is a closed subspace of M X N. Then let (m0, n0)
be an element of its closure; we will show that u(m0) — n0. If L is a linear
continuous functional on N, then Lu is continuous (u and v are transposed, modulo
the isomorphisms given by X and v). We will prove that L(u(m0) — n0) = 0 and
the assertion concerning F will follow. The functional (m, n) h> L(n) — Lu(m),
A A . A A
defined on M X N, is continuous; it vanishes on F, hence at (m0, n0), etc. By the
closed graph theorem we derive that u is continuous; similarly one may check
the continuity of v.
(3) We consider complexes of topological vector spaces (over the complex
field), namely, complexes whose components are TVS and the differentials are
continuous C-linear maps.
In this case the cohomology groups can be endowed with structures of
topological vector spaces which are generally nonseparated. A continuous morphism
between two TVS complexes is a morphism of complexes (of zero degree) whose
components are continuous and C-linear. If u : M' -* N' is such a morphism,
then the linear maps H"(u) : H"(M') -> H"(N') are continuous. The cone of
a continuous morphism is naturally a TVS complex.
Lemma 1.32. Let u : M' -> N' be a continuous morphism between two FS
complexes (the components are FS spaces). If u is an algebraic quasi-isomorphism,
then it is a topological quasi-isomorphism (induces topological isomorphism to
cohomology).
Proof. Let Zq(M') (ZQ(N'), respectively) be the space of cocycles of
dimension q of M' (N', respectively) and Hq(M') (H%N') .respectively) the corresponding
cohomology groups. Define the continuous map
y«: Z\M') © A^1 -»• Z"(N') , (m, n) t-> u (m) + d"^1 (n).
Since Hq(u) is an algebraic isomorphism, v" is surjective. The source and the target
are FS; hence v" is strict. One can conclude by means of the commutative diagram
Z\M') © N"'1 —^ H"(M-)
v" W(u)
Z"(N') —"—* H"(N')

VII. DUALITY ON COMPLEX SPACES
251
where tz is the composition of the first projection with the map Zg(M') -* H"(M')
and 7i' represents the passing to quotients.
Lemma 1.33. In the preceding statement let the complexes M' and N' be
DFS. Then the conclusion of the above lemma remains true.
Proof. One preceeds similarly, by using the properties of permanence of
the DFS spaces and the open map theorem for such spaces.
We now restate lemma 1.1.4.
Lemma 1.34. Let M' be an acyclic complex of F S or DFS spaces. Then
the transposed complex M" is acyclic.
Proof. Let L e (Mq)' be so that Ldq_1 = 0. Since L vanishes on Ker d" =
= Im d"-1, it factors through the space Im d", which is endowed with the
quotient topology from M". Since Im d" = Ker dq+1, by Banach theorem it results that
the quotient topology coincides with the one induced from Mq+1. By applying the
Hahn-Banach theorem we can find a functional L' e (M«+1)' such that L'd" = L (in
fact, the lemma is a mere consequence of 1.28 and 1.30).
Lemma 1.35. Let u: M° -* N' be a topological quasi-isomorphism between
F S or DFS complexes. Under these assumptions, the transposed morphism u' is a
topological quasi-isomorphism.
Proof. The cone of « is a complex as in the previous lemma. Its transposed
complex will be acyclic. The conclusion follows from 1.32 by the permutability of
the operations of transposition and of taking the cone.
§ 2. The construction of the dualizing complex
In the first part of the paragraph we construct a complex of 6V ^-modules for
any germ (V, x) of manifold, which will be the candidate for the stalk at the
point x of the dualizing complex Kv. For an immersion (V, x) -* (W, y) one will
deduce a natural formula of transformation between the associated complexes.
In the second part one builds the dualizing complex of an open subset of
some numerical space. We will glue the complexes punctually constructed by means
of the Frisch theorem of finiteness (V.3.1). For an immersion one will obtain a
transformation formula between the corresponding complexes.
These facts will allow the construction of the dualizing complex for an
arbitrary complex space (the third part of the paragraph).
(a) Let (V, x) be a germ of complex manifold. Denote by A the ring of
germs of holomorphic functions at the point x and let n = dim A. By m we denote
the maximal ideal of A. For an integer p, let
Z' = {ae Spec A | ht a ^ p).
In § 1 we have constructed a complex of flabby sheaves on Spec A
0 %%.IZ1 (A) - 3^1/zl (/)-...- ?C- „/z„t1 (A) = 3f- (A) - 0

252 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
(the Cousin complex of A with respect to the filtration given by codimension) and
an augmentation morphism
Since A is a regular ring, the Cousin complex of A is a resolution of A by sheaves
whose stalks are injective modules (1.14). The Cousin complex of ^-modules
associated to A, obtained by taking the global sections (in this case we actually
localize in the maximal ideal m), is a resolution of A by injective modules (1.15), which
will be denoted by L'Vx. For instance, if V is a Riemann surface, the complex
thus obtained is 0 -> é\lx -* STix/6x -> 0, where §)TLX is the ring of germs of mero-
morphic functions in x (§ 1, A, 6).
Define
KyiX = LyiX®AQyiX[n]
where Qv is the sheaf of germs of holomorphic forms of maximal degree on V.
Lemma 2.1. For any immersion of germs of complex manifolds
f
(V, x) -* (W, y) there exists a natural isomorphism
J:KVtX^Hom0(ry(ôv,x,K^J.
Moreover, the correspondence f *-*■ f agrees with the composition of immersions.
Proof. If/ is a local isomorphism, then the construction of/ is clear.
First of all we consider the case when the codimension of / is 1. Denote
A = @v,x> B = ®w,y and let P; B -* A be the surjection which corresponds to
/. Let t be an element of B which defines the subgerm Im/. If y. = y.t is the multi-
plication by t, then the sequence 0->5->5->/4->0is exact. The morphism p
nduces a closed immersion p between the prime spectra. In virtue of définitions,
the sequence of sheaves on Spec B
(*) 0 ->• B -► B ->• p* (A) ->• 0
is exact. Denote by Z" (Tp, respectively), the set of the prime ideals from A (B,
respectively), of height > p. For any integer p, p_1 (Tp) = Z"'1. In order to
construct/ we have to interpret the complex Hom^ (®v,y> L'w,y). By 1.1, the p-th
component is
Hom,(^,r3tJ,/r,t.(J))ïiHomJI(^1 © H$(B)) <* © HornB(A, H%(B)y
From the left exactness of the functor Horn it follows that HomB(A, H^{B))
is identified with the kernel of the map \i: H^{B) -> H$(B); this map can be
inserted in the exact sequence
H%~\B) -, H^ (p*(i)) ^ H$(B) 4- HfcB)

VII. DUALITY ON COMPLEX SPACES
253
given by sequence (*). Since B is regular, Hf \B) = 0 (1.11). Accordingly,
Hom^, H%0)) ^ //g"1 (p,(i)), peP\ T^\
If p $ p (Spec ^), then Spec B \ p (Spec ^) is a neighbourhood of p where p*C4) is
null; hence Hom^, H^B)) = 0.
Let pep (Spec A); thereby H"~x (Spec B, p*(A~)) can be identified with
H^1 (Spec /*, A), where a is the element of Z"'1 \ Zp so that p (a) = P (the
invariants with respect to a subspace and to a sheaf coincide with the invariants
with respect to whole space and to the trivial extension of that sheaf!). By the
antecedence it follows that © HomB (A, H$(B)) is identified with
Perp\rpH 1
® H£-\Â), which is isomorphic to r%^\z» (A) (1.1).
One thus obtains an isomorphism, which depends on the local equation t,
Af:L^^UomB(A,L^y).
By means of these isomorphisms, for any integer q we define an isomorphism
J" :K*,x^HomB(A,K^y).
We have K^x=L"v%iimV ®A QVx and HomB(A,Kjy.y)=Hom^A,L9w+/mw®Bnw.i) =
HomB(^, Lfv+fmv+1 ®BQWiy). Next, define f> by means of A, and of the
p-morphism Ù.w>y -* QK;V associating the form I-kVY | V to the form *F A dt.
If we replace t by another local parameter t', then A,- = ((t/f) \ V). Af:
this results from the fact that Ar- and A„ are obtained by means of the exact
sequences of local invariants associated to the exact sequence of the type (*) given
by the multiplications y.t and [x,-. On the other hand, as one can easily remark,
the morphism Qw>y -»• Qy,x associated to t' differs from that associated to t
by the multiplication with (t'/t) \ V. Thereby the morphisms J" are independent
of t. Obviously,/" are isomorphisms and/ = (/% is an isomorphism of complexes.
In this way/ is constructed for immersions of codimension 1.
If/ is an arbitrary immersion, then the construction of/ can be made by
the decomposition of/ into a finite number of immersions of codimension 1. The
independence of the decomposition can be proved without difficulty as soon as
one has, in the above notations, an explicit formula of identification for the
isomorphisms Hpa\Â) ~HomB(A,H%(B)), a e Z"'1 \ Z" and p e P-1(a) (deduced
from the coboundary operator). For instance, one may proceed as follows. Let
%,..., ap^1 be elements of B such that their images in Aa form a regular system
of parameters. We have the identifications
Hi~\Â) « HQ ^ ~ H"^ ((fli. • ■ ■ ' °p-i)> A°) ~ lim HP~1 (<& • • •> ûJ-iî Ao)
k

254 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
and
H$ (B) ~ H$H (£3) *. H» ((t, a,,..., a,.,), £p) ~ lim H" (tk, a\, ...,ak^; B&).
k
Consider an element £ of Hg'1 (A). It corresponds to an element of some
H"'1 (d[, ...,0%^!', Aa), hence to the cohomology class associated to an element
of the form a.a\ A •. ■ A^-i e ^Vi, • ■ -, ctP-x, Aa) (~"~A (A^1)), where ae Aa.
Let 6 e £p be such that p ((3) = a. One has d(b.d[ A • ■ • A û£-i) = 'fc ^ (>fc A
A a? A ■ • ■ A flj-i) = t(tk~^b.tk A fli A ■ • ■ A flj-i).
Let 7) be the element of H${B) assigned to the cohomology class of tk"^b. tk [\
A 4 A •.. A oJ-ieK'^.a*, ...,a*_i; 5P) in #*(**, a*, ..;, <£_i; 5p). The
correspondence Ç i-»- (1 i-»- rj) stands for the required explicitation (a
consequence of the construction of the coboundary operator!). In the above formulae
we abused notation as we denoted'by the same letters the images under p of the
elements at. _
The functoriality of the correspondence/ i-»-/ follows by using its
independence of the decomposition into immersions of codimension 1.
(b) Consider an open set U of some <£"•
Let K <= U be a Stein semianalytic compact. By theorem V.3.1 the ring 6 (K)
of germs of holomorphic functions on K is noetherian. Each maximal ideal in of
6 (K) is finitely generated, hence it is associated to a point of K; moreover, the
localization 6 (K)m is a regular ring (lemma V.4.1). Consequently the Cousin
complex associated to the ring 6 (K), which will be denoted L'K, is a resolution of <5(K)
(1.15).
Recall some facts about the analytic sets. A germ of analytic set in the
neighbourhood of K is a pair (D, M), where D is an open subset such that D => K. and M
is a closed analytic subset of D. We identify two such pairs (Dj, Mj), (D2, M2)
whenever there exists an open set D <= D1 n D2, D => K, such that M1f]D= M2 n D.
A germ of analytic set in the neighbourhood of K is called irreducible if it cannot
be written as the union of two proper subgerms (the notion of subgerm, union
and intersection of germs, ... are defined naturally).
If we associate to each germ the ideal in 6 (K) of the functions which vanish
on it, then one obtains a 1 — 1 correspondence between the germs of analytic sets
in the neighbourhood of K and the radical ideals of 6 (K) (equivalently, the closed
sets of Spec 6 (K)). In this correspondence the irreducible germs correspond to
the prime ideals. If x e K, then we denote by pKx: 6 (K) -> 6xthc natural morphism
and by pKx: Spec (SI,. Spec 6 (K) the morphism induced between the prime
spectra. If a is an ideal of 6 (K), then for convenience, we denote by pKx(c/.) the
ideal generated by the image of a under pKx. If a corresponds to a germ of
analytic set (D, M) in the neighbourhood of K, then px*(«) determines the germ of
analytic set given by M in the neighbourhood of x. Indeed, let A, . • • ,/s be a
system of generators for on;fix = pKx(fi), 1 < / < j, is a system of generators for
pKx(a) and the assertion follows from the fact that the passing from an ideal to
the associated germ of analytic set is made by taking the common zeros of a system
of generators.

VU. DUALITY ON COMPLEX SPACES
255
Lemma 2.2. Let a e Spec 6 (K) be so that pKjc(a) ^ Gx. There exist a finite
number of prime ideals [3, (3 e Spec 6X, which are minimal such that (3 => p^a).
Moreover, for such an ideal (3, ht (3 = ht a.
Proof. The first assertion is a general property of the noetherian rings,
consequence of Lasker-Noether decomposition. If a corresponds to the germ (D, M),
then the ideals [3 correspond in fact to the decomposition of the germ Mx into
irreducible components. One can easily see that all these irreducible components
are of the same dimension, equal to dim (D, M) = dim (6(K)/ol). It follows that
ht a = dim 6 (K) — dim (0 (K)/x) = dim 6X — dim (6Jp) = ht [3 for any [3.
For a Stein semianalytic compact K' <= K one can do the same for the
restriction morphism pKK> : 6 (K) -> 6 (K').
Lemma 2.3. Let K' <= K be Stein semianalytic compacts, x a point of K'
and L'K, L'K', L'x the Cousin complexes associated to the noetherian rings <3(K),
6{K') and 0X. Then, by the construction which follows, there exist natural morphisms
of complexes L'K -* L'x, L'K- -> Lx, L'K^> L'K-, which agree with pKx, pK,x, pKK, and
such that the diagram
L'K >■ L'K-
\/
l:
is commutative.
Proof. We shall indicate the construction of the first morphism. For the
other one may proceed analogously. The commutativity of the diagram can be
verified canonically.
For any p > 0 we need to construct a pX;c-morphism L"K -> Lx. We shall use
the following notations: A = 6X, B = 6(K) p=pKx, Zv = {pe Spec A\ht$ > p},
F = {«e Spec B | ht a S= p}. There exist isomorphisms
L"K ~ © H>0), Lx^ © H$(A).
aeTPXTp + ] pezl'\zl'+1
If a is an ideal of Spec 6{K) of height p, then we shall construct a pX;c-morphism
X: H&B) - © H^A).
These maps will determine the required morphism L\ -> Lx.
We first consider the case p ^ 2; this hypothesis will allow us to express
the local cohomological invariants in terms of usual invariants. By § 1, H^{B) is
the stalk in a e Spec B of the sheaf %fe (B); hence lim //{% (U, B), U being a neigh-
u
bourhood of a in Spec B. In this inductive limit we will consider only affine open
sets. We have the exact sequence
.. -* H'~\U, B) -► H'-HV \ x, B) -► Hi(U, B) -► H'(U, B) -► .. .

256
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Since U is affine, H"(U,B) =for9>l and hence (p>2), H?(U, B)^Ht'-1(U\x,B).
Therefore Hpa{B) ^ lim HP~\U \ a, B). We will define the maps
v
\u:H'-1(U\x,B)-* ® H$(A),
pezp\zp + 1
which are compatible with inclusions U <= V and thus one obtains X.
Let U be an open set which appears in the inductive limit. The set U \ a
is open in Spec B, hence there exist a finite number of elements /, e B such that
U \ a = u Dfj. The principal open sets Df are affine and T{Dfp B) = Bf .
For a finite part of them fh, . . . , jfik, Df( n . • . fl D/. = Dfi _ f.
and T(Df fik,1)) = Bft ...fik. By the Leray theorem, H^U \ a, 5) ^
-«^({D^}^ 5). Let ceHp~\U\x,B) and {&...;} be a cocycle which
represents c and which corresponds to the covering (Df^. In the case when
the ideal p(a) equals A we set ~kv = 0 (and hence X [ //£(■#) = 0). Otherwise,
let pl5 ..., Pj be the prime ideals of Spec 6X which contain p(a) and are minimal
with this property. For each p;, ht p; = hta. The open sets Dp{fj) give an affine
covering for p-\U) \ (Pi U ••• U ~PS)- The elements pfe^... jp) e AP{fh)... p(/jp
(denote by the same letter p the morphisms induced to the rings of quotients)
yield a cocycle with respect to this covering, hence an element -y in
s ^
H"~1(p~1{1U) \ [_) p^ A). For any integer q one can find an affine neighbourhood
Fofp? contained in p_1(^) \ U p"c The restriction of y to H"-\V \%, A
induces an element in H^q(V, A), hence an element ys in H$q(A).
s
Define \v(c) = £ yq. One can easily check that the definition of Xv(c) is
«-1
independent of all choice and also that the maps \v agree with the
inclusions U cz V. For p $s 2, we have thus obtained the required morphism X.
For p = 0 and p = 1, the invariants H£(B) and H*(B) can be computed by
means of suitable inductive limits and exact sequences of the form
o - H&u, B)=ra(u, È) - nu, S) - r(c/ \ a, i) - //^(c/, b) - o.
The morphisms X0 and Xj can be defined as in the casep > 2 by using these sequences
which allow us to express the invariants //- and Ha in terms of invariants H°=T.
As soon as the morphisms X are constructed, they extend as we have showed
to morphisms LPK -> LPX and it is easy to verify the compatibility with the differentials.
For an arbitrary point x e U we have the following

Vil. DUALITY ON COMPLEX SPACES
257
Lemma 2.4. The natural map of complexes
lim L'K -> L'x,
K,X£K
deduced from the previous lemma, is bijective.
Proof. We first prove the surjectivity. We will do this for degrees p > 2, the
case p < 1 being similar.
Fix p e Z" \ Zp_1 and y e H$(A). There exists a closed polydisc K which is a
neighbourhood of x, such that pkxPkx(P) = (3. For any K' <= K, K' containing x,
we also have pK'xPK*x($) = P- Then by lemma 2.2 there results that the prime
ideals a.K. = p^4(P) have the same codimension as a.
Hl(A)~ lim H£{V,A)~ lim «M^XlM).
K.peK K.peK
There exists an element ^ e ^ \ p such that the cohomology class y is represented
by an element of H"1 (Z>+ \J, Â).
If <p1; ..., <p,„ is a system of generators of the ideal xK, then one can easily
n m
see that xK $ (J D9 and Z>+ \ (3 = (J D+p (cPi); similar relations hold for any
À" <= K (x e A:'), if we replace the elements 9, by <pi = p^OPi)- Then it will follow that
the element y can be represented by a cocycle {Yh...iP}, y(l... ip e ^p B ,...p <?, > •
Choose a closed polydisc L, L a K and xel, such that the elements <\> and
Yh...iP can be "lifted" to elements ^e G(L), y(l... lp e ^L)^^,.,,^,^,,
respectively.
The family {y^...,^} defines a cocycle with respect to the covering formed
by the open sets D^KL{Vi) (it is enough to remark that the map pLx : 6(L) ->
-> <S>X is injective). One can easily show that this cocycle induces an element y in
H^JÔ{L)) and that x(y) = y. Thereby the surjectivity is proved.
The injectivity can be proved similarly.
We will define a complex L'v as follows: a section over an open set V is a
collection (sx)xev, sxeL'x, which is locally induced by an element of L'K. By the
previous lemma, the stalks of the complex of sheaves L^ are just the complexes Lx.
Denote
KÙ = LI ®s^[#i]
and call it the dualizing complex of U.
Lemma 2.5. // U and V are open subsets of some numerical spaces and f: U-* V
is a closed (or open) immersion, then there exists a natural isomorphism
f.K^Hom^if^Gy), Ky)\U.
Moreover, the correspondence f (->-/ is compatible with the composition of immersions.
]7 - c.2308

258 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. One shows that the isomorphisms fx(xe U) given by lemma 2,1 vary
"continuously" with respect to x; hence they extend to a morphism of sheaves.
Remark. We have put Hom6y(f'*(£„), K'v)\ U instead of J-KHome (j'*(6u),Kv))
(there is a constant abuse of notation in the followings). Since the ideal which
defines the immersion/ annihilates this sheaf, we may replace in fact f^1 by /*,
(c) In this section we prove the following theorem of existence of the dualizing
complex of Ramis and Ruget:
Theorem 2.6. For each complex space X, there exists a complex Kx of
<Sx-modules such that the following assertions are valid:
(i) If X is a manifold of dimension n, then the complex Kx is a resolution of
Q'x[n] and for any point x e X and for any integer p, the stalks KXx are infective
£\ ^.-modules.
(ii) ///: X -* Y is a closed (or open) immersion of complex spaces, then there
exists an isomorphism
f: Kx =* Hom3Jtft(0x), Ky) | X
In addition, the correspondence f \->f agrees with the composition of immersions.
(;'/;') If X is finite-dimensional, then the complex Kx is bounded ; more precisely,
K$ = 0Jforp < —dim X and p > 0.
(iv) The sheaves of cohomology %C(KX) are coherent. Moreover, %"(K'X) = 0
for p <-prof (£>x).
Proof. Consider an arbitrary complex space X. For an open set U of X such
that there exists an immersion <p: U -*■ V, V open set in some numerical space,
denote KÛ = Home (<f>:f(£>u), Ky) \ U, where Kv is the dualizing complex of V
defined in the preceding section, K^ is a complex of (9^-modules,
If U' is another open set enjoying the same property, then both immersions
of U fl U' derived from <p and <p' can be refined by means of a third; from lemma
2.5 we deduce an isomorphism
W.f£.\ vr\U'czKï\vr\V'.
Consider a covering of X by open sets as above. The isomorphisms x verify the
usual relations of compatibility (2,5) and hence the complexes K{j glue together
to a complex Kx, This complex is independent (up to isomorphism) of the
considered covering.
We now verify the assertions of the theorem. Let U be as in section (b). The
morphisms 6X -> L% define morphisms £l'Ux[n] -> K'x and these define a morphism
^i/W -* K-u- The isomorphisms / from lemma 2,5 (applied to open immersions)
are compatible with these morphisms. The first assertion follows from section (a)
and from the very way of constructing Kx. The second results from the construction
of Kx, by 2,5, The assertion (Hi) follows from the next lemma by passing to stalks.
Lemma 2,7, Let A be a regular noetherian local ring, L'A the associated Cousin
complex and M an A-module of finite type. Then Hom^(M, L%) = 0 for p <
dim ,4 — dimM,

Vil. DUALITY ON COMPLEX SPACES
259
Proof. We have LPA = © H%(A), Accordingly, it is enough to show
that HomA(M, HP(A)) = 0 for p = ht a < dim A ~ dim M. By means of a
composition series, we may assume M of the form A/$. We have (3 <£ a and let a e (3 — a.
Hom^Of/p^Ci)) = {z, e tf£(JÏ)|p£ = 0}. The element a is invertible in Ax and the
homotety defined by it in the ^a-module HP{A) is injective. The conclusion follows.
The last assertion will be a consequence of the corollary 3.5 from the next
paragraph and of corollary 1.1.15.
Thereby theorem 2.6 is concluded.
The complex K'x is called the dualizing complex of X.
§ 3. Theorems of absolute duality
(a) Let V be a paracompact complex manifold of dimension n. 6, D and &M (3ip'q,
respectively) stand as usual for the structural sheaf, the sheaf of germs of holo-
morphic forms in the maximal degree and the sheaf of germs of differential forms
by the type (p, q) with coefficients Cœ functions (distributions respectively). The
Dolbeault resolution
0 -»• Q -»• 3T'° —> • • • -^> §{"''' -»• 0
allows, by applying the functor TC(V, *), the computation of the invariants H'(V, Q).
The integration on V of the forms of the type (n, ri) with compact supports will thus
induce a linear form
This form multiplied by (—1)" will be denoted by Tv and it will be called the trace
map on V.
Let now gF be an (9-module. For each integer p consider the Yoneda maps
Hf(V,$)xlExTp(V; ^,fi) -+ HS(V, Q)
Hpc{V, SF)X Ext"-"(F; $, Q) -»• HZ(V, Q).
By composition with Tv: Hnc{V, Q) -> C we get pairings:
(*) H"(V, ®)xEx'2-p(V; 3F, Q) -► C
(**) HP(V, SF)xExtn-"(T;3F,n)->- C-
The invariants H'(V, fi) are isomorphic to the cohomology groups of the complex
o -> rc{v, si"'0) -^rc(v, si'"1) -»•... —> rc(v, st"*-1) -► o.

260 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
The spaces TC(V, 3i"'p) are isomorphic to the topological duals of the spaces
V(V, £>°'"-p), hence they have a DFS structure. The differentials d" are continuous in
this topology and we thus get a natural structure QDFS on H'(V, Q). One can easily
check the continuity of the trace map.
Lemma 3.1. // V is a Stein manifold, then HP(V,£l) is mill for p *- « and
the space H"(V, Q.) is separated and the map (*) defines a topological duality between
it and the space T(V, &) {the latter is endowed with its natural FS structure).
Proof. The invariants H'(V, 6) are isomorphic to the cohomology groups of
the topological complex of FS spaces
0 -► T(V, S>°'°) -%■... -^ T(V, $°>") -> 0.
This complex is in topological duality with the complex TC(V, 3C""*) and the
conclusion follows straightforwardly from 1.30 and theorem B.
The invariants H'(V, &) can be computed by means of the complex
TC(V, 3i0'*); hence they have a natural QDFS structure. Just as in the above case,
one can prove
Lemma 3.2. // V is a Stein manifold, then HP(V, 6) is null for p ^ n and
the space H"(V, &) is separated and the map (*) yields a topological duality between
it and T(V, Q.) (the latter having its natural F S structure).
Remark. If in lemmas 3.1 and 3.2 one drops the hypothesis that V is Stein,
then (in virtue of the duality between C™ functions and distributions, extended to
forms and by applying lemmas 1.28 and 1.30) one obtains topological dualities
between the separated spaces associated to the spaces (HP(V, fi), H"~"(V, 6)) and
(HP(V,6),H"~P(V,Q.)), respectively. Moreover, the separation of HPC(V, D) is
equivalent to the separation of Hn~p+1(V, &) (and a similar assertion for the second
pair of invariants). This fact is a special case of the first duality theorem and it will
be used in the proof of the second duality theorem; we will point it out in order to
make this theorem independent on the first one.
Lemma 3.3. If V is a Stein manifold and SF is an 6-module which admits of a
finite resolution with free 6-modules of finite rank, the ~Extpc(V; W, Q.) is null for p # n.
Moreover, the spaces T(V, a?) and Ext"(F; gF, Q) have natural FS and DFS structures,
respectively, such that the bilinear map (*) yields a topological duality between them.
Proof. One proceeds by induction on the length of the resolution of gF.
(b) In this section we define the trace map on a complex space. First of all
we prove the following:
Proposition 3.4. Let X be a complex space, V an n-dimensional manifold,
f: X ->■ Va closed immersion and §• e Co1i(A'). Then for any integer p there exist
natural isomorphisms
Ext?(Z; SF, Ki) * Extrp(V;f;t (SF), QK)
Ext'pT; SF, JQ~ Ext"+"(F;/, (SF), Qv)
which are functor ial in SF and compatible with the short exact sequences. Moreover,
if V c= V is an open set, then these isomorphisms are also compatible with the iso-

Vil. DUALITY ON COMPLEX SPACES
261
morphisms given by the immersion X' =f~1(V) -* V and with the morphisms given
by the inclusions V <= V, X' <= X.
Proof. We will show the way of establishing the isomorphisms from the
statement; the verification of the other assertions will be then tamely done.
Let 3* be an injective resolution of the (9K-complex Kv. This is an injective
resolution for the complex Qv[n], hence the invariants on the right hand of the
statement of the proposition are the cohomology groups of the complexes
Homc(V■,;„[&), 3'), respectively Horn (V;]*($), 3').
From the natural isomorphism
Hon^JJ^S), 3') ~ HomaJJ*{&), Hom3yV,(6x), 3'))
one derives that these complexes are isomorphic to the complexes
HomjT ; 9, Hom3v(J*(0x), 3') | X)
and
HornCT; S, Hom%0^ôx), 3*) | X).
respectively.
In order to finish the proof it is enough to show that Homsv{f^{Qx), 3') \ X
is a resolution of K'x with objects Homc(I; gF, *) and Hom(I; gF, *)-acyclic. If X
is an arbitrary ^-module, then there are canonical isomorphisms
Hom^C, Hom@vCft(3x), 301X) ~ Homs^(/»(Jt), *r),
which are functorial in %. As a consequence, the sheaves Homa (/*(<S>x)> 301 X
are c^-injective. The morphism Kv -> 3" induces a morphism
HomaJUm(6x), Kv) - HomaJJ^ex), 3")
and the proof of the proposition will follow by showing that the latter is a quasi-
isomorphism. By 1.21 it is enough to show that the sheaves Krv and 3r are
Homa (/*("*), *)-acyclic. Since f*(&x) is a coherent fiy-module, we conclude
the proof by the fact that the stalks Kv x and Srx are injective &v ^-modules for
all x e K
Corollary 3.5. Under the assumptions of the proposition, there exist iso-
ir.orphisms
Ext\Sf,Kx) si Extn+rU*(.Sf), Ov)\X,
which are functorial in SF and compatible with the short exact sequences. In particular,
W(KX) * Exr^iej, ny)\x.

262 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Corollary 3.6. Under the assumptions oj the proposition, suppose moreover
that V is Stein and that j*(@x) admits a finite resolution by free &v-modules oj
finite rank. Then jor any p > 0,
W(X, Kx) = 0.
Proof. Obviously, H"L{X, Kx) ~ Extpc(X; 6X, Kx) ~ Ext?+'(F;/„((SX), Qy) and
apply lemma 3.3.
Remark. If L is a compact of V and K =f~1(L), then, as in the proof of
the proposition, one can show that
Ext£(*; &, K'x) « Extr'(^,7*(n Sly).
Let V be a paracompact complex manifold of dimension n. Since Kv is
quasi-isomorphic to Q[n], H"(V, Q) ~ H°(V, Ky). Consequently, the trace map
can be considered as a linear functional on H®(V, Kv), which is also denoted by Tv.
We now consider a closed immersion V—^W, V and W being paracompact
complex manifolds. Let n = dim W. By applying 3.4 we get the isomorphism
HXV, Ky) (^ Ext»(K; £y, Ky)) * Ext%W;f^v), *V) ^ Ext%W ;/*(€> v), K'w).
By comsposition with the morphism
Ext°c(W;f,(e>y), K'w) - Ext%W; Gw, K'w) ^ H%W, K'w)
given by the map &w ->/.J;(£V), we obtain a linear map
TVIW: H%V, KV) - H%W, K'w).
A calculation shows that this is compatible with the trace maps Tv and Tw.
Let now X be a paracompact complex space. If U is a Stein open subset
of A'and /: U -> V is a closed immersion where F is an open set of some
numerical space, then H%U, Ky) ^ Ext%V ; flf(<3v), Kv). The morphism 6V ->
-*f*(Qv) induces a morphism
Exi%V;j^ev), K'v)^Ext%V; <?„, Ky) =* H%{V, Ky).
We thus obtain a morphism //"(£/, A"^) -> /ï^J7, A'^) and hence, by
composition with TK, a morphism
7V H°C(U, KX)-+G.
This morphism is independent of the immersion /.

VII. DUALITY ON COMPLEX SPACES
263
Consider a locally finite covering If of X by Stein open subsets which
are sufficiently sirall. If U and U' are in °TC, then the composite morphisms
n%u n u\ k'x) - h%u, k-x) ^ c
and
m'JJ n W, Kjtf - //"(£/', Ai) ^4 C
are equal. By 3.6, H\ (U n ï/', A"*) = 0. From these facts and from the
Mayer-Vietoris exact sequence
//»(£/ fl £/', Kx) - //?(£/, A*) © //?(£/', A*) - #c°(£/ u [/', A*) - //>(£/ n U', Kx),
it will result that Tv and 7V determine naturally a linear map on H%U U £/', A^-).
The above reasoning can be iterated for finite unions and this thing is
sufficient to enable us to extend the maps Tv to a linear map on H%X, Kx), denoted
Tx. This is independent of the covering IE and we call it the trace map of X.
Let SF be an analytic sheaf on X. For any integer p there exist bilinear maps
of type Yoneda
H\X, SF) X Extc-p(Jr; SF, K'x) -* H%X, Kx)
H"C[X, SF) x Fx.V\X; SF, Kx) - H%X, Kx).
By composition with the trace map Tx: H°{X, Kx) -> C we get bilinear maps
(*) H"(X, SF) x Extc-'(JT; SF, A*) -* £
(**) #*<T, SF) x Extp(A"; SF, K'x) -> C-
These are functorial in SF and compatible with the short exact sequences and with
the "localization" (i.e. with the spectral sequences associated to some coverings,
which connect the global Ext's and the Ext's associated to the open sets of the
covering) and also with the inclusions of open subsets of X.
(c) In this section we prove the first theorem of absolute duality of Ramis
and Ruget.
Theorem 3.7. Let X be a finite-dimensional complex space with countable
basis. For any coherent analytic sheaf SF on X and for any integer p there exist a
unique QFS structure on #P(A",SF) and a unique QDFS structure on Ext'^X;^,K'X)
such that the trace map Tx induces a topological duality between the separated
spaces associated to these spaces. Moreover, the separation of //P(A",SF) is equivalent
to that of Ext^-^A"; SF, Ai).

264
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
We first consider topologies on the invariants H\X, SF). Let If be a locally
finite covering by Stein open subsets of X. The complex of cochains C'Ç\l, 8F)
is naturally an FS complex. ZV{M, &) is a closed subspace of C(% SF). So
H"(X, SO ~ Zp(1f, ^)/Bf(fll, SF) has a QFS structure. We show that this is
independent of "If. Let *? be another locally finite covering by Stein open sets. There
exists the third covering ^f, "$ < "If, °W < °f. The morphisms of complexes
C\M, SO -* C'(-"VP, SF), c"(T «0 -»• C"C"V?, ^ ),
obtained by means of some refinement functions, are continuous. Then the assertion
follows by lemma 1.32. The topology thus defined is called the natural QFS
topology on H\X, gF) (according to [93] in a more general situation).
In order to prove the theorem we make some preparations.
Lemma 3.8. Let X be a complex space and SF e Coh(X). Suppose X is
embedded in a Stein manifold V by means of an immersion f such that /^.(SF)
admits a finite resolution with free &v-modules of finite type. Then Ext£(A"; SF, Kx)
is null for p ^ 0 and Ext°(A"; 8r,Kx)has a natural structure of DFS space such
that the bilinear map (*) yields a topological duality between it and Y(X, SF) {the
latter being endowed with the natural FS structure).
The proof follows by 3.3 and 3.4.
We now recall some facts about precosheaves. A precosheaf of sets (abelian
groups...) over a topological space A" is a covariant functor defined on the
category of the open subsets of X with values in the category of sets (the category
of abelian groups ...), hence a system ($>(U), py)u.uay where a)(C/) are sets
(abelian groups.. 0 and p^: a)(£/) -> sD(F) are maps (group homomorphisms ...)
such that $ = the identity and p^ = p£p^(£/ <= V c W).
If 5)11 is a sheaf of abelian groups on A", then the assignments {[/ h> Tc{U, #tl),
[/cFi->p5! = the trivial extension of sections} define a precosheaf of abelian groups
denoted by $\lc.
Let â be a precosheaf of abelian groups and "If an open covering of X.
Consider the complex C.^lf, a)) of finite chains given by the formula
cp(%m= © ®(^-„ n... n £/,.,),
(J0...-, iP)
whose differentials are defined naturally by alternate sums. Thus, one obtains
the very complex of finite chains associated to the nerve of the covering LU, which
is defined by the system given by ®. The homology groups of this complex are
denoted by //.fit, a)). The morphisms p^': â)(C/() -> â)(A") define an augmentation
morphism C0flf, -S>) = © ®(£/;) -* ®(A"), hence a morphism H0(<VL, ®) -► a)(A").
t
Lemma 3.9. Let X be a paracompact space, L\l a locally finite open covering,
and M a flabby sheaf on X. Then Hp(% S>Tic) = 0 for p^\ and H0(% SHC) =
= SHLJ(X) = rc(A", £11).
Proof. The complex C.0^> S\lc) is the inductive limit of the complexes
Cm(*Uf,SKc) where nty are the finite parts oHf. Therefore //#01E>sllc)-lim H.{%, ®lc)-

Vil. DUALITY ON COMPLEX SPACES
265
On the other hand, FC(X, Sic) = lim Tc( (J U, all). We may thus assume the
> U^Mj
covering from the statement finite, namely, L\i = (£/;)o<j<«- We may also consider
only alternate chains. For each simplex s = (/0,. - -, ip), denote Us = £/,-„ fl ...
... n Uipa.ndSH,=§mu,.Lstep.t,Sl) =_© SH,andrf:e^nf,^ll)-*^(ir,^)
dim s = p
p
be the morphism given by d\&\ls = £ (— \)k h, where yt: #ll,-0.. ,„ -> ©H/0 ..?fc ...,-„ is
the inclusion map.
One thus obtains a complex of sheaves. The inclusions a>ll; <= STc yield an
augmentation morphism <£0(f\.(, $11)-> all. In this way we get a resolution of I'll:
indeed, if x is an arbitrary point of A", then the complex S.CIf, s^L)x coincides
with the complex of chains associated to the nerve of CUX whose coefficients are
in the group SHlx, where LUX consists of the open sets U e 'U such that x e U. We
therefore have an exact sequence
o -* e„cir, cPii) -A-... -A- e0(-ïf, sii) -* s'il -* o.
The objects of this sequence are YC{X, *)-acyclic. It is sufficient to remark that
if all is flabby, then for any open set U the sheaf Sll^ is TC(A", *)-acyclic
(consequence of the long cohomology with compact supports sequence associated to
the exact sequence 0 -> SK,U -> Jtl -> SfHx\v -> 0 and of the fact that otlx\i; is a soft
sheaf). One can easily check the equality TC(A", <£.(% 5)11)) = C-Clf, Dllc) and the
conclusion of the lemma follows.
The proof of the theorem. Let gF e Coh(Y). Consider a locally finite covering If
of A' by Stein open sets. Suppose the covering sufficiently fine such that lemma 3.8
can be applied for any U e "U. This lemma applies also for finite intersections of
open sets of tSlf. The group H'{X, SF) are the cohomology groups of the complex
C'(f\.i, &) of FS spaces. By 3.8 the trace map yields a topological duality between
this complex and the DFS complex of finite chains
C.(1f, Ex&Sr, Kx)\
where Ext%S:,Kx) is the precosheaf U h> Ext°(£/; S\ K') (the corestriction mor-
phisms are defined in a canonical way). We show that the homology in the
dimension p of C.Ç\l,Ext°(®,Kx)) coincides with Extf(A*; W, Kx); then the theorem
will follow by lemma 1.28, 1.30 and 1.31. In order to prove the above assertion
consider an injective resolution 3" of Kx and the bicomplex
C" = C.p(% HomQx{W, 3")c).
We show that both associated spectral sequences degenerate. One has 'E{'" =
= C_p(% ExtXW, Kx)), hence by 3.8,
'£"• " = i ° 'f q # °
\C.^,Ext%9,Kx)) if?=0.

266
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Accordingly,
>Ep« = \ 0 if q* 0
\H-p(^,ExtXSy,K£)) if q = 0.
The second spectral sequence has the former term
"E{>o = H_p(M,Hom£x(W,Zq)c).
Since the sheaves HomQx{8f, 9q) are flabby, by lemma 3.9 we get
1 I Hom^; ^,3») if
p * 0
/>=0,
therefore,
\*x\<{X;$,K'x) if p = 0.
The conclusion will follow from the properties of the spectral sequence.
(d) In this section we prove the second theorem of absolute duality of Ramis
and Ruget.
Theorem 3.10. Let X be a finite-dimensional complex space with countable
basis. For any coherent analytic sheaf W on X and for any integer p, there
exists a unique QDFS structure on H%X, of) and a unique QFS structure on
Hxt~p(X: 3", Kx) such that the trace map Tx induces a topological duality between
their separated spaces. Moreover, the separation on H'XX, &) is equivalent to that
of Ext1-^*"; &, K'x).
For the proof of this theorem we need some preparations. Some of them
are consequences of the results in Chapter I; here we give a straightforward proof
to make the duality stated in 3.10 independent of that chapter.
Assertion 3.11. Let V be an open set of some numerical space (£.", n ^ 1.
Then H"(V,6) = 0.
This fact, which has already been applied in the proof of theorem 1.2.9,
follows from the properties of the Laplace operator (see Ch. I).
Lemma 3.12. Let V be a complex manifold and let K be a compact of
V. Then the canonical map
6{K) -* Hl(V\ K, 6)
is continuous (<S(K) is endowed with its natural LF structure and Hl(V\ K, 6)
with the topology given by the Dolbeault resolution).

VII. DUALITY ON COMPLEX SPACES
267
Proof. We must show that for any neighbourhood U of K the composite
map <£(£/) -> <S(A")-► H]{V\ K,6) is continuous. From the commutative diagram
I I
<3(/Q -* //K^X AT, <S>)
there results that it is enough to prove the continuity of the map
hence we may suppose U = V. Let (/„)„ ->/ in £(V). Consider a function
<peC™(F), equal to 1 on a neighbourhood of K. One can easily see that d"((pf,X
rf"(<p/) are cycles of rc(P\ K, &0'1) and their classes in H%V\K, G) are just
the images of the elements /„, / under the morphism £{V) -> H\{V\ K, 6). But
^"(<p/n) ~* d"(<p/) and the lemma is concluded.
Lemma 3.13. Let V be a Stein open set of some space C" and K a Stein
con-pact of V. Then HPK{V, fi) = 0 for p ^ n and the space H£{V,Q) can be
endowed with a natural FS structure such that its strong topological dual is
isomorphic to the space &{K) {the latter being endowed with the natural LF structure).
Proof. We first consider the case n ^ 2. We have the exact sequence
(*) ... <- HP(K, 6) *- HP(V, 6) <- HP(V\ K, 6) <- HP~\K, 6) <- ...
Since Hq(K, 6) = 0 for g $s 1, it follows that the continuous map
H'XV,6) <- H't(V\ K,6)
is bijective (the topologies are induced by the Dolbeault resolution).
From lemma 3.2 and the remark which follows it, there results that
H%V\ K, G) is separated and that the map T(V, Q.)^F(V\K, fi) is bijective.
From the exact sequence
(»*) ... -* Hi{v,Q)-* h\v,a) -* hp(v\ k,a)-* hpk+\v,a)-*...
we deduce that HPK(V, Q) = 0 if p = 0 and p = 1.
By (*), the continuous maps H?(V,&) <- HP(V\ K,6) are bijective for
any integer p, 2 < p < n — 1. We then deduce from 3.2 that HP(V\ K, G) = 0
for these integers. By applying again the remark from lemma 3.2 it follows that
HP(V\ K,Q) = 0 for 1 < p < n - 2 (the separation of the space HÏ(V\ K, 6)
has already been proved !). Consequently, by means of (**), HPK(V, Q) — 0 for
2 </?<«— 1. Moreover, by 3.11, one has Hl(V, Q.) = 0 for p > n.

268 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
It remains only to prove the last assertion of the lemma. By assertion 3.11
one will deduce that Iif(V\ K, 6) is separated, hence DFS. The map <3(K) ->
-> H}(V\ K, 6) is continuous (3.12) and bijective in accordance with the sequence
(*) (n ^ 2). By the closed graph theorem it will be a topological isomorphism.
The exact sequence (**) shows that the map Hn~\V\ K, fi) -> Hg{V, Q) is
bijective. The space H"'\V\K,Q) is separated (since H*(V\K,6) enjoys this
property) and it is in duality with Hl(V\ K, &).
The proof of the lemma in the case n ^ 2 will be concluded obviously. The
case n = 1 can be treated similarly.
Remark. In particular, it follows in virtue of this lemma that the natural LF
structure on G(K) is DFS. This fact can be readily extended to spaces of the form
gF(/Q, âFeCoh(F); then there result the same property for the Stein compacts
of an arbitrary complex space.
Lemma 3.14. Let V be a Stein open set of some space <£." and K a Stein
compact of V. If âF is an Q-module which admits a finite resolution with free
6-modules of finite rank, then ExtpK{V; SF,fi)= 0 for p j= n and the space
Ext"K(V; âF, O) can be endowed with a natural F S structure such that its strong
topological dual is isomorphic to &(K).
This follows by the previous lemma, through induction on the length of
the resolution.
Lemma 3.15. Let X be a complex space, K a Stein compact of X and âF
an &x-module. Suppose that X is embedded in a domain of holomorphy V by an
immersion f such thatfj^) has a finite resolution with free <3v-modules of finite
rank. Then Ext£(A"; SF, K'x) = 0 for p ^ 0 and the space Ext^; âF, K'x) can be
endowed with an FS structure such that its strong topological dual is isomorphic
to $(K).
This follows from lemma 3.14 and the remark made to proposition 3.4.
We close the preparations for the proof of the theorem by a general fact
of the sheaf theory and by indicating how the invariants H'(X, âF) topologize.
Let A" be a paracompact space, let SI = (£.). be a locally finite covering
of X by compacts and let 3\l be a sheaf of abelian groups on X.
For an integer p, denote
cp(&,mi£)= II rK.(x,m.).
dim s=p
The inclusion maps TK(X, Sic) <= TK{X, o)Tl) for K <= K' define naturally maps
by means of alternate sums.
One can easily see that in this way we get a complex C.(S\, °^sù' ^e complex
of chains associated to the nerve of the covering Si, and to the system of
coefficients K ^ TK(X, mi).

Vil. DUALITY ON COMPLEX SPACES
269
The inclusion maps rK(X, $11) <= T{X, S\l) define an augmentation morphism
c0(si, ê)iia) -* r(x, $\i).
Lemma 3.16. Let X be a paracompact space, 31 = (/Q; a locally finite
family of compacts whose interiors cover X and let Sic be a flabby sheaf on X.
Then the homology groups of the complex C.(<9î, S\l£) vanish in all dimensions
except the dimension zero which is canonically isomorphic to T{X, STi).
Proof. Let I'll -> 3* be an injective resolution. By using the spectral sequences
associated by the double complex
£"■" = . II rKs(x, s<),
dim s = p
one can easily derive that, in order to prove the lemma, we may assume S\l injective.
Let i be an index of the covering. Since %°K.(X) is injective (1.27), there
exists a morphism <p; : S>Tl -> Sfjc/oJIl) such that the diagram
0 _► %°K,(M) —» ^11
is commutative. The morphism <p : Sll -> H %°K.(S}1), defined by the morphisms <p„
is injective (the interiors of the sets K, cover XI). There exists <Jj : J\ 3<1, (Dli)-*Sil
i
such th?t <\i(f> = id. Denote by (J;; the restriction of <\i to yt°Kl($\i). The following
equality holds 1 = £ <k<P« (II ^kX311) = © ^kX®1), tne family of supports
; i i
being locally finite). Define
%■ II rKs(x,3\i)^ n rKs(x,m
dim i=p dim s=/> + l
and
Q-1:r(X,SI[L)-+TlrKXX,SK)
i
by the formulae
e/J)/„.../p+1='i14'«,?«,C/, î .- )
(Supp ^,.,9;,f/. * . )<=/£•,,.../„ + ,),
and
e_!(/) = (^;<P;(/))h respectively.

270 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Thus, one obtains a homotopy for the complex
... -► C,(a, cTia) -► C0(Si, 8lt&) -► T(X, S)\i) -► 0
and the lemma is proved.
We now consider topologies on the invariants H'C(X, SF) (X being a complex
space with countable basis and SF e Coh (X)). For any Stein compact K the space
&(K) = r(A:,âO has a natural LF structure given by the equality &(K) = lim &(U).
UziK
By a result due to Grothendieck (1.2.10 or the remark to lemma 3.13), this
topological structure is DFS.
Let a bea locally finite covering of X by Stein compacts. Since H"(K, 8F) = 0
for q ^ 1 and for any Stein compact K, H'(X, SF ) are the cohomology groups
of the complex of finite cochains C'^Si,^ ). For an integer/;, Cpc(Si, &) = © &(KS).
dim s=p
Accordingly, each C%(§1, SF) has a natural DFS structure and one can easily show
that the differentials Cpc(&, F) -»• Cpc+\Si, SF) are continuous. Thus on the invariants
H'(X, SF) a QDFS topology is obtained. This topology is independent of the
covering Si (indeed, two such covering can be refined by a third one and apply lemma
1.33) and it is called the natural QDFS topology on H'(X, SF).
The proof of the theorem. Let 8FeCoh(X). Consider a locally finite family
Si = (K;)i of Stein compacts of X whose interiors cover X. Suppose Si sufficiently
fine such that for any KeSl there exists a neighbourhood U enjoying the property
that one can apply lemma 3.15 for (U,K,$:\U). There exist natural isomorphisms
ExfK(X; &, K'x) ~ Exf^U; SF, K'x).
Thereby Ext&(^; SF, K'x) = 0 for p ^ 0. Moreover, the space Ext£(A"; SF, K'x) has
a natural FS structure and it is topologically isomorphic to the strong dual of ^(K).
These facts remain true if K is replaced by an intersection of compacts
from Si and one has comatibility with inclusions KsczKt, tczs.
The groups H'(X, SF) are the cohomology groups of the complex of finite
cochains C'c(Si, SF) of DFS spaces. In accordance with the above said the topological
dual of C'c(Si, SF) is isomorphic to the FS complex of infinite chains
C?(Si,Ext\(X;§,Kx)), where Ext|(A"; SF, K'x) is the system of coefficients
K h> Exi^X; &,KX) (the connecting morphisms are canonically defined).
As in the case of theorem 3.7, the proof ends by showing that the homology
of this FS complex is Ext'(X; SF, K'x) {modulo the sign of the dimension). This
thing follows by using the spectral sequences associated to the double complex
C"> = C- (SI, HomeJ^, Sq)&), where 3' is an injective resolution of Kx.
The fact that the duality isomorphisms of theorems 3.7 and 3.10 derive from
the Yoneda maps can be verified first on the particular dualities from lemmas 3.1,
3.2, 3.13, and for the general case one can use the properties of functoriality and
compatibility with spectral sequences.

Vil. DUALITY ON COMPLEX SPACES
271
§ 4. Duality on complex manifolds
(a) For the case of manifolds we restate the results obtained in the previous
paragraph and obtain the duality theorems on complex manifolds due to Serre
and Malgrange.
Theorem 4.1. Let X be a complex manifold with countable basis of dimension
n and Q. the sheaf of germs of holomorphic forms of maximal degree.
Then for any coherent analytic sheaf W on X and for any integer p there
exist a unique QFS structure on HP(X,S") and a unique QDFS structure on
~Ext"~p(X; SF, Q) such that the trace map induces a topological duality between the
associated separated spaces. Moreover, H"(X, SF) is separated if and only if
Fxtr^O*"; 9, fi) is separated.
Theorem 4.2. Let X be a complex manifold with countable basis of dimension
n and Q the sheaf of the germs of holomorphic forms of maximal degree.
Then for any coherent analytic sheaf SF on X and for any integer p there
exist a unique QDFS structure on H£{X, 9) and a unique QFS structure on
Ext"~"(X; SF, Q) such that the trace map induces a topological duality between
the separated spaces associated to these spaces. Moreover, H£(X, SF) is separated if
and only if Ext"~p+1(X; SF, Q) is separated.
Proof. The dualizing complex Kx is a resolution of Q[n], hence
ExtlCT; SF, Kjc) =i Exte+9(A"; 9, Q)
and
Ext|jC(A-; 9, Kx)^ Ext"lqc(X; 9, Q).
One applies then 3.7 and 3.10.
Remarks. (1) The topologies on the invariants H'(X, SF) (Hè(X, SF), respectively)
are the natural ones, defined in the previous paragraph by means of the Cech
complex associated to a locally finite covering by Stein open (respectively compact)
sets. In the next section we will show that these topologies coincide with those
induced from the Dolbeault-Grothendieck resolutions.
A similar assertion holds for the invariants Ext*, Ext;.
(2) If SF is a locally free sheaf of finite rank, then ExV(X; SF, fi)~ H\X, SF ® fi)
and Exfc(X; SF, Q) ~ H'(X, SF ® Q), where 9 = Home(9, 6) is the dual of SF;
the duality results can be expressed in cohomological terms only.
(3) Theorems 4.1 and 4.2 can be proved straightforwardly, without the
construction of the dualizing complex (which uses the Frisch theorem of noether-
ianity) and the formalism of the trace map on the singular case. In order to prove this
we need only lemmas 3.3, 3.9, 3.14, 3.16 and some simple considerations of spectral
sequence! One proceeds exactly as in the proof of theorems 3.7 and 3.10. We
give a sketch for 4.1 (according to [87]).

272 ALGEBRAIC METHOrS IN THE GLOBAL THEORY OF COMPLEX SPACES
Let 1C be a locally finite covering of X by Stein open sets. The invariants
H'iX,^) are the cohomology groups of the Fréchet-Schwartz complex C'(V,8r).
This is isomorphic to the strong dual of the DFS complex of firite chains
C.(tf, Ext?(&, Si)),
where Ext%(8r, Si) is the precosheaf U i-> Extg C(U; S", Si) (the corestriction morphisms
are canonically denned). Let 3" be an injective resolution of Si. Consider the
complex
C" = C_,(nf, Home(S=, S")c).
The associated spectral sequences degenerate. Thereby the homology of the complex
C.(nf, Ext"c(&, SÏ)) coincides with Ext'(X; §% SÏ), etc.
We now particularize 4.1 and 4.2 for the case of compact and Stein manifolds.
Theorem 4.3. Let X be a compact complex manifold of dimension n, let SI
be the sheaf of germs of holomorphicforms of type (n, 0), and SF a coherent analytic
sheaf on X.
Then for any integer p, the complex vectorial spaces H"(X, $f) and
Txt"~p(X; SF,Q) are finite-dimensional and the trace map yields a duality between
them; in particular these spaces are of the same dimension.
Proof. By Cartan-Serre's theorem (III. 2.10) the vectorial spaces H'(X,W)
are finite-dimensional. Let 'M be a locally finite covering of X by Stein open sets.
By applying 1.29 to the morphism C^M^i, &) -»• Z"(^(, ®) we deduce that
H"(X, Sf) is separated. The assertion from the statement follows now by 4.1.
Theorem 4.4. Let X be a Stein manifold of dimension n, Si the sheaf of
germs of holomorphic forms of type (n, 0), and & a coherent analytic sheaf on X.
(i) Ext?(JT; &, SI) = 0 for any p <£ n and Ext?(Jf ; Sr, SI) is isomorphic (via
the trace map) to the topological dual of the Fréchet-Schwartz space T(X, $•).
(ii) For any integer p, the space HP(X, S5) is separated (hence DFS) and its
topological dual is isomorphic (via the trace map) to Ext"~p(X; &, SÏ).
Moreover, the FS topology induced on Extn~p(X; S\ SI) from this duality
coincides with that induced from the coherent sheaf Ext"~p($;, Si) by means of the
isomorphism Exf"^; 3% Si) ^ T(X, Ext"'"{7, Si)).
Proof. The assertion (i) follows from 4.1 by theorem B. We now prove (ii).
Let (Ur)r be an exhaListion of X by relatively compact Stein open sets such
that the restriction maps T(Ur+1, 6) -> T(Ur, 6) are dense. Suppose moreover
that the topological dual of T(Ur, Ext"~p(S:, Si)) is algebraically isomorphic (via
the trace map!) to H^(Ur, &) for any r.
If 3C e Coh(.Y), then the maps T(X, 3C) -► T(Ur, T), r(£/r+1, %) -► T(Ur, 3C)
are also dense. From the topological isomorphism
T(X, %) ^ lim T(Ur, T)

VII. DUALITY ON COMPLEX SPACES
273
one will derive the algebraic isomorphism
lim (JT(Ur, ?C))' ^ (HX, ?0)'-
r
■Apply 'this to % — Ext"~r(&, Q). It results that the topological dual of
T{X, Ext"~ P(S', Q)) is algebraically isomorphic to lim Hp(Ur, 21). Consider on
Ext"^; ®, Q) ^ T(A'. Ext"-p(SF, fi)) the FS topology from Exf-"(Sf, Q), and
on the space HP(X, Sr) — lim //?(£/„ $•'), the DFS topology obtained by means
r
of the algebraic isomorphisms
HT, Erf" ''(*, Œ))' - I'm HVC(U„ Sr).
r
The pairing
HP{X, $) x Ext"-"(JT; 3-', Q) ->• C
defined by this isomorphism coincides with that given by the trace map, since it
is obtained from the pairings HP(Ur, Sr) X Ext"_"(c7r, Si, Q.) -► C- The conclusion
of (n) follows by 4.2.
It remains only to prove that for any r, H"(Ur, &) is algebraically isomorphic,
by means of the trace map, to the topological dual of r(Ur, Extn~"(SF, Q)). If §•"
is locally free then it follows easily, by (/) (there is a single, non null Ext, namely,
v
Horn (SF, Q.) ^ of® Q...). For the general case one considers a finite, resolution
of i¥ by locally free sheaves of finite rank on each Ur and one proceeds by
induction on the length of this resolution. The theorem is thereby proved.
Remark. Assertion (ii) makes theorem 1.2.1 more precise: the topology on
the invariants //*(AT, ■S') obtained by duality from the invariants Ext'(X; &,Q).=
—T(T, Ext'(Sr, Q)) coincides with the natural topology obtained by means of
a locally finite covering by Stein compact sets.
(b) In this section we will present the duality on complex manifolds following
Serre [75] and Malgrange [53].
(I) Let X be a complex manifold with countable basis of dimension n.
We still use the notation 6, §"-i, W-i, Qp. By S we mean the sheaf of germs
of C* functions. Recall the following result [54]:
For any point x e X, êx is a flat 6x-module.
Let Sr be an (S-module. By Dolbeault-Grothendieck resolution
0 -► fi" -► §p>° —> ... -^-» £"■" -> 0,
we get by tensoring ® $& an exact sequence
(*) 0 ->■ Qp ®e$f ->■ SPfi^Q^ ^—> ... -^-> &>>■" &£& -> 0.
The sheaves §">* <g>e&- fi.aye structure of ê-rnodules*, hence they are. T(X, *) and
TC(X, *)-acyclic.
18 - c. 2398

274 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
For convenience we denote:
&.<!(&) = &™ ®ô®, EP-i{W) = r~(A', ê'-i ®sSr)
and
Accordingly, H'(X,Qp®ef) are the cohomology groups of the complex
0 -► E£°(&) -%• E"-1^) -► ... ^—> £"•"(&) -► 0
and ^'(J.n"®^) are the cohomology groups of the complex
0 -► £f!(SF) -^» E?\Sr) -+ ... ^> £f"(3F) -* 0.
In particular, ff'ÇX,^) and 7/J(.V, ^) stand for the cohomology of the complexes
0 -► E°'n(S>0 -»• E"'1^) -»••..-»• £o,"(S0 ->■ 0,
0 -► £?'°(^) -* E^C^) -»•...-»• EC°'"(SF) -»• 0.
"Suppose now S- is coherent and consider TVS structures. Use of the following
result [54]:
If U is an open subset of .some numerical space and
r(u;$y-ï->T(.u,èy
is a morphism given by a holoinorphic matrix, then Im 9 is a closed subspace
{consider on § the FS structure given by the uniform convergence of functions and
all their derivarions on any compact).
Let U be an open subset of X, contained in a cart and such that one has
a finite presentation
6'\U -> 6S\V -> $\U-+0.
By'tensoring S"'9®^ one derives the exact sequence of $\U-modules
(§p*\uy -► (S"* \uy -> (&«(&))&** o.
Since the cohomology of the ^-modules is trivial, one gets the exact sequence
T(u, êp'«y -*-T(u, è1"")* -► nu, &*<(?)) -► 0,

VII. DUALITY ON COMPLEX SPACES
275'
where r(U, $"'")' and r(U, i>p'qY are free r(U, ©)-modules of finite rank;
hence they own natural FS structures. Moreover, the former morphism is
continuous and its image is closed. One thus obtains an FS structure on
r(U, §"'"(&)) and by the Banach open map theorem it is obvious that this topology
is independent of the sequence &'\U -> £S\U -> Sf\U -> 0.
For an open subset V of X we define on T{ V, $"• " ®e &) the smallest
topology making the restriction maps T(V, S,ptq®eW) -► r(U, êp'q®e &)
continuous, where U a Kis an open set as above. Thus, $p'q(W) becomes an FS sheaf.
A family of seminorms giving the topology of Ep'q(W) = T(X, êp'q®e &) is
defined in the following manner. Consider an open set U as above, an epimorphism
Gs\ t/-»SF'| U, a compact KczU and an operator of derivation D. For/e£''"'(Sr) denote
Pu.k.*.d{J) = inf^A'.oM,
where u> is an element of T(U,S<P">(6S)) whose image in T(U, êp'q(&)) under <p.
coincides with the restriction of / and p D is the seminorm '"sup" given by K
and D (one identities V(U, $p'q(£s)) with a free V(U, S)-module of finite rank ...).
The space Ep-q($?) is dense in £"">(&): indeed, any feEp">(&) arises as the limit
of a sequence (/„)„, /„ •=--• £ p;/, where (p;),>n >s a suitable partition of unity.
1=0
If SF -> (|- is a morphism in Coh(A"), then we obtain continuous morphisms
gP'^SF) -> §*"*(<§) (by means of suitable carts anH- finite presentations for SF
and ef). Moreover, the maps Ep">(Sf) -> E'"'!{<^) are strict, as follows easily from
the exact sequences
£7'.«(<S=) _► #"«(c() -»• £"'«(3C),
where ft ^ Coker (S>= -> ej).
By the local use of epimorphisms £s -* êf -* 0 one readily show that the
maps d" : f'^SF)-»• £""'+1(SF) are continuous.
In this way Ep''(Sf) is an FS complex and we can consider QFS structures
on the invariants H'(X,Çlp®g$f), peculiarly on H'(X, &). For each compact A! of
X,rK(X,§>"">(&)) is a closed subspace of E'"«(W). On ££-"(SF) = VC(X, ê<"%SF)) =
lim TK(X, &"">(&)) consider the topology of inductive limit. Ep,q (_&) is a
A'
strict inductive limit of FS spaces and the inclusion ^-'(Sr) <= E"'q{3r) is
continuous, A sequence (/„)„ of elements of ££,?(SF) converges to an element
f e Eqc-q{&) if and only if there exists a compact K of X such that Supp/„,
Supp / <= K and (/„) ->/ in Ep'q(S:). One easily checks the continuity of the
differentials Epc-q{^)^ Epciq'\W), Thus, £^'*(Sr) is a complex of locally-convex TVS.
We can consider on the invariants H'C(X, Qp®eSf), in particular on H'(X, &),
associated structures of locally-convex TVS (generally nonseparated).
Proposition 4,5 (cf. [51]). Let X be a complex manifold with countable
basis and SF a coherent analytic sheaf on X. Then the natural topologies on H'(X, 8^)
and Hc(X, Sr), obtained by means of the Cech cohomology, coincide with the topologies
obtained by means of the Dolbeault-Grothendieck resolutions.

'276
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
To prove this we need some preparations.
Let
tl d d
,400 _^ An _> ,402 ^ ...
is Is ;s
d il d
.410 _ Au -+A™-+ ...
Xs Xs Xs
A20 -> A21 -+ A* -* ...
Xs Xs Is
be a double complex of abelian groups where the rows and the columns are exact
sequence. Denote by B' the complex Ker (A0, ' -*■ A1, *), whose differential is induced
"by d and let C be the complex Ker (A''0 ->• -4"'-1) whose differential is induced
by 8.
Let p $; 0 be an integer. Consider an element çeHp(B'), where \ eZp(B').
Denote £°-p = £. Since rf£°'*> = 0, there exists frie4».rtso that rfïj0^"1 — £>■".
Let Çi./'-i = St)0-''-1. Since dl1-"'1 = 0, there exists v]1'*-2 eM1'^2 such that tfy)1'p-2 =
= ^1,i7_1. Denote £2.p-2 = §^1.^-2. we can g0 on and finally find an element Z,"-0.
Let 7] = Çp-°. The class of 7) in HP(C) is independent of all choice and .the
assignment \ h> y) defines an isomorphism HP(B') =* H\C).
Suppose now that ^p,«"are TVS and the differentials" d, 8 are continuous.
The complexes B' and C" become naturally complexes of TVS. Consider the
topologies induced on H'(B') and H\C). By the above construction there follows
Lemma 4.6. // the differentials Av-q -»• Af-^\ Av- « -V Ap+1-* are strict,
then there exist natural topological isomorphisms H'(B') ~-H'(C).
Wè shall also use the following
Lemma 4.7. Let (An)„, (Bn)n be two sequences of localfy-com'ex TVS and
fn'An^Bn a sequence of strict morphisms. Then the inorphism f : ® An -> © B„
induced by (/„),, is strict.-
Proof. Let (C„) be a sequence of locally-convex TVS. If V„ is a
neighbourhood of the origin, in C„, which is convex and disked, then © V„ is a neighbourhood
■x>f the origin in C= © C„. We "will show that the neighbourhood of this kind
constitute a fundamental system and the conclusion of the lemma will easily follow.
Let F be a neighbourhood of the origininC There exist neighbourhoods F„of the
origin in C„ which are convex and disked such. that V contains the disked convex
envelope T( U Vn) of the set U V„ in C (we have identified the elements of C„
with, the corresponding elements defined by them, in Q. The conclusion follows
by the inclusion © l/2"Fa c T( U V„).
The proof of the proposition. We first study the simpler case of the invariants
H'(X, &). Let "If be a locally finite covering of X by Stein open sets. Consider
the double complex C (ftt; BP'<i<g)e&) of the Fréchet-SchWartz spaces and the
associated simple complex C.

VU. DUALITY OM COMPLEX SPACES
27?
The connection morphisms
C'(%Sr) -* C and E°'\$) -► C
are continuous quasi-isomorphisms. The required conclusion follows by lemma 1.32*
Let now ll£ =i (f,*)re/ be a locally finite covering of X by relatively compact;
Stein open sets such that each Ut is a Stein compact. The cohomology of the
complex of finite chains C'(U, <F) coincides with H'(X, Sf). We consider on eacft
C^f, SF) — © ^(£7^...^) the topology of direct sum (in the category of locally-
convex TVS). Thus C;(-1f, SF). becomes a TVS complex. Consider on H'(X, SF) the?
induced topology and show that it is the same as the natural topology given by
the coverings of Stein compacts.
There- exis.t two locally finite coverings ST = (K[)ieIl SC = (A\)ic_r by Stein
compact sets such that the inclusions A"; c XJt <= K{ take place. The restriction maps
Q(3Ç', §•') ->■ Cc(^(, &) ->■ C(3C, $0 are continuous;, hence the morphisms induced
to cohomology-are also continuous. Since H'(CZ(3C', SF)) -* H'(.Q(3Çt &)) is a'
topological isomorphism, the assertion is proved.
Actually, we compare the topology given by C£(1C, SF) on i7c"(^, 5*0 with the
topology obtained from the complex TC(X, ï>°* • ® SF). And for this we apply
lemma 4.6. Consider the double complex of components CCÇW, &°'q® SF) on which.
we consider the topologies of direct sums (the differentials are obviously continuous?
in these topologies). For any Stein open set U, the FS topology on r(U, SF)
coincided with that induced from T(U, §°»° ®e sr)s as follows by the exact sequence
0 -> V(U, SF) -► T(U, S»'°®s SO -» Ht/, S0'1 ®e 3F)
of ES spaces and continuous maps.
The kernel of the morphism of complexes C'cÇ\t;&°>°'®e SF) -» C^f, S0'1®^)
is algebraically isomorphic to €£(% SF), and from lemma 4.7 we can deduce that
the topology of C&XÏ, SF) coincides- with the topology induced from Cê(% §°'° ® e^).
The kernel of the morphism of complexes C£% S°" ®e SF) -»• CjClC, §a"®ôSF)
is algebraically isomorphic to TC(X, S°" ®a SF). The1 maps TC(X, S0"^ SF) ->■
-* ®.T(L/;, IW®^) are continuous, since the composite maps (obtained by
restriction) r^S0''®^) -»■ ® T(Ub$P*®& Sf) are continuous (K c X being;
i
an arbitrary compact). Prove that -the topology of TC(X, l"'*®^) coincides with,
the topology induced from C°(% «""(ff)) = © r(C/j, S0'*®^). It is enough; to
construct a continuous map C°(% 8a,p(SF)) -> r,..(Z, §>0>p ®(9§r) whose composition
to the left with the map TC(X, S°" ® dW) -> C°c(% S0"®^ gives rise to the
identical map. Let (pj) be a CM partition of Unity which is subordinate to the»
covering IE. The assignment (s^ h>- S P;.s,. is the very required map. Thus the
topology of TC(X, &"'* ®0®)' coincides with that induced from the
complexes £«>• ®eSF).

"278 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
In order to conclude the proof of the proposition, it is sufficient to show
that both differentials
cpcçm. s0-" <g> $o -»• cf(ir, &«-"+i® $•), cpc(\(, s°-«® sr) -► c?+I(-ir, §°-«® sf)
are strict. For the former, the conclusion follows by 4.7 from the exact sequences
of FS spaces
V(U, ê0"® &) -»• r(£/, &°'«+1® ^) -v V(U, §0'*+2(g) SF).
As for the latter, we will construct a continuous homotopy CP+1(fl(, &0"7® S?) -*■
-> CfO^E, £0" 9® SF) and the conclusion follows. Let (p;),- be the above partition.
The map
3 = (*/. ■ ',♦>) ^ ' = ('/„...'„)• where //„ . ,■„ = Ç P; s»,... ,„
is the one required.
Remarks. 1) By the proposition it follows in particular that the topological
structure on H'c(X,Bf) obtained from VC(X, è°" ®3&) is QDFS!
2) If SF is furthermore locally free, then the invariants H'(X, SF) can be also
computed by means of the complex F(X,$f-®e&0-') ~ K0i'(s-). The topology
thus defined on H'(X,&) coincides with the existent one. A similar assertion holds
for H'ÇX,^). The proof is the same as in the proposition ([51]).
(II). Let Sf be an c3-module. For convenience, denote
K"-q(&) --= V{X, Hor»e($>\ SC'-")) = Home(S;, 5C"-"),
K^"{S-) - TC(X, Home(&, 3C"-«)) = HomejC(SF, 3C-"),
Kp.„ = Kn.q (<o) ^ r(x, &»■<') and K™ =- Kp-"(£) = fc(X, 3C-«).
These assignments are functorial.
If $? and (if are in Mod(O), then there exists a canonical <9-morphism
ff ®e ff«ms (ïf, (?) -► <$ , (/, ç) ^ ?(/).
The exterior product also defines morphisms
gP'«®e cV'«' - SC"+" '• «+■''.
Thus, one deduces for any Sr e Mod(<9) morphisms
(èp"<®e $)®e Home (S-\ Si"'-"') -► SLP+i''' «+«',

VII- PUAL1TY ON COMPLEX SPACES
279
thereby, the maps
s:E""> (Sf)®r(y,e) K?''9'(®) -> Kp+P'-3+9\
If cùsE"-" («0 and. TeKp'-«'(SF), then we set to A^e A3,+*',s+ô' instead of
£(cù0T). The operation A will be called exterior product. For any open set
U of X, (to A T)\V = (to|£/) A (r| £/). If to or T has a compact support, then
<ù f\T has also a compact support. Thus, e induces maps denoted by e too,
Epc'q{§)® K"'-^) -► KP+p'-9+9' E*-q(®)® K£'q' ($F)-+Kp+p'-9+9'.
If S*' — (9r and if we identify to with r forms {w,)1<i:Sr and T with r forms
r
(TA^Kr. then w A T = J] to; A 71,-.
/-]
If <p : Sr -> <3f is a morphism of (9-modules, then to A 9(^) = <?(w) A T for
any w6£,p" (SQ and Te Kp'■"'(§). We suppose in the following that fr'eCoh(<S)).
For p' = n — p and q' — n — q one obtains particularly the maps
££■'($0® Kn-v-n-"($) ->• A^'", £p'«(§;)® A,T/''"~W ->• A';''".
If ue^-'^) and TeK"~l'-"-"(&■) (or MeP'«(î) and Te K'^p'"-"(&)),
then we put
0,T> = ( «A T.
For J fixed in K^-"p'"-"(e) (in A?'-''"-'^). respectively), the assignment
<i)i->(w, T > yields a linear functional
Lr : £?■"($) -»• C (respectively £r : Epe-«(§) -» <C).
If TeK'c'-v-"-"(&), then by regarding Tin X"-"'"-*(SF) the corresponding
functional EP,q(&) -> C coincides with the restriction of the functional E?-"(&) ->
-> C given by T.
Lemma 4.8. The Junctionals LT : Ep'q{Sr) ^ <£. {respectively Lr : £?■ «(SF) -»• <£)
ore continuous for any TeK'c'~p-n-g(^) (respectively TeK"~p'"~«(S-)).
Proof. Let TeK?-p'"~9(&). Consider a locally finite covering nc = (f,-)?
of X by relatively compact Stein open sets and (p;); a Cœ partition of unity which
is subordinate to this covering. Suppose that É admits a finite presentation oa
each UL. By the properties of the integral it follows that
\ to A T = £ f co A (p, T)
JX i JUt

280 ALGEBRAIC METHODS IjN THE GLOBAL THEORY OF COMPLEX SPACES
(the sum is finite since Thas a compact support). Since the topology on Ep'q {&) —
— T(X, gp"(g) Sr) is defined by means of the topologies on r(Uh $p'9®e. if), we:
reduce the problem .to proving the continuity of LT when X coincides with one
of Ut 's.
Thus we may assume that we have an exact sequence of the form €r -> .? -> 0.
We obtain a monomorphism Home (SF, 3V'-p- "-«) -»• Eome{&\ 8{"-p"'-q), hence
a monomorphism A^,_,,'"-«(J)-» Acn_3''"-«(<Sr). Let J' be the image of J*
under this. One has the commutative diagram
E?'i{ôr) ». EP'"ÇW)
Lr,N /Lr
c
where the horizontal arrow is surjective. We thus reduce the question to the case
&■ = & and, by a previous remark, to the case tsr = (3. In this case the conclusion
follows from the continuity of the integral.
V
Take now Te Kn~p"'~9($;). For each compact K of X we must proye
the continuity of the composite map
rK(X, P«® SF) -> EP'"(W) -ii* C-
Let p be a C°° function on X with compact Support which is equal to 1 ou
a neighbourhood of iC. The proof is completed by the commutative diagram
TK (X, §"•« ® ff) —> £f • «(#)
Thus, we have obtained (^-linear maps T\-+ Lr which are filnctorral iri S«
i^-y. »- «(^) _► (E*<n&)y K"~p- »i« (^V> -+ c^fCS})'.
HofeoYer, we have the commutative diagram
•f v •f

Vil. DUALITY ON COMPLEX SPACES
281
(the injectivity of the second vertical arrow follows by the density of Epq(S) in
Lemma 4.9. The above maps
K"c-i>-n-i{W)^(E*'"l(S')y, K"-p'n-q{&) -»• (Epc'q ($'))'
are algebraic isomorphisms.
Proof. We first consider the case when X is an open set in C" and we have
an exact sequence of the form 6' -> 6s -*■ SF -> 0. We get an exact and commutative
diagram
o -* a-;1-"'"-«(>*) -► K^"-n-q{ls) -> a:;-"- k-«(S')
0-» (£p'«(Sr))' -*•(£''"'(6*))' -»• (£''•«(<S'))'
(for the line below we have used the exact sequence of continuous, hence strict,
maps of FS spaces, Ef,q {€)') -»• Ep'q(0s) -»■ Ef,q(Sr) -»• 0). The second and third
vertical are isomorphisms, hence the first is an isomorphism.
We show now that the map K"~''-"-q(S;) -> (E"c'q(&))' is an isomorphism.
Consider a partition of unity p = (p.), which is subordinate to a locally finite
covering 1C = (£/j)/e/ with relatively compact open sets. Let Te K"-"'"-"(¥)
be such that LT = 0. For any co eEp'q(S:), LPlr(w) = Lr(piw) = 0. Accordingly,
p-,T = 0 for any /, hence T = 0. Let now L : EP'"(f) -> C be a continuous and
linear functional. For any i the linear functional co e T(Uh $p>q® §■) m* L(p,w)
are continuous. Then there exist 7',. e rc(£/,, Hcm^S, Si"-p- "-«)) c Kn'"-n-q{è)
such that Zri(«; = L(p;co). The element J = J] T, e À?-"-"-«(SO is well defined
and L = LT. Indeed, let ue£/''(?) and L' be a relatively compact
neighbourhood of Supp co. The set J = {/ e 7| £/; n U -^ 0} is finite and the conclusion follows
by means of the relations
I(co] = L(£ p,-co) = £ L(Pico) = S Lr,.(coj = S (to A Tt =
(wA(l7-()=( (co I C/) A (£7'/) I t/-( co|C/A7,t/=(tùAr = Ir(to).
Consider now the case when X is an arbitrary manifold. Let IC = (£/,-),
be a locally finite covering of X by affine open sets where W admits finite
presentations and (p,-),- be a partition of unity subordinate to this covering. We prove
the bijectivity of the first map of the lemma. Let Te Kc"-"' "-«(Sr) be such that
Lr=0; we will show that 7=0. It is enough to show that each T\U,e K"~p- "-"(W [/,-)

282
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
is null. But LTlUi: Epc-9(8F|l7,.) ^ C coincides with the composition E^9(BF\Ui) =
= Tc(l7„ 8^(30) ^ £""(§0 -^> C and the conclusion follows.
Let now Le (E"'"(SF))'. For any / there exists T, eT(Uh Horn (3% &"-<>• "-<>))
such that LTl coincides with the composition Fc(Ut, &'q ® ®) ^- E"'« (SF) -> C-
For each pair 0',;), LTj | c/^nc/j, = LTjl UinUj, hence r,. | U, n £/,. = T; | £/,- n £/,-.
Let J be the element of Kn~"",'9(^) which is obtained by gluing together (J,.),.
We show that T has a compact support, that is Tl = 0 but a finite number of
indices. Since L is continuous, there exist C > 0 and a finite number of seminorms
/V,K°,,°,D° SUCH that
|L(co)| < C- sup />a(co), co e £P'« (S^).
a
The compacts /l" cut a finite number of open sets [/,-. If U; is not such an open
set, then one can easily see that T{ = 0. It remains only to prove that LT = L.
If co e Ev,q (&), then co = £ P,-w and the convergence is in the FS topology of Ev,q{S;).
i
The following equalities hold
I 1 I /
The bijectivity of the maps Kn'p'"-q(S:) -»• Ep"(®) can be proved in the
same manner as the particular case considered in the beginning.
Lemma 4.10. The transposed maps of the morphisms
d" : EV'q{Sf) -► £i"«+1(S;) (respectively J" : ^'"C^) -* £? 9+1(SF))
coincide with
(/■espectfve/j>
(_ l)p+«+!rf" :#"-"'"-«-l(SF) -»• /^-""'-«(SF)).
Proo/. Take T e K'c'~"-"-q-1(S:) and « e F" (ff). We shall verify that
d"a A T = (- l)p+<?+1 C co A d" T
and the lemma will be proved (the dual case is similar). Since d'(co A T) e /^c"+1"'^1=
= 0, rf(w A T) = rf"(w A T)- since ^ ^(w A r) = °> the conclusion will result
provided that the equality
d"(a A T) = d"o A T + (- l)p+«co A d'T
holds.

VII. DUALITY ON COMPLEX SPACES
283
This fact is local in nature; hence we may assume that X is an open subset of
<£" where there exists an epimorphism 6r -> & -> 0. The map Ep-q(&) -> E"-q (&■)
is surjective, hence there is a form &>' e Ep,1(6r) whose image in £"■"(&■)
is w. Denote by T' the image of T under the map JC?-"-"'"'1^) ->
^ ^'-^«-«-i^^ obviously, w A J = w' A T, d"o> A T = d"u' A 7" and
w A d"T = o)' A J7". The conclusion follows from the equality
d"(u' A 7") =d"o>' A 7" + (-lK+««' A rf'T'
(in this case w' and 7" are systems of r forms in the usual sense...).
V V
We consider topologies of strong dual on K"~p- "-"(&) and K"~v- "^\Sr) obtained
V
by means of lemma 4.9. Hence K^~v-n~q{&;) is a DFS space and its strong dual
is isomorphic to Ep*q(EF). The space £?,<7($9 is a strict inductive limit of FS spaces;
V
hence it is reflexive [68]. Accordingly, the strong dual of KK~p- "-%&) is isomorphic
to Ec^fâ). From lemmas 4.9 and 4.10 one obtains the following:
Proposition 4.11. Let X be a complex manifold with countable basis, of
dimension n. Then for any coherent analytic sheaf S on X the exterior product and
the integration of the currents of maximal degree define a topological duality between
the complexes
... -► E"'"-1^) -► E^fSr) -► £,"-«+1(S;) -»• ■.. (respectively
...->• Epc-q-\®) -► Ep-q(f) -> E°-q+1(®) ->...)
and
... *-kh-"-"-^1^) <- Kz-'-n-i(®) <-Krp-"~,i~1(h*- ...
(respectively ... *- Kr-■■• "-*+!(#) <- Kr-p-n'"(è) <- K'-p-n-q~1 (8F) <- ,. .).
(Ill) Actually, we pass to the study of the functors Ext. Recall the following
result [54]:
For any point x e X the germs of distributions in x constitute an infective
<3X- module; in particular, the stalks 3ix,q are infective démodules.
Consider the Dolbeault-Grothendieck resolution
0 -► Qp -► W-° -^> ... ^—> 31"-" -► 0.
Let JF be a coherent <?-module. For each xeX, Ext'^W, W-q)x ~. Ext'Bjc(^x, W.px-q),
Hence Extre(^,a>Lp-q) = 0 for any r > 1. By using the spectral sequences which
connect the local Ext's with the global Ext's, one can easily realize that
Ext^, 3C-«) = 0, Fxt^,c(SF, Si"-") = 0 for r ^ 1.

284
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Thereby the invariants Ext"(T; &, Q") and Ext^; S\ Qp) are isomorphic to
the cohomoiogy groups of the complexes
0 -► Kp'°(&) ^—> K"-\i) -► .. . ^-» lC-"(&) -»• 0,
0 -► A?-°(l) -i-> A^SF) -^-» ... -^-» A£-n(30 ->• 0.
In particular, Ext*(Z; S\ Œ) and Ext*(A"; &, Q) stand for the cohomoiogy of the
complexes
0 -► K"'0^) ?—> K"-1^) -►...->■ A""'"(SF) -► 0,
o -► a:?-0(sf) ^h> a?-1^ ) -►...->■ Ac"'"(l) -»• o.
These complexes are topological and we will consider the induced locally-convex
TVS topologies on Ext (A-; &, Q) and Ext^A-; 3% Q).
Proposition 4.12. Let X be a complex manifold with countable basis and
let & be a coherent analytic sheaf on X. Then the natural topology on Ext^A"; 8?, Q)
V
and Extc*(A"; SF, Q) obtained by means of the Cech homology coincides with the
topologies defined by means of the Dolbeault-Grothendieck resolutions.
Proof. Let It be a locally finite covering of X by Stein open sets which are
sufficiently small. The invariants Ext'c{X; &, f>) are isomorphic to the homology
groups of the DFS complex of finite chains C.C^f, Ext"(Sr, Q)).
Recall that £xt"c(Sr, Q) is the precosheaf U m* Ext"e,c(U; &, Q). Consider the
double complex Cp(1£, Homc{®, 3C"-"-«)) of DFS spaces, where Homc{^, 3V-") are
the precosheaves U m* A'f«(8F| U) =- Homc(C/; 3% 3V>-g). Let C. be the associated
simple complex. The link morphisms
C, -> C.(nf, £jtfc"(SS Q)) and C. -► A?-*^)
are continuous quasi-isomorphisms (they are in fact transposed of the corresponding
morphisms from the proof of proposition 4.5). Then the assertion from the
proposition with respect to the invariants Ext!(A"; Sr, Q) follows from lemma 1.32.
Recall how the QFS topology on the invariants Ext"(A"; S\ Q) is defined by
V
means of the Cech homology. Consider a locally finite family Si = (A,-),- of Stein
compacts of X whose interiors cover X. The invariants Ext"(^"; &, Q) are
identified with the homology of the FS complex of infinite chains C?(<Sl, Ext^fë, fi))
where Ext&(&, Q) is the system of coefficients K v-> Ext^A"; Sr, Q). The topology
induced by this identification is just the required one.
In the following we obtain this topology by means of the open coverings. Let
'1t=(£//)/e/ be a locally finite covering of A" by relatively compact Stein open sets such
that each U-t is a Stein compact. The homology of the complex of chains Cf flf,
Ext"{^, Q)) coincides with Ext'(T; Sr, fi) (in order to prove this, consider an injective
resolution of Q. and analyse the spectral sequence of a suitable bicomplex...). Consider
the topology of the direct product on each C*{% Extnc(®, «))= n Ext"c(\Jia...l(; %, fi).
Cf becomes~*,-TVS complex and we endow Ext"(A": Sr, Q.) with the induced

Vil DUALITY ON COMPLEX SPACES
285
topology, and show that it is the same as the already defined QFS topology. There
exist two coverings §L' = (K[);er and Si = (Kj)ier as above such that the
inclusions Kt <= [/,• <= K{ are satisfied. The natural maps
C-(&, Ext>i($r, Q)) -> Cf(W, Ext%9, fi)) -► C.*(&', £x4<3% O))
are continuous (they are in fact transposed from the maps C'c{Si', 39 -> C'^t, ^)->
-»■ C'(Si, 8r) of the proof of 4.5). The morphisms induced to homology are therefore
continuous, and since
H.(C:Xa%, Exi"A{9, fi))) - #.(<?(&', Ext\, (ff, Q)))
is a topological isomorphism, the conclusion follows.
Compare now the topology from Ext"(A"; 8r, Q) thus obtained with the to-
V
pology given by the complex K""{8r). We will consider the double complex of
components C*(% Horned, ^"'"'"X), endowed with the topology of the direct
product. The proof proceeds exactly as in proposition 4.5, by means of two lemmas
similar to 4.6 and 4.7 (we also notice that this bicomplex is the topological dual
of the bicomplex QÇU, &°'« ® §9 from the proof of 4.5 and this simplifies the
proof...).
Remark. From the proposition it follows in particular that the topological
V
structure on Ext"(A"; SF, Q) obtained from the complex K"''(&) is of QFS type!
(IV) We now establish the connection with the duality theorems. Theorem
4.1 can be immediately reobtained from 4.11 and the topological lemmas of § 1.
As for theorem 4.2 one may proceed as follows (a little different way
may be found in [28]). By proposition 4.11 one derives algebraic isomorphisms
°H?(X, SQ ^ ("Ext"-"^; 3\ Q))', "Exf'^X; ®, Q) ^ ("HRX, &))',
where a means the associated separated space and the accent is, as usual, the
topological dual; one may proceed exactly as in lemma 1.30 by using the reflexivity (in
fact the semireflexivity!) of the spaces E°'"(§9, KK'"~"(^).
The bilinear maps
are continuous. In order to see this it is enough to prove that the restrictions to
V
the spaces FK (X, &°-p ® §9 X K"'"'"^) are continuous, where K is an arbitrary
compact of X. If p is a C° function with compact support and which is equal to 1
on a neighbourhood of K, then the restriction to TK (X, &0-* ® &) x K"'"-^^)
coincides with the composition of the continuous map
TK(X, S0'" ® 8=) x K"*-i>{è) -► £°-'(ff) x A?-"-'(8F), (w, T) h> (w, pT)

286 ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
with the map
E°'»(¥) x Kc"'"-»(&) -► C, (w, T) h> [ cù A J.
However the latter is continuous (the reasoning may be made, for instance, in terms
of sequences). Thus the induced bilinear maps
H?(X, SO x Ext"-"^; 3F, Q) -► <£
are continuous. Thereby the above algebraic isomorphisms are continuous.
This is the part of theorem 4.2 which we can find here again. By means of
propositions 4.5 and 4.12 (hence practically of theorem 4.2), we deduce that °HPC(X,¥)
are DFS spaces and °Ext""''(A'; SF, Q) are FS spaces. From the closed graph theorem
it follows then that these isomorphisms are topological. Moreover, from 4.2 one
derives (replacing SF by 3F ® Q?) the following
Corollary 4.13. The morphism EP-"~\W)—> EP-%&) has closed image if

eland only if the morphism K"-"'"~%5:) —> k«-/»«-«+i (¥) has closed image.
Remark. If we confine to locally free sheaves SF, then we have algebraic
isomorphisms
Ext"e~%X; &, ÇÏ) ~ Hn-i(X, & ®e Q), Ext^A"; SF, Q) ~ H?~\X, ® ®e fi).
In this case, the results from [54] are not necessary in order to obtain the duality.
We also remark that these isomorphisms are just homeomorphisms. Indeed,
"Ext"3~\X; &, ÇÏ) is in topological duality with CH«(X, a?) and Hn~\X\ *®eQ) in
V
topological duality with Ext?(A"; W ®& Q, Q). Since
Ext«(A"; tf®l9n,Q)s H«{X, (^ ® Q)v ® Q) ~ H?(X, 8Q,
the conclusion for the first isomorphism follows by lemma 1.31 and similarly for
the second one. As a matter of fact, the reasoning may also proceed directly as in
the proofs of propositions 4.5 and 4.12.
(c) In this section we will prove a duality theorem with respect to the co-
homology with supports in a closed set. The following facts are required:
Lemma 4.14. Consider the commutative diagram of complexes of abelian groups
"I i wi
ï- 3'
A"—>B"—>C"

Vil. DUALITY ON COMPLEX SPACES
287
where [3a = 0, p'a' = 0. The morphism v induces a canonical morphism of complexes
Ker (E/Irn a -► Ker p'/Im a'
a/?rf its cone is canonically isomorphic to the cohomology of
C\u) -* C\v) -► C"(w).
The proof results straightforwardly from definitions.
Theorem 4.15. Let X be a complex manifold with countable basis of dimension
n, Y cr X a closed set, Q. the sheaf of germs of holomorphic forms in maximal
degree, S? a coherent analytic sheaf on X, andp an arbitrary integer.
Then Hpr(X, &) has a QFS structure, Ext^-"(7; &, Q) has a QDFS structure
and the associated separated spaces are in topological duality.
Moreover, H"Y(X, Br) is separated if and only if Ext"_p+1 (Y; S% Q) is separated
(Extc'(7; S\ Q) is by définition Ext© \Y,C(Y; ®\Y,Çl\ Y)).
Proof. Let 1i and f be locally finite coverings of X and X \ Y by Stein
open sets such that °f < II! n (X \ Y). Fix a function x which defines the
refinement "? < Il n (X \ Y). For any ^-module S>}1 one obtains by restriction
a morphism of complexes
c\% s>]i) 1!^ c\f, sir.).
Denote C'^f, T; Dli) = C'(T*(Sfc)) [-1], where C"(t*(£II)) is the cone of t*(STl).
We will show that 7r(C"(1l!, n?; §•)) ^ H'Y(X, &). To this aim, let 3' be an injective
resolution of S\ Consider the double complex
E»-« = C(lf, ^; 3«).
For each 9 we have the exact sequence
...-»• H"(C'(1l, °f; 3")) -► H'CU, 3") -► tff?, 3«) ->• ...,
hence
H?(C\% f; 3")) = J ry(X 3<?) if ^ = °
( 0 if p <£ 0.
By applying lemma 4.14 it follows that the spectral sequences associated to E" are
W,*)if/»=0f hence , = Wx,*) if , = 0
0 if p i= 0 1 0 if /> >/ 0
'£f*

288
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
and
= pClf, f ; 9) if q = 0^ hence //£jfç = .flP(CClf, f ; *)) if , = 0
I 0 if 9 # 0 "1 0 if q •£ 0.
Our assertion follows.
Let now a) be a precosheaf on X. By corestriction one obtains a morphism of
complexes (of chains)
Denote C.O", "If; sD) = Cm(x* (®)) [—1], where C.(t* (a))) is the cone of the
morphism T*(â>). Consider the precosheaf Ext?(&, Q) given by Ui-> Ext"(£/; S% £2).
We will show that
iip(c.(y, nc; Exrc (&, a))) =* Exirp(y; fr, n)
for any/). In order to prove this, let 5" be an injective resolution of Q. Consider the
double complex of components
£-p* = CpC?, It; Home(S^, i%).
For each 9 we have the exact sequence
... - H Jilt, Home{^, S")c) - //„(C.O>, «If; //owa(3F, £«)c)) -
^ H p_^, Home{^, 3")c)^... .
We also have the exact sequence
0 ->• Homc(A" \ Y; &, 3") ->• Horn, (A"; 8F, â«) ->• Homc(T; SF, £«) -► 0.
By applying lemma 3.9 one thus derives that
[ 0 if p # 0.
On the other hand, for any Stein open set U, Ext? (U; &, Q.) = 0 if p i= n. By these
facts and by lemma 4.14, the spectral sequences associated to the bicomplex E"
have the terms
'£2"'?=|
Ext«(7; &, Q) if p = 0
0 if p # 0,
»£-™ = |«p(c.r. if; £*£(*, o))) if ? = n
2 I 0 \iq ±n.
The conclusion follows from the properties cf spectral sequences

VII. DUALITY ON COMPLEX SPACES
289
The complexes C'(% SF) and C\°?, 3s) are FS and their topological duals
are isomorphic to the complexes C.(% Exfc{9,0)) and C.(T £x£ (ff, O)),
respectively. In these dualities the maps -r* (S=) and -r* (Ext? (3% fi)) correspond to
each other. Consequently, CÇ\t, f; 8r) admits an FS structure, C.(f, "If; £x/?(^, 0))
a DFS structure and these structures are in topological duality.
The proof of the theorem is concluded by applying 1.28 and 1.30.
§ 5. The dualizing sheaves
Let A" be a complex space and S7 e Coh (X). Denote by "\l the family of the relathely
compact Stein open sets. Let V, V e 1C, V <= [/. Consider a closed immersion
£/->£), where D is a Stein open subset of some numerical space. There exist
topological isomorphisms
H-C(U, 9) *. H-(D, 7c„ (8= ))
(it is enough to use a locally finite covering of D by Stein compacts and to consider
its inverse image under tt ...). In particular, considering the above said, and by
4.4, the spaces H'(U, §■) are separated, hence DFS.
Let D' be an open subset of D such that tt(F) = n ([/) flD'. The spaces
//C'(F, F) are topologically isomorphic to H'(D', it* (F)) and we have the
commutative diagram
H'c{V,Sr)~HmeiP',Tzt,{9))
1 1
H'C{V, &) =* H-(D, 7c» (*)),
where the vertical arrows are the natural morphisms obtained by the trivial
extension of the sections. Taking into account the calculus of the cohomology by means
of Delbeault resolutions, there results that the morphisms H'(D', t:^ (SF)) ->
-> H'(D, tt^gF)) are continuous. Then the morphisms H'(V, W) -> H'{U, SF) are
continuous.
Remark that this assertion is in fact true without any supplementary
hypothesis on U and V: the reasoning may proceed either directly by means of
Stein coverings or by applying the duality and lemma 1.31.
For any integer q we shall denote by ©«S^ the sheaf associated to the presheaf
UeM^ the topological dual of H«C(U, 8),
Vc C/M*the transposed of the morphism H«(V, &) -»• HI(U, ST).
(®*â%5>0 are called the dualizing sheaves of IF [4].
Proposition 5.1. The sheaves ®?fr are coherent and for any U £ 1C, Y(JJ, ®*§F)
is isomorphic to the strong dual of //,?(£/, &) (the topology on T is that deduced
from the coherence).
19 - C. 239S

290
ALGEBRAIC METHODS IN THE GLOBAL THEORY OF COMPLEX SPACES
Proof. Let U e "U and n: U -> (£" be a closed immersion for some integer n.
We prove that
and the first assertion of the proposition will be completed.
Let D c (£," a Stein opsn set. By 4.4, there exist topological isomorphisms
H«c(k-\D), SF)' ~ H«C{D, *,(*))' ~ ExtS"«(Z); tt, (*), Q)
^ r(A £x^-«(7t*(ff), n» ~ r^-'CD), ^Exf^^w), n»,
where we consider the strong topology on duals. These isomorphisms agree with the
restrictions, and the required conclusion follows easily. Furthermore,
T(U, S«ff) ~ T(U, n*Ext"e^:($), O)) ^ (»«(£/, IF))',
as above.
Theorem 5.2. Le? X be a complex space and f ^Oa coherent analytic sheaf
on X. Then
(/) ID'SF = 0/or 9 < proof gF and q >dim gF. Moreover, @*SF =£0 whenever q —
= prof & or q = dim gF.
(//) /"or a/y> 9, dim (â'ff) < 9 and if q = dim gF, ?A<?« dim (aW) = q.
Proof, (i) follows by the previous proposition and by theorems I. 3.6 and 1.3.7.
(n) The problem is local in nature; hence we may assume that ^ is a Stein
space and that there is a closed immersion tz: X -* (£,". If gF* = tt^gF), then
dim & = dim 3^, ^(SHSF) = ®'(S=*),
hence dim (®^) = dim (®«(^*)).
We can thus assume A' as a Stein manifold of dimension n and in this case,
S»âF ~ £x.g-?(cF, Q).
Let xeJ.We will show that dim (®*®)x < 9 and that dim (Sdim*'*fr),c = dim &x
and the proof is completed.
There exist isomorphisms (^^)x~Ext^j- «(SFX, 6X) Let p e SupptExtJ-,;'^, 6J),
hence Ext^)p((^Jp,(^)p) =* (Ext£«(ffx, <S>J)p^0. Since (5Jp is a regular ring
(hence its injective dimension is finite and equals the Krull dimension), it follows that
dim(6x)p^n~q. So dim(<9;c/p)=/7-dim(<9;c)p<(7, therefore dim(Ext"ei*{$x,6x))^q.
It only remains to prove that dim(Ext|);dim^»(S\c, 6X)) = dim§v Let us
consider an ideal p e Supp (SF ) such that dim (<3Jp) = dim §F_t; the support of the

VII DUALITY ON COMPLEX SPACES
291
localization (®x) is thereby reduced to the maximal ideal p(<9Jp. The ring (6x)p is
regular and prof (<9Jp = dim (<9v)p = n - dim (6Jp) = n - dim ®x. By applying
I. 1.16 we get
Ext^'p^((^V(^)p) « (Ext-£dta**(*„ <2>*))p *0.
Accordingly, p e Supp (Ext^;dimiir*(Sr;c, <S>J), and the proof is concluded.
Corollary 5.3. Let X be a complex space and SF ^ 0 a coherent analytic
sheaf on X. Then ¥ is Cohen-Macauley if and only if there exists an integer q0 such
that ®«& = 0 for q >£ q0.
Corollary 5.4. A complex space (X, 6) is perfect if and only if there is an
integer q0 so that the dualizing sheaves ®"6 vanish for q ^ q0.
We give the connection with the dualizing complex :
Proposition 5.5. Let X be a complex space of finite dimension and & e Coh(A").
Then
®^^Exîq\W,K'x).
In particular, ®"&x is isomorphic to the cohomology sheaf of the complex Kx in the
dimension — q.
Proof. Let 3* be an injective resolution of Kx. Then Ext^9^, K') is the
cohomology sheaf of the complex Homl9(S;,3') in the dimension —q. Accordingly,
ExtgXS?, Kx) is the sheaf associated to the presheaf U i-»- Ext^?(C/; &, Kx).
If Ue li, then the spaces H'(U, ff) are separated. By the second theorem of
absolute duality, the spaces Ext'e{U; 9, K'x) are also separated and Ext^?(£/; Br, Kx)
is canonically isomoprhic to the topological dual of H"C(U, SF). Moreover, if
V is another relatively compact Stein open set, V <= U, then the restriction map
Ext^C/; 3% Kx) -> Exte?(F; &, Kx) corresponds to the transposed map of the co-
restriction H"C{V, SF) -»• H%U, ^). This completes the proof.
Bibliographical indications
The construction of the dualizing complex and the theorems of absolute duality (§ 2 and § 3) are
presented following Ramis and Ruget [61]. Theorems of separation of the invariants H'(X, SF),
Ext*(A-; sF, Kx) can be found in [63]. The duality with respect to a morphism of complex spaces
is studied in [62] and [63].
A detailed presentation of the first duality theorem on manifolds can be found in [87].
The assertion (i) of theorem 4.4 is due to Suominen [87] and the assertion (it) to the authors [8].
The proof of the duality on manifolds by means of Dolbeault resolutions follows Serre [75] and Mai-
grange [53], [54]. Theorem 4.15 is due to Golovin [27].
The dualizing sheaves from § 5 are introduced in Andreotti and Kas' paper [4]. The same
paper introduces the dualizing cosheaves and the homology groups of coherent analytic sheaves and
provides another presentation of the duality in the singular case (namely, some spectral sequences
which relate the dualizing sheaves to dualizing cosheaves are established).

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