NORTH-HOLLAND
MATHEMATICS STUDIES
39
Introduction to Algebraic Geometry
and Algebraic Groups
Michel DEMAZURE
Ecole Poly technique
France
and
Peter GABRIEL
University of Zurich
Switzerland
N'H
cpC
m

1980
R C .2.:t 0"54
BIBLIOTECA
RE G, .......J.,C].8--6...........,
516. .,..L!i,.,:P.,,::.7:....____,
REF. 125
NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM. NEW YORK. OXFORD

?Z.. CS'-,
<D North-Holland Publishing Company, 1980
All rights reserved, No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording
or otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 85443 6
Translation or
GROUPES ALGEBRIQUES, Tome I (Chapters I & II)
Masson & Cie, Paris 1970
!D North-Holland Publishing Company, Amsterdam 1970
Translated by], Bell
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM. NEW YORK. OXFORD
Sole distributorsfor the U.S.A, and Canada:
ELSEVIER NORTH-HOLLAND, INC.
52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
lJ
I
Library of Congress Cataloging in PUblication Data
Demazure, Michel.
Introduction to algebraic geometry and algebraic
groups.
(North-Holland mathematics StUQleS ; 39)
Translation of Groupes algebriques, vol. 1.
Bibliography: p.
Includes indexes.
1. Geometry, Algebraic. 2. Linear algebraic
groups. 1. Gabriel, Peter, 1933- joint
author. II. Title.
QA564.D4513 . 516.3';' 79-28481
ISBN 0-444 -85443-6
PRINTED IN THE NETHERLANDS

INTROOOCION
A. GROI'HENDIECK has intrcxluced tv.D very useful tools in algebraic geanetry:
the functorial calculus and varieties with nilpotent "functions" . These
tools supply a better understanding of the phenanena related to inseparabili-
ty, they rehabilitate differential calculus in characteristic p t- 0 , and
they simplify in a significant way the general theory of algebraic groups;
hence we first intended to develop within the frame of schemes the classical
theory of semi-simple algebraic groups over an algebraically closed field
due to BOREL and CHEVALLEY; our purpose simply was to present the 1956-58
seminar notes of CHEVALLEY in a new light. But then we realized the in-
existence of a convenient reference tor the general theory of algebraic
groups, and the impossibility to refer a non-specialized reader to the
"Elements de Gecrnetrie Algebrique" (EGA) by GROTHENDIECK. This led us to a
considerable ITOdification of the original project and to the publication of
this introductory treatise.
In a first chapter, we develop what we need fran algebraic geanetry. In fact,
chapter I contains rrore than what is strictly necessary; it supplies a gener-
al intrcxluction to the theory of schemes fran a functorial FOint of view and
presents the fundamental notions, with the exception of those related to
ample bundles and projective rrorphisms. The matter of the first chapter is
taken alrrost ccmpletely fran EGA; but the presentation has been ITOdified in
a way we VoQuld like to justify now.
There are essentially tv.D FOints of view in ITOdern algebraic geanetry. Let
us take a simple example: If Pl,...,Pr are complex polynanials in n in-
determinates, we may assign to them, on the one hand the subset X of r
consisting of the FOints x such that P 1 (x) =. . . =P r (x) = 0 , which may be
given sane other structures: Zariski tOFOlogy, sheaf of FOl ynanial func-
tions - this is the geanetric point of view . On the other hand, we may
watch the functor assigning to every unital, ccmnutative, associative algebra
A the set X (A) formed by all xEA n such that P l (x) =. . . =P (x) = 0 - this
- r
is the functorial FOint of view . The first FOint of view is generally adopted
v

vi
INTRODUCTION
in proper algebraic geanetry; in the theory of linear algebraic groups how-
ever, the second outlook is often rrore beneficial, because it fits better
the constructions of group theory (it supplies an embedding of the category
of group-schemes into the category of group-functors, which is closed under
many constructions). Therefore, instead of defining schemes as geometric
spaces (endowed with sheaves of local rings), as EGA does it, we define them
as functors over sane category of rings. We then show that the category of
our functors is equivalent to scrne category of geometric spaces. In this way,
sane functors happen to be schemes, instead of defining a scheme up to iso-
rrorphisms; this is beneficial fran a technical FOint of view.
Chapter II then develops the general features of algebraic groups, avoiding
the more delicate problem of residue class groups and the specialized theo-
ries (ccmnutative affine groups, abelian varieties, semi-simple groups) ,
parts of which were included in the second part of the first edition.
Since the publication of the first edition, several books on algebraic geo-
metry and algebraic groups appeared. Scrne of them are mentioned in the ccrn-
plementary bibliography. They all adopt the geanetric FOint of view. There-
fore we hope that a text-book presenting the fundamentals of the functorial
approach may still be useful.
I
r,
This second edition reprcxluces with scrne minor changes in chapter I,  2 the
first Tho chapters of the first French edition. The translation is due to
J. BELL, the typing to Mrs. R. WEGMANN. To both we express our best thanks.

CONTENTS
INI'RODUCTION
PRERB;dUISITES AND REFERENCES
COMPLEMENTARY LITERATURE
GENERAL CONVENTIONS
v
x
xi
xiii
CHAPTER I Th'TRODUcrION TO A.I.G:BRAIC GECMETRY
1
 1 THE LANGUAGE 1
1 Geometric spaces 1
2 The prime spectrum of a ring 4
3 Z-functors 11
4 The gecmetric realization of a :l-functor 19
5 Fibred products of schemes 28
6 Relativization 32
 2 QUASI-COHERENT illDULES; APPLICATIONS 42
1 Sheaves of rrodules over a geanetric space 42
2 Direct and irwerse images of quasi -coherent sheaves 48
3 Faithfully flat quasicanpact rrorphisms 53
4 The functorial FOint of view 58
5 Affine rrorphisms 63
6 Closed embeddings 67
7 Ei11beddings 77
8 An affineness criterion for schemes 80
9 Transporters 82
 3 ALGEBRAIC SCHEMES 85
1 Finitely presented rrorphisms 85
2 Algebraic schemes 90
3 Constructible subsets of an algebraic scheme. Flat rrorphisms 95
4 Monomorphisms of algebraic schemes 105
5 The Krull dimension of a noetherian ring 110
vii

viii
CONTENTS
6 Algebraic schemes over a field
 4 SMXY.rH IDRPHISMS
1 The ITOdule of an embedding
2 The rrodule of differentials
3 Clean rrorphisms
4 Sm:x:>th rrorphisms
5 Proof of the EmX>thness theorem
6 Etale schemes over a field
 5 PROPER IDRPHISMS
1 Integral rrorphisms
2 The valuation criterion for properness
3 Algebraic curves
CHAPTER II ALGEBRAIC GROUPS
 1 GROUP SCHEMES
1 Group-functors and group schemes: definitions
2 Examples of group schemes
3 Action of a k-group on a k-scheme
 2 LINEAR REPRESENTATIONS
1 Definitions
2 Linear representation of affine groups
3 Existence of linear representations (in the case of a
base field)
 3 HOCHSCHILD COHCMOLCGY FOR GROUP SCHEMES
1 The Hochschild ccmplex and the exact coharology sequence
2 Extensions and cohanology of degree 2
3 Coharology of a linear representation
4 Calculation of various coharology groups
 4 DIFFERENTIAL CALCULUS ON GROUP SCHEMES
1 Infinitesimal FOints of a group-functor
2 Examples
3 Infinitesimal points of a group scheme
4 The Lie algebra of a group-scheme
116
122
122
126
132
135
144
153
159
159
162
168 
173
173
173
182
198
208
208
212
222
226
226
230
233

244
244
246
250
254

 7
CONTENTS
ix
5 Differential operators
6 Invariant differential operators on a group scheme
7 Infinitesimal groups
262
269
278
 5
LOCALLY ALGEBRAIC GROUPS OVER A FIELD
1 The neutral canponent, etale groups
2 Srrooth groups
3 Orbits
4 The group of rational points over an algebraically
closed field
5 Haranorphisms of algebraic groups
282
282
287
291
293
299
 6
THE CHARACTERISTIC 0 CASE
305
1 The enveloping algebra and invariant differential operators 305
2 Relationships between groups and Lie algebras 309
3 The exr::onential map 315
THE CHARACTERISTIC P t- 0 CASE
1 The Frobenius rrorphism
2 The pth-r::ower operation in Lie (G)
3 Lie p-algebras
4 Groups of height ;2 lover a field
321
321
324
326
334
FUNCTORIAL DICTIONARY
INDEX OF NOTATIONS
TERMINOI.J::X;ICAL INDEX
337
344
351

PREREXJUISITES AND REFERENCES
"
,
,The following items are suPFOsed to be known:
{
a) The "Elements de matMrnatique" by N. BOURBAKI, especially the chapters I
to V of his Ccmnutative Algebra. We refer to it by giving first the name of
the book, then the number of the chapter, of the paragraph ... (for instance
Alg. camm., III,  2, no 4).
b) A primer of the theory of sheaves, including the paragraphs 1 and 2 fran
chapter II of [3] .
c) A good grounding in categories and hanological algebra, which may be
found in [2], [3], [ 4] and [5]. References to these treatises make mention
of the authors, chapters and paragraphs (for instance CARTAN-EILENBERG,
chap. XVII,  7) .
In order to refer to number X. Y of paragraph Z of chapter T of the present
treatise, we simply write X. Y if the reference takes place in paragraph Z
of chapter T ; otherwise, we write  Z, X.Y if the reference takes place
within chapter T , and T,  Z,X.Y if it takes place in another chapter (the
reference (2.3 and I, 2, 5.6 and 5.7) means for instance: see number 2.3
of the present chapter and number 5.6 and 5.7 of paragraph 2 of chapter I) .
We collect in a functorial dictionary, to which we refer by means of "dict.",
sane starrlard definitions and notations of category theory freely u
throughout the book.
Modulo these references, and with the exception of some very peculiar comple-
ments, for which a reference is given within the text, all definitions and
proofs are canplete.
[1] N.BOURBAKI, Elements de matMrnatique, He:r:mann
[2] H.CARTAN-S.EILENBERG, Hanological algebra, Princeton University Press,
1956
[3] R.GODEMENT, Theorie des faisceaux, He:r:mann, 1958
[4] S.MACLANE, HOIIDlogy, Springer, Grundlehren, Band 114, 1963
[5] B.MI'ICHELL, Theory of categories, Academic Press, 1965
x

CCMPLEMENI'ARY LITERATURE
BOREL, A.
Groupes algebriques lineaires, Ann. of Math. 64, 1956.
BOREL, A.
Linear algebraic groups, redige par H. Bass, Benjamin, 1969.
CARTIER, P.
Groupes algebriques et groupes formels, Conf. au col1. sur la theorie
des groupes algebrigues, Bruxelles, 1962.
CHEVALLEY, C.
Theorie des groupes de Lie, tame II, Hennann, 1951
CHEVALLEY, C.
Classification des groupes de Lie algebriques, S8minaire 1956-58,
multigraphie, Paris, Secretariat mathematique.
DEMAZURE, M.
Schemas en groupes reductifs, BulL soc. Math. France 93, 1965.
DEMAZURE, M., A. GROTHENDIECK et a1.
Schemas en groupes, SSRinaire de geanetrie algebrique 1963-64, IHES,
Bures-sur-Yvette.
DIEUDONNE J. et A. GROTHENDIECK
Elements de geanetrie algebrique, Pub1. Math. IHES, nos. 4,8,11,17,20,
24,28,32,.. .
GABRIEL, P.
Des categories ab:'5liennes, Bull. Soc. Math. France 90, 1962.
GRCYI'HENDIECK, A.
Sur quelques FOints d' algebre hanologique, Tohoku Math. J. 9, 1957.
GROTHENDIECK, A.
Fondements de la geanetrie algebrique, Extraits du s8minaire BOURBAKI,
multigraphie, Paris, Secretariat mathematigue, 1962.
GRaI'HENDIECK, A. et a1.
S8minaires de geanetrie algebrique du Bois-Marie, multigraphies, IHES,
Bures-sur-Yvette.
xi

xii
Ca.1PLEMENTARY LITERA'IURE
HARTSHORNE, R.
Algebraic Geanetry, Springer-Verlag, 1977 .
HUMPHREYS, J.E.
Linear Algebraic Groups, Springer-Verlag, 1975.
MUMFORD, D.
Algebraic Geanetry I, Canplex projective varieties, Springer-Verlag,1976.
RENTSCHLER, R. et P. GABRIEL
Sur la dimension des anneaux et enserobles ordonnes, C.R. Acad. Sc.
Paris 265, 1967.
SERRE, J.P.
Groupes algebrigues et corps de classes, Hennann, 1959.
SHAFAREVITCH, I.R.
Basic Algebraic Geanetry, Springer-Verlag, 1974
j
>('!

GENERAL CONVENTIONS
In the present treatise  fixed universes .y. and y such that INEQ and
yEY are supposed to be given. We replace the tenu "set" by the tenu "class",
reserving the name "set" to the elements of the universe ::l: for instance,
y is a class, not a set, whereas IN and 1! are both classes and sets. A
set will be called small if it has the sarre cardinality as sane element of
Q : for instance, IN is a small set, whereas 11 is not.
If C is a category, Ob C and F 1 C represent respectively the class of
objects and the class of rrorphisms of C. We simply write cEC instead of
cEOb C ; if a,bEC, we denote by C (a,b) the class of arrows or rrorphisus
fran a to b. Similarly, when C is an abelian category, (a,b) is the
group of Yoneda-extensions of b by a of order n. We denote by
!:,,,,,1'£E... the categories of sets, rronoids, groups, ccmnutative
groups, unital COImU.1tative rings, toFOlogical spaces ... belonging to y.
Unless otherwise stated, we reserve the appellation rronoid, group... to the
objects of ,... In particular, unless we expressly state the contrary,
we supFOse all the considered rings to be ccmnutative and unitaL If A '
Mod A represents the category of A-ITOdules belonging to y; if MA' we
set  = A(M,A) .
A rronoid, group, ring, ITOdule is called small if the underlying set is
so. We give a special name to the (unital, ccmnutative) rings belonging to
Q , calling then ITOdels . Consequently, a ITOdel is a small ring, and every
small ring is isanorphic to sane ITOdel, without necessarily being a ITOdel it-
self. The full subcategory of  formed by the ITOdels is denoted by l1..
If k, we write  for the category of associative, ccmnutative, unital
k-algebras; similarly, if kEf1,  represents the full subcategory of 
formed by the k-algebras having a ITOdel as underlying ring.
Now let us reassure the readers frightened by universes: the part played by
y is ccrnpletely secondary, and we could easily do without by using the
axianatic of Bernays--GOdel.
xiii

xiv
GENERAL CONVENTIONS
on the one hand, we intend to study the category  of functors fran  to <'i
 and the rrorphisms between tv.D such functors should form a set; for that
reason, !! should not be to large. But on the other hand, we \\Ould like to
apply to ITOdels the usual constructions of carmutative algebra: residue class
rings, rings of fractions, completions... For this purpose it would be enough
to assume that for any ITOdel R, every ring with cardinality smaller or
equal to (Card R) IN is isarorphic to sane ITOde1. We could have ensured this
condition by fixing an infinite set E and calling ITOdel any ring sUPFOrted
by a subset of E IN . We have not chosen this way, because many Inathematicjans
are accustaned to universes by nCM, and also because we \\Ould like to use
freely direct limits in the category of ITOdels.
,{


CHAPTER I
INI'roDUCTION TO ALGEBRAIC y
 1 THE LANGUAGE
Section 1 Geanetric spaces
1.1 Definition: !2 geometric space E = (X, r!J x ) consists of a
toFOlogical space X together with a sheaf of rings r9 x such that, for
each xEX, the stalk 19 x ,x (or simply rJ x ) of 4 at x is a local

By abuse of notation, we shall ofter). write X instead of E. The unique
maximal ideal of Ox will be denoted by m x and the residue field Ox/
by K (x) . If s is a section of (Qx over a neighbourhood of x, the
canonical image of s in () will be denoted by s and called the genu
x x
of s at x; rroreover, the canonical image of sinK (x) will be called
the value s (x) of s at x . This value is thus zero iff the genu of s
at x lies in m
x
1. 2 Example: Let X be a toFOlogical space, and let L9x be the
sheaf of germs of continuous ccmplex valued functions on X. For each xEX,
the stalk If) is local and its maximal ideal is the set of germs of func-
x
tions which vanish at x.
1.3 Example: Let (X, &X) be a geanetric space, and let P be a
subset of X, endowed with the induced toFOlogy. Let i: P + X be the
inclusion map; then the restriction of l?x to P , written t9 x lp, is by
definition the inverse image i. (t!Jx) (dict.) .
1

2
ALGEBRAIC GEOMETRY
I,  1, no 1
Accordingly, if xEP , we have (xlp)x = {)X,X . We call (P, tOxlp) the
geometric space induced on P by (X, X) . If P is open in X, then
(P, 0xlp) is called an open subspace of (X, OX)
s(x) of
o , there is a section t of
r!J X over X. If xEX and
Ox over a neighbourhood of x such
1 for all FOints y lying in scrne
For example, consider a section s of
that s t = 1 . It follows that s t
xx yy
neighbourhood of x , so that the set of xEX such that s (x) of
o
is open.
Such an open set is called a special open set and is written X
s
1.4 Definition: ?2 rrorphism of geanetric spaces f : (X, iJ x ) ->-
(Y, i!J y ) consists of a continuous map f!2: X ->-y and a hanorrorphism of
sheaves of rings ff of rJ y into the direct ima g e f. ( .J X) of .J x such
that, for each xEX , the han anorph ism f : (!J f (x) ->- 12 induced b y ft:
x x
is local, i.e. satisfies
f (m )cm .
x y x
We shall often write f for f. If U is an open subset of X and V
is an open subset of Y containing f (U) we write f: I!}y (V) ->- tO x (U)
for the ring haranorphism induced by ff

CanFOsi tion of rrorphisms of geometric
Geanetric spaces and morphisms between
by Esg. A rrorphism of geanetric spaces

embedding if f
spaces is defined in the obvious way.
them thus define a category, denoted
f: X ->- Y will be called an open
induces an isanorphism of X onto an open subspace of Y
1.5 Example : If r!J x and r2y are the sheaves of germs of canplex
valued continuous functions over X and Y, each continuous Iffip f: X ->- Y
defines a rrorphism of geometric spaces: with the above notation, we need
only set fV (s) = s of' , where f' : U ->- V denotes the map induced by f.
U
1. 6 ProFOsition : If 1. is a category such that Ob T and Fl 1:
are in V, each functor d: T ->- Esg has a direct limit.
.- .......

I,  1, no 1
THE LANGUAGE
3
Proof: It is sufficient to show first that any family of geanetric spaces
has a direct sum, and secondly that any pair of morphisms
f,g : (X, lOX) :::: (Y, i!J y )
has a cokernel. Now the direct sum
(S, O s ) = li (x" 1!J x )
iEI 1 i
has as its underlying space the tOFOlogical
c9 S I Xi = r.9 x , . The cokernel (Z, t}z) of
1
sum of the X. and we have
1
(f, g) is constructed as follows:
a) Z is the cokernel of the continuous IffipS f and g in the category
of tOFOlogical spaces, and is therefore obtained by identifying in Y the
points f(x) and g(x) for each xEX;
b) if p: y + Z is the canonical projection, each open set We Z
-1 -1-1
determines two open sets V = P (W) , and U = f (V) = g (V) ; then
(!/z (W) is the ring of all sE <9y (V) such that f (s) = g (s) . The
restrictions rj Z (W) + &z (W') are induced by those of iDy, and the
canonical projection (Y, y) + (Z, <9z) is defined by p and the inclu-
sions r{;: i!}z (W) + l!Jy(V) The only tricky FOint in the proof is in show-
ing that the stalks of r2 z are local rings, and this is done as follows.
,n W V V
With the above notation, let wE V z (W) , v = Pv(w) and u = fu(v) = gu(v) ;
,n -1
since the hananorphisms f : V f ( ) + are local, we have f (V v ) = u
-1 x x -1 x -1 -I u
(1.3); similarly g (V) =u , so f (V) =g (V) and V =p (W'),
v u v v v
where W' is an open subset of W. If zEW' , the inverse of the genu of
-1
w at z is thus the genu of (vIV) ; on the other hand, if z' ,
-1 v 1
p (z) does not meet V and v vanishes at each FOint of p - (z) . Fran
v
this we infer the following facts: first, if w,w'E Oz (W) have non-inver-
tible germs at z , then p (w) and p (w') vanish at each FOint of
p -1 (z) ; hence p (wtw' ) also vanishes and so w + w' is not invertible;
therefore t.? z is a local ring. And secondly, if w vanishes at z , then
r{; (w) vanishes at each FOint yEp-l(z) ; thus Py ()z + rJ y is a local
hanorrorphism.

4
ALGEBRAIC GECMETRY
I,  1, no 2
1. 7 Example: SupFOse that t!J x and {)y are the sheaves of germs
of ccxrplex valued coptinuous functions over X and Y and that the rrorphisms
f , g are defined by ccrnFOsi tion with the underlying continuous maps. Then
r2 z may be identified with the sheaf of germs of ccxrplex valued continuous
functions over Z.
1.8
Remark: It can be shown that, if 1- is a category such that
Ob 1: E 2. and Fl! E .:::: ' then each functor d : .! ....  has an inverse
limit.
Section 2 The prime spectrum of a ring
2.1 We write (): Esg--;.An for the functor such that
-- -'
t)(X) = t!Jx(X) for each geanetric space X, and 19(f) = f (1.4) , for
each rrorphism f : X .... Y of geanetric spaces.
/
Spectral Existence Theorem: For each ring A, there is a qeanetric space
Spec A and a haranorphism CPA: A.... V(Spec A) satisfying the condition
(*) below:
(*) If X is a gtric space and  : A.... cO (X) is a ring haranorphism,
there is a unique rrorphism f : X.... Spec A such that  = !9(f) A
A ) I?(X)
j (f)
t9(spec A).
Such a pair (Spec A , A) is evidently unique, since it is the solution of
a universal problem. This universal problem means that the map ft.>(f)A
is a bijection Esg(X,Spec A) -+ An(A, e>(X)) . Instead of a proof, we merely
,...... .......
describe the pair (Spec A , A) and give the inverse of the bijection
f &(f) A .
,(

I,  1, no 2
THE LANGUAGE
5
Description of (Spec A, A) : The FOints of Spec A are the prime ideals
of A (Alg. ccmn. II, 4, no. 3). If fEA and p;:Spec A , we call the
canonical image of f in the field of quotients of A/p the value of f
at p; if a is an ideal of A, we denote by D(a) the set of FOints of
Spec A where at least one element f of a does not assume the value 0 .
The subsets D (a) of Spec A are the open sets of Spec A .
Let S (a) be the set of all sEA which do not assume the value 0 at any
FOint of the open subset D(a) of Spec A . Thus S(a) = S(b) if
D(a) = D(b) . We obtain a presheaf of rings over Spec A by setting
F(D(a)) = A[s(a)-l] (Alg. ccmn. II, 2, no. 1) and defining the restriction
hanarorphisms in the obvious way. If a is the ideal generated by the single
element s, and if As denotes the ring of fractions of A defined by the
multiplicatively closed subset {1,s,s2,s3,...} , then it is easy to verify
that the canonical map A ->- A[s(a)-l] is bijective. In particular,
s
F (Spec A) may be identified with A (by setting s = 1) . The structure
sheaf of Spec A is now defined to be the sheaf associ.ated with F. The
stalk of this sheaf at p is the local ring A = A[ (A-p) -1] . Finally, we
p
let A be the canonical map of F (Spec A) into the ring of sections of
the associated sheaf.
We must now describe the inverse  g of the map f  i!J(f) A . Let
 : A ->- O(X) be a haranorphism and let xEX. By definition, g(x) will
be the inverse image of m x under the cat1p:)si tion
 can
A<9x(X) t9x
-1
The map g is obviously continuous: if a is an ideal of A, g (D(a))
is the set of FOints of X at which at least one element of  (a) does not
vanish; the ccrnFOsition
./1 can /1 -1
AvX(X)vx(g (D(a)))
thus factors through A [S (a) -1] , which defines a rrorphism F ->- g. (<9 x )
(of the presheaf F into the direct image of Ox under g) , and thus the
required rrorphism 12 Spec A ->- g. ( t9 x) .

6
ALGEBRAIC GEOMETRY
I, 9 1, no 2
2.2 Example: Let X be a geanetric space. If we set A = t.? (X)
and  = IdA in 2.1, we get a unique rrorphism 1jJx : X .... Spec 0(X) such
that 19(1jJX)A = IdA' In virtue of 2.1, 1jJf assigns to each xEX the
prime ideal of t!) (X) consisting of all s such that s (x) = 0 . The
f () e
rrorphism 1jJi Spec D (X) .... (1jJx ). (tD X) is constructed as in 2.1.
2.3
Definition: For each ring A the geanetric space Spec A is
called the prime spectrum of A.
Of course, Spec A "depends functorially" on A; if  : A .... B is a ring
haranorphism, we write Spec  Spec B .... Spec A for the unique morphism
satisfying B = (Spec )A This morphism is defined explicity as fol-
e -1
lcws: the map (Spec )- underlying Spec  sends q onto  (q) ; if
a is an ideal of A, we have
)
((Spec )g)-l(D(a)) = D(B(a))
and the canFOsition
A -.t.,. B --0. 19 Spec B (D (B (a) ) )
factors through A[s (a) -1] . As a varies, we thus obtain a morphism
F----.>(spec ).(CJSpec B) ,
fran which we derive the required morphism
(Spec )f : CJ spec A ----JOo(Spec ). (U spec B)
In particular, if sEA and  : A .... As is the canonical map, Spec  is
an isanorphism of Spec As onto the open subspace (Spec A) s = D(As) of
Spec A .
2.4
For each ideal a of A, set V(a) = (Spec A) - D(a) .
For each subset P of Spec A , the closure P of P thus coincides with
V(2 p ) .

I, 9 1, no 2
THE LANGUAGE
7
If  A + B is a hananorphism and b is an ideal of B, it follows that:
(Spec )g(V(b)) = V(-l(b))
For, if la denotes the radical of a (Alg. ccmn. II, 92, no. 6), we have
g - n -1 _ -1 n
(Spec) (V (b)) - V(pEV(b) (p)) - V( (pEV(b)P))
= V(-l(v'b)) = V(/-l(b)) = V(-l(b))
(Alg. comm. II, 94, no. 3, corr. 2 of prop. 11).
In the particular case
-1
b = 0 , we see that V( (0))
is the closure of the
image of Spec B . For Spec  to be daninant (that is, for the irnage of
(Spec ) to be dense), it is thus necessary and sufficient that -1(0)
be a nilideal.
2.5 If A is the ring ![T] of polynanials in a variable T, each
haranorphism  : [T] + V(X) is determined by  (T) , which can be arbi-
trarily chosen in D(x). It follows that ([T] , t?(X)) may be identi-
fied with c?(X). Applying the adjunction formula
(X, Spec [T]  ([T], t?(X))
established above, we see that t.? (X) may be identified with the set of
morphisms of X into Spec [T] . This justifies the following
Definition: If X is a geometric space, a morphism
 : X + Spec [T]
is called a function over X; the dng O(X) is called the ring of func-
tions over x.
2.6 ProfX)sition: For each ring A, the hananorphism
A : A + {) (Spec A) of 2.1 is an iscrnorphism .

8
ALGEBRAIC GECM.TRY
I,  1, no 2
Proof: Set X = Spec A . We show rrore generally that the presheaf F of
2.1 assumes the same values as the associated sheaf tO spec A over the
special open sets X f = D(Af) , fEA . Since XiIXg = X fg for f,gEA, it is
sufficient to show that whenever X f is covered by X f ,... 'X f ' we have
1 n
an exact sequence
v
F (X f ) --) TT F (X f . )  .n: F (X f . f .) ,
1 1 W 1,J 1 J
where u, v, w are defined by uta) = (a.), v((b.)) = (b..) and
1 1 1J
w( (b,)) = (c..) , a. , b., and c., denoting, respectively, the restric-
1 1J 1 1J 1J
tions of a, b i and b j to X f , ' Xf,f. and Xf,f. . Since
1 1 J 1 J
X f . = Xl1X f . = X ff , and F (X f ) = Af '
111
it is sufficient to show that the sequence
u n 
Af -;. . Aff. -;.
1 1 W
TIA
, , ff.f.
1,J 1 J
is exact. To see this, set C = Af' B = V Aff. . Then B is fai thfull y
1
flat over C (Alg. Caml. II,  3, prop. 15 & eeL), and TTAff'f' may be
1,J 1 J
identif ied with
(T!. Aff ° Aff ) -+ BOB
1,J i C j C
v and w being identified with the maps b Hb Oland bt-7010 b . Exact-
ness' follows from lemma 2.7 by setting M = C = Af
2.7
Lem11a: Let C be a ring , M  C -mcxlule and B a faithfully
flat C- algebra . Then the sequence of C -mcxlules
o
-;. M -;. MB
-;.
M'eBB
-;.
M0B0B0B
C C C
-;.
... ,
where
() (m) = mOl and
()
i=n
a (m 0b 1 0.. .0b) = L (-1) i m 0b 1 0.. .0b ,010b ' 1 0.. .0b ,
n n i=O n-1 n-1+ n
if n > 0 is exact.

I,  1, no 2
THE LANGUAGE
9
Proof: Since B is faithfully flat over C, it is enough to show that the
sequence
dO 0B  0B "2 0B
o -;. M0BM0B0B--?M0B0B---;.
C C C C C
is exact. But, if we set
sn (m 0b 0 0.. .0bn+l) = m 0b 0 0.. .0bn_10bnbn+l
(n  0) , we have
So ( a 00B) = Id
(<1 0B)s +s 1 (0 l OB) = Id .
n n n+ n+
2.8
Corollary: The functor A Spec A is fully faithfuL
Proof: Set X = Spec B in Theorem 2.1. The map
(A, B) -;. g(Spec B, Spec A)
is the composition of
(A, B) : !!:(A, B) -;. (A, <?(Spec B))
with the bijection
An (A, r!J (X)) -+ Esg (X, Spec A)
"""' --
of 2.1. It is therefore itself a bijection.
2.9 Definition: A geometric space X is called a prime spectrum if
the rrorphism 1jJx: X -;. Spec O(X) of 2.2 is an isarorphism . X is called a
spectral space if X has an open covering by prime spectra.
-1
When X = Spec A , it follows from 2.1, 2.2 and 2.6 that 1jJx = (Spec A) ,
so that X is a prime spectrum. Since the special open subsets of Spec A
are prime spectra (2.3) and form an open base for Spec A , we see rrore
generally that each specqal space has an open base consisting of prime
spectra. It follows that each open subspace of a spectral space is a spectral
space.

10 AIGEBRAIC GEOM8I'RY I,  1, no 2
2.10 Recall that a torological space X is said to be irreducible if
it is non-empty and each finite intersection of non-empty open subsets of
X is non-empty. For example, for each torological space X and each roint
xEX , the closure {x} of x in X is an irreducible closed subset of X
ProFOsition: If X is a spectral space , the map x 1+ {x} is a bijection
of X onto the set of irreducible (;-,-ub subsets of X .
Proof: In the case where X is a spectrum, the proFOsi tion follows fran
Alg. ccrmt. II, 4, no. 3, cor. 2 of prop. 14. This special case immediately
implies the general case.
If F is an irreducible closed subset of X and if x is the unique FOint
such that F = {x}, x is called the generic roint of F .
2.11
Examp le: For each family (S.) ' EE of copies of Spec Z we
11 ."..
for the direct sum li s. . To each geanetric space X and
iEE 1
write Ez
,..,
each rrorphism
f . x +il s
. iEE i
we assign a map g: X + E
such that g(x) = i
if xEX and f(x)ES, . The
1
is continuous if E is
, the canonical isanorphism
that the induced rrorphism
map g
is locally constant, that is to say, it
-1
Xi = g (i)
(2.1) shows
assigned the discrete torology. If
Esg(X. , Spec Z) -+ An (Z , 19(x.))
1 - 1
f i : Xi + Si is determined by i
jection Esg(X,E')-+ Too (X, E)
.- Z ......
-
and Xi ; the map
h--7g
is thus a bi-
A spectral space X is said to be constant if there is a set E and an
isanorphism
X -+ E'
Z
-
2.12 Example: Let k be a field and X a Boolean space , that is a
torological space with a base of canpact open sets. Let <Dx be the sheaf
of rings which assigns to each open subset U of X the ring of locally


I,  1, no 3
THE IANGUAGE
11
constant functions over U with range k. For each xEX , we have
i!J = k.
x '
U + Spec \O(U) of
is a s[Je\::tral
for each canpact open subset U of X the rrorphism \jJu:
2.2 is invertible (Stone)*. It follows that Xk = (X,tJ x )
space.
2.13 Remark: The theorem and remarks of 2.1 signify that the functor
Spec : An° + Esg is the right adjoint of V c : Esg + An Q . It thus transforms
  ..............- "'-
direct limits of rings into inverse limits of geometric spaces. In particular,
for each diagram of rings of the form B 'k A  C , the canonical rrorphism
Spec B @A C ----;7(Spec B S A Spec C)with comFOnents Spec (in l ) and
Spec (in 2 ) is invertible.
Section 3 -functors
3.1 Definition: !-functor is a functor fran the category of mcxlels
£2 ,into the category of sets 1E.. The category of -functors is denoted by
ME
3.2 Notational conventions: If </>:R+S is an arrow of 1:1 ' if
 and xE (F) , we write  (x) ,x S ' or simply x for the image of x
under the map ;i(</» ;i(R) + (S)
If f: X + Y is a rrorphism of  and xE (R) , we write f (x) instead
of (f (R) ) (x) for the image of x under the map f (R) :  (R) + Y. (R) . If
Y I is a subfunctor of , f -1 (' ) denotes the inverse image of :!:' in ,
Le. the subfunctor X' of X, satisfying '(R) {xE(R) f(x) EY' (R)}
for each REM
If AEAn, we write
-
 A for the functor represented by A:
for REJ:1. If A is a rrodel, we say that  A is the
ring A. With this terminology, an affine scheme is
( A) (R) = (A,R)
affine scheme of the
* See, e. g. J. L. KELLEY, General TOp:Jlogy, Chapter 5, exercise S,
Van nostrand, 1955

12
AIGEBRAIC GECMETRY
I,  1, no 3
thus simply a representable functor. If f: A -r B is a rrorphism of .!::1.,
Sp f : Sp B -r  A is the functor haranorphism which assigns to each REM
the map  t-7 .f of (B,R) into (A,R)
If RE£::, XEME and pE (R) , we write p:tt-:  R -r  for the rrorphism of
functors which sends E( Sp R) (S) onto (X ()) (p) = PS E(S) . We know that
the map P t---'> pfF is a bijection of X (R) onto ME (Sp R, X) . The inverse
- .."......- - -
map sends a ( Sp R,) onto a b = (a (R) ) (Id R )
3.3 Example: The functor Q which assigns to each REJ;:. its under-
lying set is called the affine line. For each Z-functor X, the set
.... -
(,Q) carries a natural ring structure: if ,1j; E (,Q) , we set
((+1/J) (R)) (x) = ((R)) (x) + (1j; (R)) (x) and
((.1/J) (R)) (x) = ((R)) (x).(1/J(R)) (x) for R and xE(R) . A morphism
 E ME (X, 0) will be called a function on X The ring of functions on X
""'....- -
is denoted by <9()
If Z[T] is the ring of polynanials with integer coefficients in a variable
""
T , and if RE£i, an element  E ([T], R) is determined by (T) ; thus
the maps  I----;>  (T) enable us to identify Sp ],[ T] with o. Accordingly,
Q is an affine scheme.
3.4
Example: Let n, r be, two integers  0 ; the Grassmannian
is the functor n r which assigns to each R the set of direct factors
, n+r
P of rank n of the R-mcdule R If : R+S is an arrow of tl,
gn,r () assigns to P the image of S 0 R P in sn+r under the map induced
by . If n=l, G l is called the projective space of dimension r and
- ,r
is denoted by 1'r . The functor 1'1 is called the proj ecti ve line.
3.5 Example: Let X be a geanetric space. The l-functor
R 1---;0  (Spec R, X) is ca11ed the functor defined by X and is written
X . If A is a ring, (Spec A) may, by 2.8, be identified with the
functor RAn(A,R)
-
Accordingly, we have a canonical isanorphism

I,  1, no 3
THE LANGUAGE
13
 A -+ (Spec A) .
If I is an ideal of A, we can interpret the functor (D(I)) in a simi-
lar fashion, where D(I) is the open subspace of Spec A consisting of all
points where at least one sEI does not vanish (2.1). For if
 E (A) (R) = (A,R) , we have (Spec )-l(D(I)) = D(R(I)) by 2.3. It
follows that Spec  factors through D (I) if and only if R = R (I) . We
see accordingly that 9. (D (I) ) may be identified with the subfunctor
(A)I of Sp A satisfying
(A)I(R) = {:A+RIR=R(I)}
for each RE£1. We call (A) I the sub functor of Sp A defined by I .
3.6 Definition: Let  be a - functor and let !I be a sub functor
of X. We say that Q is open in  (or is an open subfunctor of 2i.) if ,
for each ITOdel A and each f: Sp A +  ' the sub functor f -1 (U) of
A can be defined by an ideal I of A (3.5).
A morphism : '!.- +  of  is said to be an open embedding if i is a
monanorphism and the image-functor is open in X.
If f E m.( Sp A,) , set a = !p E X(A) (3.2). The subfunctor rl(U)
of  A is such that, for each R, i- l (U) (R) is the set of :A+R
for which aREQ(R) . We can thus reformulate the above definition by saying
that Q is open in  if, for each AE:1 and each aE (A) , there is an
ideal I of A satisfying the following condition: for each arrow :A+R
of !1' we have aREQ (R) C  (R) iff R (I) = R .
3.7
Example: Let X be a geanetric space, Y an open subspace of
x . Then 2Y is an open sub functor of 2.X . For if a: Spec A + X is an
-1
element of (2X) (A) , a (Y) is an open subset of Spec A and is therefore
of the form D(I) for some ideal I of A. This ideal I satisfies the
conditions of 3.6.
We infer fran this that, if A is a ITOdel , a subfunctor U of  A is
open iff U is of the form (A) I. For since

14
ALGEBRAIC GEOMETRY
I,  1, no 3
( Sp A) I -+  (D (I)) , (A) I is open in  A for each ideal I. The con-
verse is established by setting  = Sp A and f = I in definition 3.6.
3.8 Example: If A, fEA and if q:A+A f is the canonical map,
then Sp q :  Af + Sp A is an open embedding whose image functor is
-tS!?A ) Af . For instance, in the case where A = lCT] and f = T ,  l[T]
and Sp l[r]T =  ][T,T- l ] may be respectively identified with the affine
line 2. (3.3) and the subfunctor 11 of 2. which assigns to each RE.!;:l its
set of invertible elements.
More generally, if  is a -functor and i:]{-7Q. is a function on X, we
write f for the inverse image f- l (l1) ; we shall say that f is the sub-
functor of  where !: does not vanish . This subfunctor is open ( the inverse
image of an open subfunctor is an open subfunctor).
3.9 Example: Let Q be a direct factor of rank r of the group
];.n+r For each R we identify R 0zQ with its image under
n+r - "" n+r and . te f - th . 1 . . f n+r
R + R 'O'z we wrl 'TT R or e canonlca prO]ectlon 0 R
n+r-
onto R / (R 0,Q) . Let QQ be the subfunctor of gn,r (3.4) which
assigns to each RE the set of (R-linear) canplements of R 0zQ in
R n + r . We claim that U Q is open in G , that is to say, for ach REM
- -n,r ..-
and each PEG (R), there is an ideal I of R such that, if :R+S
-n,r
is an arrow of !1, we have S (I) = S iff S 0RP = (S;n,r ()) (P) is a ccxrple-
ment of S 0zQ in Sn+r .
In order for S 0 R P to be a canplement of S 0zQ , it is necessary and
sufficient that the map V s : S 0 R P ->- Sn+r /S 0zQ induced by 'TTS be bijec-
tive; since the danain and the range of V s are finitely generated projec-
tive modules of the same rank, this holds iff V s is surjective, i.e. iff
Coker V s -+ S 0 R (Coker V R ) = 0
Since Coker v R is a finitaly generated R-mcxlule, this last condition is
equivalent to S (I) = S , where I is the annihilator of Coker v R in R
(Alg. camm. II,  4, no. 4, props. 17 and 19).

I,  1, no 3
THE LANGUAGE
15
Now consider a basis
e l ,e 2 ,...,e n + r of
zn+r
....
over
z
such that
Q =:B l. e i
l>n
If R is a ITOdel and P a ccmplement of R 0zQ in R n + r , we have the
identi ties
10 e.
1
p. + L: a" 0 e ,
1, 1J J
J>n
for i<n, P ,EP and
- 1
that specifying P
a, .ER . Since the P , form a basis for P, it follows
1J 1
is equivalent to specifying the a,. , so that Y Q is
nr 1J
an affine scheme (3.2) isarorphic to the prcxluct Q . Hence lI Q is a
representable open subfunctor of 9n,r ; we express this by saying that Y Q
is an affine open subfunctor of G .
-n,r
Finally, we assign to each subset I of {l, ... ,n+r} of cardinality r
direct factor Q I of zn+l consisting of the sums In.e, such that
1 1
n i = 0 for iI. For each field KE£:!, the exchange theorem says that
G (K) is the union of the sets Yn... (K) . We express this fact by saying
-n,r l
that the sub functors U _ Q of G cover G . More generally, we make
I -n,r -n,r
the
the
3.10 Definitj,on: Let  be a l.- functor . A family (i)iEI of sub-
functors of X is said to cover .is. if , for each field K, X (K) is the
union of the sets i (K)
For example, if Y is a geanetric space and (Y i ) iEI is an open covering
of Y, then (Yi) iEI is an open covering of SY Le. a covering consist-
ing of open subfunctors. In particular, if A is a ring and (fi,xi)iEI
is a finite family of pais of elements of A such that
:1 = I x.f,
iEI 1 1
then (D(Afi))iEI is an open covering of (Spec A)
(frx i ) iEI a partition of unity in A.
Sp A . We call
3.11 Let X be a -functor, R a model and (fi'Xi)iEI a partition

16
ALGEBRAIC GECME.'I'RY
I,  1, no 3
of unity in R (3.10) . We associate with these the sequence of sets
(*)

u 11 v ,-rr
!(R)  ' (Rf ) ==:::.",(R f f )
l , l,J .,
l W l ]
defined as follows: if a, (resp. a,,) denotes the canonical map of R
l Jl
into Rf. (resp. of Rf. into Rf' f.J ' we set
l l l ]
pr,ou = (a l ') , pr. ,oV = X(a,,)opr, and pro .ow = X(a..)opr, .
l l,J - Jl l l,J - lJ ]
Definition: a) A Z- functor X is called local if, for each R and each
- 
partition of unity (fi,x i ) iEI in R, the sequence (*) above is exact .
b) X is called a scheme if X is local and has a covering (i) iEI of
affine open sub functors i indexed by a set I belonging to the fixed
universe U.
-
The full subcategory of  forrned by schemes will be written Sch.
We observe immediately that an open sub functor J of a scheme X is itself
a scheme. For, with the notations used in the definition, choose ITOdels Ai
and isanorphisms U. :  A. ->- X. . By 3.7, the open subfunctor u -:-1 (X J1Y)
-l l -l -l -l-
of  A l , is of the form (Sp A.) I where I, is an ideal of A, . As s
-- l ill
runs through Ii ' the affine open subfunctors (Ai) s of  Ai cover
 A. ) I . If X, denotes the image-functor of (Sp A.) under u, , the
l i -lS - l S -l
the family (is) forms the required covering of Y by affine open sub-
functors.
We shall henceforth call an open subfunctor Y of a scheme X an open sub-
scheme of X.
3.12 Example: For each geanetric space T, the functo r ET is locaL
For if we replace X by EiT in the sequence (*) ,  (R) , X (R f ,) and
- l
X (R f ' f .) may be identified with Esg (Spec R, T) .. Esg (D (Rf.) , T) and
- lJ - _A l
Esg(D(Rf,) n D(Rf,), T) , given that D(Rf,) n D(Rf.) = D(Rf.f.) and that
___ l ] l ] l ]
Spec Rf. is the scheme induced by Spec R over the open set D (Rf i ) . Thus
l
the exactness of (*) means simply that a morphism m : Spec R ->- T is de-
termined by its restrictions to the open sets D (Rf i ) and that these restric-
tions satisfy the usual matching conditions.

I,  1, no 3 THE IANGUAGE 17
It follows from this that T is a scheme if T is a spectral space and is
also small (Le. the set underlying T is small and, for each open subset
T' of T, ciJ T (T') is small).
3.13 Example: The grassmannian G is a scheme. For by 3.9 it is
- -n,r
sufficient to show that G is a local functor; if PEG (R), the
-n,r -n,r
sequence
P--DPf !T,Pf f '
l i l,J i j
with the obvious arrows, is exact (set M = P , C = R , B = DR in lerrrna
l fi
2.7) . Since P fi C R:r and P fifj C R:j this simply means that P is
the inverse ima g e in Rn+r of the su1:mcdule TTP f of TTRn f +r, in other
l i l i
words, that PEG (R) is determined b y the P f E G (R f ), or that
-n,r i -n,r i
the map u in (*) (3.11) is injective when X = G .
- -n,r
Suppose now that we are given the J, E G (R f ) and we identify the family
l -n,r i
(J.) E D G (R f ) with the prcduct ITOdule J = TG-, over the ring
l l -n,r ill
B = TIR fi . Suppose that v(J) = w(J) : we have to show that there is an
I E G (R) such that J = utI) . Now the assum p tion v(J) = w(J) means
-n,r
that in+r (J) and in2+r (J) generate the same B Q9RB-subnodule K of
(B Q9R B ) n+r ; i t fo1.ows that, if ini denotes the i th canonical inj ection
of B into B Q9R B Q9 (i = 1, 2, 3) , the B Q9RB Q9RB-subnodule L of
(B Q9R B Q9RB) n+r , which is generated by int+r (J) , is independent of i
Using the notation of Lemma 3.14 below, if we let u, and u denote the
l ]
maps induced by d1: 1 + r and d  n+r , then this lerrrna establishes the existence
l ]
of an R-subnodule I of R1+r such that J = TJI f . It thus remains to
l i
show that I is a direct factor of W+ r of rank r ; and this is the con-
tent of lerrrna 3 .15 below.
3.14 Let us intrcduce sane terminology which we shall use in the
following lerrrna: Let h: R ->- B be a ring haranorphism, M and R-ITOdule,
N a B-ITOdule. We say that a map f : M ->- N is adapted to h if we have
f(m+m') = f(m) + f(m') and f(rm) h(r)f(m) for all m,m' EM, rER , and
if f induces an isanorphism B Q9if1 + N

18
ALGEBRAIC GECME:I'RY
I,  1, no 3
Lerm1a: Let R be a ring , and let B be a faithfully flat R- alqebra.
SUPFOse we are given ITOdules J ,K,L  B, B @R B , B @R B @RB respectively ,
and maps u' ()
u 
JKL

adapted , respectively , to the ring hamanorphisms
d'O
dO 
BB@Bd'lB@B@B,
 l RRR

2

such that dO (b) = b @l , d l (b) = 1 @b , dO (b <Zic) = b @c @l , di (b <Zic)
b @l @c , and d 2 (b <Zic) = 1 @b @c for b,c E B . Assume also that
uou o = uiuo' uOu l = u 2 u O and uiul = u 2 u l . Then if I = Ker(uO'u l )
the inclusion map of I into J induces an isanorphism I @RB -+ J .
Proof: Inspect the diagram
i@B uO@B
"-
J@B ::. J@B ; K}"
R R U l li9B
v1 vol
v'o
u'
1
J  K ) L
1d1 u l VII u' > IW
2
j ddJ
J ) B  J ) B @ B @ J ,
dl&J > R R
where i is the inclusion map, j the canonical injection, v,vo,v' 0 and
v l are induced by i,uO'u'O and u l ' and w is the composition
B @ B @ J B @Vl> B @ K
v'2 L
> ,
v:2 being induced by u:2 . By lama 2.7, the horizontal sequence at the
bottan is exact. Since v l and w are bijections and w(dO&J) = u'lv l '
w(dl&J) = U;?l ' the second horizontal sequence is exact.

I,  1, no 4
THE LANGUAGE
19
Finally, since va (ud 2lB ) = uivo and va (U l i5?>B) = u:zv o ' the bijections V o
and v l induce an isanorphism v of I@RB = Ker(uO@B,ul@B) onto
J = Ker(ui,u:z) .
3.15 L€rm1a: Let R be a ring , M a finitely presented R -ITOdule,
I a submodule of M and B a faithfully flat R- algebra. If B @RI is a
direct factor of B @RM , then I is a direct factor of M.
Proof: It is sufficient to show that the canonical map
 : H(M,I) + H(I,I)
is surjective or, equivalently, that B @R is surjective. This follows fran
inspection of the carmutative square
B @Han(M,I)
R Li
HCXTL (B @M,B @I)
H R R
B
) B @ H(I,I)
1/J RJ;
") H(B I,B I) ,
in which all the arrows are the "obvious" ones. For since B @RI is a direct
factor of B @RM , 1j; is a surjection; on the other hand, since B @R1 is
finitely presented over B, I is the same over R; accordingly j is a
bijection, as is i, and B @R is surjective.
3.16 Remarks: One can avoid using lerrrua 3 .14 by employing the descrip-
tion of quasicoherent ITOdules over Spec R given in  2, no. 1. Anyway, we
shall find this lerrrua useful later.
Consider, on the other hand, the subfunctor G' of G which assigns to
n,r n,r
each ITOdel R the set of free direct factors of R n + r of rank n. This
of 3.9, but it isn't
functor is covered by the affine open subfunctors U Q
I
locaL
Section 4
The geanetric realization of a -functor
4.1
Proposition: The functor S :  +  has a left adjoint.

20
ALGEBRAIC GEOMETRY
I,  I, no 4
Proof: We sketch a proof of this particular case of a well known. theorem of
Kan. Let X be a l-functor and let !1p be the category of f-ITOdels: an
object of this category is an !, -ITOdel , Le. a pair (R, p) consisting of a
ITOdel R and a pEP (R) a morphism of (R, p) into a second F-model
(S,a) is defined by a hananorphism (R,S) such that  (p) = a (Le.
such that ptto   = a:/F) . If  : (/ +  denotes the functor
(R,p) 1+ Spec R , we set If I = l.iID  .-It thus remains to construct, for
each XE a bijection
Hf,X) : (II ,X) -+ !:1£2(f,X) ,
each RE!i, a map 2(r)
canFOsition
.£'(R)
f: Ifl+x be a rrorphism; we must define, for
+ (eX) (R) . If pEI(R) (g(R)) (p) is the
which is functorial in X. Let
i(p) f
Spec R --7 l.?;m <1:  X ,
\
where i(p) is the canonical rrorphism of '1'(R,P) into l  . The maps
g(R) define a morphism g:+9X and we set g = (,X) (f)
It remains to show that the map (f,X) is bijective. To this end let
o
of : (J +  denote the functor (R, p) Sp R . It is well known and
easy to-verify that the rrorphisms p.:fF: of (R, p) + f induce an isanorphism
l.?;m of -+ [ . Now  (f,X) has been defined. in such a way as to make the
squares
(F,X)
(lfl,X)(f,X)
"",(i(p) ,X) l 1!'!:(p#,exJ
a
((R,p), X) P ) (oF(R,p), X),
carrnutative, where a p (X) (R) +  ( R, EX) is the canonical isanorphism.
Since I!' I = l.?;m  and .£' +- l.?;m of '  (f ,X) is obtained fran the bijec-
tions a by passage to the limit over the objects (R,p) ; hence  (,X)
P
may be identified with the inverse limit of the bijections a , and is there-
p
fore a bijection.
If  = fu2 A , an f-ITOdel is simply a ITOdel carrying an A-algebra structure.

I,  1, no 4
THE LANGUAGE
21
In this case .tip coincides with  and its initial object is the pair
(A , IdA) ; accordingly we have a canonical isanorphism i (IdA) : Spec A -+
IAI .
4.2 If f:E->-F is a rrorphism of , we write
unique rrorphism of  which makes the squares
Ifl:IEI+!F!
- - -
for the
Esg ( I F I , X)
-- -
(I!I ,X) 1
Esg(IEI,x)
- -
 (f ,X\ ME (F ,SX)
1!,PX)
(E,X)
- ) (,X),
carmutative as X runs through . We can obtain I! I
lows: with the notation of 4.1, if !:Jf :  ->- lip denotes
- - -
(R,p)  (R ,!(p)) , we obviously have "(:Jf)Q =  ; consequently,
induces a rrorphism l  ->- 1.1m  ' and-thi is III .
explicitly as fol-
the functor
f
Definition: If F is a - functor (resp. if ! is a rrorphism of ) , we
call If I (resp. I! I)
functo r I?I :Ifl
the geanetric realization of E' (resp.!)
is called the geanetric realization-functor.
The
If F is a -functor, the symbol I!' I denotes, depending on the context,
either the geanetric realization of f, or the underlying space of this
realization. A point x of this underlying space will be called a point of
E: ; we write simply xE!, for xEIE'1 . We shall take care not to confuse the
points of E.' with the elements of !' (R) for RE:£::. We shall also call the
topological space which underlies the geanetric realization of E: the space
of points or spectrum of E.' . A subset P of E' will then by definition be
a set of points of f ; we write peE: for P C IF I . Moreover, if g:.f->9.
is a rrorphism of ME and Q is a subset of , we let 2:. (P) (resp. g -1 (Q) )
denote the image of P (resp. the inverse image of Q) under the map I <I I 
induced by . Similarly, g(E') denotes the image of lEI under 12:.1
this image is thus a subset of G which is not to be confused with the
image-functor of 5l which is denoted by  : this image-functor assigns
to each REM the image of the map g(R) :(R)(R)

22
ALGEBRAIC GE<]\1ETRY
I,  1, no 4
Henceforth we shall say that a rrorphism i: of !:Pi is surjective (resp.
injective , open , closed ) if the continuous map I fl e is surjective (resp.
injective, open, closed). We also call If I  the map underlying f and de-
note it by f .
Finally, we write <!IF for 0 1fl and call t!J F the structure sheaf of the
Z-functor F
-
4.3 Corollary: For each - functor  and each ring A there is a
canonical isanorphism ( , A) -+ (A ,rJ(F)) .
Proof: It follows fran 3.5 and 2.1 that there are canonical isanorphisms
( , A) -+ (g ,(Spec A)) -+ (IXI ,Spec A) -+ (A ,Ji( If I )) .
When A=CTJ, ( , A) is just c9(f) and (A ,J1( Igl)) may be identi-
fied with iJ( If I ) . We thus infer the existence of a canonical isanorphism
19 (g) -+ tJ(II) and hence the required isanorphism. This latter may be defined
 explicitly as follows: to each  E  ( , A) we assign the haranorphism
1/J:A+d(F) such that the map (1/J (a) ) (R) :  (R) ->- R sends xE (R) onto
((x) ) (a) for each aEA and REM.
In particular, if A = 19(£:) , we write 1/JF:! + Sp L9(£:) for the rrorphism
associated with IdA via the canonical bijection (tP() ,rJ(!)) -+
ME (g , Sp r.9()) .
4.4 Now consider a -functor  and a geometric space X . We write
'ji (f) : .£' +  I  I and cjJ (x) : I SX I + X
for the images of the identity rrorphisms of 1.£'1 and X under the bijec-
tions  ( ' I f I ) and  (X, X) -1 of 4.1. Let  I be the full subcategory
of  consisting of the functors !' such that 'ji (f) is invertible, and
let ' be the full subcaegory of  consisting of all X such that
cjJ (X) is invertible. It follows fran the well-known relations between cjJ
and 'ji that I? I and  induce an equivalence between kE2' and  I .

r

I,  1, no 4
THE IANGUAGE
23
We may thus manipulate the objects of these categories either geanetrically
(be regarding them as belonging to ') or functorially (be regarding them
as belonging to ME') .
For example, if A and R are ITOdels, the map
't'( Sp A) (R) : (A,R) ->- (Spec R,Spec A)
is precisely the map f f--+.Spec f , which, by 2.8, is a bijection. It follows
that Sp A belongs to  ' , so that Spec A -+ I Sp A I belongs to  I
We now describe further objects of ME' and Esg' :
,.... -
Canparison Theorem: a) Let X be a geanetric space. Then cj> (X)
is invertible whenever X satisfies condition (*) below:
IXI ->- X
(*) there exists an open covering (Xi) iEI of X by prime spectra Xi
such that IEU and (J(x.) is isanorphic to a ITOdel for each i
... - 1
b)
for
Let F be a Z- functor . In order for I I I to satisfy (*) above and
- --,owtI
't' (!') :!' ->- .II to be invertible , it is necessary and sufficient that
F be a scheme.
The functors I? I and  induce quasi-inverse equivalences between the cate -
gory of schemes and the category of geanetric spaces satisfying (*) .
The proof of this theorem is deferred until 4.16, when we have a second des-
cription of the geC:metric-realization-functor at our disposal.
In the sequel we shall often make implicit use of the canparison theorem by
arguing as if a scheme were a geanetric space. Thus, for example, we may
write f:X->-Y or X for I I : IHx.1 or II . When confusion is possible,
we notify the reader by using a phrase like: "Taking the geanetric viewr::oint".
4.5 Let !. be the full subcategory of fields of . Given a functor
H:K->-E , we recall that the direct limit lim H is the quotient of the dis-
- ->-
jOint sum 1CEt(K) by the smallest equivalence relation containing all
pairs (a,b) for which there is a rrorphism f :K->-L such that aEH (K) ,
bEH(L) and b = (H(f)) (a) . If x is an equivalence class modulo this rela-
tion, we write Hx for the subfunctor of H such that, for each KE!.,
H (K) is the set of aExfIH (K) . Then H is the disjoint sum of these sub-
x
functors H and each H is indec anpo sable, that is to say, it is not the
x x

24
ALGEBRAIC GEOMEI'RY
I,  1, no 4
disjoint sum two non-empty subfunctors. In this situation we shall call the
H the indecanFOsable c anFO nents of H, and we may identify lim H with
x +
the set of indecanr:osable canFOnents.
4.6
FOsable.
Example: Each representable functor of K into E is indecan-
4.7
Exarrple: Let X be a geanetric space and let H = (X) 1!5. be
the restriction of the z-functor EX:!:f+ to 15.. For each KEK , we thus
have
H (K) = Esg (Spec K, X) = 11 An (K (x), K) ,
- x-
where x runs through the FOints of X whose residue field K (x) is iso-
rrorphic to a ITOde1. Hence, since H is the direct sum of the indecanFOsable
functors represented by the K (x) , we see that liID H may be identified
wi th the set of xEX such that K (x) is iscroorphic to a ITOdel.
4.8 Exarrple: Let A be a ITOdel and let H be the restriction
(Sp A) IE. of Sp A to !. If KE, an element xEH (K) , Le. a horncrnor-
phism of A into K, is determined by its kernel p which is a prime ideal,
and by the induced hananorphism x : Fract(A/p) + K . The map x (p,x )
p P
is thu  a bijection of H(K) onto the disjoint sum llK (Fract (A/p) , K) so
p-
that e indecanposable canFOnents of H are represented by the residue
fields Fract (A!p) and liID H may be identified with the. set of prime
ideals of A.
4.9 Proposition: For each !- functor I., the underlying set of the
geanetric realization IE:' I of ! is canonically iscroorphic to liID (F 119
Proof: If we let X denote the set of FOints of a geanetric space X, we
have, by 4.1, 4.8 and the cCXImUtativity of direct limits, the following
canonical bijections:
IE:'I = ( l ) = l
- (A,p)E
lim lim (Sp A) (K) .:;. lim F (K) = lim (F I K) .
--..,. -----> - -->'-.....
K (A,p) K
(Spec A)  -+ lim lim ( Sp A) (K) -+
 ---'J>
(A,p) K

I,  1, no 4 THE LANGUAGE 25
4.10 We now consider a l-functor ! ' a subset P of F and the sub-
functor I p of l: defined by letting, for each REM, f P (R) be the set of
pE!.'(R) such that
 (p) E U (FIK) (K)
xEP - - x
for each haromorphism  : R+K into a field KE,l5. (in the notation of 4.5).
Clearly
F pl K = U (F I K) ,
-".. xEp-"'x
so we can recover P fran F via the formula P = lim (F I K) .
-P ->- -P-
If i:Q+£, is an arrow of ME and if Q = !-l(p)CI1 ' we verify that
-1
gQ =! (£'p) (3.2) .
4 .11 Example: If X is a geanetric space and f = X, P may be
identified by 4.7 with a set of points xEX whose residue fields K(X) are
isanorphic to ITOdels. Thus !.'P may be identified with SP , where P is
assigned the toFOlogy induced by X and the restriction G7 X I P of cJ x to
p . In particular, if P is an open subset of X, !' P is an open subfunc -
tor of Sp A
4.12 ProFOsi tion: Let !:. be a !- functor. Then the map P}----? F P in-
duces a bijection between the open subsets of I!.'I and the open subfunctors
of F.
Proof: Let P be a subset of IFI, (R,p) an object of  and Q the
inverse image of P under the map i(p) : Spec R ->- lE'I (4.1). By definition,
P is open in IXI iff Q is open in Spec R for all (R, p) E J:1g . Also,
£'p is an open subfunctor of F iff p#-l (fp) is an open subfunctor of
Sp R for all (R, p) . Since Q is open in Spec R iff ( Sp R) Q is an
open subfunctor of Sp R (4.11), and since we have further p#'-l(!.'p) =
( Sp R)Q (4.10), we see that P is open in I!I iff !'P is open in_ F
Since P = ljrn (fpIK) for each subset P of I!'I ' it remains to show that
each open sub functor U of F is of the form E'p for same subset P of
 . To prove this observe that M2 (K) is a subset of Ei!- (K) which is
saturated with respect to the equilence relation defining lim (FIK) ; if
-+ - ',MoO'
we set P = ljrn (!:!IK) , it is easily shown that E=f p .

26
ALGEBRAIC GEOYIETRY
I,  1, no 4
4.13 ProFOsition: The following conditions on a !- functor F are
equivalent:
(i)
F is local.
(ii) For each ITOdel A, the presheaf Ui---)( ( Sp A)U'!:) is a sheaf
of sets over Spec A .
(iii) For each l- functor G the presheaf U 1->-  (%,f) is a sheaf
of sets over II .
Proof: (iii) => (i) . Usin g the notation of 3.11, let U. = (Spec R) f .
1 i
Then E:(R fi ) is k!( (R)Ui' f) so that exactness of sequence (*) of
3.11 means that one determines a section of the presheaf UI(( Sp R)U' !:)
by specifying the sections over the open sets U i ' provided these sections
satisfy the usual matching conditions.
(ii) => (iii) . If (R,p) E.t: G ' and if U is open in II set
V = i(p)-l(U) (4.1). We thus-have isanorphisms
(U'!) -+ ( ( Sp R)V' D -+  (( Sp R)V'!) ,
(R, p) (R, p)
which shows that the presheaf U t---;> (<'?u,f) is an inverse limit of sheaves
(namely, the direct images in 101 of the sheaves defined by F over the
spaces Spec R) ; it is therefore itself a sheaf.
(i) => (ii) : If U is open in Spec A and (U i ) iEI is an open covering
of U, we must show that the sequence
(l)
ME(SU, F) ->- TTME(SU. ,F) :t IT ME (Su.nsu " F)
........ - - . - - - 1 - .......- - 1 - J
1 1,J
is exact. We can, rroreover, restrict our attention to sufficiently fine cov-
erings, so let us assume that U i = (Spec A)fi' fiEA. Let <I>:A->-B be a
rrorphism of M such that B = L B<I> (f.) , that is, an elffi\ent of (U) (B)
- 1
If g.=(f.) , V, = (Spec B) i is the inverse image of U, under Spec 
1 1 1 gi 1
and f (B gi ) may be identified with (i' K)
It follows then fran (i) that the sequence
ME (Sp B, F) ->- TTME (SV" F) ->- TTME (sv.nsv., F)
......"'" - - 1"'- - 1 - -+ i,j........ - 1 - J
is exact (this is clear when I is finite; if not, pass to the limit over
the finite subsets of I) . Since U ,
SU, and SU . nsu ' may be identified
-1 -l-J
and SV, nsv , as (B, ) runs through
- 1 - J
with the direct limits of  B ,
SV,
- 1

T
:f
I,  1, no 4
THE IANGUAGE
27
the objects of I2su ' we see that (1) may be identified with an inverse
limit of exact sequences, and so is itself exact.
4.14
IXI
Proposition: Let X be a .t,- functor. The Structure sheaf of
is canonically isarorphic to the sheaf of rings U L?(F) (3.3) .
-U
Proof: The presheaf U 1+ V(E'u) is a sheaf in virtue of the fact that
Q -+  (Spec [T]) is a local functor (3.12 and 4.13). If AE, ! =  A
and U is a special open set of the form (Spec A) f' fEA, we have canonic-
cal isancrphisms Sp Af -+ E' U and
Gi spec A (U) -+ Af -+ !1([T], A f ) -+ (£'u'9) = L?(!u)
These isanorphisms induce the required canonical isanorphism when f =  A
Now suppose that ! is arbitrary. Let U be open in II I and let V be
the inverse image of U under i(p) : Spec R 1+ 1£:.1 for (R,p) E lJp (4.1).
By 4.1 and 4.10, we have
E'u -+!U x  -+ Xu X  of -+  (E'u x Sp R) -+  ( Sp R)V
F F - (R,p) F (R,p)
In view of the definition of t?lfl ' we obtain fran this the required isanor-
phism:
g1(U)  t9 spec R(V) -+  (( Sp R)V' 2) -+ (  ( Sp R)V' 2) -+
(R,P) (R,P) (R,p)
.!!(U,.Q) = D(£,u) .
4.15 Corollary: If Y is an open subfunctor o f , II is an open
subspace of II (1.3) .
Proof: This follows immediately fran the description of the geanetric reali-
zation given above (4.9, 4.12 and 4.14).
4.16 Proof of the ccmparison theorEm: For each geanetric space X
satisfying condition (*) of 4.4 and for each xEX , the residue field K(X)
is isanorphic to a model, so that (X) : lxl 1+ X induces a bijection of
the underlying sets (4.7). If (U i ) is an open covering of X by prime

I',
28
ALGEBRAIC GEOMETRY
I,  1, no 5
spectra then <I> (U .) : ! SU. 1 ->- U, is an isorrorphism by 4.4 and I SU, I is an
1 -1 1 -1
open subspace of l.ex! by 4.15. The tOFOlogies of lxl and X, and also
their structure sheaves may thus be locally identified via <I> (X) thus
<I> (X) is invertible and a) is proved.
By 3.12, it remains to show that the condition in part b) of the comparison
theorem is sufficient. Let !' be a scheme, (U.) ' EI an affine open covering
1 1
of F such that IEU, and (U,,) an affine open covering of u,nu, . Then
- - -lJa -1 -J
the prime spectra I.!hl yield an open covering of LI , so that II
satisfies condition (*) of 4.4. We show that '¥) :! ->- I!'I is inver-
tible by displaying the inverse '¥. of '¥ (f) ; one can define a rrorphism
'¥' : .e IF _ I ->-!' by specifying rrorphisms '¥  : S I U ,I ->- F such that
1 --1
'¥ I (s l u.. I ) ='¥ I (S l u., I )
1 - -lJa J - -lJa
for all i, j, a . Since '¥(U.) : U. ->- slu,1 is invertible (4.4), we need
-1 -1 - 1
only set '¥ equal to the canFOsition of '¥(u.)-l with the inclusion
1 -1
rrorphism of  i into  .
Section 5 Fibred products of schemes
5.1 Let f:X->-Z and 9:.:l->- be rrorphisms of . Recall that the
fibred product functor  X z'L satisfies (! x '!!.'£) (R) = !(R) >< '!!.(R),£(R)
for each REM. It follows inmediately that 25' X Z'!. is open in  X Z'£. if
X' is open in X .
ProFOsition: If X, Y and Z are schemes , so is  X Z'!... .
Proof: It follows easily fran the definitions that X x zl is l=a1. Let
(} be an affine open covering of , let i :;.  X ZE and
Y. :; Y x Z Z, be the inverse images of Z , in X and Y , and let (X,)
-1 - -1 -1 - - -10.
and (Y. S T be affine open coverings of x, and y, . Then X. X Z y, S
-1 -1 -1 -10. i-1
coincides with (io: X zY) n( X'ZliS) and is therefore open in  X zY
Also if 2S ia :; Sp R - iS :; Sp - Sand i :; Sp T , then obviously
ia X z. '£i:;' Sp (R <ZiTS) , so .that the X, X Z y, S form an affine open
-1 -10. _i- 1
covering of 2S X z¥.

r

I'
tt
I,  1, no 5
THE LANGUAGE
29
More generally, if (X, , f k ,) is a finite diagram of schemes, the inverse
-J - J
limit functor can be constructed with the help of fibred prcxlucts. This in-
verse limit is therefore a scheme; in particular, if ,y:y is a double
arrow of , the kernel functor Ker tE,) , which assigns to each RE£'l
the set Ker ( (R) , ':': (R)) = {xE(R) I':! (x) =  (x) } is a scheme.
With the assumptions of the foregoing proposition, it follows easily from
2.13 and 4.15 that, for each pair of rrorphisms d:T+I1 and e:T+I1 of
 such that 1!ld=Ile, there is a unique h: T ->- I .x zxl such that
d=19xlh and e=liyl h . In other words, (I)( zxl , 12xl, Iii) , is a fibred
prcxlct of the diagram I 11111  II gll X I in the-category  . More general-
ly, the restriction to Sch of the functo r I?I :ME->-Esg ccmnutes with finite
..... -
inverse limits.
5.2 With the assumptions of proFOsition 5.1, we now examine the
spectral space I x zxl in more detaiL Let x be a FOint of 1251 ' K (x)
its residue field and- E(X) :  K(X) ->-  the following morphism: IE (x) I
carries the unique FOint w of Spec K (x) onto x, and I E (x) I : 0 ->- rJ
w x w
IE (x) I is a mono-
is the canonical projection of iJ onto K (x) . Clearly
x
rrorphism of  .
Corollary: With the assumptions of proFOsition 5.1 , let x, y, Z be FOints
of , '!, z such that ! (x) =z=g (y) . Then the rrorphism
E:(x) X E(y) : Sp K(X) x Sp K(Y) ->- X X Y
E(Z) Sp K(Z) - Z -
induces a bijection of the set of prime ideals of K (x) @K (z) K (y) onto the
set of FOints tE X z'!. which are projected onto xEX and yEX.
Proof:
First recall that Spec K (x) X Spec K (y)
Spec K (z)
may be identified
with Spec K(X) @ K(Y) (2.13).
K(Z)
Moreover, E (x) X E (y)
E(Z)
a rronomorphism; the following lEmTta implies that the induced map is injec-
tive. Finally, if t E  x Z¥. is projected onto x and y, the canFOsitions
19x E (t) I and I iyE (t) I factor through I E (x) I and IE (y) I . Consequently
- -
E (t) factors through E (x) )t E (y) and t belongs to the image of
E (z)
IE (x) E(Z) E(y) I .
is a f ibred prcxluct of monanorphisms, thus itself
$
i

30
I,  1, no 5
ALGEBRAIC GECMErRY
5.3
tive (4.2).
Lemna: If f:X->-Y is a rronanorphism of schemes , f is injec-
Proof: Let x, u satisfy !: (x) =!" (u) =y. Let
P l ,P 2 :  (K(X) @ K(U)) ->- 
- - K (y)
be the rrorphisms
and K (u) into
E: (y) , we have
induced by E: (x) , E: (u) , and by the canonical maps of
K (x) @ K (u) . Since fE: (x) and fE: (u) factor through
dy)
fPl=fP2 ' whence P l =P2 . Thus if
K(X)
we have
t E Spec (K(X) @ K(U)) ,
K(y)
x=p (t)=p (t)=u .
-1 _2
5.4 Corollary: With the assumptions of proFOsi tion 5.1, let
i!,i!,'!:C2 and ( )C2 be the underlying sets of II, II, II and I  I
Then the map
(£Cxz ) ->- y:f} which sends tE   onto (2 x (t) '!y(t)) is surjective .
Proof: This follows fran 5.2 and fran the fact that K (x) @ K (y) t- 0 .
K(Z)
5.5
Corollary: With the assumptions of proFOsition 5.1 , if
9::->- is surjective (4.2), so is 5l x :    ->-  .
Proof: This follows irmlediatel y fran 5.4,
5.6 We have just described the FOints of X)( Y . To describe the
- Z -
local rings, let 11 be an affine open subset of the-scheme K, let j be
the inclusion rrorphism of U in X and let xEU. If p11: <9(U) ->-rl is the
- - - x x
restriction map, write E: :  <!J ->- X for the comFOsition
x x-
u -1
 P;C \jJu j
 t!J x  spJ(g)  g (4.3)
Evidently E: does not depend on U.
x
ProFOsition: Let X be a sche'ne , x a FOint of  and P the set of
x
FOints y such that xE {y} . Then the morphism E: :  <!J ->- X induces an
x x-
isorrorphism of Spec () onto the geometric space (p , l? x lp) (1.3).
x x x

r

>:i
:
I,  1, no 5
THE LANGUAGE
31
proof: Observe that P consists of all FOints tEX for which xEll.
x -
Such a FOint t belongs to each open set containing x . We may replace .K.
by an affine open set, so we may assume that K =  A . The proFOsition now
follows fran Alg. ccmn. II, 2, no. 5, prop. 11 and fran the description of
local rings in Sp A
(2.1) .
5.7 ProFOsition: Let !:->- and '1:¥->- be rrorphisms of schemes ,
x,y and z FOints of , and  such that !(x)=z=sl(y) . Let Q be the sub -
set (4.2) of    consist of FOints t whose projections  and 
onto  and X - satisfy xE{ t x } and yE {} . Then
E x E :.eE.c9 X .eE.l9 -+ X;.<Y
x Ez y x Oz y -  -
induces an isanorphism of
Spec tJ 0.2
x 19z y

Spec rJ x: Spec r.!)
x spec J y
z
onto (Q ,OXxyIQ) (1.3).
-z-
Proof: write P (resp. P ,p) for the geanetric space induced by II
x y z
(resp. by II, II) on the subspace consisting of all FOints s such that
xEW (resp. yETS},zE{S}) . Since I Z xl is the fibred product of II
and II over II':' in the category  (5.1), evidently (Q 'xxylQ) may
-z-
be identified with the fibred product P x z P y The proFOsition -now follows
from 5.6 and 2.13.
5.8 Let !:->- be a morphism of schemes and let y be a FOint of
Y. Since E (y) :  K (y) ->- X. is a rronanorphism, the same holds for the
canonical projection E (y) : (Sn K (y) )xX ->- X . We may thus identify
X  y--
(K(y))0 with the image-functor of -E(y)X ' which we write !-l(y) and
- --1
call the fibre of ! over y. The set of FOints of ! (y) is a subset of
X (4.2).
ProFOsition: The tOFOlogy of the space of FOints of f- l (y) is induced by
that of I  I If xEX, f (x) =y and 19. is the local ring of X at x ,
then the local ring 0; .!-l(y) at i: canonically isanorphic o -
Ji)myJJ x .

"
ii
..Ii
'il
,il
i:1
:11
:i
i l
]1
i'
I
!II I
U
i I
32
ALGEBRAIC GECMETRY
I,  1, no 6
Proof: We may reduce everything to the case in which  = Sp A, J.... = Sp B ,
1. being induced by a rrorphism  :B-+A of .t:.. Then I ( Sp K (y) )y1 is just
SpeC(K(Y) Q!)BA) , Le. the prime spectrum of the ring of fractions of A/(y)A
with respect to the multiplicatively closed subset  (B'y) . The assertion
about the toFOlogy now follows fran Alg. CaTlffi. II,  4, cor. to prop. 13.
The second assertion follows fran the canonical isomorphisms
(K (y) Q!) A)  (B / m) Q!) A  A 1m A ;
B x YB x x yx
Y
it can also be deduced fran the description of the l=al rings of a fibred
prcxluct derived in 5.7.
Section 6 Relativization
6.1 Let g be a l.-functor and let Me be the category of g-ITOdels
(4.1). An - functor is a functor  of 1:1g into E.. For instance, if
A= (R, p) EJ:1 s ' the functor B f---? £J S (A,B) , which is represented by A, will
be written. f. Each represen:;;mle .e.-functor, Le. one isanorphic to a
functor of the form SA, will be called an affine g- scheme.
If k is a ITOdel, and  =  K , 1:J s coincides with the category  of
k-ITOdels. An -functor is in this case called a k- functor. If A is a k-model,
Le. a k-algebra belonging to the fixed universe £., we write kA for
eEA , and speak of affine k- schemes instead of affine S-schemes. In parti-
cular, when A is the algebra k[T] of FOlynanials in T, kA is canoni-
cally isanorphic to the k-functor .Qk which assigns to each k-ITOdel R its
underlying set. For each k-functor K, a rrorphism !:->2k is called a func-
tion on 2S.. The set of these functions is written (9k () and carries a
k-algebra structure: addition and multiplication are defined as in 3.3; if
XEk and fEUk (X) , Af satisfies ((A) (R)) (x) = A (f (R) ) (x) for each RE
and each xE2S. (R) . We call .Qk the affine k- line.
In the case k = Z , we have M =M and the k-functors coincide with the
.,.".. AtoK I\oIIot
Z-functors considered so far.
...
6.2 If S is a -functor, the theory of S-functors reduces imnedi-
ately to the theory of -functors. For let /S be the category of !- functors

r
I

;)
I,  1, no 6
THE LANGUAGE
33
over : an object of this category is a rrorphism p:-+e of  with tar-
get ; a rrorphism of !?:-+ into g:X-+e. is a conmutative triangle of 
of the form
f
X > Y
--
2,
CanFOsition of these triangles is effected in the obvious way. The category
/9 is related to the category lJsf,; of 9-functors via the functor
is: /9 -+ 1is which assigns to P:-+9 the 2-functor
is(p) : (R,p) (/9) (p#,p) ,
where p#:  R -+ 2 is as usual the rrorphism canonically ass=iated with
pE.e(R) (3.2).
ProFOsition: The functor is: / -+ s:s:. is an equivalence of categories .
Proof: We merely give a functor je.: 11e.:[ -+ .w/ which is a quasi-in-
verse for is. Let l' be an -functor and let PT:Z:!'-+9 be the image to be
defined of  under js; we have (z:!') (R) =4'E(R,P) for RE, where the
sum is taken over all pE (R) . Then -PT (R) : (z 'l) (R) -+ E- (R) maps 1:. (R, p)
(which is contained in the disjoint sum-1:.(R;P)) onto pE2,(R) .
6.3 If T is an 2,-functor, we shall call Z T the under lying
,g;- functor of T and PT: zT+ the structural projection . For example, if
 = Sp k, RE and AE, then # (A) (R) is the disjoint sum of the
sets  (A, R) , where R is assigned all k-algebra structures canpatible
with the given ring structure. This disjoint sum may be identified with
I:1(A,R) , where  denotes the underlying ring of A. We thus have a canoni-
cal isomorphism ; (kA) -+ ZA .
-
We frequently define an .e-functor l' by giving zT and PT . For instance,
- -1
let :-+ be a rrorphism of schemes, y a FOint of X and p:!. (y)-+
Sp K(y) the canonical projection (5.8). By abuse of language, we call the
-1
K (y) -scheme 1:. such that z1' =! (y) and PT=P the fibre over y; this
.... -1-
K (y) -scheme will also be denoted by ! (y) when no confusion with the ter-
minology of 5.8 is possible.

34
I, . 1, no 6
ALGEBRAIC GEOMETRY
Similarly, if K and  are .e.-functors, we deduce fran 6.2 that the follow-
ing diagram is cCXImUtative (where the canFOnents of u are Z pr 1 and J;,P! 2) :
z (25 II) , ;..- > (zX) x (z';)
- 
S
In general, given an -functor ! we shall carry over implicitly to ! all
those results and definitions which apply explicitly to z. Thus we shall
say that r is local if z is local, that T is an 2 -cheme if zt is a
scheme, that a subfunctor -g of T is open if z!:! is open in z1' '"'; etc.
r-breover, we set 11:1 = Iz1:1 , and call 11'1 the- geanetric realiation of
T . Finally, in sections ere k is constant and no confusion is likely,
we shall employ an abuse of notation and write Sp A or &(X) for kA
or &k (X)
If kEJ::, we wr i te
k-schemes, i.e. the
 for the full subcategory of !:.\. formed by the
( Sp k) -schemes.
With each rrorphism of -functors f: S 'S there is ass=iated
a functor S ,];->t1 s f2. called the base restriction and simply denoted by Sf ,
- -
although it depends primarily on J: if T' is an '-functor, and if
(R,p)S' we define
6.4
(sl") (R,p)
U :t' (R,a)
:f(o)=p
where the sum is taken over all oE.e' (R) such that i (o)=p . We thus have a
canonical isanorphism
T' -+ (T')
z- Z S-
- -
which makes the following square conmute:
z'
P T .[
S'
 z (st')
Mo } sT I
> S
f
It follows that if T' is a scheme, so is st' .
When
call
s' = Sp k' , .e. = k and i = Sp  , we write k' for r' , and
T' the k-functor derived from T' by the restriction of scalars 
k-

I,  1, no 6
THE LANGUAGE
35
For example, if AE!:\:, we have k(,A) :; (kA) , where kA is the under-
lying k-rrodel of A.
6.5 'Ib each rrorphism f:'->-.e we assign a functor l:1s ,] called
the base extension functor and simply denoted by ? S ' , although it depends
primarily on !: if 1: is an S-functor and REM, we define
(1'S') (R,a) = 1'(R,!:(a))
where aE' (R) and f(a)E.e(R) . We thus have, by definition,
Z(1$,)(R) =4-1'e.(R,a) =1J-1'(R,!(a)) ,
where a runs through S' (R) . If in (e) denotes the canonical image of
- a
eE'!'(R,!(o)) in the disjoint sum-Y-1:(R,!(a)) , we obtain a bijection of
z(1's,) (R) onto ]'!') (R) X.e(R)e' (R) by sending ina(e) onto (in!: (0) (e), 0) .
We therefore obtain a canonical isomorphism
z cr S ,) :; ( z1')l,  ' ,
""'" - 1"""'_
whose second canFOnent is the structural projection Pl's,: z (:!,s,) ->- S' . Fran
this and 5.1 it follows that g  is a scheme , so is - I-. -
tor
?
's'
g : . ->- is a rrorphism of l- functors , the base
is right ,djOint to the base restriction functor ?
extension func-
ProFOsition:
Proof: If T' is an S' -functor and T is an -functor, we must define a
bijection
x(:!" ,'!') : .£1s(s'!" ,) :; !:1s,,E('!" /r s ,) ,
which is functorial in T' and  . Now a rrorphism <:J:st' is a family of
maps
g(R,p): LL T' (R,a) ->- :r(R,p) ,
- !(o)=p-
where REli, pE(R) and oE' (R) . With the above notation x (:!" ,T) assigns
to g the rrorphism Q:'!"->-,!,s, which sends eE'!'.' (R,o) onto
2(R,(0)) (ino(e)) . If we identify (er') with J.'r' and z(ts,)
(z:!')!<S' , as explained above, h is the unique rrorphism such that
zh: ; (s'!'.') ->- (z'!') )(S. has ccmFOnents z2: and PT' 0
...... -- .... - --
with
When S '
Sp k'
 = Sp k and ! = Sp  , we write '!'k' for Ts'

36
ALGEBRAIC GEOMEI'RY
I, . 1, no 6
Thus we say that 1'k'
we have 1'k' (R) = l' (kR)
particular, if A,
is derived fran l' by extension of scalars . If RE.tk, ,
, where kR is the underlying k-albebra of R. In
R, and ! =  A , we have
k' (R) = A (A'k R ) -+1:\. (AQ9 k', R) ;
k
and we infer the existence of a canonical isanorphism
(kA)k' -+ eP k . (A  k') .
In virtue of this fact we occasionally write r Q9 k k' for 1: k , , even when
l' is not of the form A . The rrorphism associated with I Q9 k' by the
canonical bijection - k
x(:!' k', :!') : tk(k(:!' k'), T) -+ ,](! k',1' k')
is then denoted by Pk : k cr Q9kk') -+ l' and is called the structural projec-
tion.
6. 6 The base extension functor ? S' of 6.5 also FOssesses a right 1
adjoint TI : kJ _ s' -;- LJ, called the Well restriction or direct image
S'js -
functor. - - It is given by the formula
( I,A! ') (R,p) = l1s.((eP.e(R,P))" !') ,
where .!'Et1s'' (R,p)E]:1s' and (fu:?S(R,p))S' is obtained by base extension
- -
from the e.-functor represented by (R, p) . Given JE.t1s, we can then define a
bijection
q:!','!:'): lJs'£;(:!'s":!") -+ .k1 s £:cr ,g'r') ,
which is functorial in 1: and l' : with any J::!,s'-+'!" , (,,!:') associates
the maps ! (R, p) -;- ( sl, / '!:') (R, p) assigning to TE (R, p) the can[Dsites
- - :/f:
TS' g
( SP s (R,p)) s' 1's' !'
- -
In the case where S = Sp k ,
J. /k instead of r:J. For
SPk ' (A Q9kk') , so that we get
(g<;r') (A) -+ l' (Al59 k k') .
s' = Sp k' and 1 = Sp  , we simply write
A=(R,p) , (S(R,p))S' is then identified with

I,  1, no 6
THE LANGUAGE
37
In this case (T,1") can also be defined as associating with g the family
of cc:mFOsites
T(in ) 'I (A @kk')
T(A) - \ '!'(k(A@kk')) > ' (A@kk') ,
where AEk\.
f

\1.
r.
r4
ft
I
[
!
1:;
[
ProFOsition: Let :k-;-k' be a rrorphism of ITOdels and supFOse that the
k-ITOdule k' is projective and finitely generated . Then
a) if T' is an affine k'- scheme, ')k :r:" is an affine k-scheme;
b) if T' is a k'- scheme and if , for each finite subset P of T' , there
is an affine O pe n subscheme U' of T' such that ECU' ' then TT T'
k'/k
is a k-scheme.
Proof: SUPFOse first that l"=kA where A = k' (E @kk') is the symnetric
algebra of a k' -ITOdule of the form E @kk' . If R, we then have canoni-
cal isorrorphisms
(gT')(R) =t\.(A ,R@kk') -+.(E@kk',R k') -+(E ,Rk')-+
(E " R) -+ t\(.:2k(E @k')' R) ,
where . is the k-ITOdule dual to k' (Alg. II, 4, no. 1, prop. 1 and
no. 2, prop.
2). It follows that
JJ:., k' (E k'))
-+ SP k (S k (E@')) .
- "" k
In the case in which I'=k,A, where A is an arbitrary k'-rrodel, -let I
be the kernel of the canonical homomorphism of  (A @kk' ) into A . Then
A may be identified within 1\ I with the amalgamated sum of the diagram
S k ' (A@k') S k ' (I @k')k'
- k  k
where a is the canonical map and B (I @kk') = 0 . Since n is a right
adjoint functor, it cCXImUtes with inverse limits, and so gSPk.A, the
fibre product of affine schemes, is itself an affine scheme.
Now for b)
is open in
: clearly R :£'
 ' , TT U . ' s
k'/k _ l
is local whenever T' is. Furthermore, if U'
open in :!": for consider the morphisms
f: SP. A -;- TT T' ,
----K k'/k -
&k: epk B -;- kA

38
ALGEBRAIC GEOMEI'RY
I, 'l, no 6
and the rrorphism
f': Sp . (A @k ') -+ T'
-k k
associated with ! by the bijection  (kA,1") -1 defined above. Clearly
!: o(k) factors through /k lI' iff !' 0 (eI>k' ( @kk') ) factors through
U' . By 3.7, this latter condition is satisfied iff
B @ ((A @ k') II) = 0 ,
A k
where I denotes the ideal of A @kk' def ining the open subscheme ! ,-1 (!/J
of SPk,(A@kk'). Since (A@kk')/I is a finitely generated A-ITOdule, this
is equivalent to saying that B (J) = B , where J is the annihilator of
(A@kk')/I in A.
This enables us to construct for each xE k ! IJ? ' , an affine open subscheme 11.
of kk T' such that xE . For x is the equivalence class of an element
ET'(K@k') = (TTT') (K, p ) C z (n  T' )( K )
- k kj'k- -_kk
where KEt: is a field and pE}j(k,K) (4.5 and 4.9) . Since Spec K @kk'
has only a finite number of FOints, there is an affine open subscheme 11.'
of T' such that IlI'l contains the image of the rrorphism
II: Spec K k' -+ 11'1 .
It na.v suffices to set 11 = 0J 1J.'
6.7 Example: Let k l ,... ,k n be n copies of k and set
k'=k l x ...x k . If we assign k. the k'-algebra structure derived fran the
n l
i th canonical proj ection pr.: k '-+k, , there are canonical rrorphisms
l l
E i : 01 r -+ :!'k i
such that Ei (R) is the map
T (R @pr . ): T (R @k ') -+ T (R @k, )
- k l - k - kl
for each R . If l' is a l=al
definition 3.11 that the rrorphism
functor, it fol1a.vs immediately from
 T -+ TITk, whose i th canFOnent is
l

I,  1, no 6
THE LANGUAGE
39
1\ is an isomorphism (apply definition 3.11a) to the partition
(e' .,e' ') 1 ' of unity in R@ k k'suchthate' l ,=J& k e l .,wheree.Ek'
l l ::;ln l
and pr. (e,) = 0, ,)
] l lJ
TT
In this example, we see that k/'k 1'. is a scheme whenever T is a scheme.
But we also note that the functor g does not preserve open coverings:
for let T i be the subfunctor of T such that ti (A) = !(A) if Ae i = A
and T, (A) = rj; if not. Then the T, fo:r:m an open ccvering of T , while
-l -l
g !i = rj; for each i if n > 1 c
6.8
We now return to the diagram of functors:
M O
7:
<
ME
....-
?

I ?I
considered in section 4. Given a geometric space T, we define the category
W'T of geometric spaces over T: a geanetric space over T is a rrorphism
of  with target T . A rrorphism of T with danain p :X->T and target
q:Y-+T is a ccmnutative triangle of the form
f
x / y
T
Canposition of rrorphisms is defined in the obvious way.
The category T is ccnnected to .t1 ST via a pair of mutually adjoint
functors
I?IT'T : s:t T
To each object p:X->T of T we assign the ST-functor .e with struc-
tural projection i'p:i'X-+fiT (6.3) . Conversely, if XE,t1ST' !EI T : !EI-+T
is the morphism assigned to the structural projection P f : _ zf-+T by the
-1
bijection  (z£"T) of 4.1. With the above notation, the bijection
....

40
ALGEBRAIC GECMEI'RY
I, . 1, no 6
 (E,X) enables us to associate the ccmnutative triangle (2) with the CCXImU-
tative triangle (1) below, where f' (zf,X) (f)
(1)
f
II'I > X
If
T
(2)
f'
;\  :
ST
Thus we obtain a bijection T(['P): T( IfIT,p) .:; l1.e(f. ,¥) byassign-
ing to each triangle (1) the rrorphism 9:K-"' such that z<r-=!' It follows
directly fran the canparison theorem (4.4) that the functo I ? IT and .!?T
induce quasi-inverse equivalences between the categories of ST- schemes and
the full subcategory of T consisting of the p:X such that X satis-
fies condition (*) of 4.4.
If kE£:'! and T = Spec k , we shall also write I ? I k ' Sk and k for
I? IT' ST and W T ; a geanetric space p: X ->- Spec k over Spec k will be
called a gecrnetric k -space; by the spectral existence theorem, to specify
such a space is equivalent to specifying X and a structure of sheaves of
k-algebras over Ox . Hence, if RE..I\, ('kp) (R) may be identified with the
set gk (Spec R, X) of fEg (Spec R, X) such that U(f): D'(X) ->-R is a
k-algebra homomorphism.
6.9 Example: Let kEJ:1 be a field and let X be a Boolean space
(2.12). The sheaf (!Ix defined in 2.12 carries the structure of a sheaf of
k-algebras in a natural way, and this defines a rrorphism p: X'k ->- Spec k .
We shall say that  =k (p) is the Boolean k- functor associated with X. If
R ' the map ! 1-+ J of k(Spec R, X'k) in (Spec R, X) is clearly
bijective (see also 2.11). Thus we get a canonical isancrphism
 (R) .:; !2r (Spec R, X) .
If k is an arbitrary model, we define the Boolean k- functor associated with
X by the formula  (R) = !SJ? (Spec R, X) , for RE,..,I\. It X belongs to the
fixed universe J! ' the k- functor  is a scheme: in fact, as we have
 = (XZ)k ' it is sufficient to show that Xz is a scheme. But Xz is
obviously local; also, if U is open in X, -U z is an open subfutor of
is....an affine scheme whenever U
Xz . It is thus sufficient to show that U z
{; a compact. Boolean space. In this case 11
is the topological inverse

I,  1, no 6
THE LANGUAGE
41
limit of its discrete quotients V. For each RE£1., we therefore have iso-
rrorphisms
U! (R) =  (Spec R, U) .:; lp. '!'sP (Spec R, V) = lp. V z (R)
is therefore
of the rings
into Z
.....
is an affine scheme and rJ(v]) is the ring 1,V of all
 . Hence Uz: is an inverse limit of affine schemes, and
itself affine (2.13). Its ring of functions is the direct limit
zV , in other words, the ring of locally constant maps of U
,..
By 2.11, V =SV'
Z - Z
maps of V.... into
6.10 Example: Let E be a set with the discrete toFOlogy and let
E:E7(k) be the map which assigns tc each xEE the constant map with
value x of Spec k into E . For each k-functor X, we define a canonical
map
i(E,X): E(R ,X) 7 E(E,X(k))=X(k)E
.;..k..... -k MooI
which assigns to each !:  7,K the canFOsi tion
E(k) !(k) (k) .
We claim that if 2S is local, i (E,) is a bijection. For if eEE, we have
{e}k .:; SPk k so that i({e},) is invertible for each X. In the general
case, the {et k form a covering of  by disjoint open subfunctors. The
assertion therefore follows from 4.13.
If  is a local functor, we write YLC:(k)k7 for the morphism that
i ((k) ,) sends onto I (k) EE ((k), (k)) . We shall say that  is k -con-
stant (or simply contant) - if there is a set E and an isanorphism .:;.
If Spec k is connected , this is equivalent to asserting that Y x be inver
tible.

i'
i'
I';
1";;"
11;
'Ii
k
!i
Ii'
'Iii
ill
 2
QUASI -COHERENT MJDULES; APPLICATIONS
Section 1
Sheaves of ITOdules over a geanetric space
1.1 Let X be a geanetric space, and let vIG be a sheaf of modules
over the sheaf of rings  (or simply over X) . For each open subset U,
of X, v!&(U) is then by definition a ITOdule over the ring Clx(u) ; more-
over, the transition maps cfC (U) -+ clt(V) are canpatible with the ring hano-
rrorphisms rJ X (U) -+ cJ x (V) . Hence, if xEX, the stalk ,,/( of c/{; at x is
a module over the l=al ring f!l ; we set J6' (x) = eft @ ," K (x) . The category of
x x v x
all sheaves of ITOdules over U x will be denoted by X .
Now let f:X-+Y be a rrorphism of geometric spaces. If vi{, (resp.#) is a
sheaf of abelian groups over X (resp. over Y) , we write f.  for the
direct image of cIfC (resp. f. lfj for the inverse image of uf/). Thus we have
f. (./C) (V) =(f-l(V)) where V is open in Y , and f. W1 x = (x) for xEx
When vI0 , f. (<4) carries the structure of a sheaf of modules over the
sheaf of rings f. (&X) . To see this, notice that for each open subset V ofi
X , cAb (f -1 (V) ) is a module over 11 X (f -1 (V) ) and this ITOdule structure is
"functorial in V" . Fran the hcmcrnorphism f!: J7 y -+f. (tO x ) , we derive for
f. (cd) , by restriction of scalars, the structure of a sheaf of modules over
LJ y . The sheaf of ITOdules thus defined will be written f*) , and we shall
call it the direct image of the sheaf of ITOdules c46 .
1.2
left adjoint .
Proof: Let c/f6 and # be sheaves of ITOdules over i.Px and &y . Clearly
f. (oM naturally carries the structure of a sheaf of modules over f. (cJ y )
Considering the geanetric space X' = (X,f. (Jl y ) , we have a canonical bijec-
ProFOsition: The direct image functor f*:  has a
tion.
1/JW,.4&) : (#,f. ()) -+, (f. (#) ,.4&) ,
which assigns to v:.¥'-+f.)
canFOsition
the rrorphism u:f. (vfj such that u is the
x
''If (x)
) f. (vt') f (x)
can.
) Jt
x
f. (J')x={(X)

I,  2, no 1
QUASI -COHERENT IDDULES
43
for each xEx The proFOsi tion thus follows fran the existence of a canoni-
cal bijection
Mod X ' (f" (J) ,cIt) -+  (c..9)(1 0 f" (of') , .It) ,
....- M.;W1( "ff" (V. )
Y
which sends h:f" (vf)->-J0 onto the rrorphism
hI : J x 0 f. (dI) ->- u1;{;;
f. (V y )
such that h(s) = h' (1 0s) for each section s of f. (J') .
1.3 Definition: If f:X->-Y is a morphism of geanetric spaces and
vr is a sheaf of mcxlules over J}y, the sheaf of ITOdules tJ19f. (0£/. (.vJ
over &X is called the inverse image of c/' under f and is written f* (.f)
We FOint out that, if xEX, we have
f* iJ1  0 cA(
x X,x 4,f(x) f(x)
Thus the inverse image f* (J1 of / as a sheaf of modules over X is not
to be confused with the inverse image f. (J1 of the underlying sheaf of
groups.
1.4 ProFOsition: Let A be a ring and X = Spec A . For each
A-ITOdule M, there is a sheaf of modules M over e7 x and an A- linear map
 : M->M(X) satisfying condition (*) below:
M
(*) If ."r is any sheaf of mcxlules over dJ x  CP:M+.N1X) is any A-linear
map , then there is a unique rrorphism 1/J:vY such that cp = 1/J (X) 0 cpM

M ) of"(X)
I X)
M(X) .
By definition, cA1(x) carries an  (X) -mcxlule structure. This induces an
A-ITOdule structure in virtue of the isomorphism A: A ->- iJx(X) of  1, 2.1.
The proof of the proFOsition is similar to that of  1, 2.1. We confine our-
selves here to constructing M and M. As in  1, 2.1, we first define a
presheaf M over Spec A by setting

44
ALGEBRAIC GEa.1ETRY
I,2,nol
M( D(a)) = M[s(a)-l]:::: M0A[S(a)-l] ,
A
for each ideal a of A; in this way we assign to each open subset D (a)
of Spec A a module over the ring A[S(a)-l] ; for example, for a special
open set X f with fEA, M(X f ) may be identified with the ITOdule of frac-
tions M f of M with respect to the multiplicatively closed subset
(1,f,f 2 ,f 3 ,...) ; in particular M(Spec A) may be identified with M. Let
'"
M be the ass=iated sheaf of the presheaf M. The action
11 (D(a)) : A[S (a) -1] x M[S (a) -1] -+ M[S (a) -1]
induces, by passage to the ass=iated sheaves, a morphism
,-./ ,n  "'"
11 : V x x M -+ M
"'"
which defines the structure of a sheaf of modules on M. We define M to
be the canonical map of M  M(X)
-'
into M(X) .
We FOint out that the functor M 1--+ M is exact, in as much as it is the
1
canposition of the exact functor M \--+ M with the "associated sheaf" functor.
This result could also have been derived fran the structure of the stalks of
,.J
M ;
rv ,.v
if pEX, the stalk M of M at p is the ITOdule of fractions
p
M = M [(A-p) -1] .
p
(
!
1.5
rrorphism.
Corollary: With the assumptions of prop. 1.4, M is an iso -
Proof: To prove this corollary, we refer back to  1, 2.6. It follows fran
the exactness of the sequence
M(X f ) -+ T!M(Xf.) :r!M(Xf.f.)
1 1 liJ 1J
ass=iated with each fEA and each covering of X f by special open sets
X fi
1.6 proFOsition: Let y:B->-A be a ring hanorrorphism . For each
A -ITOdule M (resp. each B -=ITOdule N) , there is a canonical isanorphism
f"V A../ ,"-J ,.....J
(BM)  (Spec y) * (M) (resp. (A 0 B N)  (Spec y) * (N)) where BM is the
B -ITOdule derived fran M by restriction of scalars .

""'-
I,  2, no 1
QUASI -COHERENT MJDULES
45
Proof: Let X = Spec A, Y = Spec B f = Spec Y . By 1.4, there is a unique
rrorphism 1jJ: (BM) -+ f* (M) such that \jJ (y) 0 BM = M . This rrorphism is an
isanorphism: if sEB, we have
(B M )"-" (Y s ) :; (BM)s :; My(s) :; M(f-l(y s ))
In order to obtain the isanorphism
i : (A 0 N)'" :; f* (N) ,
B
,.J
we observe that the functor N i-+ f* (N) is the canposition of t:w:::J functors
which are both left adjoints. It is therefore a left adjoint for the canFO-
sition of the corresFOnding right adjoints, that is, for the functor
y t-+ B1(X) . The same argument may be applied to the functor N.-+ (A 0BN)
When i is given an explicit description, we see that, for each p E Spec A ,
i is the canonical isomorphism of
p -1
q = y (p)
(A 0 B N) onto A 0 N where
p P"4.:J. q
1.7 Recall that, if M is an object of a given category, we write
M (I) for the direct sum of a family of copies of M indexed by the set I .
Definition: Let X be a scheme . A sheaf of ITOdules cAb over C/x is called
quasi -coherent if , for each xE, there is an open neighbourhood V of x
in I X I, sets I, J belonging to the chosen universe U, and an exact
- - '. 
sequence of  " of the form u I) -+ rJ J) -+ )G Iv -+ 0 , where c1&lv denotes
the restriction of cl& to V.
We write  I for the full subcategory of  I formed by the quasi-
 J_
coherent sheaves of ITOdules. Similarly, if A is a ITOdel, A denotes the
category of small A -ITOdules . If MA' M is a quasi -coherent sheaf of
ITOdules over Sp A . To see this, notice that there is an exact sequence fran
A of the f orm A (I) -+A (J) ->M+O , with I,JEQ. Since the functor M 1-+ M
cCXImUtes with direct limits, this exact sequence is transformed into the exact
sequence
II)(I) -+ (J (J) -+ M -+ 0
V x X '
where X = Sp A .
Conversely, if there is an exact sequence
C{ (I) (J) f _ 0 ,

46
ALGEBRAIC GEOMEITRY
I,  2, no 1
then vf is of the form M: indeed, for L, L' EA ' the canonical map
A (L,L') -+ (L,L') -=-s the canp:>sition of the bijection g!--+ L'0 g of
A (L,L') into A (L ,L' ()) (1.5) with the bijection ........ 1/J obtained
by setting M=L and r4I ='L' in prop. 1. 4. This proves that the functor
M >-+ M is fully faithfuL In particular m is of the form 11, where
I1:A(I) -+ A(J) is A-linear. Thus we have <AI:; (Coker 11)""
1.8 A scheme X is said to be quasicanpact if its space of FOints
is quasicanpact.  is said to be quasi-separated if its space of FOints is
quasi-separated, that is, if the intersection of two quasicompact open subsets
is quasicampact. For this condition to be satisfied, it is sufficient that
there be an open covering of  by affine open subschemes i such that
x,nx, is qua sican pa ct for each (i,j) . For x,nx, ma y then be covered b y
-1 -J -1 -J
a finite number of affine O pen subschemes x, ' 1 ; if sEJi(x,) and tE£J(X,) ,
-lJ -1 -J
(X,) n (X,) t is the union of the affine open sets (X, ' 1 ) t and is therefore
-1 s -J -lJ S
quasic anpa ct. Hence the (X, ) form an open base whose pairwise intersections
-1 s
are quasicanpact, and the assertion follows.
ProFOsition: Consider a cartesian square of schemes
f
X
£ 1
> .1:
1 
Sp 
Sp A ) Sp B
where A,BE£i-, A is flat over B, and  is quasicanpact and quasiseparted.
Then for each quasicoherent sheaf of ITOdules Jt over Y, the canonical map
A 0 .,.f(Y) :; * (oM ()
B -
is bijective .
(Clearly we write here f* iA0 () instead of I f I * (.f) ( I X I) , see  1, 4.2).
Proof: Let (y,) (resp. (Y. ' 1 ))
-1 -lJ
(resp.Y,ny.) . Let X,=f-l(y,)
-1 -J -1 - 1
gram
be a finite affine open covering of X
-1 ,
and X" l =f (y" 1 ) . Then we have the d1a-
-lJ - 1J

I,  2, no 1
QUASI -COHERENT MJDULES
47
A @vY(Y)
BF
;;. A @ (IT vf(Y,))  A @
B fv -1 a B
({;ivf"(Y ijl ) )
lw
b
? TJf* (/') (X. ' 1 ) ,
1Jl - -lJ
f* (of) (X) -----7 TT f* (.)1 (X,)
- - i--1
in which the arrows are the obvious ones. (For example, the canFOnents of
index (i,j,l) of a and b are induced respectively by the inclusions
X. ' 1 C x. and x, . 1 ex,) . The first line is exact since uV is a sheaf and
-lJ -1 -lJ-J
A is flat over B; since !* (v¥) is a sheaf, the second line is also exact.
Now, b y 1.7, we may assume that each y, is sufficientl y small for rY I Y, to
-1 -1
be of the form N i with NiC1(Yi) . By 1.5 and 1.6, it follows that
A @ vf/(y .) -+ A @ N. -+ f* (J) (X,) ;
B -1 B 1 - -1
whence, since the set of indices {i} is finite,
A @ (IT cf(Y , )) -+ n (A @ # (Y , )) -+ TT f* (f) (X,)
B i -1 i B -1 i- -1
thus v is invertible; so also is w; and so is u.
Remarks: With the above notation, supFOse that the B-ITOdule A is projec-
tive and that  is quasicanpact, but not necessarily quasiseparated. Then
the canonical map
A  (1jIcf(¥ijl)) + ljIA J'(Yijl)
is injective and v is invertible. Hence u is invertible. Similarly, if
A is a finitely generated projective B-ITOdule, our proFOsition remains true
without restriction on X . For v and w are then invertible irrespective
of the set of indices {i} and {ijl} .
1.9 Corollary: Let X be a scheme and let vf' be a sheaf of
ITOdules over X. Then rAt is quasi -coherent iff, for any affine open sub -
scheme 11. 2£  and any fEJI(y) , J/' () is small and the canonical map
cY(g:J f+ J'(!:J f ) is bijective .
Proof: The last condition on u1' simply tells us that the restriction of Jr
to 11 is identified with the sheaf of ITOdules 4(11)'" derived fran the
cY(T}) -ITOdule vf'"(!,J) . This implies, by 1.7, that J is quasi-coherent.

48
ALGEBRAIC GEDMEl'RY
I,  2, no 2
Conversely, if cAf is quasi-coherent, J(y) is clearly small. We have further
C#(W f = D(!:J) f @ $(!I)cf(u) = G(1J f ) @19(Q)vYl.!!) (l, 2.6). Setting =,  u)
and A=iJ(f) in proFOsition 1. 8, we get c/"(1J) f = A @Ef'1) = JI(f) .
1.10 Corollary: (Structure theorem for quasi-coherent sheaves )
For any ITOdel B, the functor M >->- M of 1.4 is an equivalence of the cate-
gory of small B -ITOdules onto the category of quasi -coherent sheaves of mooules
 Sp B.
Proof: By 1.7 we have already shown that M is quasi-coherent if M '
,...,
and that the functor M>--+ M is fully faithfuL Conversely, by corollary 1.9
applied to 1J =  B , we see that any quasi -coherent sheaf of m:xlules ,AI" is
---'
of the form M, with M = df( Sp B)
1.11
Corollary:
Let  be a scheme and f :J(, -+.Y a rrorphism of
X If J( and .I' are quasi -coherent, so are Ker f
sheaves of ITOdules over
and Coker f.

Proof: Let 11. be affine and open in K. By 1.10, flY is "isanorphic" to 1
"v
an arrow g for sane arrow g of  (t1() . We thus have isanorphisms
Ker flQ -+ Ker g -+ (Ker g) , Coker flu -+ Coker g -+ (Coker g)""" .
I /I
1.12 Corollary: Let K be a scheme and O-+j( -+Jt -+dt -+0 an exact
sequence of quasi-coherent sheaves of ITOdules over . Then , if U is affine
and open in , the sequence O-+"K' (Q) -+)((y) -+Jt'((Q) -+0 is exact .
Proof: We need only consider the case in which  = Q =  A, AE£i. By 1.10
and 1. 4, the functor M >->- M with danain A and target the category of
quasi-coherent sheaves of ITOdules over K has for its quasi-inverse the func-
tor <-+J;"(1J) . This last functor is thus an equivalence; in particular, it
is exact.
Section 2
Direct and inverse images of quasi -coherent sheaves
2.1
ProFOsition: r.et f:X-+Y be a morphism of schemes . If
.fE1 is quasi-coherent , so is f* if)

I,  2, no 2
QUASI -COHERENT MJDULES
49
Proof: Let Y be open in I X I , let .!J be its inverse image, and let
c:J: 11-+Y be the rrorphism induced by! There is a canonical isarorphism
t* (#1111 -+ 2* (,,(1 Y) . Hence, if I, JE,£ , each exact sequence
!2 (I) -+ cJ (J) -+JI' l v -+ 0 induces an exact S equ ence
V V -
J (I) -+ lJ- (J) -+ f* (J') I U -+ 0 (observe that g* (r.fl (I)) -+ rjJ (I))
11 11 - -y U
With the notation of 2.1, t*  is in general not quasi-co-
 ' if cfl is quasi -coherent over  : take for  the direct
of sane copies of X , for f the "ccx:liagonal" rrorphism and for
JC an inverse image * r,.f) , where vf is quasi-coherent over  . We clearly
have *) =,.//1 , the prcxluct being taken in I. It is easily seen that
such a prcxluct is in general not quasi-coherent if I is infinite. Neverthe-
2.2
herent over
SlID1 y(I)
less, we shall see that the direct image of a quasi -coherent sheaf is quasi-
coherent under sane general conditions to be specified below.
Definition:
Let f:-+'I be a rrorphism of schemes .
a) f is said to be quasicanpact if the underlying continuous map of f
is quasicanpact , that is if , for each quasicanpact open subset V of
II , f-l(V) is quasicompact .
b) f is said to be quasi-separated if the diagonal morphism
0'(./:/  -+ " with canFOnents I and I is quasicanpact .
If X has a coveril1g consisting of open affine subschemes Xi such that
t-l(X i ) is quasicanpa.ct for each i, then ! is quasicanpact. For let
(X. ,) be a finite covering of f- l (y,) by affine open subschemes; for each
-J - -1
affine open subscheme  of ¥i' ! () is then the ill1ion of the affine
open subschemes ?ij , hence quasicanpact. The open V such that i- l (V)
is quasicompact thus form an open base; and the assertion follows.
Applied to the diagonal rrorphism 0X/y' this assertion says that f is
quasiseparated if Y can be coverErl by affine open y, and each f- l (Y,)
-  - 
by affine open X., such that x,. n X'l is quasicompact for each (i,j ,1)
-J _l-J -
(For observe that X" nx. 1 = 0 x/y (X. ,x y X ))
-J - _ _ -J _i-il
Finally we can say that i is quasicanpact and quasi-separated iff Y can
be covered by affine open y, such that f- l (Y,) is quasicanpact and quasi-
- - -
separated (1.8) .

50
2.3
f is
ALGEBRAIC GEOMETRY
I,  2, no 2
f g
ProFOsition: COnsider the diagram of schemes X -> Y .... y' If
quasicanpact (resp. quasi-separated ) then the canonical proiection
f y ': "'y' -> X' is quasicanpact (resp. quasi-separated ).
Proof: SUPFOse for instance that
can be covered by affine open y
-1
(X, .) be an affine open covering
-lJ
f is quasiseparated; then y' (resp.)
(resp. i) such that 2" (.i)qi . Now let
of f-l ( y, ) t
_ -1 ; se
X, = X. ''' y y
-lJ -lJ _i- 1
then we have
x,nx" l
-lJ - 1
(X. .n X. l )x y
-lJ -1 Y.-1
-1
since
-1
x. ,n X' l = 0X ! y(X. ,x X'l)
-lJ -1 1JYi 1
is quasicompact, as well as the morphism
Hence f y ' is quasiseparated (2.2).
Similarly, if f is quasicompact, so is f y ' .
y  ----?y,
-1 -1
(see 2.2), so is X ,nx.
-lJ -lJ
2.4 PrOFOsition: If 5!:-> is a quasicanpact and quasiseparated
rrorphism of schemes , the direct image 2" *  of a quasicoherent sheaf of
ITOdules J is quasicoheren t. r-Dreover , the functor  I induced by
9:* on quasicoherent sheaves preserves filtered direct limits .
Proof: We first show that, for each affine open subscheme V of Z and
each sEV() , the canonical map
(CJk (J') (y)) s -> 2"* if) (y s)
is bijective. By replacing X by <I -1 (::!') , we may assume that Z = .Y = Sp B
and set A = B . We then have
s
(* (J1 (D ) s :;: A  y(!:)
and the first assertion follows from 1.8 and 1.9.
I
!
i
,I
As for the secorrl assertion, it is enough to consider the case where Z is
affine. It is then enough to show, that for quasicoherent sheaves on a quasi-
compact and quasi separated scheme y , the functor vY -> J'iX) preserves

I,  2, no 2
QUASI -COHERENT MJDULES
51
filtered direct limits. But, using the notation of 1.8, the functors
4-'> cf(y.) and )/-,> vt(y, ' 1 ) clearl y P !l:"eserve filtered direct limits. Thus
-1 -lJ
our assertion follows once again fran the canonical exact sequence
o -'> c.f (y) -'> TTJ(y,) -'> IT 4'(y, '1)
i 1 ijl 1J
2.5 We now supFOse  to be quasicanpact and quasiseparated . Then
the following conditions on a quasicoherent sheaf of modules 016 over X are
equivalent:
(i)  can be covered by affine open subschemes U such that Jt (Q)
is a finitely generated &(Y)-module;
(ii) for any affine open subscheme U of ]: , JC(y) is a finitely
generated x(Q)-modUle;
(iii) if a quasicoherent sheaf J is the union of sane directed set
of quasicoherent subsheaves A/', , then any morphism f: fi-'> / factors through
1
sane vV, .
1
Proof: (ii) =i>(i) is clear, and (i) =? (iii) follows easily fran 1.10
(show that the restrictions
show that (iii) implies
f I Q factor through sane )I", I U) . In order to
1
(ii) , let 1.;l:y-'> be the inclusion rrorphism and
let (N.)
1
We then have, by 1.10 and 2.4
be a directed set of suhnod.ules covering sane rJl X (Q) -ITOdule N.
lim ITOd, ( ) ((U),N,) ;:: lim Q:nO(L j (J!,U*(N,)) ;:: -= f J(  (Ni)) -
-r """""'v X Q - 1 r!: - 1 I
I(J/'*(Y);i)) ;:: d ,4,*(N)) ;:: (Q) (Q) ,N) .
Thus we get lim mcx:J IlL ( ) (A'(U) ,N,) = ITOd,.1_ (U) (Jt(U) ,lim N.) for all N and
---;> ........vx U - 1  VX --7 1
(N i ) . This is clearly Equivalent to (iiT:
2.6 Let  be quasicanpact and quasi separated . A quasicoherent
sheaf fi over X satisfying the equivalent conditions (i) - (iii) of 2.5
will be called finitely generated .
If  is any quasicanpact open subscheme of  and rp any quasicoherent
sheaf of ITOdules over U, it follows directly fran 2.4 that there is a quasi-

;',
:1
I,
' I i .. 1
i :'
i
'I I "
', . 1 1
ii
i,111
! ". ' I I
i I
Ii II
52
ALGEBRAIC GEOMEl'RY
I,  2, no 2
coherent sheaf Jt( over X with J£ 1 u ;; j) (take for instance * (Jl , where
12:-+- is the inclusion rrorphism). But there is a better result, which says
that J£ may be chosen to be finitely generated if '? is. This clearly fol-
lows fran the
ProFOsition: Let  be a quasicanpact and quasiseparated scheme and let U
be a quasicanpact open subschene . For any quasicoherent sheaf 'P over 
and any finitely generated quasicoherent subsheaf rfl of '1' I!Z ' there is a
finitely generated quasicoherent subsheaf vY of P such that cJC =vfl!Z .
Proof: Let 11 1 ,... '!:!:n be affine open subschemes of  which, together with
y , cover  . Our proposition is trivial for n=O . For n)O let ' be the
open subscheme covered by .!!'!:!:l'... 'n-l . By induction on n we may suppose
that there is a finitely generated quasicoherent subsheaf,y' of 'J'I' such
that,j(,= JI'!!:! . It is clearly enough to extend Ai' I'n!:!n to sane finitely
generated quasicoherent subsheaf of 'Y l u which, matched together with vV'
-n
over x'nu , will supply us with the required .Y. Replacing X by U arrl
- -n - -n
U by x'nu , we reduce the proof to the case in which X is affine.
- --n
In this case, let :-+- denote the inclusion morphism. The inverse image
Q. of *  under the canonical morphism P -+- }l* (Y I!:!) is quasicoherent by
2.4 and satisfies Q IQ=cd. By 1.10 Q. is the union of the directed set of
its finitely generated quasicoherent subsheaves A'" . Thus J( is the union of
the restrictions vV I!:!: and is equal to sane /I!L by 2.5 (iii) .
2.7 For any scheme  the category I of quasicoherent sheaves
over X is closed in  !  I under kernels and small direct limits. Hence
it is an abelian category with exact filtered direct limits. If  is quasi-
compact and quasiseparated, it follows fran 2.6 that the finitely generated
objects of  j generate this category. This implies the existence of a
generator in  I' since the isarorphism classes of finitely generated
quasicoherent sheaves over 2S. may clearly be indexed by sane small set.
In other words, if  is quasicanpact and quasiseparated, we can apply to
I the general results .known for Grothendieck's AB5-categories with
generators. For instance, if ! is a category with small rrorphism sets
!(x,y) , then any functor F:I -+- !.. preserving direct limits has a right

I,  2, no 3
QUASI -COHERENT MJDULES
53
adjoint. This holds in particular if  is the full subcategory of II
formed by the sheaves :f such that j"(U) is small for any open u=11 .
Taking for F the inclusion functor, we infer that any '" E! may be assigned
a quasicoherent sheaf v-tA c together with a rrorphism q :J(qc-+cAf enjoying the
'o(f
following universal property: for each quasicoherent vI(, and each  : cit-+ #"
there is a unique 1jJ: rf(-+ c¥qc such that 1jJ= .
We may in fact give a direct construction for Jr qc : Assume first that # is
of the form cAI = y*) , where Y:Y-+ denotes the inclusion of an affine open
subscheme. Then we set J:lc =:c(y) and write q,t':,tIc-+.t for the unique
rrorphism such that q;t(y).t (V) = I (V) (1.4). It is then easy to show that
(* ci:J c ) , Y * (qt) ) is a solution of o universal problem.
In the general case, when vfEK , consider a finite open covering (X.)
(resp. x, ' 1 ) of X (resp.  x,nx.) . If vi (resp. v ijl ) is the-clusion
-lJ - -1 -J - - ,
morphism of X. (resp. of X. ' 1 ) into X, we set vY, = V * l(vf!x,) (resp. JI. ' 1 =
. . -1 -lJ - 1 _ 1 1J
vJl ifl ijl)) . In this case we obtain a canonical exact sequence of 
U vr v"" //
vf"TTJ( -----'-'--TIT V'Vijl
i 1 ijl
Fran our previous remarks, we get solutions
(p,p) = (Tf0iqC,TTqu) and (Q,q) = (!J.fijl , TT 1 ' J ' l q"r 1 ' J ' 1
1 i i 1Jl
of our universal P cmblem relative to IT", and IT cY. ' 1 . We ma y therefore
i 1 ijl 1J
define rrorphisms v, w: p==:; Q by the conditions qv=vJfP and qw=w ",P . It
thus remains to set ur qc = Ker(v,w) and to define qvf: c -+,)/ by the con-
dition uJq,v- = pu , where u : Ji qc -+ P is the inclusion morphism. The pair
rj;qc : qet) is the required solution of our universal problem.
Section 3
Faithfully flat quasicompact rrorphisms
We give here another extremely useful property of quasicanpact morphisms. We
first make a new definition.

54
ALGEBRAIC GEOMETRY
I,  2, no 3
3.1
Definition:
A rrorphism of schEmes ! :->- is said to be flat at
f :r!) f ( ) ->-tJ makes t9. a flat ITOdule over JJ f ( ) .
-x x x - x x
- -
it is flat at each xq, and faithfully flat if it
a point xEX i f the map
It is said to be flat if
is flat and surjective.
In the case of affine schemes, Sp  :  A ->- Sp B is flat (resp. faithfully
flat) iff :B->-A makes A a flat (resp. faithfully flat) module over B
(Alg. Catrn. II,  3, prop. 15 and cor.). For a morphism of schemes f:X->-Y to
be flat, it is therefore necessary and sufficient that the following condition
be satisfied: if 1l,Y are affine and open in ,'£ and !(ll)cy, then tJ(g)
is a flat module over tJ(y)
3.2
ProFOsition: Consider the cCXImUtative square of schEmes below ,
which assume to be Cartesian . Let x' be a point of ' and x=J2 (x I ) . If
f is flat at x, f' is flat at x' If f is flat (resp. faithfully flat ),
f' is flat (resp. faithfully flat ) .
p
X'
f'l
y '
q
) X
If
> Y
Proof: By  1, 5.7,
y'=f' (x') and y=f(x)
flat over tJ. if ()
y x
f' by  1, 5.5.
, is a ring of
x
. Hence  I
X
is flat over tJ .
y
fractions of r!) ,0.n () , where
y v y x
is flat over J) ,0!).J , which is itself
y y x
Also if f is surjective, so is
Remark: We occasionally make use of a converse form of the foregoing propo-
sition, namely, if !' is flat at x' and if S!. is flat at y'=f' (x') then
f is flat at x=p(x') For t2, is then flat over rY. , where Y---<I(Y')=(x);
x y
by the proposition r!J, is flat over If) ; so t!J is flat over & (Alg.
x x x y
camm. I,  3, no 4, rem. 2) .
3.3 Let k be a ITOdel and  a k-scheme. We say that X is flat
(resp. faithfully flat) over k . if the structural projection
!?: '#,0. ->-  k is flat (resp. faithfully flat) .
ProFOsition: Let k be a ITOdel , Z and r two quasicanpact and quasisepa -
rated k- schemes. If Z is flat over k and if 0(Y) is a flat k-ITOdule,
-

->.
I,  2, no 3
QUASI -COHERENT MODULES
55
then the canonical map c1()l8ikt9() ->- J() is bijective .
Proof: Of course  x  denotes the prcxluct in the category  . Let (.ei)
and (Z. ' 1 ) be finite affine open coverin g s of Z and Z . nz, . In the dia-
-lJ - -l-J
gram
,j ()I8ic9C)----7 (TTJi(.) )18ic9(y) ===+( TT .fl(z. 'l)I8iJ'(Y)
u r iv ( k ij ('J k -
iJ(l!x¥)---7TT(z,xy) > IT(z..xy)
i -1 - > ijl -lJl -
the two lines are exact. We see that v
(resp. w)
setting B=k and A = (!!i) (resp. A =
follows that u is invertible.
ffi, ' 1 ) )
1J
is an isanorphism by
in proposition 1.8. It
3.4
ffqc descent theorem: If f:X->-Y is a faithfully flat quasi -
compact rrorphism of schemes and if 2f 1 ' P.!:2 are the canonical prolections
of the fibre prcxluct "y onto its factors , then
Ill If I
IXI t II ) I1'!
Ip.!"21
is an exact sequence of qeometric spaces .
Proof: By definition 3.1 If I is surjective. By  1, 5.4, if x,x'E satis-
fy (x)=f(x') , there is a point ZE0X such that x = Pl(z) ,
x' = pr 2 (z); since the converse is obvious, we may identify the underlying
set of  with the quotient of the underlying set of X obtained by identi-
show that Y carries
fying P!"l (z) with Pf 2 (z) for zE0X. It remains to
the quotient topology and that the sequence
i!J 1' ->- i* (&) :to * ("0) ,
where 'I = ipl = tp!:2 ' is exact. We prove the second assertion first:
-1
For each affine open sub scheme V of Y, if we set Q = i (Y) , we must
show that the sequence
JI(y) ->- cY(]}) :t J7 (]} V l})

56
ALGEBRAIC GEOMETRY
I,  2', no 3
is exact. If Q is affine, this follows from  1, lemma 2.7 by setting
e=t9(y)=M, B=(Q) . In the general case, by 2.2, there is a finite affine
open covering (\:Ii) of .; if Q' is the disjoint sum of the 9 i , U' is
affine and the canonical rrorphism of 9' into  induces a diagram
u v
t9 (V)  t9(U) ====:3. c9(U x U)
 u' i: :,1;
J(y) 7 $(9') ====:';J1(U'x U')
w' ,- v-
such that u'=iu,
, ..
Jw=N 1 ,
j\=V'i . The bottan sequence is exact since U'
is affine; the top one is likewise since i is injective.
Finally, I X I carries the quotient topJlogy. For if F is a closed subset
of X, we show below that
f-l( (F) ) = f-l(f(F))
if F is saturated, we therefore have
and
f-l( f(F )) = F = F
f (F) = £ (( 1 ( f (F) ) =! (F)
It follows that f (F) is closed.
We now show that f- l ( f (F) ) = f- l (f (F)) , where F is closed in II . By
substituting y.. for X and U' for X we may assume that X =  B ,
 = p A and f = p  . Now let a be an ideal of A such that F=V (a)
-1 -
( 1, 2.4) and b =  (a); we have f(F) = V(b) (l, 2.4). From the exact
sequence O+b+B+A/a we derive, by means of the flat extension  :B-+A ,
A @ b .:; M (b)
B
Ker in l
where
of the
in l A -+ A(A/a) satisfies in l (x) = x@l . By  1,2.4, the closure
image of Spec (in l ) is thus
V(A(b)) = f-l(V(b)) = f-l( f(F) )
It therefore renains to show that the image of Spec (in l ) coincides with
f- l (f (F)) , ant this follows fran  1, 5.4 applied to the fibred proouct of
the diagram -Lxp(A/a) : indeed, if we set .f = p(A/a) , the image

I'!"
I,  2, no 3
QUASI -COHERENT MJDUIES
57
of
11C fl
in II
equals that of
IX l x I F I , which is f-l(f(F))
- I- - -
3.5
COrollary: COnsider the ccmnutative diagram of schemes below. If
q is faithfully flat and quasicompact , and if
f x Y : Xx Y -+ X'" Y
-- -- -E-.-
is invertible , so is f.
f
X > r
Y ) .!!
9
Proof: If f x Y is invertible, so is
- z-
f x (Y x Y)
-z -z-
-+ (f x Y) x (Y x y)
-z- Y -z-
- - -
Thus, in the diagram below, the first two vertical arrows are invertible, so
therefore is the third provided the two horizontal sequences are exact. But
this follows fran theorem 3.4 ITOdulo the well known identifications
x x (y x y) .+
-  -.9-
(X x Y) x (X x Y)
-z- X -z-
- - -
and
X'x (Y x Y) ::; (X'xY)x (X'xY)
- Z - Z - - z- X'- z-
X x (Y x Y)   PEl > X x Y > X
- Z -z- X x Rr 2 ) - z- P.!" 1 1,
- z - IfY
fx(YxY)
-z - Z -
t X'xRr l
X'x (Y x Y) - Z - > X'xy x'
>
- z -z- X'''Rr ) - z - PEl
-  -2
3.6
COrollary: COnsider the conmutative square of schemes below . If
i is a monomorphism and .f is quasicanpact and faithfully flat , then there
is a ill1ique rrorphism j :-+ such that =i and li.

58
AlliEBRAIC GEOMEl'RY
I,  2, no 4
f
 -=-----". X
t i ly
ZT
- -
Proof: With the notation of 3.4, we have P! 1 = !p 2 whence P! 1 =
Y!PE 1 = YP!" 2 = !l}P!" 2 ; hence .YP.!' 1 = .YPE 2 since 1: is a rronorrorphism. By
3 .4 and  1, 4.4, there is a unique rrorphism j such that .':.1 = j  ; hence
 = !- = Y ' so that  = v
Section 4
The functorial FOint of view
We now develop a purely "functorial" theory of quasicoherent modules on schemes
and show how this new notion overlaps the preceding "geanetrical" definition.
4.1
Let  be a l-functor and let !1 be an -functor ( 1, 6.1).
For any REM and any pEE (R) , call t1(R, p) the fibre of !i over p.
SUPFOse each fibre (R,p) to be given an R-mcxlule structure in such a way
that, for any EJ:J.(R,S) , the induced map 1jJ: t1(R,p) ->- t!-(S, (p)) is - linear ,
which shall mean that 1jJ is additive and satisfies 1jJ (Am) =  (\) 1jJ (m) for all
m(R,p) and \ER. We then say that !i, together with these module struc-
tures, is an  -ITCdule . If t!- and  are two -ITOdules, a rrorphism
fEJ:J e £; (t1,) is called linear or an  module rrorphism if
f(R,p): t1(R,p)->- (R,p) is R-linear for all (R'P)e. These definitions ob-
viously give rise to a new category, which we denote by S: the abelian
category of f?-ITOdules.
An -ITOdule  is called quasicoherent if its fibres (R,p) are small and
if, for any Et1(R,S), the iriduced map t!-(R,p)Q9RS ->- M(S, (p)) is invertible.
r-breover, if (R,p) is a projective R-ITOdule of finite rank (resp. of rank
n) for each (R, p) , we say that Ii is a vector bundle (resp. a vector bundle
of rank n) over  The full subcategory of S formed by the quasicoherent
-ITOdules will be denoted by 's -
4.2
First examples:
a)
Take t1(R,p) = R n with the usual R-ITOdule
will be written Q and called the
structure for any (R, p) . This M

I,  2, no 4
QUASI -COHERENT OODULES
59
trivial vector bundle of rank n. For the underlying -functor ( 1, 6.3)
n
we clearly get z!:1 = f'xQ where Q is the affine line ( 1, 3.3). Thus M
is a scheme if S is one.
b) Let A be a ITOdel and let M be a small A-ITOdule. Set e. =  A and
Ma (R, p) = M @AR for all pE!:1(A,R) We thus get a quasicoherent e.-mcx1ule
Ma ' and the functor M 1-+ Ma is clearly an equivalence of A (1.7) onto
S ; the functor  I->- (A,IdA) is quasi-inverse to it.
c) Let f:.s be a morphism of 1-functors. For any.s-module !:1, the
e.'-functor S ' derived fran  by base extension ( 1, 6.5) is clearly
assigned an '-ITOdule structure. This '-ITOdule !:1 s , is quasicoherent or a
vector bundle, if  is so.
d) The base extension functor ?f": f' ->- .e.' defined in c) has a right
adjoint. In order to see this, we only note here that, for any EITOdule M'
and any (R, p) S ' the set
(Elle.M') (R, p)
t1 s ']( ( SP s (R,p)) S., !:1')
considered in  1, 6.6, carries a natural R-rrodule structure. In fact, any
X E ( sl')S M') (R,p) is a function assigning sane X(,p') E N'(R',p') to a
pair (,P') E r:1(R,R').x.s' (R') such that .s() (p) = (R') (p') . We define
addition and scalar multiplication by the formulas (Xl+X 2 ) (,p')
Xl (,p') + X2('P'):' and (AX) (,p') = (A) .X(,p') . That the so defined Weil
restriction functor '/1 : ..s' ->-.s is right adjoint to
? e. ' : .s ->- .s' may be proved as in  1, 6.6.
We shall see later that the Weil restriction of a quasicoherent ITOdule over
.s' is in general not quasicoherent over s.. Nevertheless, this is obviously
true if both SandS' are aff ine schemes.
4.3 ProFOsition: Let S be a - functor and let  be a quasi-
coherent  -ITOdule . Then li is local ( 1, 6.3) iff S is. r-Dreover , g .l1.
is a vector bundle, then M is a scheme iff S is.
Proof: Clearly the structural map E:zr:t+
.s is a retract of zr:1 and thus is lQ;al or
true of !:1. Conversely, first supFOse .e..
of definition  1, 3.11 with  = z . Let
( 1, 6.3) has a section so that
a scheme provided the same is
to be local; we use the notation
E; ,EX (Rf ' ) be elEments such that
1 - 1

60
AlliEBRAIC GECMEI'RY
I,  2, no 4
v((S,)) = w((.)) , and set 11, =P (.) E S(R f ) . Since 11 and 11 ' are
1 '0 1 1 '0 1 - - i i J
assigned the same image 11" in S (R f f ) for each i, there is a unique
1J - i j
nEe (R) with image n i in(Rfi) . Since  is quasicoherent, (Rfi,ni)
and M(R f f ,n, ,) are identified with M(R,n)0 R R f' and M(R,n)0 Rf 0 R R f.
- i j 1J - 1 - R i J
Hence by lerrma  1, 2.7 applied to the case C=R, B = TJR f and
1 i
M = (R,n) , there is a unique S E (R,n) c (R) having si as image in
I1(R fi ,n i ) c (Rfi) for each i
Suppose finally that  is a scheme and that M is a vector bundle. Since
I1 is known to be local, it is enough to show that, for any affine open sub-
scheme U of  ' the Q-mcxlule !:1IQ derived fran  by the base extension
!I ->-.e (4. 2c)) is a schEme. The proof is thus reduced to the case in which
e. is of the form 2. A and  of the form Ma (4. 2b)) . If M is free of
finite rank, we are through by 4.2a). In the general case M is projective,
Le. a retract of sane An. Thus ! = 'J:..Ma is a scheme as a retract of
SxO n = (An) (4.2a)).
- -  a
4.4
Let tl and N be vector bundles over S ; we call I1 a sub-
bundle of , if for any (R, p) EJ:1 s the fibre !1(R, p) of M is a direct
surrmand of !,!(R,p) as an R-module. Vector bundles and subbundles arise
naturally in the study of grassmanians ( 1, 3.4, 3.9 and 3.13). We clearly
get a subbundle ! of rank n of the trivial bundle of rank n+r over
 n,r be setting 1'(R,p) = pcRn+r for all R and pEG (R). This is
-n,r,
the so-called tautological bundle over G
--n,r
ProFOsition: Let  be any l- functor. By assjgning to each rrorphism
:->gn,r the vector bundle E  .e derived fran the tautological bundle
by the base extension f , we get a bijection fran ME (S,G ) onto the set
- .............., - -n,r
of subbundles of rank n of the trivial vector bundle of rank n+r over S .
Proof: Tautology: A rrorphism i assigns by definition to any OE (R) a
direct surrmand of rank n of R n + r :
Notice that the preceding proposition is often given another equivalent formu-
lation: if M is an -ITOdule, just call a section of !i any section ° of
the structural proj ection I'M: zt1+ ( 1, 6.3). This section ° assigns to
each (R,P)E£"J.i2 an element -;(R,p) E (R,p) . We shall say that the section
0l,...,Ot gener2te , if 01 (R,P)'...'Ot(R,p) generate the R4TIOdule

I,  2, no 4
QUASI -COHERENT MJDULES
61
£1(R,p)
On ->- M
-S -
for each (R,P) . Equivalently, this means that the induced rrorphism
is an epirrorphism of S
Now consider the vector bundle T' of rank rover G defined by
n+r -n,r
'!" (R, p) = R I:!' (R, p) . The images of the natural basis elements of R n + r
give us n+r sections El,...,E n + r of T' . For any rrorphism !: ,r
of  ' we denote by El'... , E n + re the induced sections of 1' (set
EiS(R,p)=E i (R,fp) E :!,'(R,fp)S(R,p) for any (R,P)E) . We thus assign to
any g a vector bundle of rank rover S together with n+r generating
sections. Conversely, a vector bundle 11 of rank rover S and n+r
generating sections ° 1 ,... ,on+r of l:1 detennine a rrorphism En,r
assign to any (R, p) Etl S the R-mcxlule of all relations between
0 l (R,p),..., ° + (R,p)-E M(R,p) 
n r -
4.5 We still have to relate the functorial to the geanetrical pJint
of view: first consider a geometric space X together with a sheaf of ITOdules
JC over V x . We get a module dr; over the associated -functor EX (l,
3.5) by assigning to any p: Spec R ->- X the ITOdule of sections of p*) .
In other words we set
(SiC) (R, p) = p* (J() (Spec R) .
When X is the geanetric realization of a scheme  , p* (Jto) is quasicoherent
over Spec R if J(, is so over X . In this case, it follows directly from
1. 6 that J£ is a quasicoherent f'X-ITOdule.
Conversely, let !'! be a ITOdule over sane -functor .x . Assign to any open
subfunctor y of X the rrodule () of all sections of I (the -module
derived fran !'!" by the base extension y->-y) . This obviously provides us
with a presheaf of ITOdules over 11'1. The associated sheaf of ITOdules will be
denoted by II . Notice for instance that IMal = M if X = Sp A and
MEA (1.4 and 4.2b)). This implies in the general case that I!'!" I is quasi-
coherent if Y is a scheme and N is quasicoherent.
ProFOsition: For any scheme , the functors J£ ->- S,{( and M ->- IMI provide
quasi-inverse equivalences between the category  of quasi=herent
ITOdules ove r !. and the category  I X I of quasicoherent sheaves of
rrodules over I X I . -
Proof: By construction we have .ft  I  J( I for any sheaf of modules,)( .

62
AlliEBRAIC GECMEI'RY
I, . 2, no 4
Conversely, let !:1 be a quasicoherent -ITOdule. If  = Sp A is affine
there is, by 4. 2b), exactly one isarorphism i:: I:1::; e I I inducing the identi-
ty on I:1 (A, IdA) ;;; (e 1I:11) (A, IdA) . In the general case, we may thus assign to
each affine open subscheme y of  an isanorphism : I:1IQ::; (e 1t11) Ig .
As I:1 and e IJiI are local by 4.3, these  may be tched together and
provide us with a canonical isarorphism I:1::;- e 1I:11 .
4.6 Let f:X->-Y be a rrorphism of geanetric spaces and vr be any
sheaf of mcdules over CJ y . For any : Spec R ->- X we have by definition
(f*J') (R,p) = (p*f*f) (Spec R) and (.v)SX(R'PJ = ((pf)*J) (Spec R) . Thus
 (f*.(J and (eJ) SX are canonically identified.
For direct images the situation is rrore involved. Let  be a sheaf of ITOdules
over iJ x and let a: Spec S ->- Y be a rrorphism with S. By definition
we have e(f*.J() (S,a) = (a*f*.4) (Spec S) , whereas ( l/Jy e.k) (S,a) may be
described as the set of maps assigning to each cCXImUtative square
p
') X
lf
> Y
Spec R
'111
Spec S
a
a section of p* () , which of course has to be functorial in (R,p,'TT) .
Thus the canonical maps (a*f*utC) (Spec S) ->- (p*J() (Spec R) provide us with
a canonical rrorphism
j : f*J( ->- eIY e JC
But this j is not an isarorphism even if X, Y are geanetric realizations
of schemes and vi{; is quasicoherent. Indeed, consider the case where
x=lxl , Y=lxl and f=lfl . Then the above-mentioned caTlIT!Utative square is
mapped into the pull back square
liS)f X
Spec S
PQ
l
a
) Y
where (Spec S)y X is the geanetric realization of the schEme (S)X 

I,  2, no 5
QUASI -COHERENT-M)DULES
63
Therefore ( s/y 4) (S,a) may obviously be identified with the set of
sections of - pr (uiC) over (Spec S)y X , or equivalently with
(prl*pr 2 ciC) (Spec S) ; rroreover, j (S,a) is induced by the canonical rrorphism
a*f*,At; -+ pr 1 *pr 2 c4C .
The question whether j (S,a) is an isanorphism plays sane role in algebraic
geanetry. A simple example where this is so is given in 1.8. A simple
counter-example is obtained as follows: consider the pJint w of the J.,-func-
tor P l (the projective line) assi gn ed to the subspace F (1,0) E P l (F) of
- """P --p
F 2 , where F is the prime field of characteristic p > 0 . Let X be the
-p ....p
open subspace of I..\ I ccrnplementary to w and let f: X -+ Spec  and
a: Spec F -+ Spec Z be the canonical rrorphisms. It can then be shown that
"'"'p -
f * (t2 x ) (Spec Z) = c!J x (X) ;; JJ p (P l ) ;: Z hence (a*f * <2 x ) (Spec F ) ;; F , where-
- _1 - ,... -p "'p
as pr* 2 (J x ) ((Spec F )( X) is F LT ] , the ring of pJlynanials in one variable.
"'p -p
4.7 When X is a schane and M is a quasicoherent -ITOdule, we
shall sanetimes simply write  instead of II , thus identifying quasi-
coherent -ITOdules with quasicoherent sheaves of ITOdules over II .
Notice also that the definitions and statements of this section may be easily
extended to the relative case of k-schemes, with k. We entrust this task
to the reader.
Section 5
Affine rrorphisms
5.1 Definition: !::. rrorphism i:-+ of Z-functors is said to be
affine if , for each ITOdel R and each morphism 9": Sp R -+  , the fibred
product Sp R Y  is  affine scheme .
If  is a !-functor, we say that a -functor X
structural projection r:-+e is affine.
is affine over S
if the
5.2
Example: If Y is an affine scheme, f is affine iff X is
an affine scheme. We see that the condition is necessary by taking g to be
an isanorphism; conversely if X = Sp B and  = Sp C , the fibred product
Sp R Y  may be identified with Sp (R'if) , which is an affine scheme.

,
64
5.3
is X
ALGEBRAIC GEOMETRY
I,  2, no 5
ProFOsi tion: If f : x+y is af fine and if Y is  scheme, so
Proof: We show first that  is local, making use of the notation of def.
3.11 of  1. Let i E X(R fi ) be elements such that v( (i) )4-J( (i)) ; set
,=f(.) . Since . and , have the same ima g e in Y(R f f ) and Y is
1- 1 1 J - ij
local, there is a unique E¥ (R) whose image in Y (R f ) is  . for all
- i 1
iEI . If Pi:R....Rfi is the canonical map, (Pi'i) belongs to the fibred
product ( Sp R 1 29 (R fi ) defined by #: Sp R ....  . Since Sp R Y  is
representable, hence local, and the (p.,.) satisfy the usual canpatibility
1 1
E ( Sp R Y ) (R) which maps onto
 is the unique element of  (R)
conditions, there is a unique (p,)
for each i. Obviously p=I and
image in (Rfi) is i for all i
(Pi'i)
whose
This shows that  is locaL We obtain an open covering of  by affine
schemes by considering the fibred products  R Y  attached to the open
embeddings g: Sp R ....  for RE.£1.
5.4
Let  be a kfunctor. An - alqebra is a pair (,a) consistiJig
of an .e-module !1 and a rrorphism of .e-functors a:!y""!2. such that, for any
REl;! and any PEe (R), a (R, p) is the multiplication of an R-algebra structure
(associative, cCXImUtative, with unit) canpatible with the R-mxlule structure
of (R,P) . The e.-algebra (,a) is said to be quasi-coherent if A is
quasi-coherent (4.1). If (12,13) is a second -algebra, a rrorphism of (,a)
into (,13) is defined to be a rrorphism of -functors f:A....B such that
f(R,p) is an R-algebra hananorphism for each e-ITOdel (R,p) . Henceforth we
shall simply write 1': instead of (,a)
As a first example we have the .e-algebra 9!'i defined by .9.e(R,P) = R . More
generally, if X is an S-functor, the Weil-restriction TT O x clearly bears
- - 25/S --
an -algebra structure. This e-algebra is quasi -coherent iF X is affine
over S .
ProFOsi tion: If  is  1- functor , the functor   11 Q is an anti-
equivalence between the category of e.- functors , which  affine over  ,
and the category of quasi -coherent e- algebras .
Proof: Our prop::Jsition is a direct oonsequence of the definitions. We can
associate with each quasi-ooherent -algebra !': an !'i-functor Sp !': which we

I,  2, no 5
QUASI -COHERENT-M)DUIES
65
call the spectrum of :
( Sp N (R,p) = ((R,P), R) for each (R,p)Et1 S .
The R-functor
l: SP R....
( ?:) R deduced fran ?: by the base extension
satisfies
( Sp A)R(R' ,) =.£%. (A(R' ,p), R')
:+ M ,(R' Q9A(R,p), R') :+ M (A(R,p), R') .
.;,, R ""
This means that ( Sp A)R is represented by the R-model b(R,p) . Hence
Sp A is affine over S . Moreover, if we set X = Sp A (! 10 ) (R,p) is
- - - - - - XIS -
ide11tif ied with the algebra of functions of (A) R ' i. e . with  (R, p)
This means that the canposition (g Q?) 0 Sp is isomorphic to the identity
functor. Finally, we also have o( Q?) :+ Id because (RQ) (R,p) is
is the R-algebra of functions of the fibred product of
# Px
Sp R P ) S .( - - 'f:..
Being affine, this fibred product is identified with SP ( J/J Q) (R,p)
5.5 When the -functor  is a scheme, we may interprete the pre-
ceding prop::Jsition in the following way. Supp::Jse  is affine over e.. Then
the direct image (Px) * (J1 x ) of J x under Px: Z0.... is quasi-coherent by
2.4. Moreover, the mOrphi J7 S .... (p) * (iJ x ) indu;;ed by Px assigns (Px) * (J x )
the structure of a sheaf of algebras over  (by definition a she..af of - alge =-
bras uf; over  asociates with each open subset u=1 el an associative and
ccmnutative algebra ci(U) with unit; rroreover, the restriction maps
d(U) .... vt(V) are ring haranorphisms and are canpatible with  (U) ....  (V))
Corollary: If  is  scheme , the functor (P)*(J0) is an antiequiva-
lence between the category of S-schemes , which are affine  e , and the
category of quasi -=herent sheaves of algebras over E .
Pr=f: We only have to show that the rrorphism j: e (p) * (J7 0 ) .... /J 9 de-
fined in 4.6 is an isomorphism. In fact, as we know already that
g Qx is quasi-coherent, it is sufficient to canpare S (x) * (Ji x ) (R,p) and
( Q) (R, p) when l: Sp R .... S is an open embedding. In this Case both
algebras are identified with J( R x ):
- S

,
66
ALGEBRAIC GEOMETRY
I,  2, no 5
According to 5.4 we get an antiequivalence, which is quasi-inverse to
 (PX) * (J1 x ) by assigning as follows a -functor Sp vG, called spectrum ,
to any -quasi-coherent sheaf of algebras vi over  : If REJ:)., ( Sp d) (R)
is the set of pairs (p, A) consisting of a pE (R) and a rrorphism of sheaves
of algebras A:ut ---7' P(tJ R) ; if :R + R' is an arrow of 1:2, (Spd) ()
maps (p, A) onto ((p) , p* (11) 0 A) , where 11 is the morphism
(Spec )f 0spec R + (Spec )*(c?spec R') .
The maps (p,A) 1-+ P define the " canonical pro-jection " 12
sp d-+ 
5. 6 Corollary: L€t : + be  morphism of local l.- functors and
(Xi)   covering of !: If , for each i, f induces  affine 
h ' f -l
P sm i: _ C'r:"i) + Xi ' then ! is affine .
In particular, vector bundles over a scheme X are affine over Y .
Proof: With the notations of 5.1, it is sufficient to sha.v that the canoni-
cal projection Sp R Y  + Sp R is affine, assuming that the canonical pro-
j ections S R)( C 1 (Y ,) + S R)( y,
-.P X - -  X-
are affine. We may thus supIXJse straight away that X = Sp R , hence that 
is a scheme. Replacing (i) by a finer covering if necessary, we may assume
further that each y, is an affine open subscheme of Y. Then f -1 (Y , ) is
- - - 
aff ine by 5.3. Being local and being covered by affine open subfunctors, 
is a scheme.
The direct image d = * (c9 X ) is a quasi-coherent sheaf of algebras over Y ,
since this is true for each dl ¥ i . As  and Sp ut are both local functors,
the canonical isanorphisms i- l (¥ i) +  cJ(r 1 (¥ i)) = Sp d(y i) induce an
isanorphism  +  d (cf, the argument of  1, 4.16).
5.7 Corollary: L€t  -L   X be  diagram of schemes such that
S!. is faithfully flat and quasicanpact. If the canonical projection
fX:    + X is affine , so is .
Proof: By restriction to an affine open subschane of , we reduce directly
to the case in which Z is affine. We must now show that X is affine. If
(i) is an affine open covering of Y, we may replace q by the canIXJsition

'-
I,2,n06
QUASI-COHERENT-MJDULES
67
Y'yZ
- - -
where Y' is the disjoint sum of the Xi .
SupIXJse therefore that '!: and z: Z X are affine and set k=J() , B=J()
and C = tP(zy) . Since S! is then affine and surjective, the same holds for
the canonical projection S!:   X ->-  ; therefore, since   X is quasi-
compact, so is X. Hence there is an affine open covering (i) of X
-1
since SIx (i) is affine
-1 -1 - t
g (ij) = SI (i) yq j)
is affine, hence quasicanpact. It follows that x,nx, is qua sicanpact and
- -J
lerrrna 1.8 implies that there is a canonical isc:morphism
tJ() B = J()J(X) -+ tJ(z:r n = C .
It remains to show that the rrorphism 1)!x:  ->- Sp U(X) (1, 4.3) is inver-
tible. Now we have isorrorphisms
( rJ()) Z X -+ (G9()B) -+ Sp (J(¥)) ,
which allow us to identify 1/J x Z X wi th the rrorphism
1)!XXY :  Z X ->- Sp tP(zX) .
-z-
But this latter is invertible; so therefore is 1)!x (3.5) .
Section 6
Closed embeddings
6.1 Defini a) Let i:->-¥ be  rrorphism of l- functors . We say
that ! is  closed embedding if  is affine and if , for each A and
each rrorphism 2:: Sp A ->- 1 , the 
J( A)
J\Sp A) ->- J(Sp A x X)
- y-
is surjective .
b) If  is  subfunctor of X,  say that X
inclusion rrorphism is  closed embedding .
is closed in Y if the

68
ALGEBRAIC GECMETRY
I, 9 2, no 6
If Y is a scheme and  is a closed subfunctor of Y , 2S is a scheme by
5.3. Accordingly we call  a closed subscheme of Y .
SuPFOse that in statement a) , 'i! is of the form !..!:. Then the morphism
:  A ->-  A X  with canFOnents Id Sp A and b- is a section of %sp A
Hence &(s) J(! A) =Id , so that J(!sp) is bijective, whence .:>=!-;.
and h = 'i.fl A . This shows that .!?- is uniquely determined by 'i!' so
that a closed embedding is  rronanorphism .
Now sUPFOse we are given a monanorphism of -functors !:2<:->- . By arguing as
in 9 1, 3.6, we -see that ! is a closed embedding iff for each AE!j and
each o.E (A) there is an ideal I of A satisfying the following condition:
for each arrow  :A->-R of !2, a R  (R) belongs to the image of 2S (R) in
Y(R) iff (I) = 0 .
6.2
Example: If, in 6.1,  is an affine scheme, then a necessary
and sufficient corrlition for f to be a closed embedding is that  be an
affine scheme and &(!) be surjective. The necessity of the condition is
verified by taking g:  A ->-  to be an isanorphism ; the converse is ob-
vious.
If I is an ideal of AEt: and :A->-A/I is the canonical map, the functor-
image Y(I) of Sp  is such that, for each R, we have
Y(I) (R) = {f£:1(A,R) I  (I)=O} .
It follows that the map I H-Y(I) is  bijection of the set of ideaJs of A
onto the set of closed subschemes of  A
6.3 Example: Set X = 9r,n-r x 9s,n-s ' r ,; s (9 1, 3.4). Write
F (r,s) , or simply f , for the subfunctor of Y satisfying
-n -
g(A) {(P,Q)E9 r ,n-r(A)x9 s ,n_s(A) I PCQ}
:hen F is closed in J . For let 0.= (P ,Q)
;J in A n + r
and let Q'
be a canplement for
; let Pi'...' Pn, be the proj ections onto Q' along Q of a
system of generators of P. SUPFOse that Pi = (a li' a 2i , . . . , a (n+r)i) . With
the notation of 6.1, the Condition R@PCR@Qisequivalent to  (a l . .)= 0
A A J
for each (i,j) . It thus suffices to set 1= L Aa" (6.1).
Jl
i,j

I, 9 2, no 6
QUASI -COHERENT-M)DULES
69
6.4 It is obvious that the canFOsition of two closed embeddings is
a closed embedding. Similarly, if in the diagram -L¥y' of  f
is a closed anbedding, then the canonical projection fy.: ¥. ->- Y' is like-
wise. Finall y , g iven a P crojective S y stem of closed embeddi ng s f. :X.->-y, ,
-l -l -l
the projective limit
1 im f. : 1 im x. ->- lim Y.
 -l  -l -l
l l l
is a closed embedding. We prove the last assertion: if AEtJ and
a E   i (A) , let a i EYi (A) be the projection of a with iooex i. By
6.1, there is an ideal Ii of A such that, for each :A->-R, the relation
aiRE%i (i) is equivalent to  (Ii) = 0 . Hence a R belongs to the image of
lim x. (R) iff  (I.) = 0 for all i In order to satisf y the criterion of
-l l
6.1 it thus suffices to set I = I I.
i l
6.5
Example: Let
l$r,$.. .$rs$n be an increasing sequence of inte-
(n; r l ,...,r s ) with coefficients in REM is a
gers. A flaq of nationality
sequence
(P i '''. ,P s ) E G (R)x...X G (R)
-r l n-r l -rs,n-r s
such that Pi C P 2 C... C Ps . The subfunctor !'n(rl,...,r s ) of
G x ...x G formed by these flags is called the scheme of flags
-r l ,n-r l -rs,n-r s
of nationality (n; rl,...r s ) . It is a closed subscheme of
G r n - r x ... x G r n r For let F" be the subfunctor of this last scheme
- l' 1 - s' - S -lJ
formed by sequences (P l ,...,P s ) such that P,CP. (i<j) . Then F" is
l J -lJ
the inverse image of F (r"r,) under the canonical projection of
-n l J
.:; x ..." G onto G x G ,
-rl,n- r l -rs)n-r s -ri,n-ri -rj)n-rJ
is a closed subscheme, and so is I n (r l ,...,r s ) =
By 6.3 and 6.4
(\,F. '
l<J-lJ
F, ,
-lJ
6.6
ProFOsition: If f:X->-Y is a closed embedding , f is a haneo -
morphism of II onto a closed subset of 1'[1 .
Proof: Let  :K->-L be an extension of small fields and let p be an element
of 'irK) . It follows easily fran 6.1 that PL E (L) belongs to the image of
:f (L) iff P belongs to the image of f(K) . This implies that flK is an
- No<
isanorphism of XIK onto the union of a collection of indecomposable campo-
- ...
nents of YIK (9 1, 4.5) , so that f is injective (9 1, 4.9). To canplete
- "" -

70
ALGEBRAIC GECMEIT'RY
I,  2, no 6
the proof, we may assume that  is a closed subscheme of Y and that f
is the inclusion morphism. We show that each closed subset P of II is
closed in II . By the definition of I¥! (1, 4.1), we must show that
g-l(p) is a closed subset of l.ep AI = Spec A for each AE£:: and each mor-
phism g: Sp A ->-  . Now 2:- 1 (P) is closed in g-l(IXI) , which is easily
-1 --
seen to be the set of FOints of g (X) (use the fact that XIK and
g -1 (X) I K are both sums of collecion of indecanFOsable can;n;nts of Y I K
 p""AIK respectively). Since g-l(X) was assumed to be of the form ....
- "" -1 -1 --
Sp(AjI) , g (IXI) = Ig (X) I = Spec (A/I) is closed in Spec A , and the
- - - --
proposition follows.
Corollary: If X is a closed subfunctor of :f. and Q is an open subfunc-
tor of , there is an open subfunctor y  :x: such that .Q. =  n:L .
Proof: By the proFOsition there is an open subset P of I Y I such that
IQI = P n II . If we set :Y=p (l, 4.12) , V nX and U have the same
FOints. Thus U = V nx .
6.7 In general the map   II of the set of closed subfunctors
of Y into the set of closed subsets of II is neither injective nor sur-
jective. For if I and J are two ideals of A, (I) and 'y(J) (6.2)
have the same Wlderlying space iff If = yJ , which can occur without I
being identical with J. This shows that  1-+ I I is not necessarily in-
jective. On the other hand, let T be the geanetric space whose Wlderlying
set consists of two points 0, 1 , whose closed sets are , {O} , {O,l} ,
and such that the restriction &T({O,l}) ->- cJT({l}) is the identity map of
the prime-field with p elements. If Rf£j, (T) (R) is empty if p f 0 in
R and may be identified with the set of closed subsets of Spec R if p=0
..n R. It can be shown that  and ET are the only closed subfunctors
-.Jf ST it follows that the map   II is not necessarily surjective.
6.8 Let f:->-'£ be a closed embedding and let (A,p) be a -ITOdel
( 1, 4.1). As a quotient-ring of A -+ cJ( Sp A) , t5'( Sp A Y ) bears a natural
A-algebra structure. M::>reover, the induced :f-mcxlule Qx/ : (A, p) \0+ J( A Y )
is clearly quasi-coherent (4..1) . We say that Q/:x: is a quasi-coherent -
Y- algebra . In fact. it follows immediately fran the definitions that Ge map
 t+ Q/:x: is a bijection between the closed subfunctors X of Y and the

I,  2, no 6
QUASI -COHERENT-MODULES
71
quasi -coherent quotient--algebras of 9 y / y . For schemes this statement may
be reformulated as follows in terms of sheaves of mcxlules:
SUPFOse that Y is a scheme and that i :+ is a closed embedding . If Y
is affine and open in ¥ and if <]:Y+¥ denotes the inclusion-morphism, defi-
nition 6.1 a) ensures that lilf(y) : JyCY) + cPxC(l(y)) is surjective. Hence
I :f: If: cP y + i * (Jl x ) is an epirrorphism of sheaves. Set "f = Ker I :f: I f By 1.11
and 5.5, - ] is ;:Iuasicoherent. By prop. 6.6, the support supp(t1 y /J) of
t!J / '1, that is the set of yEY such that (& I::/) 'f 0 , coincids with the
y - y y
fuage of fe . Hence fe is an isanorphism of I  I onto the geanetric space
(£' , (JJ/'Y) 1£:) , where uJ/J) I!' denotes the restriction of J/J to
!' = SUpp(c.?y/J) J and fe - is the map underlying f.
Conversely, let 1 be a quasicoherent ideal of CJ x (Le. a quasicoherent
y-sutmodule of cJ y ) . For each .<if: Sp A +  ' the sequence
g* (1) + g* (12) + g* (d /1) + 0
- - ¥ - '!
derived fran 0 + J + (j y + Uy/J + 0 is exact. Hence the canonical map
of A:;" <;[* (J7 y ) ( Sp A) into B = <1.* (I1/J) ( Sp A) is surjective (1.12). Since
 A Y ((J/J) may be identified with Sp B by lemma 6.9 below, it follows
that the onical projection p 19v/1 :  (c9 /1) + X is a closed embedding.
The image-functor of ? rJy/1 will be written yeJ) and called the closed
subscheme o f Y defined-by J With the above notation, by 5.5, we may
identify Iy(:r) I with (F ,t1 y /f) IF) , where F = supp(61/])
Fran these results, we deduce in particular the
ProFOsi tion: Let  be a scheme . Then : a) The map J I->- Y (1) is a bi -
jection of the set of quasicoheren t - ideals onto the set of closed subschemes
of 1" ;
b) A morphism of
I:f:lf: JJy+!*(Jl x )
space of FOints of
schemes i:+Y is a closed embedding, iff
is a sheaf epirrorphism and tc f2 is a hanecrnorphism of the
X onto a closed subset of IYI .
Proof: It rerrains to prove b) . The condition is necessary by 6.6 and 6.8.
Conversely, if the condition holds, the kernel j of 1% I :f: is a quasicohe-
rent ideal (2.4 and 1.11). Setting F = Supp(y/J) , it follows that K in-
duces an isanorphism of 11{ I onto the geanetric space (F, U 1 / 1 ) I F)

----------
72
ALGEBRAIC GECME:I'RY
I,  2, no 6
6.9
Lerm1a: a) If !:t-+- is a rrorphism of schemes , and ..t 
quasi-coherent sheaf of algebras over S, we have a canonical iSanorphism
' 'S .t + Sp f* (.-6 .
b) If R and B, we have a canonical isanorphism 9-: Sp B + Sp B ,
,../
where B denotes the sheaf over Spec R associated with B (1.4)
Proof: a) For any R, (f,?' Xs ,d) (R) is the set of pairs
(p , ((p) ,Je)) , where pEs' (R) ,-and where Je:..i -+- f*P: (p R) is a rror-
phism of sheaves of algebras over S . If Je' is associated with Je by the
bijection
E(vi,!*P:((p R)) + f,?t (!*) , p:(cJ R))
of 1.2, our canonical isanorphism sends (p, ((p) ,Je)) onto
(p,Je') E ( f* vi)) (R) .
b) Let i:R-+-B be the canonical isanorphism. For any M ' (M) sends
1/JE ( Sp B) (M) onto (1/Ji, ;J;) E (B"(M) (1.6) . The isanorphism  therefore
satisfies p u = Sp i
- B-
6.10 Corollary: Let :-+-X be a rrorphism of local - functors . If ,
for each i, ! induces a closed embedding i: !-lC'i) -+- Xi ' then f is
a closed embedding.
Proof: By 5. 6 we need only consider the case in which Y is a scheme. Then
f is affine by 5.6, and it remains to show that lilt': t9 y -+- f(t?) is an
epirrorphism of sheaves. But this follows fran the fact that the restrictions
of I i I  to the open sets I y i I are epirrorphisms (6.8).
6.11 We return now to an arbi trary -functor :! . By 6.4, the inter-
section of the closed subfunctors of X containing a given subfunctor  is
again closed. This intersection will be written  and called the closure
of  in ¥ . If i:-+-X is a rrorphism of  and  is the image-functor
Im f , we alsc say that  i is the closed image of f. We now consider
several examples of closed images:
Let P be a subset of Y. With each field KEt! and each element pEY(K) ,
such that III maps Spec K into P, we associate a copy (K,p) of
Sp K . Let


I,  2, no 6
QUASI -COHERENT-M)DULES
73
 = ck,pl) "f>.(K,P)
and let f:X-+-Y be the rrorphism whose canponent of index p is
# - - -
p : (K,p) -+- Y . Write P red for the closed image of f; it is the smallest
closed subfunctor ! of  such that P c If I .
We write simply X red for 1lred and say that ed is the reduced part
of . We say that  is reduced if Y red = Y . For each subset P of X,
P ed is reduced and has the following property: if T is a reduced Z-func-
r __
tor, each mrophism t:T-+-Y such that !('!') c P factors through P red .
6.12
If Y is a -functor and
P a subset of Ixi , !Predl obvious-
II By 6.7, it may happen that
ly contains the closure P of P in
13 of I P red I . This cannot occur, however, if X is a scheme:
ProFOsition: Let Y be a scheme , P a subset of !I and j the sheaf of
ideals of L9X such that
J (U) = {sEJJ y (U) I s (x) =0, VxEU np}
for each open subset U of Ixl . Then : a) 1 is quasicoherent ; b)
y (1) = ed c) the space of FOints o f ed is the closure 13 of P
in IYI
Proof: If J is a quasicoherent sheaf of ideals contained in J , then ob-
viously P c Supp(JJ/J) , and so P red C yr.;p (6.8). Assertion b) thus
follows fran a) . Similarly c) follows fran a) since we obviously have
p = Supp (J y /1) . We now prove a): by 1. 9, it is enough to show that, for
each affii1e open subset U of I I and each fE J(U) , the canonical map
i: J(U)f -+- J(U f ) is bijective. Now we have J(U)f C C9(U)f and iJ?(U){+ (;?(U f )
hence i is injective. Conversely, if x=g/r E 1(u f ) , gf is annihilated
on P n U since g is annihilated on P n U f and f on U - U f . The
equation g/r = gf/f n + l implies that x belongs to the image of i.
6.13
Corollary: A necessary and sufficient condition for a scheme
 to be reduced is that the local rings 0y,yE!'1 be reduced (Le. do not con-
tain any nilFOtent elements apart fran 0 ).
Proof: Set P = IYI
- -
nilpotent elements of
in 6.12. It then suffices to show that 1 consists of
y
(J . Now if U is affine and open and contains y
y -

74
ALGEBRAIC GEDMEI'RY
I,  2, no 6
and if fJJ() is annihilated at each FOint of Q, then f is contained in
every prime ideal of 19(9) , and is therefore nilFOtent. If f is the genu
y
of f at y, f E'1 implies that f is nilFOtent. Conversely, if  = 0,
n Y l' Y Y
then (fIV) = 0 for a suitable open neighbourhood V of y . Hence
flv E j(V) , so that f  J.
y y
6.14
Corollary: Let k be a perfect field ,  and Y two reduced
k- schemes. Then  x  is reduced.
Proof: In accordance wi th  1, 6.3, X is called reduced if z'f= is reduced.
By  1, 6.3, there is an isanorphism
z('f=xn -+ (z'f=) S x k (Z¥)
- -  -
By  1, 5.7, the local rings of Z ('f= x X) are the rings of fractions of the
rings [Jx iSikt.?y , xE, yEy . Si-;'ce <J x and  are reduced, they are
contained in prcxlucts of fields (the local rings of iJ. and r2 at minimal
x y
prime ideals). By Alg. VIII,  7, no. 3, tho 1, it follows that tJiSi k iJ is
x y
reduced, so that each ring of fractions of r.f}@ k t.?isreduced.
x y
6.15
We now turn our attention to the closed image of a morphism of
schemes f:'f=-+-I. Since the functor M of 1.4 ccmnutes with inductive
limits, the sum of a family of quasicoherent sheaves of ideals of Jl y is
again quasicoherent. Each sheaf of ideals of c?y ther efore ,contains a largest
quasicoherent sheaf of ideals. It follows that Im f = 'lXi) (6.8), where
J is the largest quasicoherent ideal contained in the kernel of the morphism
Ii I I: tJx. -+- !* (J}) induced by f.
This may be simplified when  is quasicanpact and quasiseparated. For by 2.4
!* () is then quasi coherent, so that '1 = Kerl!l
ProFOsition: Let T!'  be a diagram of schemes , where g is flat
- =1-
and f is quasiccrnpact and quasiseparated . Then we have Im !Y' =  (Im!)
Proof: Let  and . be affine open subschemes of y and y' such that
g(') C V Let 9 = !-l(y) and U' = !7(') . By 1.8, the canonical map
cJ(') iSi(/'(y)J() -+- J(!t)
is bijective. By varying y and V' , we derive a canonical isanorphism

I,  2, no 6
QUASI -COHERENT-MODULES
75
* (!* (Jl X ))
 ( d
+ t y ' y';c X)
* - y-
I!I! and '
f
IJ y . 1-, this canonical isanorphism
(see 1.6). If we set 
allows us to identify
g* () !:* (J!y) + <I* (* (J7))
with
' 12y. + t y . (.fly." X)
- - * - y-
Hence Ker . -+ Ker 2* () -+ g* (Ker ) , since the functor g is exact. So
by (6.9)
-1 -1 ----
Im fy. = '!(Ker ') = g (Y(Ker )) = g (Im f)
6.16
In the following, we require a statement analogous to prop. 6.15
in the case where ! is not quasicanpact and quasi separated . By way of can-
pensation assume provisionally that we have y = Sp B, BE M . If
vII= Ker I f If, it is clear that the largest qua:icoent shea of ideals J
contained in cIf/ satisfies J = vf(y)"" . Therefore Im! = 'i(":!) = y(vf())
Let (a) be an affine open covering of  and let to. :a + be the rrorphism
induced by f. If we set vV = Ker I f If, then Ji = n cJf/ , cI(y) = nvY (y) .
a -a a a a a -
Now if S:B+B' is a ring hananorphism which makes B' a projective B-rocxlule,
then
B'@ (nvV (y)) -+ n(B'@# (Y))
Baa a Ba-
If  = Sp B' and g = Sp S , it follows
Y(Q B' @BvVa (Y)) is the smallest closed subscheme of
V(B'@ B vV(y))=g-l( Imf )= Im f Y ' .
- 0.- - a a
that g -1 ( Im f ) = V(B'@ v?f(y)) =
- B
y' containing each
But this smallest closed subscheme is precisely Im f y ' . This is a particular
case of the following
ProFOsition: Let  be a quasiccxrpact quasiseparated scheme and let .9::T+.!:
be an affine rrorphism of
(Y.) of y such that
-l ----= -1 -
then Im fy I = 'I (Im)
schemes. If there is a finite affine open covering
('I-l(i)) is a projective Jl(¥i) -rnodule for all i,
for each rrorphism of schemes f:+X:.
The proof of this result in the general case is sketched in 6.17. We shall
use this proFOsition in the following form: Let k be a ITOdel and K a
k-mcxlel which is projective over k as a ITOdule. Then for each quasicanpact

76
ALGEBRAIC GEOMETI'RY
I,  2, no 6
quasi separated k-scheme Y and each rrorphism of k-schemes f : X->-Y , we have
( Im  )K = Im K .
6.17 To sketch a proof of prop. 6.16, let cItI be the kernel of I! I! ,
let vV qc be the largest quasicoherent ideal contained in Y and let
qui' : .r qc ->-vf be the inclusion rrorphism. Evidently (c ,q,r) enjoys the
following universal property: for each quasicoherent sheaf of ITOdules ,A{,
over Y and each rrorphism :vf(->-J there is a unique 1/J:J(->-,f'qc such that
q.,v 1/J =  . For since ,r c c9 Y ' Im  is quasicoherent, hence is contained in
..r qc . We show first that if Y is quasiccmpact and quasiseparated , this
universal problem has a solution whenever i/V is a X-module satisfying the
following condition: (*) for each open V in y, J(V) is small ( the in-
clusion functor of m:E I X I into the category of sheaves of ITOdules over Y
satisfying (*) has a riqht ad"ioint) . /'
Assume first that oA" is of the form y* ( ), Y:Y->-¥. being the inclusion rror-
phism of an affine open subscheme and ,t a y-module. We set flc= f(y)""
and write qx: .t qc ->-X for the unique rrorphism such that
qt (y)t(y') = Id,t(y) (1.4). It is then easy to show that (v*c) ,v*(qz..))
is a solution of our universal problem.
In the general case, when vi' is a y-ITOdule satisfying
a finite affine open covering (y, ) (resp. (y" 1) ) of
, "1 -1 -lJ
If v 1 (resp. v 1J ) is the inclusion rrorphism of y,
- i ijl 1
Y _ ' set vV. = V * (J' I Y,) (res p . Jf. ' 1 = V * (.J''\Y, ' 1 )
1 - -1 1J - -lJ
a canonical exact sequence of -rrodules
(*) above, consider
Y (resp. of Y ,flY,)
- -1 -J
(res p . y" 1 ) into
-lJ
In this way we obtain
ufI
u"" n y
> ,
i 1
v/  TT I/'
w,y- ' ijl VV ijl
Fran our previous remarks,
(1T;( (d'[.¥i)qc) , lJy;(qu I y,)) (p)
1 1 w- 1
and
IT ijl 1/ 1 qc IT ijl _ /)
(i J 'l y* ((<I' X ijl ) ), i J 'l Y * (qvf/ I Y,,)) - (cr,q)
-lJl
are solutions of our universal problem relative to TJ  and IT c{jl . We
may therefore define rrorphisms v,w:   Q by the conditions qv = vvi' P
and qw = wvVP . It thus remains to set ",fqc = Ker(v,w) and to define

I,  2, no 7
QUASI -COHERENT-IDDULES
77
qJ' : J(qc ->- vY by the condition uq,f" = pu , where u :jAc ->- P is the
inclusion rrorphism. The pair (#qc,q/) is the required solution of our
universal problem. Returning nON to the proof of 6.16 in the general case, it
only remains to ShON that
g* ( (n vf) qc) = (n g* lvV') ) qc
- a a a a
when uV is a family of quasicoherent sheaves of ideals of Y. To prove this
a
we need merely verify that the construction of (n J'::) qc above "ccmnutes"
a a
with the change of base functor 5[::t->-:::
Section 7
EInbeddings
7.1 Definition: An embedding is a canFOsition of arrONS <;! o! '
where f is a closed embedding and 51 is an open embedding.
If Y is a Z-functor and X a subfunctor of 'i ' we' say that X is locally
closed in Y if the inclusion rrorphism of  into Y is an embedding. If
Y is a scheme, a locally closed subfunctor of Y is called a subscheme . By
6.1 and  1, 3.11, each subscheme for a scheme is itself a scheme.
Consider, for example, a scheme Yanda locally closed subset P of Y
(Le. of IXI) . Set U = Ixi - (P-P) . Then ¥u is an open sub scheme of Y
( 1, 3.11 and 4.12). When no confusion is possible, we write P red for the
intersection of the family of closed subfunctors of Xu whose space of points
contains P. We may characterize P red as the unique reduced sub scheme of
I whose space of points is P.
The following assertions are inmediate consequences of the properties of open
and closed embeddings: an embedding is a rronanorphism; the canposition of two
embeddings is an embedding (cor. 6.6); for each diagram X  Y $ y' , where
f is an embedding, the canonical projection
¥' : y¥' ->- Y'
is also an embedding.
7.2
PrOFOsition: Let Y be a scheme and X a subscheme such that
the inclusion rrorphism f of
X into Y
is quasicanpact . Then 
is open

78
ALGEBRAIC GECMETRY
I,  2, no 7
in the closure X of X in Y (6.11).
Proof: Let U be open in  and supFOse that  is closed in U . Since
f:+Y is a monanorphisrh, o/y=  + y is invertible; hence ! is quasi-
separated (2.2). By 6.15, fi2 coincides with the closure of r\Q in U ,
Le. with X; but xnu is open in X. This canpletes the proof.
7.3
ProFOsition:
Consider the diagram of schemes X t Y  Y' where
- - - /-
quasicanpact. If the canonical projection Y:
then f
g
is faithfully flat and
 y Y' + 't is an open (resp. closed , resp. quasicanpac t) Embedding ,
is an open (resp. closed , resp. quasicanpact ) embedding .
Proof: If Iy. is an open Embedding, it follows fran the equality
g-l(!()) = !. Op:') (1, 5.4) and the fact that !y' (yyr) is an open
subset of Y' that f () is an open subset of 1'" (3.4). If 1';' is the open
subscheme of y such that I . I = ! () , and if f': X+X' is the morphism
induced by ! , then !' y 1:' is invertible; hence so is f' (3.5).
Now supFOse that !Y' is a quasiccxrpact (resp. closed) embedding. Then f
is quasicanpact and quasi separated since t"y. has these properties (cf. the
argument of 5.7). If "= Im! ' and if !" :+" is induced by ! , then
-1 ----- -1
g (") = Im!yr (6.15). It follows that !"X" (")
(resp. an isanorphism). Hence f" is an open embedding
is an open embedding
(resp. an isanorphism) .
7.4
Definition: A morphism !:+y of  is said to be separated
if the diagonal rrorphism o/y:  +    is a closed embedding .  - functor
X is said to be separated if the unique morphism Ex:  + SP ! is separated .
Referring back to 6.1, and recalling that
is determined by  and '!, we see that
and each pair of arrows (,) of Sp A
a morphism T('::',y) : Sp A + ¥
f is separated iff for each AEM
,Mo
in X such that =, Ker (l,'y)
is closed in Sp A . Fran this and the definition we irrmediately infer the
following assertions: each monanorphism is separated; if, in the diagram
X t y  y' of ME, f is separated, then so is the canonical projection
- - - ,...........-
Iy.: Y' + y' ; if 9: a!. is separated, so is
rrorphisms is separated.
f
a prcduct of separated
7.5
Pror;osition : An affine rrorphism f:X+y is separated.

I,  2, no 7
QUASI -COHERENT-MJDULES
79
Proof: Let g,y: Sp A ::::  be a double arrow satisfying !\!=t: . By defi-
nition, (A) y is affine, so that there is a cartesian square of the
form
fu2.B
 I 1
fu2.A
J!l'
> X
1 
) y
Y:!.
and a double arrow 0.,6: B =::A such that '( a) = , '!!. ( Sp 6) = v and
a = 6 = IdA. Since we have
Ker(:g,y) = Ker( a,  6) -+ (Coker(a,6))
Ker(1,l,y) is represented by A/I, where I is the ideal of A generated by
the elements a (b) -6 (b) , bE B . Hence Ker (p,y) is closed in  A .
It follows in particular that f:+Y is separated if X and X are affine .
By setting ¥ =   ' we infer that an affine scheme is separated .
7.6 ProFOsition: a) Let :+y be a separated rrorphism (resp. 
morphism of schemes ) and u,v:S=::X a double arrow of ME such tha t fu = fv
---- .........".,.. ----
Then Ker (1,l,y) is closed (resp. locally closed ) in .
b) Let f:X+Y and g::¥+ be rrorphisms of m:; . If gf is a closed embeddinq
and 5L is separated , then t: is a closed embeddinq . If Y.'Y'E are schemes
and gt: is an embeddinq , then :t is an embeddinq .
Proof: a)
Let w: S+ X " X
- - -Y-
be the rrorphism with canp::ments ,Y and let
6 be the subfunctor of  y such that 6 (R) = {(x,x) IxE X (R)} for all R
-1
Then Ker(1,1,Y) = Vi (6) . If f is separated, 6 is closed in y , so that
Ker(':!,,:,:) is closed in  . If  and Y are schemes, 6 is locally closed
in  X  (by the lsrrna below); hence Ker (1,1, y) is locally closed in S .
b) For the first assertion, observe that f is the composition
h
XX"Y
- - z-
(gf)y
-- - , y
where h has canFOnents I and f. Now h is derived fran
o¥/: y+ y  ¥ by the change of base i i Id : ]5  ¥ -?¥  l' ; since o¥/ is
a closed embedding, it follows that f is the canposition of the two closed

80
ALGEBRAIC GECMETRY
I,  2, no ,8
embeddings !:: and () Y . The proof of the second assertion is similar.
Lemma :
For each rrorphism of schemes f:X-;Y, the diagonal oC/X:  ->-  y2S.
is an embedding.
Proof: Let U and Y vary through the affine open subschemes of  and
X. satisfying !: (11) c:y . Then oX (Y y 11) =!l and the morphism Q. ->- 11 Y II
induced by O!X coincides with the morphism induced by the map aSib ->-a.b ,./
of c?(U) @ J(V) iJ(U) into u(U) . Since this map is surjective, (6.10) implies
that 01¥ induces a closed embedding of X into the open subscheme of
 y  which is covered by the .!! V Q .
7.7
Corollary: The canFOsition of -(:\.oX) separated rrorphisms is sepa-
rated.
Proof: SupFOse
(':!):  A ::: 
closed in Sp A
that Ker(,y)
Sp A
!:->- and g:->- are separated. Let A be a mcdel and
a double arrow such that CJ': = 'E . Then Ker (!,!y) is
since g is separated. Since f is separated, it follows
is closed in Ker (fu,fv) . Hence Ker ,y) is closed in
7.8
and
Example: The projective s pa ce P , the grassmannian G
- - -n - -n,r
the flag scheme In (r l ,... ,r s ) are separated schemes . For the "diagonal" of
G x G coincides with F (r,r) , hence is closed by 6.3. Since a
-r,n-r -r,n-r -n
prcxluct of separated functors is separated,
G x ...)( G is sepa-
-rl,n-rl -rs,n-r s
rated. By 7.7 and 7.4, a subfunctor of a separated functor is separated. The
assertion therefore follows by 6.5.
Section 8
An affineness criterion for schemes
8.1 Affinity theorem: Let X be a scheme and let :; be a nilpotent
quasicoherent sheaf of ideals of J X . If the closed subscheme y Cf) of X
defined by 1 is affine , then so i X.
Proof: First recall that 'If(')) has the same space of FOints as  and its
structure sheaf is c!J x /1. Since y(;;n) =  for sufficiently large n, it
is enough to show by induction on n that V (ljn) is affine. Now YOn)

I,  2, no 8
QUASI-COHERENT-MODULES
81
coincides with the closed subscheme Y(ljn/{'+l) of YC;rn+l) Since 7 n /:f+ l
has vanishing square, we may assume straight-away that '1 2=0 Set ¥='! (1) ,
and supFOse provisionally that the map J(!) induced by the inclusion rror-
phism !:.Y-+ is surjective. Since Kert!J(f) = 1'(X) has vanishing square,
Spec (,?(!) induces an isanorphism of the underlying tOFOlogical spaces of
Spec uQ9 and Spec J(X) . Now in the conmutative square ( 1, 2.2)
1/J!yl
"7 Spec J7(y)
1  t9(!1
) Spec vQP
!YI
I!l
II
1/Jlxl
1/J I y I ' I! I and Spec (f) are all haneanorphisms, and therefore so is 1/J 1  I
To show that 1/J I  1 is an isorrorphism, it remains to prove that 1 KI and
Spec t.?() have "the SaITe" strucb..lXe sheaf, that is to say, that the canoni-
cal map t!J(X) -+ c9(X) is bijective for each sE GJ(X) . This follows fran
- s -s
the diagram
o  /j (Y) ---'" lJ(X)
0.1 s pf s
o 1(Y ) L?(X )
-s -s
r9(f)
- s
1 ()(Y) _ 0
- s
<1(t s ) Yl
) J(y )  0
-s
where since ¥ is affine and '1 may be regarded as a quasicoherent sheaf of
ITOdules over ¥, a and y are bijections. Finally J!(f) and -J(f) are
-s
surjective. We prove the latter contention, the proof of the former being
similar .
For each sEtiJ(Y) , write X for the open subscheme of X \\hich has the
- -s -
same space of pJints as Y ; then J (X) may be identified with 1 (Y ) ,
-s -s -s
regarding 1 as a quasicoherent sheaf of ITOdules over Y. If aEJ7 CD , there
is a partition 1 = \ l x,s. of unity in cJ(Y) and elements a, Et9 x (X )
Ll= l l - l -s'
() - l
whose images in lY. Y (Y ) are the restrictions alY . Also, a" =
_ -si -si .. lJ
a, ! X - a, I X is a section of 1 over X n X = X . Since the
l -SiSj J -SiSj -si -Sj -SiSj
family (a, ,) is obviousl y a l-cocycle of '1 for the coverin g (Y ) , the
 -
equation Hi ( ( ! Y I ) ,1) = 0 , established below, implies that there are
s'
b. E 1 (X ) such t a., = b, Ix - b.1 X . It follows that the restric-
l -Si lJ l -SiSj J -SiSj
tions of a,-b, and a,-b, to X are the same, so that there is
l l J J -si Sj

82
ALGEBRAIC GEOMEI'RY
I,  2, no 9
a' ED' ()
such that a' Ix = a.-b, . Hence &(!) (a') = a .
-si l l
8.2 LEmPa: Let A be a ring , M  A -mcxlule , 1 = I=lxisi a
 ti tion of unity in A , and let J = Spec A . Then the cohanoloqy groups
of M with respect to the covering (Y,) satisfy H O ((Y ) ,M) :;. M and
-; - Sl si
H l ( (Y s.) ,M = 0 if i>O .
l
Proof: It suffices to show that the sequence
/
0--+ M(Y) -+ TTM'(Y s .J -+ IT M(Y nY ) -+ T:T 1 M(Y nY ny )...
l l i,j Si sj l,J, si Sj Sl
associated with M and the covering
(Y ) is exact. But this sequence is
si
the same as the one obtained by setting C=A, B = ITA in  1, 2.7.
l si
Section 9
TransFOrters
k denotes a ITOdel throughout this section.
9.1
Given ,XE !1 k Jii. ' write H (,) for the k-functor satisfying
( H (,!:)) (R) = (R,XR)
for REA
Given lC,,cl1k and ! E!\!f( , H (,)) , set
!x = ((R)) (x)   (R'R)
for each RE £:\ and each xE  (R) . This enables us to associate with f an
arrow g: XxX -;-  such that (g(R)) (x,y) = (!x(R)) (y) for REJ:\ ' xE(R)
and yE X (R)
Proposition: The map i,,:f I--+:I defined above is a bijection
( , HS!!k (X ,)) :;. (lC x X, )
Proof: We merely give the inverse bijection: it assigns to g:  x X -;- Z an
arrow f:  -;- H (y,) such that (x (R')) (y) = (R')) ("y) if
R, xEiUR), R' and yEY(R') .
9.2 Corollary: If ,El\ H (X ,) is canonicallv isanorphic
to the k-functor R I-+(R) >< Y, Z) .

I,  2, no 9
QUASI -COHERENT-MODULES
83
Proof: Simply take the ccxrposition of
morphism
i with the canonical iso-
R,¥,
( HS!\. (¥ ,)) (R) ->-l\( R , H9!!.k (¥'))
9.3 Example: If AE.&\, H (kA,) is canonically isanorphic
to the k-functor R t-+  (R C:9 k A) in view of the canonical isanorphisms
( H!!\ (kA, )) (R) ::;. l\,B; ( (kR) (.eI2k A ) , ) ->- l\ (fu2k (R A) , ) ->-  (R A)
Fran this and  1, 6. 6 we deduce the existence of a canonical isanorphism
HQ!!\ (A, ) ::;. ;f;IA
By Prop. 6.6 of  1, we arrive at the following criterion:
H (kA,) is a scheme if the following three conditions hold : Z is a
scheme , A is a finitely generated projective k -module , and for each finite
subset P of Z
there is an affine open subscheme
U of Z
such that
pc 1111
9.4 Definition: Let ..E:  -;-  be a morphism of l:\!, i:'-;-
a rronanorphism , and ..E':  -;- H (:J ,) the rrorphism canonically associated
with .!2. by Prop. 9.1. The transporter of  into . relative to J2. ,
written Transp (Y, Z ') , is the pull- back of the diagram
p
-r>' HS!\. (¥,)
---7 H C',) <: H (,')
For each REt&, ( Transpp (¥'')) (R) may thus be identified with the set
of arrows :  R -;- £C.. such that ..E' factors through H (¥,iJ . Mcdulo
prop. 9.1, the existence of such a factoring means that the canFOsition
xId k
k R"Y ) x  ----';> 
factors through . .
9.5
ProFOsition: Let :'-;- be a closed embedding  k- functors
and Y a locally free k- scheme , that is to say , a scheme having a covering
by affine open i whose algebras of functions are free k -modules . Then
H (,) : H (¥,') -;- H9!!.k (¥') is a closed embedding .

84
ALGEBRAIC GECMEI'RY
I,  2, no 9
Proof: SupIXJse first that '!.. = S B where B is a free k-mcxlule. By 9.3,
we must show that the canonical rrorphism B ->- UB is a closed embedding,
Le. that the following condition is satisfied (6.1): for each AE and
each aEWB(A) = E-(BiSiA) , there is an ideal I of A such that, for
each haranorphism  :A->-R of J:\ ' the element a R of k. (B iSi R) belongs to
Z' (BiSiR) iff  (1)=0 . Now '->- is a closed embedding, so if A and a
are as above, there is an ideal J of BiSiA such that, for each  as
above, a R belongs to . (BiSiR) iff (BiSi) (J) = 0
is a smallest ideal I of A such that BiSiI:J J
; since B is free, ther
The conditions I C Ker 
and JCBiSiKer  = Ker(BiSi) are accordingly equivalent, which proves the
first assertion.
In the general case, by 6.1 it is enough to show that, for each 2: sPkM x ->- 
where ME  ' 1: = Trans p p (¥ , ' ) is represented by a quotient of M. If
f\ denotes the restriction of 2 to the affine open subscheme SM x i of
Sn k M x _ Y, T. = Transp (Y, , Z ' ) is represented by a quotient M/m, of M,
= -1 p. -1 - 1
\' _1
and n T. by M/ L .m. . It is now sufficient to show that T = nT, , Le. that
i -1 1 1 - i -1
the morphism c:L:  '!:{  ->-  induced by .e factors through E-' . But this
-1
follows fran the fact that q (Z' ) is a closed subscheme of nT, \( Y con-
- - i -1
taining each
(nT,)x Y.
i -1 -1
9.6
corollary: If k is a field , Y is a scheme and i:'->- 
closed anbedding of k- functors , HS!!k (¥,!) is a closed embedding .
9.7
COrollary: With the notation of deL 9.4, if X. is a locally
free k- scheme and i is a closed embedding , Transpp (¥'Z_') is closed in Z
9.8 COrollary: If Z is a separated k- functor and X is a locally
free k- scheme ,  (Y ,) is separated .
Proof: By  1, 6.3, X is separated if J:..X is separated. By 7.4, 7.5 and
7.7, this is the same as saying that the structural proj ection EX: X ->- fu2. k
is separated. Now apply prop. 9.5 to the diagonal morphism of X into Xx. X

1.1 Definition: An A- algebra B is said to be finitely presented
if it is isanorphic to the quotient of an algebra of polynanials A[X l ,...,x n ]
by a finitely generated ideal I .
'I . i ... , .. .
II
.,
'
I
!
I
i
!
i
"i'
9 3 ALGEBRAIC SCHEi'1ES
Section 1 Finitely presented rrorphisms
,
I
"
,I
I
!
When A is Noetherian, in particular when A is a finitely generated algebra
over ,!, each ideal of A[X l ,... ,xn] is finitely generated, so that an
A-algebra is finitely presented iff it is finitely generated.
, 1.2
LEmma: Let
(A) be a directed system of rings, with direct
a
finitely presented A-algebra , there is an inde,'{ a
A - alqebra B such that B is isanorphic to
a a
limit A. If B is a
- -
finitely presented
A iSlA Ba
a
Proof: Suppose that the ideal I of deL 1.1 is generated by the FOlynanials
P l' . . . , P r . Choose a so that the image of Aa in A contains all the
coefficients of the FOlynanials P. . If Q l ,...,Q EA [X l '...,X J are
1 ran
mapped onto P l ,...,P and generate the ideal I , set B = A [X l ,...,x ]/1
r a a a n a
For example, we may take (A a ) to be the system of all finitely generated
subrings (Le. ,&-subalgebras) of A. It follOlNs that B is finitely presented
over A iff there is a finitely generated subring Ao  A and a finitely
generated Ao - algebra Bo such that B:+ AiSl A Bo .
o
1.3
LEmma: a) If B is finitely presented over A and if C is
finitely presented over B, then C is finitely presented over A
b) If  :B->C is a surjective haranorphism of finitely presented A- algebras ,
then the ideal Ker  of B is finitely generated .
Proof: These assertions follow easily fran lerrroa 1.2. For example, we prove
(b) . Let A be a finitely generated subring of A and C a finitely
o 0
generated A -algebra such that C:+ AiSlilC . With the above notation, let
o . .'0 0
P be the canonical projection of A[X l ,...,x n ] onto B = A[X l ,...,X n ]/I
If A is sufficiently large, there are ,E C such that
o 1 0

86
ALGEBRAIC GECMEI'RY
I,  3, no 1
10 Ao i =  (p(X i )) ; if Ill'... ,Il s generate Co over Ao' we have fX)lynanial
relations
101l. =Q,(101,...,lA0)
Ao J J Ao 0 n
with coefficients in A; thus, if Ao is sufficiently large, we have
Il. = Q'( l '...' ) , so that the , generate C . Also, if P l ,...,P
J J n lor
generate the ideal I , we have P. ( 1 '...' ) = 0 for sufficiently large
J n /
A . Under these conditions, the haranorphism of A [X l '...,X J onto C
o 0 n 0
which sends x, onto , factors through a hcm::xrorphism  of B =
l l 0 0
A [X l '...,X ]/(P l ,...,P ) onto C . Therefore  = 0 and Ker  is the
o n r 0 -Ao
image of Aff (Ker o) in B, and is hence finitely g2nerated.
o
1. 4 LEmma: Let A' be a faithfully flat A-algebra. Then an A- alqebra
B is finitely presented iff A'0 A B is finitely presented over A' .
Proof: The latter condition is obviously necessary without restriction on
A' ; we show that it is sufficient. Let B' run through all the finitely
generated subalgebras of B; then we have
lim A'0 B' = A'0 lim B' :+ A'0 B
-;. A A -;. A'
so that, since A '0 A B is finitely generated over A' , A '0 B' = A '0 B for
A A
same subalgebra B' . The assumption that A' is faithfully flat over A
then implies that B' = B , so that B is the quotient of an algebra
A [Xl ' .. . ,xnJ by an ideal I . Let I' run through all finitely generated
ideals contained in I . Then we have
A' B = A'[Xl,...,Xn]/(A' I) ,
and A'0 A I is finitely generated (1.3). Since
A'iSi I = lim A'0 It
A -;. A '
we have A'iSi I = A t 0 I' for sane I' , whence I I'.
A A
1.5 Lemna: Let B be an A-algebra and let 1 = Ix,f, be a parti-
l l
tion of unity of B. If B fi is finitely presented over A for each i,
then B is finitely presented over A.
Proof: Let Bo run through the finitely generated A-subalgebras of B con-
taining the f i and the xi. The equalities


I,  3, no 1
AffiEBRAIC SCHEMES
87
1:tm B Ofi = B fi imply B Ofi = B fi for sane Bo
hence
TIB = TT B = D"B@ B = B@B (DB f )
i ofi i fi' l Bo ofi 0 l 0 i
and, since TJ B f is faithfully flat over B , we have B=B . This implies
l 0 i 0 0
that B is of the form A[X l ,...,x J / I . If Q. , p, are representatives
n l l
of x., f. in A[X l ,...,x ] and if I' is the ideal generated by
l l n
1 - I-Q.P, , then (A[X l ,...,x ]/I') p is finitely presented over A By
III n i
lemma. 1.3, the kernel (1/1 · ) p. of the map
l
finitely generated for each i,
(A[X l , ...,X ]/1' ) p -;. B
n i fi
hence so are 1/1' (Alg. ccmn. II,  5
is
no. 1) and I .
1.6
Definition: A rrorphism of schemes :f.:-;. is said to be locally
finitely presented if , for each point XE, there are affine open subschemes
U of  and Y of .x such that XE, f (x) E.Y
a finitely presented algebra over 0(';:}
f () cy and 1J(!l) is
f is said to be finitely presented if f is quasicanpact , quasiseparated
and locally finitely presented .
1.7
ProFOsition: If :B-;.A is a rrorphism of mcxlels , the following
assertions are equivalent .
(i) A is a finitely presented B- algebra.
(ii)   is locally finitely presented .
(iii) Sp  is finitely presented .
Proof: By  2, 2.2 it is clear that (ii) <=> (iii) . Also (i) => (ii) is
triviaL We prove (ii) => (i) ; set  =  A, ..x. =  B and X = .eE. 
If x,  and Y are as in 1.6, there is a tEE such that f(x)EYtCV .
Thus
U(c l C:t) n ) = O(Q (t)) = tJ(Q)  (t)
hence ()(Q(t)) is finitely presented over UCD t = Bt ' so also over B .
Now substitute Q (t) for Q; we may then assume that Y='L in the notation
of 1.6. In this case, there is an sEA such that xEx cu ; \\hence
-s -
J!(X )=l?(U) =A , and A is finitel y P resented over B. By coverin g "f>.
-s - s s s
with finitely many of these "f>.s ' (i) follows fran lemma. 1.5.

88
ALGEBRAIC GEa.1ETRY
I,  3, no 1
1.8
Corollary: Let :+ be a locally finitely presented morphism ,
affine open sub schemes of  and '.!. such that .f(U')C'
is a finitely presented algebra over J)(V') .
U' and V'
Then 01(2' )
Proof: By 1. 7, it is enough to shaw that the rrorphism !' :Q'+Y' induced by
i is locally finitely presented. If xEQ.' and if U,V are chosen as in 1.6,
there is a tEl!J(y) such that ! (x) E Yt c yn Y' . Then
19(!-1 (Y t ) n Q) .:; CJ(Q) t
is finitely presented over t!J (Y t ) . M:>reover, there is an sEOq:J) such that
x E 1!s c f- l (y t )nQn11.'
Since 19(12 s ) = L9(Q) ts' t/(Qs) is finitely presented over J7(Yt) , and we
have xE Qs C 11.' , (x) E Y t c' and (Qs) C Y t .
1.9 Corollary: A morphism of schemes f:X+Y is finitely presented
iff _ y and X can be covered by affine open subschemes Y. and X., such
- - -1 - -lJ
that:
a) for fixed i, there are only finitely many !5 ij , and they cover
-1
f (i);
c)
for each pair
rj(y.) .
-1
(i,j,.Q,), X. ,nx,o is quasicc:mpact ;
-lJ - 1)(,
(,i,j) , c1(ij) is a finitely presented algebra over
b)
for each triple
Proof: This follows imnediately fran 1.8 and  2, 2.2.
If !J(Y,) is a noetherian rin g , so is &(X..) b y c) . In this case the
- 1 -lJ
underlying toFOlogical space of ij is noetherian. Each open subscheme of
x,. is therefore quasicanpact, so that condition b) is implied by c) if
-lJ
19(i) is noetherian for each i.
1.10 PrOFOsition: a) The canposition of tv.D locally finitely pre-
sented (resp. finitely presented ) rrorphisms is locally finitely presented
(resp. finitely presented ) .
b) In the diagram of schemes !}': . , if f is locally finitely pre -
sented (resp. finitely presented ) then the canonical projection

I,  3, no 1
ALGEBRAIC SCHEMES
89
:[:X-' :   . ->- T is locally finitely presented (resp. finitely presented) .
c) If ':£0:[: and g are locally finitely presented (resp. if got is finite-
ly presented and if ':l i s quasi separated and locally finitely presented ), then
! is locally finitely presented (resp. finitely presented ) .
prcbf: b) follows inmediately fran 1.6 and  2, 2.3. Assertion a) follows
fran 1. 8 and the fact that the canposition of tv.D quasiseparated morphisms
:->-:¥ and --':l::¥->- is quasiseparated . For 0"f>./: K ->-   K is the canposition
of 6yX: K ->- "f>.  K (which is quasicanpact) and the inclusion rrorphism
¥. ->- K] , which is derived fran 0X/%:.::¥ ->- 1'r by the "change of base"
i 1 f : !"f>. ->- :¥:¥ This inclusion morphism is therefore quasicanpact, and
so is oi'l:...
It remains to prove c) . We prove the "resp." part. If fE.@2: (,:¥) and
gE  (:¥, ) , f is the canposition
h (g:[:)y
X--=--;'X x y  y ,
- -z- -
where h has canponents Id x and f. Notice that, since gf is finitely
presented, so is (<I)1'.. By a) , it is enough to show that h is finitely
presented. Now h is derived fran 0y:/%:.: 1'. ->- 1'11' by the "change of base"
f  l' :  l1' ->- l'  1'. Since 0y:j is quasiseparated (it' s a monanorphism:)
and guasicampact, it remains to show that 0y:/ is locally finitely presented.
For this purJ:Ose we rray assume that  and  are affine. In this case, we
must show that the kernel of the canonical rrap of (J(1')00\')L? (1') onto r.J (¥)
is finitely generated; and this is clear, since it is generated by the
g .01 - 10 g. , where (g,) 1 , is a system of generators of 0CO over
1 1 1 515n
()()
1.11 ProFOsition: Consider the diagram of schemes !   1'.' ,
where <J is faithfully flat and quasicanpact . If the canonical projection
!1',:   1'.' ->- y:' is locally finitely presented (resp. finitely presented) ,
then :[: is locally finitely presented (resp. finitely presented ) .
Prcof: One easily shows that if :l: y ' is quasicanpact and quasiseparated,
so is f (cf. the argument of  2, 5.7). It therefore remains to show that
if y' is locally finitely presented, so is !. To see this, let Q and

90
ALGEBRAIC GEOMETRY
I,  3, no 2
'{ be affine open subschemes of  and Y. such that i (9) cy, let (. i)
be a finite family of affine open subschemes of Y' covering g -1 (V) . By
1.8, 0(Yi x v 9 ) is finitely presented over (,Q('{i) for each i, hence
1]0(Yi) l?D iJ (Q)
is finitely presented over
over rj C:D
T!(.Q(V)
1 -1
and, by 1. 4 , U CY.J
is finitely presented
1.12
A rrorphism of schemes
for any point xE  ,
X , such that xE Q ,
i:K+'f is said to be locally finitely
generated if,
there are affine open subschemes
and r!)(U)
U of
X and V of
- -
finitely generated algebra over [J(y) . We say that i
if i is quasicanpact and locally finitely generated.
! (x) E '{ ,
f(U)CV
is a
is finitely generated
In statements 1.7, 1.8, 1.10a), b) and 1.11, "finitely presented" rray be re-
places by "finitely generated", whereas 1.9 and 1.10c) should be replaced
by statements a) and b) below.
a) A rrorphism of schemes f:X+Y is finitely generated iff there are cover-
ings of '£ and  by affine open subschemes y, and X., such that con-
-1 -lJ
ditions a l ) and a 2 ) hold:
a l ) for any
a 2 ) for any
algebra over
i , (X..) is a finite covering of f- l (Y.)
 - 
(i,j) , 0(x. ,) is a finitely generated
-lJ
0(Y, )
-1
b) If 2 is locally finitely generated, so is K. If gof is finitely
generated and <J is quasi separated, then f is finitely generated.
Section 2
Algebraic schemes
Throughout the rest of  3, k denotes a model.
2.1
Definition: A k- scheme 1!: is said to be locally k-algebraic
(resp. k-algebraic) if the structural morphism .PK zK + eE k is locally
finitely presented (resp. finitely presented ) .
J

I,  3, no 2
AIG:BRAIC SCHEMES
91
Any rrorphism of k-algebraic k-schemes :£;->-!: is finitely presented by 1.10c) .
If 2:Y'->-Y is a second morphism of k-algebraic k-schemes, the pull back
!( y!:' is a k-algebraic k-scheme by 1.10a) and b), since the structural pro-
jection Ex x Y' is the canposition
- '1-
z'J X px
z  z!:' ----=--=.,. z£; Sp k
/ - -". -
Hence a finite inverse lit of k- algebraic k- schemes is k- algebraic . The
same result holds for locally k-algebraic k-schemes.
By 1. 9, a k-scheme £; is k-algebraic iff X has a finite affine open cover-
ing (X.) such that the x, nxo are quasicanpact and 0(X,) is finitel y
-J - J -" -J
presented over k for each j . If k is a noetherian ring, the k-scheme
X is locally k-algebraic (resp. is k-algebraic) iff the structural morphism
Px : z ->-  k is locally finitely generated (resp. is finitely generated) ,
t is iff 1:; has an affine open (resp. open and finite) covering (i)
such that r!!(X,) is a finitely generated k-algebra; in that case, Spec (X,)
-l -l
is a noetherian topological space; hence each open subset, and in particular
X. n X. is autanaticall y qu asican pa ct.
-l - J
Each open sub scheme y of a locally k-algebraic k-scheme X is locally
k-algebraic. If, in addition, X is k-algebraic, k-algebricity of Y is
equivalent to quasicanpactness.
By abuse of language, we shall sanetimes confuse "algebraic" with "k-algebra-
ic".
ProFOsition: A k-scheme X is locally k- alqebraic iff for any directed
system of k-ITOdels (AN) , the canonical map lim X(A ) ->- X(lim A) is bi-
 ->--0. -->-0.
jective.
Proof: When X is affine, the proposition reduces to the fact that a k-mcxlel
B is a finitely presented k-algebra iff for any directed system (A a ) the
map lim M (B,A ) ->- M (B,lim A) is injective. The proof of this well known
-+ N.'k a .wk -+ q.
fact is entrusted to the reader.
Suppose now first that the maps  : lim X(A ) ->- X (lim A) are alwa y s bi-
->- - a - ->- a
jective. Let .Q be an affine open subscheme of 1:; , set B = (j un ,
A = liID Aa ' and denote by Po. : Sp A ->- Sp Aa and PaS :  AS ->- Sp Aa the

92
ALGEBRAIC GECMEI'RY
I,  3, no 2
transition rrorphisms. By hYFOthesis any f:B->-A is induced by sane
-1 -1
g : Sp A ->- X, and the relation p (g (U)) = Sp A implies the existence
- - a - -a - - -
of a partition of unity 1 = LX. y . in A, where the Y l ' are the images of
1 1
sane elements in A vanishing outside g-l(U) . Such a relation must exist
a - - -1-1
already in AS for sane S > a . Hence we obtain EaS (5L ()) = Sp AS '
which means that f:B-+A is induced by sane f' :B->-Aa ' or in other words that
the rrap lim M (B,A ) ->- M (B,A) is surjective. As M (B,A) and M (B,A)
--+- M-.-k a .w:k kij( a ,...A.;k
are identified with subsets of 2:C(A a ) and (A) , this map is even bijective.
Hence B is finitely presented and  is locally k-algebraic.
Conversely, supFOse that  is locally k-algebraic. We first prove that 
is injective. Let f,g : Sp A ->- X be two rrorphisms such that fp = gp .
- - - a ->- - --a --a
We want to prove that !PaS = JPaS for sane S  a . By taking sane partition
of unity 1 = LX. y . in A it is easy to reduce the prcof to the case, in
1 1 a
which f and g are factored through affine open subschemes U and V of
- -1 -1
 . Then we can prove as above that PaS (<;r (12)) = Sp AS for some S  a
We may therefore supFOse that !:! = "i . As &(!:!J is a finitely presented
k-algebra, the rraps ()() '&(9) : cJ(Y) :::: Aa are equalized by sane Aa ->- AS
For such a S we obtain !PaS = 912aS .
Consider finally a morphism b : Sp A ->-  There exist a partition of unity
1 = LXi Y i of A and affine open subschemes i of , such that be I Sp Ax,
1
is factored through sane h, :  A ->- X, . For sufficiently large a, x.
-1 Xi -1 1
is the image of sane elements in A , which we still denote by x. . More-
a 1
over, as A -+ lim(A) , E. is induced by sane h . : Sn A  X. . B y
xi ->- a xi 1 -0.1  axl -1
the first par t of the P !roof h, and h, coincide or> Sp (A) . for
, -0.1 -u.J -  xixJ
large a and thus define a morphism .b : lli2 Aa ->- K such that 0a P a = !2. .
2.2 Corollary: Let (k a ) be a directed system of ITOdels with direct
limit k. f, for sane index a,  is a locally ka - alqebraic scheme and
X a quasicanpact , quasiseparated ka -scheme , the canonical map
lim t\ E (X 181 k , Y 181 k S ) ->- E (X 181 k, Y 181 k)
S  B - ka S - ka ....... ka - ka
is bijective.
Proof: Let (X,) (resp. (X.. 0)) be an affine, open and finite covering of
-1 -lJ;<,
J

I,  3, no 2
"\
ALGEBRAIC SCHEMFS
93
K (resp. of x.nx.) . Set A. = l?(X,) and A. '0 = [,?(X. '0) . The assertion
-1 -J 1 -1 1J", -lJ'"
follows fran proFOsition 2.1 by taking direct limits in the exact sequence
M E(X 0 k S ,Y 0 k S )-+ lTY(A. 0 k S )-+ .TToY(A. '0 0 k S ) .
"'"XC - ka - ka i- 1 ka -+ 1,J,"'- 1J'" ka
2.3 Corollary: With the assumptions of 2.2 , let b:->- be a morphism
of algebraic ka- schemes . If h 0ka k is invertible (resp. a monanorphism , 
open embedding , a closed eTIbedding ), then there is S > a such that .b.  kS
is invertible (resp. a monanorphism , an open embedding, a closed embeddi) .
Prcof: If h 0 k is invertible and
 h . ka h I Y 0 k X r>, k
1S a morp lsm : - k S ->- - k S
a a
S is sufficiently large, by 2.2 there
such that h' 0 k = (h0 k)-l . Since
k k
( (b  kS) b') 0 k = Id and
a kS
(b' (h  kS)) 0 k = Id ,
a kS
we have
( (b  kS) b')  ky = Id and
a S
(h' (h 0 k S ))0 k = Id
- - ka kS y
for sufficiently large y .
If h 0 k is a rronano rp hism, it is enough to apply the above to the diagonal
-ka
rrorphism 0y../¥:  ->-   (which is invertible iff .b. is a rronanorphism) .
Now supFOse that h '1t k is an open embedding. By the above argument,
h  kS is a monanorphism for sufficiently large S; we may thus assume
a
straightaway that  is a subfunctor of y., b.- being the inclusion morphism.
Let (i) be a finite affine open covering of . If  t k is open in
Y 0 k , for each i there are functions f, o,...,f, E r!J(Y,)0 k such that
- k,., '''' lIi -1 R:
(X iT y, )0 k is the union of --
- -1 k"
the open subschemes (i '1t) fij of i t k . For sufficiently large B,
the f.. are the ima g es of functions g ., E 12 (Y , ) 0 k s . Let Z be the open
1J 1J -1 ka -
subscheme of Y 0 k s covered by the (y, 0 k S ) . Since Z 0 k = X <;9 k ,
- ka -1 ka gij - ks - Ka
by the first part of the proof we have  ly = "f>. '1t  for sufficiently
large y . This implies that  t ky is open in  t ky .
When  t k is closed in  t k , we choose
kernel of the canonical map
c?(¥i) k ->- cJ (( nXi)tk) ;
f H ' . . . , fir' to generate the
1

94
ALGEBRAIC GECMETRY
I,  3, no 2
Let !i be the closed sub scheme of }:9ka k6 whose algebra of functions is
the quotient
(J1(¥i)  k 6 )/(gil,...,gir,) ,
a 1
the g., being defined as above. We have
1J
T. Q:9 k = Q:ni) k ,
-1 k6 a
so that
T,  k = (X ny, )  k
-1 6 y - -1 a y
for sufficiently large y. Hence (X nY,)<3J k is closed in Y. Q:9 k
- -1 Ka Y -1 K
all i, so that 2S   is closed in Xi  ky (2, 6.10).
for
2.4 Proposition: Let (k a ) be a directed system of ITOdels with
direct limit k. For each algebraic k- scheme T there is an index a and
an algebraic k -scheme T such that T is isanorphic to  Q:9 k
a -a - k
a
Proof: Let (!i) be a finite affine open covering of l' ; let
ij ij , i
f l ,..., f ( . ' ) E v (T) be functions such that T, nT, is covered by the
r 1, J. - -1 - J
open family (T1)ft j , 1 5  5 r(i,j) . By 1.2, for sufficiently large 6
there are algebraic k-schemes 1' and functions gj such that
T i Q:9 k ::;. T. and ij Q:9 1 = fij .
-6 k -1 g k 
6 6
Let T ij be the open sub scheme of 1'1 6 ' covered by the open family
-6
By 2.2 and 2.3, for sufficiently large a there are isanorphisms
ij : T ij Q:9 k ::;. T ij 0 k
'I' -6 k a -6 R a
6 6
(T1 6 ') ij
- g
such that ij k is the identity of
a
T ij Q:9 k = T i nT j = T ij Q:9 k
-6 k - - -6 k
6 6
If a is sufficiently large, ij induces an isanorphism
 ij . (Tijn T U ') <Si k ::;. (Tjin Tj) 0 k
'I' . -6 -6 k6 a -6 -6 R6 a
for which

I,  3, no 3
AIG:BRAIC SCHEMES
95
"j.Q, "ij = "H
'¥i 0 '¥.Q, '¥j
for all
(i,j,.Q,) . It is then sufficient to take for T the k-scheme ST'
-a --a '
is the spectral space obtained by rratching together the
along the open subspaces I ,!,j{a I via the isanorphisms I  ij I
where T I
. -a
IT; 0 k I
-" RS a
2.5 Corollary: A k- scheme  is algebraic iff there is a finitely
generated subring ko  k and an algebraic ko - scheme o such that X
is isanorphic to o(2)kok
2.6 Corollary: Let :-7- be a morphism of algebraic ka - schemes .
li h  k is an eTIbedding , so is h a kS for sufficiently large S
Proof: We may assume that h is an inclusion rrorphism (2.3); since  t k
is quasicanpact, there is a finite family (Xi) of affine open subschemes
in   k such that the ( k) n Xi are closed in };:i for each i, and
cover   k . The open subscheme  of X t k covered by the Xi is an
algebraic k-scheme. For sufficiently large S, there is an algebraic
kS-scheme
Z'
such that
Z' (2) k :; Z
- RS -
furthermore, for suff'!i-ciently large (3 there are morphisms
h l :  (ts -7- Z' and h 2 : . -7-  tts
such that b}l = h  kS (2.2); for sufficiently large S, h l is a closed
embedding, h 2 an open embedding by 2.3.
Section 3 Constructible subsets of an algebraic scheme. Flat rrorphisms.
3.1 Let X be a toFOlogical space. We shall say provisionally that
a subset U of X has the property C if the intersection of U wi th each
quasicanpact open subset of X is quasicanpact. We say that a subset P of
X is constructible if P is a finite union of sets of the form U n C V ,
where U and V are open subsets of X with the property C . Since any
constructible set clearly has the property C, we see that an open subset U

'!j
96
ALGEBRAIC GECMETRY
I,  3, no 3
of X is constructible iff U has the property C .
Fran
P = ls;s;n (uinCv i )
it follows that
C p = n( C uo UV,)
ill
so that [p is a finite union of sets of the form
V 0 n... n v, n C (U 0 U... U U 0 )
J l J s 1 1 1r
Hence if P is constructible, so is Cp . It follows that the family of con-
structible subsets of X is closed under finite union, finite intersection
and canplementation.
3.2
If X is quasiccxrpact and quasiseparated, the constructible
open subsets of X coincide with the quasicanpact open subsets. If X is
a noetherian space , that is, if each family of open subsets has a maximal
member (under inclusion), then every open subset of X is constructible. It
follows that the constructible subsets of X are then precisely the finite
unions of locally closed subsets.
ProFOsition: The following conditions on a subset P of a noetherian space
X are equivalent :
(i)
(ii)
P is constructible.
For each irreducible closed subset F of X such that P n F
is dense in F , P contains a non-empty open subset of F .
Proof: (i) => (ii) : Suppose that
P = ls;n Pi
where Pi is locally closed in X for each i. If P nF is dense in F ,
P on F is dense in F for at least one i; :for such an i, P ,n F is
1 1
locally closed in F , hence of the form U nK where U is open and K is
closed in F . Since Po = F , we have K = F and Po contains the open
1 1
subset U of F.
(ii) => (i) : Since each decreasing sequence of closed subsets of X is

..
I,  3, no 3
ALGEBRAIC SCHEMES
97
(ultimately) stationary, we may argue by noetherian induction by assuming
that the implication (ii) => (i) holds within any closed set strictly con-
tained in X If X = A UB is reducible, with A and B closed, and if
P satisfies (ii) , then P nA and P nB also satisfy (ii) ; in this case
P nA and P nB are unions of sets which are locally closed in A and B,
hence also in X, and so the same applies to P . If X is irreducible and
P is not dense, apply the induction hYFOthesis to P . If, on the other hand,
P is dense, P contains an open set U. Then P-U satisfies (ii), and is
therefore constructible by the induction hYFOthesis. It follows that
P = U U (P-U) is constructible.
3.3 We now apply the results of 3.1 and 3.2 to the geometric real-
ization II of an algebraic k-scheme X. Since X is quasiccxrpact and
quasiseparated, the constructible open subsets of X (Le. of I I ) are pre-
cisely the quasiccxrpact open subsets. Furtherrrore, if k is a noetherian
ITOdel , then II is a noetherian topological space . To prove this, observe
that for each affine open subscheme Q of K, l?(QJ is a finitely gener-
ated k-algebra, hence is noetherian. It follows that Igi = Spec tY(Q) is
noetheriani since II is covered by finitely many such IQI, II is
also noetherian.
ProFOsition: Let  be an algebraic k- scheme and let P be a constructible
subset of X. Then there is an affine algebraic k- scheme Y and a rrorphism
f:Y+X such that P=f(Y)
Proof: SUPFOse first of all that P is the union of two constructible sub-
sets P l and P 2 . SUPFOse also that we have constructed two rrorphisms
1=:1 :l+]: and J2:¥2+ such that P l =:1=1 (l) and P 2 =%2 (1 2 ) . Then P is
the image of the map underlying (t"1,t"2): .¥fLX 2 + K . Accordingly, we may
confine our attention to the case in which P is of the form I Q I n C I YI
covering Q by a finite open family (a) and replacing P by the
I Qa I n P , we reduce to the case in which U is affine. We can then cover
T.J. n,,- by the special affine open subschemes Qfl'"'' Qf n with
f l ,...,f E O(U) . It is now sufficient to set
n -
 = SPk (0(Q)) /h?(Q) f i ) ,
1
and to choose for l the rrorphism induced by the canonical projection of

98
ALGEBRAIC GECMETRY
I,  3, no 3
LD(Q) onto JJ(Q) ILi J(lI) f i .
3.4 corollary: Let x be a FOint of an algebraic k -scheme  and
let P be a constructible subset of . Then P is a neighbourhood of x
iff P contains each point y such that x is in the closure of {y} .
Proof: The condition is obvious 1 y necessary. Conversely, supFOse the con-
dition holds; we may then assume that  is affine; furthermore, since P
is constructible, there is an affine algebraic k-scheme  and a rrorphism
i:+ such that P = f(r) . If
Ex k 19 x + X
is the rrorphism defined in  1, 5.6, the space of FOints of (epk tD x ) x¥. is
anpty, i. e.
(2 0 t!J(Y) = {O} ,
x <9 () -
or 1 0 1 = 0 . Since (!) = lim 01(x) , as s runs :through the functions
19(X) x + - s
not vanIshing at x, we have 10 j(X) 1 = 0 in at least one of the rings
t1q)s00()0CD . Hence the underlyi;:g space of s)( is empty, which im-
plies that Isl is contained in P (l, 5.4).
3.5 Larma: Let A be a finitely generated algebra over an inte -
gral danain B. If,. M is a finitely qenerated B -ITOdule , M is free over
g
B for sane 0 'f g E B .
g
Proof: The following elementary proof is due to Dixmier. Clearly we may
sUPFOse that A = B(T l ,... ,Tn] is an algebra of FOlynanials and that M is
generated by a single element m (replace M if necessary by the cyclic
quotients of some composition series and take for g the product of the
elements of A associated with these different quotients) . For any
_.n I I V v) \In
v = (v l '...,v ) EN set v = V l +...+V and T = T l ...T Further-
n ,.. n n
rrore, let A:W + N be the bijection such that A (v) < A (11) if
(Ivl,vl,...,v:) : (1111,11 1 ,...,l1 n ) in the lexicographic ordering on 1f+l
If E, E W is such that TEi: T, , we then clearly have that A (v) <A (11)
l  l
implies A (WE,) < A (11+E:.) . Furtherrrore, for each r=A (I1)Etl, set
l l
M = L BTvm and let I be the ideal of B annihilating Mr+iMr
r A(v)<r 11

....
I,  3, no 3
AIG:BRAIC SCHEMES
99
(this B-ITOdule is generated by the residue class of T l1 m). We have clearly
I1+Ei - I BT I1 C T M - T Ei M C M
IBT m-T, m " () - ' () ' ( ) .
11 l 11 l 1\ 11 1\ 11 1\ I1+Ei
Hence I C I , and rrore generally I C I if 11 < P in the prcxluct
11 I1+Ei 11 P
order of W . But this order has the property that, for any subset S of
W , the subset S of minimal elements of S is finite (otherwise take any
o
infinite sequence of distinct elements in S and construct successively
o
infinite subsequences such that the first canponents increase, then also the
second ones .... contradiction ). Take now S = {11 EJtf : I fO} and
11
Orfg E n Es I . Then g annihilates all the quotients M 1 1M such that
-1 11 11 r+ r
A (r)ES Le. such that M + l /M is not free. Hence the B -ITOdule
r r g
(M 1 ) I (M) is cyclic free or zero for all r . Therefore M is free.
r+ g r g g
3.6
Generic flatness theorem: Let ,  be alqebraic schemes
over a noetherian ITOdel and let f:X-'.y be a daninant rrorphism , that is to
, such that the imaqe i () is dense in X . 'Then , if X is reduced ,
there is a dense open subscheme V of Y such that f induces a faithfully
flat rrorphism of f- l ('{) onto V.
Proof: (Of course, y is said to be dense in Y if the space of FOints of
y. is dense in IYI.) Let Y l ,... 'Y r be the irreducible ccmponents of Ixi
(Alg. ccmn. II,  4, no. 3, prop. 10). By replacing - X separately by open
subschemes Z. such that I z. I = y, -u '..;.. Y , and  by the inverse images of
-l -l l J,l J
these open subschemes, we may assume that Ixi is irreducible. If we then
replace X by a (dense) affine open subscheme, we may assume that  = S B ,
where B is an integral danain. Let "f>.l'... 's be an affine open covering
of X if we replace  by a smaller affine open subscheme and suppress
sane of the i ' we may assume that each i is daninant over Y . Thus if
there exist non-empty open Yic  such that the rrorphism fran -l (Y i ) i
into Y i induced by .f are faithfully flat, then we may set y = niY i . We
are therefore left with the case in which  = A and K is defined by
a rronanorphism :B+A, which is necessarily injective ( 1, 2.4). To canplete
the proof, it DCM suffices to set M=A in lEm11a 3.5 above and take for V
the special open subscheme Xg' thus taking the open subscheme  (g)
!-l(y)
for

100
ALGEBRAIC GEOMETRY
I,  3, no 3
3.7 CorollaJ:y : If k is noetherian and if f :X->-Y is a daninant
rrorphism of algebraic k- schemes , then the image ! () contains a dense open
subset of I I .
Proof: Apply 3.6 to the rrorphism t red : lCred->-X red induced by f. Since
the toFOlogical spaces II and I!:'red l are the same, as are II and
I red I, t: () contains the open subset I y I constructed in 3.6.
3.8
In the three following corollaries, we do not assume the ITOdel
k to be noetherian.
CorollaJ:y :
If i:->-X is a rrorphism of algebraic k- schemes , and P is a
constructible subset o f 2S , then the image f (P) of P is constructible .
Proof: We first assume, that k is noetherian; then P is a finite union
of locally closed subsets Pi and we may assume straightaway that  coin-
cides with the reduced sub scheme carried by one of the Pi. Now let Y' be
an irreducible closed subset of Ixi it is enough to show that, if
:f () ny I is dense in Y', then :f () contains an open subset of Y' (3 .2) .
If X' is the inverse image of Y' in 12S1 ' we have ! (XI) =! () nY' . If
X' and Y' are the reduced closed subschemes carried by x' and Y' , it
is thus sufficient to apply Cor. 3.7 to the rrorphism t' :'->-Y' induced by
i .
In the general case, we may assume that 1I=p by Prop. 3.3. By Props. 2.2
and 2.4, there is a finitely generated subring ko of k and a rrorphism of
algebraic ko -schemes to: o ->-x o such that t may be identified with
io0ko k. If g:i£->-7J¥.o is the canonical projection, we thus have
t:() = S!.-l(io(o)) by  1, 5.4. Since to(o) is constructible, so is
-1
S!. (io()) by de£. 3.1.
3.9 CorollaJ:y : For each morphism of locally algebraic k- schemes
f:X->-Y , the following assertions are equivalent:
(i)
! is open .
(ii)
For all y,y'EJ... , such that
-1
and each xq (y) there is
the closure of {x '} .
Y is in the closure of
-1
xq (y I ) such that x
{y' }
is in
,.,r;'(' ."
>"
!
::

r
'\
I,  3, no 3
ALGEBRAIC SCHEMES
101
(iii)
For each x, f induces a suriection of
Spec t!) i (x)
Spec {) onto
x
Proof: (ii) <=> (iii) : This follows imnediately fran  1, 5.6.
(ii) => (i) : If V is an open quasiccxrpact subset of X
structible by 3.8 . The assertion then follows fran 3.4.
:f(V) is con-
(i) => (iii): If not, set y=f(x) . Let y'
does not belong to the image of Spec,j) .
x
the unit of dy' )@  is zero. Assuming,
y x
affine, we have
J) = lim D(x)
x ->- - S
S
be a FOint of Spec J) which
y
Then Spec (K (y' )@() ()) = .0 , and
u y x
as we may, that'i and X are
where s (x) 'f 0 . It follows that the unit of K (y' )@ cD(X) c9(D s is zero for
sane s, Le.
I!I-l(y')nlsl = .0
so that Y ' jZ f (X) , a contradiction.
-s
3.10
Corollary : A flat morphism of alqebraic k- schemes is open.
Proof : With the notation of cor. 3.9, we must show that the maps
Spec J)x ->- Spec JJ f (x) induced by ! are surjective. Now tV x is flat over
cOf(x) , hence faithfully flat (Alg. corrm. I,  3, prop. 9). The assertion now
follows fran Alg. ccmn. II,  2, no. 5, cor. 4 of prop. 11.
3.11
Corollary : If k is a field and , y are k- schemes, the
canonical proiection J2!" 1: )( X ->-  is open .
Proof : Clearly we may reduce the problem to the case in which =l?Jl'
.xB , A,B E . If B is finitely generated over k, ASJkB is finitely
presented over A and 3.11 follows fran 3.10. In the general case it is
enough to show that if
has finitely generated
the image of ( ;>< 'io) s
remains to show that the
sEASikB , then pr l ()( ¥) s is open in II Now B
subalgebra Bo such that sEASikBo; setting ¥o=Bo'
in I I is open by our previous remarks. It thus
image of in I  I are the
same. To see this, write

102
ALGEBRAIC GECMETRY
I,  3, no 3
_ p-l (x) :; Sp (K (x) iZIB)
- k s
and
_ P O -l (x) :; er>(K (x) iZIB )
k 0 s
since the canonical map
-1 -1
p (x) 4- P (x)
-0
(K (x)iZI k B) 4- (K (x)iZI k B) is injective, the morphism
o s s
is daninant for each x (l , 2 .4). This ccxrpletes the proof.
3.12 corollary : For each ITOdel k and each rrorphism of locally
algebraic k- schemes f:X4-, the ]X>ints xEX, where ! is flat , form an
open subset U of .
Proof: Evidently we may assume that  and  are affine, with algebras A
and k. Let p' be a prime ideal of A such that A I is flat over k
p
for each prime ideal pCp' , Ap is flat over ApI , hence also over k.
By 3.4 it is therefore enough to show that U is constructible. By our pre-
vious remarks, U contains a point p E Spec A if U meets the closure
{p} . Hence, if k is noetherian , it is sufficient (3.2) to show that if
pE U , then U contains a non-empty open subset of {p} .
Now if q=!(p) , by 3.5 (or 3.6) there is an sEk-q such that (A/qA) is
s
flat over (k/q) . Since we are assuming that
s
k k
Tbr l (A,k/q)p = Tbrl(Ap,k;q) = 0
and since Tbr(A,k/q) is a finitely generated A-ITOdule (see 3.13 below),
there is a gE A -p such that
k k
0= Tbr l (A,k/q)g=Tbr l (Ag,k/ q )
If p' is a prime ideal of A such that p'={:J and sg' , and if
q'=!(p') , we have accordingly
- kql
o - Tbr l (Ap.,kq'/') ,
and A ,/q_,A, is flat over k .lq_. . By Alg. ccmn. III,  5, tho 1 and
p-qp q-q
prop. 2, A, is flat over k, , which proves the corollary when k is
p q
noetherian.
Now assume that k is not noetherian; choose P E: S pe c A so that A is flat
- - p
over k. With the notation of 3.14 below, this result implies that there is
atE A -p such that A t is flat over k . If t I is the image of t
o 0 0 0

I,  3, no 3
ALGEBRAIC SCHEMES
103
in A, then p E (Spec A) t, and At' is flat over k, which canpletes the
proof in the general case.
3.13
Lermna: Let k be a noetherian ring , A a finitely generated
k- alqebra, M (resp. N) a finitely generated A -ITOdule (resp. k -ITOdule ). Then
k
Tor. (M,N) is a finitely generated A -rrodule for each i.
1.
Proof: Since A is a quotient of an algebra of polynanials k[X l ,...,x ] ,
- n
we may, by replacing A by k[xl'... 'X n ] , assume A to be flat over k.
To prove the lEm11a, we need only calculate Tor (M,N) using an acyclic

resolution of M by free finitely generated A-rrodules.
3.14 Lerrma: Let A be a finitely presented algebra over a ring k
and let p be a prime ideal of A such that A is flat over k. Then
p
there is a finitely generated subring ko of k, a finitely generated
ko - algebra Ao and a ring horromorphism o : Ao ->- A such that
a)
o (E;a) = E;o (a)
for
E;Ek and aEA ;
0- 0
b)
c)
the map A o Q9k k ->- A
-1 0
if P o =o (p) , A
oPo
induced bv o is bi lective i
is flat over k
o
Proof : By 1.2, there is a finitely generated subring k l of k, a finitely
generated k l -algebra Al ' and a hCX!lCID:)rphism l : Al->-A satisfying con-
ditions a) and b) above mutatis mutandis. Let ko be a finitely generated
subring containing k l ' set A = A 1 Q9 k k and let  be the hCX!lCID:)rphism
o 1 0 0
induced by l. Set Pl = ? (p) and let q, %' ql be the inverse images
of p, Po' Pl in k, ko' k l :
If 7 A ) 1\
It , .1
Pi I ' I7 t
Il' 11
ql 1 %  q
Set B=A B =A Bl=A1Pl ' £=k £ =k £l=k l
P o 0Po q o OCJo ql

104
ALGEBRAIC GEOMETRY
I,  3, no 3
Since we have £ 
o = Tor l (B ,£/ql£) = l Tor l (Bo'£o/ql£o)
and since Torl(Bl '£l/ql£l) is a finitely generated Blule we may, by
taking ko sufficiently large, assume that the canonical image T of
Torl(Bl ,£/ql£l) in TorO(Bo ,£iql£o) is zero. In this case, ko satis-
f ies the condition of the lEm1la:
For if
u V
D l ---» Cl-----;.B l  0
is a presentation of the £l-ITOdule B l ' then
uiSi£ viSi£
D l 181 £ 0 ,. C l 181 £ 0 ) B l 181 £ > 0
£1 0 £1 0 £1 0
is a presentation of the £o-rrodule B l iSi£l£o
£0
Tor l (B l 181 £o'£o/ql'£o)
£1
Hence
is a quotient of
Ker(u 181 £ /ql£ ) :; Ker[u 181 (£/ql£l) 181 (£ /q £ ) ]
£1 0 0 £1 £l/ql£l 0 1 0
 Ker[u 181 (£l/q l £ l )] / 181 (£o/ql£o)
£1 £1 ql£l
£
o
(obfe that £l/ql£l is a field:). Since Tori (B l l£o'£o/ql£O) and
Tori (B l ' l/qrel) are quotients of Ker (u l £o/ql £0) and
Ker(u l£l/ql£l) , it follows that the image of
£1 £
Tor 1 (B l '£l/ q l£l) in Tor l o (B l l£O'£o/ql£O) generates this last £o-rrodule.
Since B is a ring of fractions of B l 181 £ , we have
o £1 0
£
o
Tor i (B ,£ /q l '£ )
000
£
.::;: B 181 Tor l o (B l 181 £ , £ /q l £ )
o Bl  £0 £1 0 0 0
1
hence the canonical image T generates the Bo-rrodule
£
o
Tor l (B ,£ /q l £ )
o 0 0
which is therefore zero. Since B /q l B is a ring of fractions of
o 0

I,  3, no 4
ALGEBRAIC SCHEi'-1ES
105
(B/qlB l ) 0 (.Q'iql £0) ,
£l!ql£l
it is flat over £o/ql £0' , so that, by Alg. conm. II,  5, tho 1 and prop. 2,
B is flat over £
o 0
Section 4
M::manorphisms of algebraic schemes
4.1 ProFOsition : Let xE and yE¥ be points of tw:> k- schemes
X and .1; we assume Y to be locally algebraic over k.
a) It f,<J:?f=if are two rrorphisms of  such that i(x)=gJx)=y , and if
the induced maps f,g:: J '*& coincide , then there is an open subscheme !I
-x x y x
Qf "f>. such that x ElI and :fIll = -9:111 .
b) If :..9y  Jl x is a k- algebra haranorphism , then there is an open sub -
scheme Q. Qf  and a rrorphism i: 11-+1' of t.\ such that i (x) =y and
:!x =
Proof:
a): We may assume without loss of generality that X and Y
are affine. Thus let =kA and X=kB, where the k-algebra B is gene-
rated by b l ,... ,b n . Then we have i=k and <ySl?k1/J , with ,1/J E (B,A)
Since A = lim (I....A , and  (b. ) and 1/J (b, ) have the same image in A
x -+ Sh S 1 1 X
for each i, there is an sE A , s x , such that  (b i ) and 1/J (b i ) have
the same image in A s for each i. Now set U = X -+ SP,.A
- -s -,.. s
b): We may again assume that =kA and ¥=kB . Let sE A
be such that s(x),o 0 . Let p:A -+() and q:B-+{) be the canonical maps,
s x y
and sUPFOse that the k-algebra B is defined by generators b l ,... ,b n and
relations P l (b l ,...,b n ) = ... = P t (b l ,...,b n ) = 0 . Since x is the direct
limit of the rings of fractions A , there is an s and elements
s
a l ,...,a n EA s such that P l (a l ,...,a n ) =... = p t (a l ,...,a n ) = 0 and
p(al)= (q(b l )), ..., p(an)= (q(b n )) . Hence there is a haranorphism 1/J:B-+A s
such that ,I, (b, ) =a. , and we need onl y set U=X and f = Sn ,I, to canplete
 1 1 - -s -  
the proof.
4.2
Corollary: Let f :X-+Y be a morphism of locally algebraic

106
ALGEBRAIC GECME:I'RY
I,  3, no 4
k- schemes and let x EX. If the map fx: Jhx) +0 x is bijective , there are
open subschemes .1I and y.. of "f>. and 'i such that xE U
induces an isanorphism !I:+ Y. .
f (x) E Y and f
Proof: If y=f (x) , by 4.1 b) there is an open subscheme V' of Yanda
rrorphism g:V'+X such that yEy' , g(y)=x and g =t- l . By 4.1 a), we may
_ - - - -y x
take '1' so small that !<;r coincides with the inclusion rrorphism '1'+X .
Set !J,=(l ('1') and let !' :!J'+y'" be the morphism induced by !. By 4.1 a)
there is an open subscheme !J of g' such that xE Q and g (!' I!J) coincides
-1 -
with the inclusion rrorphism of Q in X. Now set y = <I (!J)
4.3
Definition: Let f:+¥ be a morphism of schemes and let
xE  We say f to be a local embedding at x if there is an open sub -
scheme .1I of K such that xE.K and % IQ is an embedding . f is said to
be a local embedding if it is a local embedding at each xE 
For instance, let A = Z[U,V]/(W) , B == Z[u,V]/((V-u 2 +U)V) , and 2£ == Sp A
and X = Sp B . For each mcxlel M, we then have
and
(M)
X(M)
2
{(x,y)EM
2
{(x,y)EM
xy=0}
2
(y-x +x) y=0}
y
f
m >
Q
n i<. IC
The rrorphism f:+¥ such that t. (M) maps (x,y) E (M) onto
2
(x-y, y +y) E X (M) is a local embedding, but not an embedding. For the map
t:(Z) sends (l,O)E(Z) and (O,l)E(Z) onto (l,O)El'(:J) , so that .t,
is not a monanorphism. On the other hand, let a,b (resp. u,v) be the images
of U,V in A (resp. in B). Then we have f = eE  , where :B+A is such
that  (u) = a-b and cP (v) = b 2 +b . Now one verifies easily that  in-
. .  [ -1 J d  "' b b -l ]
duces Sur]ectlOns of B onto A + Z a,a an A +"L , .
u a -1)

F'"
I,  3, no 4
ALGEBRAIC SCHEMES
107
Moreover, in the ring of fractions A l - a + b we have
a = (u) + 
(l-u)
and
b = (v)
(l-u)
which shows that  induces a surjection of B l - u onto A l - a + b . Since the
open sub schemes a  and l-a+b cover , it follows that f is a
local embedding.
4.4
with 
if f
-
If f:+X is a morphism of k-schemes, we say, in accordance
1, 6.3, that f is a local embedding at xE  (resp. a local embedding)
: 'j, + z¥ is a local embedding at x (resp. a local embedding) .
Proposition: Let  be a locally alqebraic k- scheme and f:+X a rrorphism
of k- schemes . Then K is a local embeddinq at a point xE X iff
f : t9 f ( ) +t2 is surlective .
-x _ x x
Proof: SUPFOse that fx is surjective; we may assume that "f>. and X are
affine. If a l ,... ,an generate the k-algebra I2 k () , then we have
(b,)P'(s.) = a,/l E19 k (X) , where =i.1. k (f) , s.,b.E{) k (Y) and
1'¥ 1 1 -x - 11 -
s, (f(x))fO . Hence t. (a.(s,)-(b,)) = 0 for sane t,E{) k (X) for which
1- 11 1 1 1 -
t,(x)fO. If we set s=s l ".s and t=t l ...t(s) the equations
1 n n
(bi)N(Si) = ail already hold in t9k()t . Since t/l belongs to the sub-
algebra of t9k()t generated by the ai/l, t/l is of the form (v)/(s)n
where vE Uk (¥) . Hence  induces a surjection of Uk ('f) sv onto t9 k () t '
which implies that f is a local embedding at x . The converse is obvious.
4.5
Lerm1a: Let X be an irreducible scheme (Le. such that the
toFOlogical space II is irreducible ) and f:+X an in'jective local
embedding . Then f is an embedding .
Proof: Let xE X , let !:!x and Y x be open subschemes of  and Y such
that xEQx' f (Qx) C Y x and that f induces a closed embedding of  into
Y x Since 1Qx l is dense in II , f(Qx) is dense in f()nIYxl since
f (Q) is closed in I Y x I ' we have f (llx) = f (ii) n I Y x I . Since f is in-
jective, we have Illx l = !-l(yx) , so that f induces a closed embedding of
 into the open subscheme of X covered by the Y x (2, 6.10).

108
ALGEBRAIC GEOMEIT'RY
I,  3, no 4
The irreducibility assurrption is essential for the truth of the above lemma.
For let X' be the scheme satisfying
2
!;.' (M) = {(x, y) EM: xy=0 and 1 +y is invertible}
for each ITOdel M. The rrorphism 1': ?;-+ of example 4.3 then induces an in-
jective local embedding !:' :'-+' but f is not an embedding.
4.6
ProFOsition:
SUPFOse tha t k
is noetherian. If f:X-+Y is a
- ---
rronarorphism of alqebraic k- schemes , then there is a dense open subscheme
U of X such that !:Ig is an injective local embedding .
Proof: First of all consider a morphism of schemes <I:-+:£. Then 9: is a
monarorphism iff the diagonal morphism 011': -+ )('r is an isanorphism. If
K is a field, :!: = Sp K and  = Sp A , this last assertion holds iff the
map a 181 b -+a.b of A Q9kA into A is bijective, so that A={O} or A=K
It follows that a monanorphism of a nonempty scheme into the spectrum of a
field is an isanorphism .
In the general case, if <i! is a monanorphism, <i!. is injective ( 1, 5.3).
If xE , set y-(x) and '= Sp K(y) . The canonical projection
-1
g': g (y)-+ X' is a monanorphism. If l?x is the local ring of  at x ,
- - -1
that of  (y) at x is 0 x /J x m y ' so it follows fran the above that 
induces an isanorphism K (y) .+ oj / rJ m . If <fl is Artin, this implies that
x x Y x
g :,J -+ J} is surjective. For if J}' and m' are the images of () and
-x Y x Y Y Y
m in r!J , then K (y) .+ rJ /,j m is equivalent to CJ = J). + m'  , that
Y x x xy x y y x
is to r!) = r!)' by Nakayama' s lemma (m' being nilIX'tent) .
x y y
This results applies when <,Fzt: and x is the generic point (l, 2.10) of
an irreducible canponent of II . For by  1, 5.6 " Spec L9x then has just
one point, so that Ox is Artin. The assertion now follows fran 4.4.
4.7 Corollary: With the assumptions of prop. 4.6, there is a
dense open subscheme V of Y such that f -1 (y) is non-empty and t: I f -1 (y)
is an embeddinq.
Proof: Let xl' . . . ,x n be the generic points of the irreducible ccmponents
of II and let y, = f(x,) . write y,  y, if y, is in the closure of
l - l l J l
yj . Since f is injective, we may assume Yl to be maximal with respect to

I,  3, no 4
ALGEBRAIC SCHEMES
109
this ordering. Then there is an open subscheme y 1 of r such that
ylEIYll and yllYll if i > 1 . If u l '''.''1n are the generic points of
the irreducible canponents of I!; I - 1 Q I (4.6), then y i is distinct fran
:IJul),...,i(um) . If :,[1 is small enough, IYll does not meet the image of
II - 1111 . L€rnma 4.5 then implies that the restriction of f to g-l (:,[1) is
an embedding; for 1 g -1 (Y 1) 1 is open in {xl} and is therefore irreducible.
This proves the corollary when Y l is d ense. If not, consider the open sub-
scheme Y' of X such that IY'I = IXI-{Yl} . Then apply the preceding con-
struction, using the morphism g': !-l (Y' )-+Y' induced by i ... .
4.8 Example: Observe that a monanorphism of locally algebraic
schemes over a field k is not necessarily a local embedding. For sUPFOse
that k is of characteristic '12 ; take for 1i and .x the algebraic schemes
such that
15 (M) {tE M : l-t is invertible}
X(M)
2
{(u,v)EM
322
u -=u -v }, M
Themorphism f:X-+Y such that !(M)(t) = (l-t 2 , t(1-t 2 )) for each M is
a monanorphism, but it is not a local embedding at the FOint of 15. associated
with the element -1 of X (k) . (To prove that K is a monanorphism, show
that X/Y= 0 (cf.  4, below); by the theorem of  4, 3.1 below,
oX/X : X -+ J5. "'l is then an open embedding, and all that remains is to show
that o"!:/X is bijective, i.e. that :f (K) is injective for each field K)
o
Vi}.
I
I
j' ;;
I
I
I
x
-1

110
ALGEBRAIC GEOMEI'RY
I,  3, no 5
Section 5
The Krull dimension of a noetherian ring
5.1 Definition: Let T be a toFOlogical space . The dimension of
T is the supremum of the lengths n of all chains F 0 t F 1 'i ...  F n of
irreducible closed subsets of T.
Given xE T , the local dimension of T at x is the infimum of the dimen-
sions of open neighbourhoods of x in T .
We write dim T (resp. dim T) for the dimension of T (resp. the local
x
dimension of T at x). If T= , we set dim T = -00 .
For each ring A, the dimension of the to]XJlogical space Spec A is called
the Krull dimension of A and is written Kdim A . By  1, 2.10, Kdim A
is the supremum of the lengths n of all chains PO... ¥ Pn of prime
ideals of A.
5.2 ProFOsition: Let B be a ring and let A be a subring of B
such that B is an integral extension of A. Then Kdim A = Kdim B
Proof: If qo  ...   is a chain of prime ideals of B, then
q. i n A 1- q,nA by Alg. carm. V,  2, cor. 1 of prop. 1 so that
1+ 1
Kdim A > Kdim B Conversely, if PO...  Pn is a chain of prime ideals
of A, then we can construct inductively a sequence of ideals qO,ql... of
B such that q,C q, l and Anq. = p, (Alg. ccmn. V,  2, cor. 2 'of tho 1).
1 1+ 1 1
Hence Kdim B > Kdim A
5.3 Corollary: Let k be a field and let A be a finitely gene-
rated k- algebra without zero divisors . If Fract (A) denotes the field of
fractions of A, and trkdeg Fract (A) the deqree of transcendence of
Fract (A) over k, then
Kdim A = trkdeg Fract (A)
Proof: Let po...  Pr be a chain of prime ideals of A. By the normali-
zation lerma (Alg. comn. V,  3, no. 1), there are elements a l ,... an of A,
algebraically independent over k, such that A is an integral extension of
B=k[al,...,an] and PinB is the ideal of B generated by aO=O,
a l ,... '(i) (where h(i) is an increasing function of i). Since

r
I,  3, no 5
ALGEBRAIC SCHEMES
111
PinB of PHlnB (loc. cit.), it follows that r.::.n, and hence Kdim A =
Kdim B ::: n. On the other hand, since the prime ideals (a l ,. .. ,an) of B
form a chain of B of length n, we have Kdim A = Kdim B > n .
5.4
Corollary: With the assumption of 5.3, for each prime ideal
P of A we have
Kdim A = Kdim Ap + trkdeg Fract(A/p) .
Proof: Using the notation of 5.3, set r=1, PO={O} and Pl=P. Evidently
Kdim A > Kdim A + Kdim (A/p) , since the second summand is just the su p remum
- p
of the lengths of prime ideal chains containing p. On the other hand, by
the "going down theorem" (Alg. ccmn. V,  2, tho 3), there is a chain of
prime ideals Qh(l):JQh(l)_l:J...:Jql of A such that Qh(l)=P and
Q, n B = (a l ,...,a.) for each i; hence Kdim A > h (1) Moreover, since
1 1 P -
A/p is an integral extension of B' = k[(l)+l'." ,an] , we have
trkdeg Fract(A/p) = Kdim(A/p) = Kdim B' = n-h(l) j
so that
n = Kdim A = h(l)+(n-h(l)) < Kdim A + Kdim(A/p)
p
5.5 We now turn to the problem of calculating the Krull dimension
of certain rings which are not covered by corollary 5.3 a1:ove. For this pur-
pose we shall use another formulation of the Krull dimension:
Let E be a partially ordered set. If a,bE E , write [a,b] for the set
of x E E such that a < x < b. We shall assign to E a quantity dev E ,
called the deviation of E: this will be a natural number or one of the
symbols -00, +00 . To define dev E , we determine by induction on n the
partially ordered sets E for which dev E < n: if E is discrete (Le.
a < b implies a =b) , set dev E = -o:J ; if E is Artin (Le. each strictly
decreasing sequence fran E terminates) set dev E::: 0 ; now suppose that we
have determined the partially ordered sets F such that dev F < n-l ; then
set dev E < n if each decreasing sequence ai' a 2 ' . . . from E such that
dev[a, l ,a, ] < n-l for each i is finite. Finally, set dev E = +00 if,
1+ 1 -
for each n ErN it is not the case that dev E < n .
For example, dev E = 0 means that E is Artin and not discrete. Accordingly

112
ALGEBRAIC GEOMETRY
I,  3, no 5
we have dev IN = 0 , but dev 'Q{ = 1 and dev IQ = +00 (where IN , :i!' and IQ
are assigned their natural orderings) .
5.6
We now list sane elementary properties of the deviation function:
a)
If f:E+F is a strictly increasinq map between partially
ordered sets (i.e. a < b implies f (a) < f (b)) , then dev E :: dev F . For
the truth of the assertion is obvious when dev F = -00 j supp:>se it holds
when dev F < n j we prove it for the case dev F = n . Let a l ,a 2 ,... be a
decreasing sequence fran E such that dev[a' +l ,a, ] > n ; by the induction
11-
hYFOthesis, we then have dev[f (a Hl ) ,f (a i ) ]  n , so that the sequence
f(a l ),f(a 2 ),... is finite. Hence a l ,a 2 ,... is finite.
b)
If E,F are non-empty partially ordered sets , then
dev (E x F) = sup (dev E,dev F) . We show that dev (E x F) :: sup (dev E,dev F) by
induction on the j:B.ir (dev E,dev F) (the reverse inequality follows direct-
ly fran a)). 'Ib this end supp:>se that dev E :: dev F = n ; the assertion is
obvious when n = -00 . If not, let xl' x 2 . . . be a decreasing sequence fran
Ex F such that dev[x' +l 'x, ] > n for each i . Let x, = (a, ,b,) j if the
11- 111
sequence x l 'X 2 ... were infinite, then we 'MJuld have dev[a i + l ,a i ] :: n-l
and dev[b, l ,b,] < n-l for sufficiently large i. By the induction hYFO-
1+ 1 -
thesis, this would imply
dev[XH1,Xi] = dev(f a H l'a i ]x [bHl,bi]) :: n-l .
c)
Let E be a partially ordered set and let Sc (E) be the set
of infinite sequences e l ,e 2 ,... fran E such that
sufficiently large n. If (e,), (f,) E Sc (E) , set
1 1
for all i. Then we have
e is constant for
n
(e,) < (f,) if e, < f,
1 - 1 1 - 1
dev Se(E) = 1 + dev E .
'Ib prove this we argue by induction on n = dev E . The assertion is obvious
if n = -00 , so supp:>se that n>O . If e=(e,)ESc(E), set e = lim e .
- 1 00 n-+oo n
Let a l = (a li ), a 2 = (a 2i ) ... be a decreasing sequence fran Sc (E) such that
dev[a' +l ,a, ] > n+l for each i. Were the sequence infinite, we would ha';e
1 1-
dev[a (i+l) oo,aioo] < n for sufficiently large i; choose jo so that
a" = a. and a ( ' +l) ' = a ( ' 1) for j >j o . Let
1J 1 00 1 J 1+ 00

I,  3, no 5
ALGEBRAIC SCHEMES
113
f : [ai+l,ai] + EjOX Sc[a(i+l)OO,a ioo ]
be the map (e,)  (e l ,...,e, , (e, 1 ,...)) . By (a) , (b) and the induction
l JO JO+
hYFOthesis we have
devla i + l ,a i ] 2 sup(dev E , devSc[a(i+l)oo,a iro ] ) = n .
This shows that Sc (E) < 1 + dev E . The reverse inequality is clear if
n = dev E = 0 , so suppose that n is finite and > 0 . Then there is an
infinite decreasing sequence
b l ,b 2 ,... fran E such that
i j if a.E Sc(E)
l
> n by the induction hYFOthesis,
is such that a. .=b,
lJ l
so that
for
dev[b' +l ,b, ] > n-l for each
l l-
each i, then dev[ai+l,aiJ
dev Sc(E) > n+l .
d)
Let Cr (E) be the subset of Sc (E) consisting of increasing
sequences. Then dev Cr (E) 1 + dev E . For by a) we have
dev Cr (E) 2 dev Sc (E) 2 1 + dev E; the reverse inequality is proved as in
c) . If E is noetherian, each increasing sequence fran E belongs to Sc(E)
and so to Cr(E) .
5.7 Given a ring A and a mcdule M , let dev M be the deviation
of the set of suhrodules of M , ordered by inclusion. Then dev M = -00 iff
M={O} . If sA is the underlying A-ITOdule of A, we write dev A instead
of dev sA (although we are only considering cCXImUtative rings here, do not
be misled into thinking that the notion of deviation is useless in the general
case:) .
a)
If N is a sul:mcdule of M, then dev M = sup (dev N, dev M/N) .
For the map PI-->- (P nN,p/p nN) is strictly increasing, so
dev M < sup(dev N, dev M/N) by 5.6 a). The reverse inequality is obvious.
b) If A is noetherian and M is a finitely generated A -mcdule ,
we know that there is composition series M = MO ::JM1::J ... ::JM n = {O} such that,
for each i, M./M' +l is iscxrorphic to A/p. for sane prime ideal p. of
l l l l
A (Alg. COIID1. IV,  1, tho 1). By loco cit., tho 2, we have dev M =
sup dev (A/p) , where p ranges over the minimal prime ideals containing
Ann(M) .

114
ALGEBRAIC GEC1-1ETRY
I,  3, no 5
5.8
PrOFOsition: Let A be a (cCXImUtativ noetherian ring . Then
Kdim A = dev A .
Proof:
To show that Kdim A :: dev A , we may assume that dev A < -too j it is
then enough to show that dev A/p > dev A/q whenever p and q are prime
ideals such that p ¥ q . For this PurFOse we may assume that p={ O} j let
s Eq, s'fO. Then we have an infinite sequence AAsAs2 ... such that
i i+l i i
dev (As /As ).:. dev (As /qs ) = dev (A/q) and the assertion follows.
We show finally, by induction on Kdim A , that dev A :: Kdim A . The inequal-
ity is obvious when Kdim A = -too j suppose then that Kdim A is finite and
> 0 . Let S be the set of sEA which do not belong to any prime ideal p
such that Kdim A = Kdim(A/p) . If M is a finitely generated A-mooule such
that s-\i=O, then we have
dev M = sup dev(A/q) < sup Kdim(A/q) < Kdim A
q  AnnM
(apply 5.7 b), the induction hYFOthesis and the fact that Kdim A/q < Kdim A
if qAnnM,henceif q meets S). Consequently, if IlI2... is a
sequence of ideals such that dev(Ir/Ir+l) .:. Kdim A , then
-1 _ -1 -1
S (Ir/Ir+l) - S Ir/ S Ir+l 'I 0 .
Since s-l A is Artin (Alg. cam\. IV,  2, prop. 9), the sequence is finite.
5.9 Corollary: If x belongs to the radical of a (cCXImUtative )
noetherian ring A, then Kdim A :: 1 + Kdim(A/Ax) .
Proof: Let Gr(A) be the graded ring associated with the (Ax)-adic filtra-
tion of A. To each I of A assign the graded ideal
Gr (I) = EB (I nAx n ) / (I nAxn+l)
n
By Alg. comn. III,  3, tho 2 and prop. 6, the map II->-Gr(I) is strictly
increasing. Accordingly we have Kdim A :: dev F , where F is the set of
graded ideals of Gr (A) . Since Gr (A) is obviously a quotient of the graded
algebra (A/Ax) [T] , where T is an indeterminate, we have dev F :: dev F' ,
where F' is the set of graded ideals of (A/Ax) [T] . But if E is the set
of ideals of A/Ax, clearly F' ::;. Cr (E) . The corollary now follows fran

,..
I,  3, no 5
ALGEBRAIC SCHEMES
115
5.6 d) and 5.8.
5.10
Corollary: Let A be a noetherian local rinq with maximal
ideal m, and let n be a natural number . Then the following assertions are
equivalent.
(i)
(ii)
Kdim A < n
There is a sequence a l ,...,a n of elements of m such that
the rinq A/i Aa, is Artin.
1
Proof: (ii) => (i) : 'Ihis follows imnediately fran 5.9 by induction on n
(i) => (ii): This is obvious when n = 0 j if n > 0, m is not a minimal
prime ideal (Alg. comn. IV,  2, prop. 9). Hence there is an element a l of
m which belongs to no minimal prime ideaL By the definition of Kdim A
and 5.9, we have, setting A=A/Aa l , Kdim A = Kdim A - 1 . Arguing by in-
duction on n, we may assume that we have proved the existence of a sequence
B. 2 ,...,a n from m/Aa l such that A/EAa i is Artin. If a 2 ,...,a n are the
representatives of a 2 ,...,a n in m, the sequence a l ,a 2 ,...,a n satisfies
(ii) .
5.11
Corollary: With the assumptions of 5.10 , we have
Kdim A. < Cm/m 2 :A/m]
Proof: If a l ,...,a n are the representatives in m of a basis for m/m 2 ,
then m = LAa, (by Nakayama' s lemma).
1
5.12
Corollary: Let A and B be noetherian local rinqs such that
A CB . If the maximal ideal m of A is contained in the maximal ideal of
B , then
Kdim B S. Kdim A + Kdim B/Pm
Equality occurs if B is flat over A.
Proof: By induction on Kdim A . For each ring C , let rC denote the set
of nilpotent elements of C; we then have r A =rB nA . Hence we may replace
A by A/r A and B by B/Br A ' which thus enables us to reduce the problem
to the case in which r = 0 . If Kdim A = 0 , the assertion is then trivial
A

116
ALGEBRAIC GEOMEI'RY
I , . 3, no 6
since m = {O} . If Kdim A > 0 , let x Em be a non-zero divisor of A
(Alg. COIm1. IV,  2, prop. 10) We then have Kdim A/Ax = Kdim A - 1 (5.9
and Alg. ccmn. IV,  2, prop. 10) , Kdim B/Bx  Kdim B-1 arrl
Kdim B/Bx ::: Kdim(A/Ax) + Kdim(B/Bm) by the inductive hypothesis; the re-
quired inequality follows. If B is flat over A, x does not divide 0
in B and all the inequalities above may be replaced by equalities.
5.13
Corollary: If A is a noetherian local ring with maximal ideal
m , then
Kdim A = Kdim A ,
where A is the canpletion of A in the m- adic tor;oloqy .
5.14
then
Corollary: If k is a field and T l ,... ,Tn are indeterminates,
Kdim k[[T l ,... ,Tn]] = n .
Proof: k[[T l ,...,T n ]] is the completion of the localization of k[Tl,...,T n ]
at the ideal (T l ,... ,Tn). The result now follows from 5.3, 5.4 and 5.13.
Section 6
Algebraic schemes over a field
Throughout this section k denotes a field
6.1
Given a ITOdel R and an R-scheme "f>., we write dim  or
dim X for diml?;.1
x-
local dimension of
and dim x I ?;.I , and speak of the dimension of X or the
X at x.
Dimension theorem: Let x be a r;oint of a locally alqebraic k -scheme 
and let Xl'... , X r be the irreducible comr;onents of I ?;.I passinq throuqh
x . Then
dim X = sup dim X, = Kdim t9 + tr k deg K (x) .
x- li$r l x
Proof: Clearly dim X = sup,dim X. . Le:-. p. be the prime ideal of () x
x- l X l l
which is carriEd onto the generic ]X)int of Xi by the map
IE: I: Spec  + Ixi (1, 5.6 and  1,2.10). If we identify X, with the
x x - l
closed reduced subscheme of X carried by X. (Le. the subscheme X, ed
- l l r
of  2, 6.11), () /p, is the local ring of X, at x. By  1, 5.6 we see
x l l

I,  3, no 6
ALGEBRAIC SCHEMES
117
that Pl'... 'Pr are the minimal prime ideals of Ox . It follows that
Kdim () = sup. KdimW /p,) , so that it is enough to show that dim X,
x 1 X 1 X 1
dim X. = Kdim(G) /p.) + tr k deg K (x) .
1 X 1
We may thus assume that X is irreducible and reduced. If X is affine, the
theorem follows fran 5.3 arrl 5.4. If not, let w be the generic ]X)int of
II . For each non-empty affine open subscheme Q of , we then have
dim 9 = trkdeg K (w) , whence dim!; = sup dim Q = trkdeg K (w) , which re-
duces the general case to the affine case.
6.2
Corollary: Let K I k be an extension of fields and let  be
a locally algebraic k-scheme. Then a) dim X = dim X0 k K
]X>int x E"f>.0 k K , if Y is the projection of x in X
dim x Q. Q9 k K ) .
b) for each
we have dim X =
y-
Proof: Since dim X = su p dim X and the canonical projection
- x x-
p: 2' (0 k K ) -+ 7J is surjective, we see that b) iinplies a). Now to prove b) .
Let !;1'... 'n be the reduced sub schemes of !; carried by the irreducible
ca1l]X)nents containing y; the irreducible CCITI]X)nents of ):'; (7) kK containing
x are then irreducible ca1l]X)nents of the schemes X.0 k K. Moreover, if Y ,
-1 1
is the g eneric ]X)int of X, , then there is a ]X)int x. Ex 0 k K such that
-1 -1
P (x,) = y . and x E {x, } ("forget" the irreducible canponents of X ° k K not
1 1 1 -
j:B.ssing through i and apply 3.11). The irreducible ca1l]X)nents of !;0 k K
j:B.ssing through xi thus contain x and are irreducible can]X)nents of
X, ° k K . It is therefore enough to show that dim X. = dim X  for each
-1 -1 -1
irreducible CCITI]X)nent 2:';i of 2:';i 0 k K (6.1). For this purpose we may obvious-
ly assume that X, is affine with algebra A; several cases then arise:
-1
If K is a pure transcendental extension of K, A0kK is a ring of frac-
tions of an algebra of ]X)lrnomials over A, and hence is an integral danain.
Accordingly !::;i 0 k K is irreducible and by 5.3
dim !::;i ° k K trKdeg Fract (AI5'\K) = trkdeg Fract (A) = dim ):';i
If K is an algebraic extension of K and if w' and w are the generic
]X)ints of
K(W') is a
arrl we have
X and X. , then p(w') =w because p is open (3.11). Since
1 1
quotient of K (w) (7) kK, K (w' ) is an integral extension of K (w)
dim!::;i = trkdeg K (w') = trkdeg K (w) = dim !;i

118
ALGEBRAIC GEa1ETRY
I,  3, no 6
Finally, in the general case, there is a pure transcendental subextension K'
of K such that KIK' is algebraic. Then dim X = dim x. @ K' = dim X,
-l -l k -l
6.3 Corollary: Let i:->-X be a rrorphism of locally alqebraic
k- schemes , let x be a point of  and let Y=i (x) . Then we have
dim X < dim Y + dim f- l (y) .
x- - y- x
Equality occurs when i is flat at x .
Proof:
By  1, 5.8, if i!) and i!) are the local rings of  and X. at
x y
is the maximal ideal of () , 12 /,j m is the local
y x x y
. By 5.12, Kdim 0 < Kdim rj + Kdim(u /rJ m) , whence
x- y x xy
tr k deg K (x) < dim Y - tr k deg K (y) + dim f -1 (y) - tr ( ) deg K (x)
- y- x- K Y
x and y, and if m
y
ring of i- l (y) at x
dim X -
x-
(6.1) and the required inequality follows. When f is flat at x, the in-
equalities may be replaced by equalities (5.12).
6.4 Corollary: If i:->-X is a rrorphism of locally algebraic
k-schemes, the function x 1-4 dim f -1 (f (x) ) is upper semi -continuous.
x- -
Proof: For any e ElN' , let
Take any x in the closure
X
e
X of X
e e
scheme of X carried by the irreducible C anpo nents X l ,...,x. of X
- s e
passing through x. For the rrorphism !': . ->- ! (. ) r ed irrluced by f we
have by 6.3
be the set of xEX with dim f- l (f(x))> e
- x- - -
and define ' as the reduced sub-
dimxi-l! (x) .:. dimxi,-lf' (x) .:.-dim i (') r ed + dim '
In order to prove that x EX, it is therefore enough to show that
e
-dim f' (')red+ d' .:. e . For this purpJse take Xi such that
dim X' = dim x, . By 3.6 there is a tEX nx, such that f' is flat at t.
- l e l
Hence
dimti.-l!(t) = -dimi(t) i(')r ed+ dimt K '
-dimi(t) i (') red+ dirnX i :::. -d (') red+ d:irrQ';'
On the other hand,
dim Z > e lie in
all the irreducible components Z of f-l(f(t))
 ' so that dimti,-l! (t) = dimtf-l! (t) -> e-
with

r
I,  3, no 6
ALGEBRAIC SOiEMES
119
6.5
Corollary: Let  and J... be locally algebraic schemes OVer
k . Then a) dim X x Y = dim X + dim X
projections
x and Y onto X _ and _ y , then dim X;<. Y = dim X + dim Y
- z- - x- y-
b) if z is a p::>int of x X with
 Since dim  x X = sup dimJC x l' , we need only prove b) . We may
assume that X and X are affine, and by 6.2, that k is algebraically
closed. By 6.1, we may replace z by a p::>int of W. We may accordingly
assume that z is closed (Le. associated with a maximal ideal of O(><. 1'))
By Alg. corrm. V,  3, prop. 1 (iii), it follows that k = K(Z) , whence
k = K (y) . If £: ;<.  --;- X is the canonical Projection, we then have
f-l(y) -+ (X)Xy( K(y)) -+  . Hence by 6.3 dimz><X = dimyX + dim z f-l(y)
dim Y + dim X
y- x-
6.6
Prop::>sition: If X is a locally algebraic k -scheme , then a
point x of X is closed iff the residue field K(X) of x is a finite
extension of k.
Proof: Let 1L be open and affine and contain x, let A be its algebra
of functions and let p be the prime ideal corresp::>nding to x . If K (x)
is finite over k, A/p is a field (A/p CK (x)) , so P is maximal and
x is closed in !I for any !I Conversely, if x is closed, P is maximal.
Since U is algebraic over k, A is a finitely generated k-algebra. The
proof is ccmpleted by applying Alg. ccmn. V,  3, prop. 1 (iii).
6.7
Corollary: If X is a locally algebraic k -scheme , each locally
closed point of X is closed.
6.8 Definition: A p::>int x of a k -scheme X is said to be
rational if its residue field K (x) is identical with k .
Clearly one obtains a bijection of  (k) onto the set of rational p::>ints of
 by assigning to each sE2S (k) the image of the unique p::>int of Spec k
under the rrorphism I s#l: Spec k ->-jxl. Accordingly, we simply write "f>.(k)
for the set of rational p::>ints of X.
6.9 Prop::>sition: If k is algebraically closed , and if  is a
locally algebraic k -scheme , the map PP n(k) is an isanorphism of the

120
ALGEBRAIC GEOME:rRY
I,  3, no 6
lattice of closed sets (resp. open sets , constructible sets ).Qf.. lli.1 onto
the lattice of closed sets (resp. open sets , constructible sets ) of the sub-
space  (k) of X.
Proof:
First consider the case in which P runs through the closed subset
of II
of (k)
. We construct an inverse map by assigning to each closed subset F
its closure F in Ixl . For F = FrqJk) , so it is enough to prove
that P = pnl;(k) if P is closed in II . If P of PrK(k) then there is an
affine open !:! in  such that I!! I meets P but not P n  (k) . Then P
contains a point x which is closed in !! (that is, associated with a
maximal ideal of \9 ({,!) ). By 6.6 and 6.7, it follows that x E!; (k) , a contra-
diction.
The assertion a1:out the lattice of open sets follows fran the a1:ove by passage
to complements. Finally, it is clear that each constructible subset of K(k)
is of the form P nK(k) , where P is a constructible subset of II
Accordingly, we have to show that if P and Q are constructible in K,
then pn(k) = Qn(k) implies P = Q . By setting U=P-Q, or U=Q-P
we reduce the problem to proving that U n"f>.(k) = (jJ implies U=(jJ for con-
structible U. But this holds when U is locally closed, and hence without
restriction on U .
6.10 Remark: Now that we know that under the assumptions of prop.
6.9, the lattices of open sets of Ixi and X(k) are isomorphic, we see
that the theories of sheaves over Ixi and K(k) are equivalent: explicitly,
each sheaf  over IKI is associated with the sheaf r- J over X(k) such
that y-J(U n  (k)) = T (U) where U is open in I  I . Since any rrorphism
of locally algebraic k-schemes sends rational points onto rational points, it
follows that, if k is algebraically closed, the functor  - ((k), 1.9' x)
which is defined on the category of locally algebraic k-schemes and takes its
values in k is fully faithfuL The geanetric spaces of the form
((k), t9') where  is a separated algebraic k-scheme are the algebraic
sets of Serre.
If k is not algebraically cosed, we obtain analogous results by replacing
 (k) by the set of closed points of IK I . Finally, proposition 6.9 remains
true when K is an arbitrary k-scheme, provided one replaces K (k) by the

I,  3, no 6
ALGEBRAIC SCHEMES
121
set of locally closed points of II . Unfortunately, a rrorphism of k-schemes
does not necessarily send locally closed points onto locally closed points:
6.11 Let k again be arbitrary, and let k be the algebraic
closure of k.
Corollary :
A rrorphism of alqebraic k- schemes f:X: 4 ¥. is suriective iff
i (k) : K. 0<) ->- ¥. (k) is suriecti ve .
Proof: Since the canonical projection of ZC'(@kk) onto Z¥. is surjective,
it is enough to show that i@J< is surjective. But this follows fran 3.8
and 6.9

 4
SM:XJ.I'H MJRPHISMS
Section 1
The mcdule of an embedding
1.1
Let i: 6 + I be an embedding of schemes and let '{ be an open
subscheme of .x such that 1: is the cOITl]X)si tion of a closed embedding
j :X+V and the inclusion morphism of Y in I. If j is the kernel of the
rpism }: cJv+j* (J7 x ) induced by i, it is clear that the closed sub-
scheme 'fyj) f Y depends only on 1 and not on y. Accordingly, we
denote this closed subscheme by y 1 and call ¥ J- the first neiqhbourhood
of  (or of i) in . Moreover, !2:¥;i: + denotes the inclusion rrorphism
and ;1,1 : +.x i the morphism such that ;i: = ;i: 2 :h. l . We then r.ave ¥ i = Y j and
1. 1 =1 1 .
The construction of y, is obviously functorial in i. For each ccmnutative
-1:
square
;1.'
:> Y'
(*)
,r
X
1 g
> X
i
such that i and ;i:' are embeddings, there is a unique rrorphism b:¥l +¥1
such that D;i,i = 1 1 :f and 9'2 = 2b . We say that h is induced by L and
'1 ; if 5! (resp. !) is the identity rrorphism, we say that 11 is induced by
:f (resp. ) .
We make analogous definitions for embeddings of k-schemes.
1.2
Lemma: Let k be a ITOdel , i: +Y and j: +1'- embeddinqs of
k- schemes . Then the morphism
U : (Y x T) , . + (Y x T), '
- - -!l x Jl - - J,X 1.
induced by i 2 )< ; 2 : Y. x T. + Y X T is invertible .
__...L -1-']--
Proof: We .i1m1ediately reduce the proof to the case in which I = kA ,
T = B , where !. and i are defined by the canonical projections of A
J

I,  4, no 1
SMXJl'H MJRPHISMS
123
and B onto the quotients A/I and B/J. Then u is induced by the canoni-
cal isanorphism of
(A @ B) / (1 2 @ B + I @ J + A @ J2)
k k k k
onto
(A/I 2 )  (B/J 2 ) / (1/1 2 )  (J/J 2 )
1.3
j* (1/,l)
We now reinstate the notation of 1.1.
The quasicoherent -mcdule
and is denoted by w,
1,
invertible. Also, if
12. such that ') (Q) is a
is called the ITOdule of the embedding i
Clearly the canonical isanorphism If /1 2 -+- j* (w.) is
- 1
each point v E'1 has an affine open neiqhbourhood
finitely generated ideal , then the equation w, =0 implies that 1- is an
!
open embedding . For if x E.;>; and v=j (x) , then (111\ = 0 implies that
'1 = 0 (by Nakayama' s lemma), and 11 U = 0 for a suitable open neighbourhood
v
U of v. The ITOdule w, in a certain sense thus measures the extent to
1-
which the embedding !. fails to be open.
Now let '1' be an open subscheme of X' (1.1) contained in 2,-1 () such
that i' is the canposition of a closed embedding ;j,' :. -+-Y' with the
inclusion morphism of Y' in . . Then 2. induces a rrorphism of y..' into
Y which for simplicity we also denote by 'I. If ":1' is the kernel of
"f ,n " ( .11 ) and l ' f 11 () ' th ' 1 , h ' th
1 -: v v ' -+-1* !;Ix' 0.:.1 -+- V lS e l11C USlon rrorp lsm, en
2* (a) :* (1) -+-;, factors through l' and induces a cCXImUtative diagram
2T2'1)2'r)O
o -----> j ,2__ j, ---+ Ij '/1,2 -->- 0
We shall say that the morphism l'*(r) of i'*CJ*(1/1 2 ) = f*(w i ) into
2'*(1'/1,2) = w" is induced by f and g

1.4
Lemna: If the cCXImUtative square (*) above is Cartesian,
then so is the square

124
AI£EBRAIC GEOME:rRY
I,  4, no 1
, ,
?
Y'
, 1 2
y'
r
Y.
-1
i 2
;> y
and the induced rrorphism i* (Wi) ->- wi' is an epimorphism .
Proof:
(*) is Cartesian iff .<j' is the image of g* (j) in r!J v '
But
if q is an epirrorphism, so are p and r.
With the assumptions of lemma 1.4, !* (wi-) ->-wl is an isanorphism whenever
q is a monanorphism. The latter holds when 3: is flat , and also under the
conditions of lemma 1.6 below.
1.5
Consider the diagram of schemes
, i
zzLxy
-0 - -p--
where f':!:. = 1% ' O is a closed subscheme of E- defined by a -ideal ;;
of vanishing square, and i is the inclusion morphism. By 2, 7 .6b),.:!: is
an embeddin g ; rroreover, Z, = Z and the mcdule w. of j ma y be identified
-J ] -
with :f. Defining 20=<11-, we examine the morphIsms !:->-I such that
f' = <1 and the square
;1
' I'
20 r
;:;
j,
) X
cCI11!!lutes.
Lerrma: The map which assigns to  the morphism go (wi) ->-:f induced by <Io
and ! is a bi 1 ection of the set of i:  ->- X such that £t = 9- and i<1o = i
onto Z (rO (c.J i ) /f) .
-0 -
Proof: For simplicity identify the space of points of ;:;. with that of Xi
by means of ;[,1: ->- Xi . Accordingly if: JJ y . ->- U x is an epirrorphism, with
- f .1111 -J, - f
kernel w, and (pi- 2 )-: vy->-CY y , is a section of 1-i . We may then
  -J,

I,  4, no 1
SM:XJI'H MJRPHISMS
125
identify t?Yi with JJEBW in such a way that 1 and .P2 are defined
respectively-by the canonical projection of Ox EBw i onto c?x and the in-
clusion morphism of t?x in 0XEBWi . This do;;e, ince we ve Z ,=, f
- - - ]
necessarily factors through the first neighbourhood Y i j we agairl write f
for the induced morphism of  in ¥ i . The underlying map of this ! coin-
cides with that of <;1"0 and the hanan;;rphism it:: JXEBW i ->- <;1"0* (z) satisfies
gflt?x = :l. (since p=) and gf(w i ) C O*(:;) (ince <20 =,6) . It
follOws that f is uniquely defined by the morphism  E (W i '<;1"o* (1))
induced by ff. The rrorphism o (w) ->-:; of the lemma is obvi;;usly assigned
to  by the canonical bijection
 (W,<;1"o* q)) :; o (:10 (w j ) ,1)
( 2, 1.2). Since  is arbitrary, the lemma follows.
1.6
Lerma: Consider the diagram of schemes
i
x' --.L..- X --=--->- Y
-
P
such that
P:i =I . Let Y' be the fibre prcxluct X xtf=-' , SL:l' ->-X and
the canonical proiections , and ;i':]<' ->- Y' the morphism with cOITljX) -
and I. . Then each of the squares in the diagram
£':y' ->-"f>.'
nents g
x' J,i ' ' ' . ) X' p' ) X.
f1 1 2 l If
J:C. ., y, ., y :> X
!1 -1 1 2 P
is cartesian and the morphism f* (wi) ->- wi' induced by % and Sf is invert-
ible.
Proof: Observe that, by  2, 7.6b), i is an embedding: the right hand square
is cartesian by construction, and so is the middle one by 1. 4. By reduction
to the category of sets we see that the square with sides f,.2' J. 2 :!:i =!'
and J.21 =  is cartesian. Therefore so is the left-hand one. Identifying
X with Sp (tJLCEBW) as explained in 1,5, we deduce that Xi is

126
ALGEBRAIC GEn1EIRY
I,  4, no 2
 :f*(t9E9W) .+ Sp (u,?;,E9f*(Wi))
and the last part of the lemna follows.
Section 2
The rrodule of differentials
2 .1 Definition: Let .9::  -+ E. be a rrorphism of schemes ; for y
take the fibre prcxluct  x  , for i the diaqonal rrorphism
cS/: X. -+  xf}'f . The quasicoherent J1 -rrodule wi is then denoted by
?/e and called the ITOdule of differentials of .1'; over f}.
When k is a ITOdel and 1'. is a k-scheme, we also write 1'/k for the
rrodule of differentials associated with the structural projection
121': 7£'r. -+ Sp k . We then say that l5/k is the ITOdule of differentials of T
over k.
2.2 ProFOsition: Let .9:: -+ be a morphism of schemes and le t
e: S -+ X be a section of Sl.. Then there is a canonical isanorphism
* (15/e) .+ w
Proof: Apply lemna 1.6 to the diagram
 015/
S
;;. X <::
E:l
."
!;)(S 'f
and to the cartesian squares
S
 1
g
l
g
> X
- l ('Id)
!2!1
X
Ovs.
) Xxx
-s
> X
For example , if  is the spectrum of a field k and x is the image of
2
Spec k under , then * (:1S/) may be identified with m/mx ' where
;J
Ii

I,  4, no 2
m is the maximal ideal of t!1 .
x x
2.3
SMCOl'H MORPHISMS
(**)
q' r
y.
Consider the corrmutative square of schemes
' 1 9
]d
'
J 
We write Y/]d: y* W.?:/) -+ .?:. /E.'
y x1JY : K' xl!'' -+ .?: x (1.3) .
u=Id and  if X' = S'x X
- S ]d - i?- ,
for the morphism induced by ::?;'-+ 
We also write  /g if Y=I, ;!/
'1'=%, and ,y=%
ProFOsition: If the square (**) above is cartesian , the rrorphism
Y/]d: y* (2:;/) -+ , /. is invertible .
We now apply the results of 1.5 to the diagram
°X/S
) X ---=--=+- X)( X
---S-
!2!1 -
Proof: Apply lemna. 1.6 to the diagram
°X/S
x' -L.. X --=--=...- X x X
- --S-
pr l
2.4
o

, Z
g:
127
and
if
We may assume that .9. 0 is a closed subscheme of .9.. defined by a -ideal:;
of vanishing square; set 20 ='i1I . Then with each 1/J E.90 ('i1o (/p)' '(/ )
there is associated a rrorphism of schemes J:  -+ :?; xS such that !2!1! = c::J:
and !i = o?;/l'c::J:O. The first condition rreans that f has -'I as its first
canponent. Denoting the second ca1l]XJnent of f gy .9:. + 1/J , the condition
!i = o£.-/pgO means that 2:2 = (c::J:+1/J) 1. .
Module 1.5, 2+1/J = 2 may be explicitly constructed in the following way.
Set 0 = o£.-/ ; the space of ]XJints of the first neighbourhood (25)( p9 0 may
then be identified with that of  and the structure sheaf of (2:; x Ef-) 0
with 0xE9x/s ' so that 0 1 and l O 2 corresFOnd respectively to the
canonicl p-;:'ojection of JE9 /p. onto t9 x and to the inclusion rrorphism

128
ALGEBRAIC GECMErRY
I,'4,n02
of 19£; in <D(B/!,> .
The rrorphism @'202: o.....£; is then associated with a rrorphism t9"f>..... J1(B/e.
of the form xt----?(x,dx) , where d:t9x....x/S is called the universal deriva-
tion ot\elative to Ei . Under these-conditions, 2: +1jJ has the same under-
lying map as 'J, and 12 +1jJ I!: c!J x .... 2* (z) is the rrorphism
f --
xl->-!2:I-(x) +(dx) , where E  (/...s'2*(1)) is assigned to 1jJ by the
canonical bijections
(/.e'9* (1)) :; 1lf9 9 ('1* (D1</) ,1) :; £o (9"0 ('Vf'),])
We deduce fran 1. 5 the
ProFOsition: Let £     be rrorphisms of schemes and let .£0 be the
closed subscheme of  defined by a £ -ideal :J of vanishinq square . Then
the map 1jJ.-+ 2 +1jJ is a bijection of z (<;):* (DX!S) ) onto the set of
rrorphisms .<J':"" such that g =g and- '1-'1:[10
We leave it to the reader to verify that (<J+1jJ)+ 1/J 1= '1 + (1jJ + 1jJ , ) and [+ 0 = g
2.5 When  and .e. are affine, we may reformulate 2.4 in the
following way: supFOse that f! =  k and  = Sp A , where k,A E!i . We
write A/k for the ITOdule of differentials of the k-algebra A , Le. the
ITOdule of sections of x/S over £; . By definition, we then have
2 --
A/k = 1/1 , where I is the kernel of the k-algebra honanorphism
aiZlb ......a.b of AiZlkA into A . Moreover, the universal derivation of Ox
into /E (2.4) induces a map of A into A/k which we denote by d.
This map d sends a E A onto the residue class mod l of 1 iZI a - a iZll E I
Proposition: The map d:A ....A/k is a k-d erivation . For each k-d erivation D
of A into an A -ITOdule M, there is a unique A- linear map f: A/k.... M
such that D = f 0 d .
Proof: Uniqueness follows fran the observation that since the elements
1 iZI a - a QY 1 obviously generate I , the da generate A/k as an A-ITOdule.
As for existence, write E:AiZlkA.... M for the map aiZlb ....a.Db ; this map
vanishes on 1 2 and induces the required map f on 1/1 2
This last assertion ties up with prop. 2.4 in the following way: Let P :A->C
be a k-algebra hararorphism, J an ideal of C of vanishing square and
I

I,  4, no 2
SMOOTH IDRPHISMS
129
p:C-+C/J the canonical projection. The k-algebra hananorphisms p' :A-->C such
that pp=pp' are then of the form p' = p + D , where D is a k-derivation
of A into the A-ITOdule J.
2.6 The upshot of ProFOsition 2.5 is that (A/k' d) is a solution
of a universal problem. Thus we may identify (A/k' d) with  solution
of the same universal problem. A few examples:
a) Let (Xi) iEI be a family of indeterminates; set A = k[ (Xi) ]
Let M be the free A-module generated by a family (i)iEI and let D:A
be the derivation such that
DP = \' oP . ,
L - l
i (IX,
l
Since (M,D) is obviously a solution of our universal problem, we see that
A/k is the free A-module generated by the dX i = i .
b) Again let A be arbitrary; let (a,) ' EJ be a family of ele-
J J
ments of A and let a be the ideal of A generated by these elements.
Let M be the (Ala) -ITOdule Alk/ (A/k +LjAda j ) and D: A/a.... M the map
derived fran d by passage to the quotient. Then (M,D) is a solution of
the universal problem with respect to Ala and k, so that
 (A/a) /k :; A/k / (A/k +  Ada j ) .
J
c) Consider a multiplicatively closed subset S of A. Let M
-1 -1 - -1 -1 -1
be the SA-module S A/k.... S A @ A A/k and D: SA.... S A/k the map
2
a/s ...... (s (da) - a (ds) ) /s . Then (M,D) is a solution of the universal problem
with respect to S -lA and k, whence
- -1
s-lA/k .... S A/k.
Fran this it follows in particular that for each k-scheme  and each FOint
x of , the stalk
indeed, if we assume
(X/k) x of X/k my be identified with JJx/k :
X -to be affin;;;, 0 may be obtained by localizing the
x
algebra t9)
d) Consider the corrmutative square of rings

130
ALGEBRAIC GEX:M8l'RY
I,  4, no 2
1/J
) B
r j
 e
i r
k

The Iffip a d1/J (a) is a k-derivation of A into B/f and by 25 induces
a k-linear map 1/J/: A/k ->- B/f . In the case where B = A<Si k £ and 1/J,j
are the canonical IffipS, 1jJ/ induces an isanorphism : A/k<Sike ::; B/£
(2.3). Moreover, one verifies directly that the B-ITOdule A/k <Si k f and the
derivation a<SikX ->-da<Sikx form a solution of the universal problem with
respect to B and £.
2.7
ProFOsition: Let k be a ITOdel and let K be a locally
alqebraic k -scheme . Then for each affine open subscheme  Qi K, the
t9 k (y) -ITOdule 2C/k (y) is finitely presented .
Proof: We have /k (y) ::;  ()k (y) /k . Since LDk (Y) is of the form
k[X l ,...,x n ]/(a l ,...,a r ) (3, 1.7), we need only apply 2.6a) and b) to ob-
tain the desired conclusion.
2.8 In Section 4 we shall need a corollary to Prop. 2.7 which we
now prove; namely, let C be a k-ITOdel, J an ideal of C of vanishing
square, p:C-+C/J the canonical projection, and Il E1£(C/J) . Supp:>se that we
are given a partition 1 = t =1 X' f. of unity in C and sonie E; l ' q:; (C f . )
 II l
such that Pf' (E;i) E2S(Cf./Jf.J coincides with the image of Il for each i
l l l
Then there is a E; E2S(C) such that p(E;) =1l .
To prove this assertion, let E;., be the image of E; . in
# lJ l
 = Il * ( X/k and let U. and U.. denote (Sp C) f and
_ -l -lJ -- i
of which are open in Sp C . Since E;,. and E;.. have the
-- lJ Jl
X(C f f /J f f ) , by 2.4 there are 1jJ" E nn:'J. ( I U, "J \ U, ,) such that
- i j i j lJ ;';;';;"-1!ij lJ lJ
E;. = E;. +1/J., . Clearly the family (1jJ. ,) is a l-cocycle of the sheaf
Jl lJ lJ lJ
J"': !:!:..- WI!J,JI!:!:) for the covering (Qi) of Sp C . If we can show that
Hl( (U,), 'F') = 0 , then it will follow that "'.. = "'. I u, , -"', I U" , where
-l lJ l -lJ J -lJ
1jJ. E nn:'J wlu.,Jlu.) . If E; satisfies E;# = d+1jJ. , then we have
l ;;w;..i -l -l l l l l
E;  . = E;  . in the above notation. Since X is a local functor, there is a
lJ Jl -
E; E1£(C) with .iJnage q in (Cfi) for all i. This E; is the required
(Cf'f') ; set
l J
(Sp C) f , f ' , both
-- l J
same image in
j


I,  4, no 2
SMOOTH MJRPHISMS
131
solution.
It therefore remains to show that Hl( (Qi) ,1"') = 0 . By  2, 1.10 and 8.2,
it is enough to show that T is quasicoherent. To do this we choose a cover-
ing of Sp C formed by affine open subschemes .!:! such that the images Il # ()
are contained in affine open subschemes V of X. By 2.7 ,
 (y) :; /k (:{) iJ() (Q)
is then a finitely presented 19(11' -ITOdule. It now follows fran Alg. ccmn. II,
 2, prop. 19(i) that the sheaf
Tly W -rw(n-I, JI)
is quasicoherent.
2.9
ProFOsition: If f:X-rY and .9.:-r are rrorphisms of schemes,
then the sequence
* W'j)
/ /
, / ----L £;/
 0
of  is exact .
Proof: First we reduce the problem to the case in which  = Sp A, r = Sp B ,
E = Sp c, ! = Sp , 2 = Sp 1/J . It is then sufficient to show that, for
each A-ITOdule M , "the sequence
o -+ H WA/B,M) -----+ H WA/C,M) ---->- H (B/C  A, M)
is exact. But this is a consequence of the fact that the sequence
0-----+ Der B (A,M) -----+ Der c (A,M) ---->- Der c (B,M)
is exact, where Der k (K,N) is the ITOdule of derivations of the k-algebra K
with values in the K-ITOdule N.
2.10 Let k be a ITOdel, £;
residue field K (x) . Write 'yk (x)
over K(X) .
a k-scherre, x a pJint of  with
for the vector space W X / k ) x 0 LD K (x)
- X
PropJsition: If X is a locally algebraic scheme over a field k and x
is a point of X, then [/k (x) : K (x) ] ? dimx .

132
ALGEBRAIC GEOMETI'RY
I,  4, no 3
Proof: Let k be an algebraic closure of k and let x be a p:>int
of 'Z = Q\ k with projection x. It follows fran 2.3 that
8/k (x) = /k (x) K t) K (x)
We also know that dim X = dim..x (3, 6.2) . Re p lacin g X b y X and k
x- x- -
by k, we may assume k to be algebraically closed. If the required in-
equality is false, S"2/k will be generated by less than n = dimx sections
over sane open neighbourhocxl U of x. Choose a closed p:>int y in U
which belongs to all the irreducible canp:>nents of  passing through x.
Then we have dimx:S dim  = Kdim  and S"2 X / k (y) -+ m j m 2 . But this contra-
Y2 y - Y
dicts the inequality [mjrry: K(y) ] .2: Kdim {jy (3, 5.11) .
Section 3
Clean rrorphisms
3.1 Cleanness theorem. Let !: -+-":f be a locally finitely presented
rrorphism , x be a ]:Oint of X and let y = i. (x) . The followinq assertions
are then equivalent :
(i)
S"2 /y (x) =  / 0 K(X) = 0
2:C X cJ x
(ii)
There is an open subscheme Q of  containing x such that
the diagonal rrorphism o!'!/i.!:! -+- .!:!)( yQ is an open embedding .
(iii)
There is an open neighbourhocxl JI of x in  which has
the following property : for each scheme E and each pair of rrorphisms
2,!::=tQ , the equations .9:1 red = !:IEred and (flQ)Sl = (!IY)]:1  Sl =.h.
(iv)
For each local ring C , each ideal I Qf.. C of vanishing
square and each ccmnutative square

o
!xr
cf)
y
T 
> c
1jJ
in which  and 1jJ are local harrcrnorphisms , there is at rrost one horocmor-
phism X: r!J -+-C such that 1/J =Xf and can 0 X = .
- x -x -

I,  4, no 3
SMOOTH MJRPHISMS
133
(v)
Ox/my 19 x is a separable finite algebraic extension of K (y)
Proof: We Iffiy assume that  = S!2 A and X = Sp k . Since A/k is a
finitely generated A-ITOdule, the equation S"2/X(x) = 0 is equivalent to
(X/Y) x = 0 , or again, to the existence of an open neighbourhocxl U of x
sch that S"2/.¥ I Q = S"2W = 0
(i)==)(ii) : Replacing  by the open subscheme Q above, we may assume
that A/k = 0 . Our statsnent then follows by 1.3.
(iiJ===:Xiii): Let .!!):E +2 Xyll be the rrorphism having canponents  and E
Then !!lIEred factors through 0!JIY. Since 0U/Y is an open embedding and
Ered has the same underlying space as E , .!!l- factors through U/!,
whence 2 = b. .
(iii)==}(iv): If X,X' are two haranorphisms satisfying assertion (iv) ,
apply (iii) to the pair of canposite rrorphisms
Sp X'
Sp C
E:
 SP Jx2
Sp X
(iv) ' (i):
Let C be the local ring at x of the first neighbourhocxl
o of K in x'vX. Then we have C = t9 x Eb (%/:g)x (2.4); let I = (/.¥)x
hence C/I = Ox and we may set  = Id, 1/J =2x' where ?J:?5 0 + l' is the
canonical rrorphism. The local haranorphisms X'X': c!) ===*C associated with
x
prlo02,pr2002 : Xo=4X then satisfy assertion (iv). Thus we have X =X' .
Since the 0Ix -ITOdule W x / y ) x is generated by the elements dl; = X' (I;) - X (I;) ,
I; E:t.?x (2.5) , it folls- that mx/y)x = 0 .
(v)===} (i) :
By 2.6 we have
(S"2/.¥) x  K (y) ::: S"2 B / K (y)
where B = ,j /m  . If B is a separable finite algebra over K (y) ,
X Y x
B 0 k B is semisimple and the kernel J of the canonical map satis-
fies J 2 =J. Hence S"2 B / k =J/J 2 =0 and (S"2 X / Y )x=O
(i)(v):
Set Y'=K(Y)
-1
and X' =! (y) . Then we have

134
ALGEBRAIC GECMEI'RY
I,  "4, no 3
 /(X) = B/K(Y)  K(X) = '/' (x) ,
with the above notation. Replacing 2:; by X' and X by Y' , we may then
assume that  = Sp k for some field k . By 2.10 , we then have dimx = 0 ,
so that K (x) and t!} are finite algebras over k (3, 6.1 and 6.6). If
x
k is an algebraic closure of k , it remains to show that L1 x <Si k k is a pro-
duct of copies of k . Now, for each FOint
x EX = X x Spk
- - Sp k-
with projection x, we have
Z/k (x) = /k (x) KX) K (x) = 0
Since x is a closed FOint of X, we have K (x) = k . By 2.2, it follows
that 0 = X/k (x) = mx/m ' whence m x = 0 and k = J x
3.2 Definition: Let !:2:;-+Y be a rrorphism of schemes and let x
be a FOint of X. Then ! is said to be clean ( non-ramified) at x if
there is an open neighbourhood Q Q! x such that !I is locally finitely
presented and !¥ (x) = 0 . !t", rroreover , f is flat at x, ! is' said
to be etale at x.
By 3.1 and  3, 3.13, the set of FOints of x at which f is etale (resp.
clean) is open. We say that f is etale (resp. clean) if !. is etale (resp.
clean) at each FOint of  If k Et1 ' we say that a rrorphism i of k-schemes
is etale (resp. clean) if :l is etale (resp. clean). We aiso say that a
k-scheme X is etale (resp. clean) if the structural projection
p: .;s, -+ Sp k is etale (resp. clean). Finally, if A E.£;\, we say that A
is an etale (resp. clean) k-algebra if the k-scheme SPk A is etale (resp.
clean) .
3.3
bf =bg: . If
PrOFOsition: Given a diagram of scheme s
h. is clean , Ker (,9) is open in  .
 b
-+X-+z
2
where
Proof: Ker ( ,9) is the inverse image of the diagonal of X ><z ¥. under the
rrorphism -+ xz'i. with canponents !. and 2... Now apply 3.1(ii).
Corollary: Let E: -+  be a clean rrorphism and  a section of E Then
 is an open embedding .

I,  4, no 4
SMCXJrH IDRPHISMS
135
Proof:  induces an isanorphism of  onto Ker (!:?12, I) .
3.4
ProFOsition:  "f>.JLx Z be a diagram of schemes and x
a FOint of X. If E is etale at x, then the map
(u/)x: *(/)x  (/)x is bijective .
Proof: We may assume without loss of generality that  is etale. Now con-
sider the diagram
° x/z 11 Z 1;1.
X -- XxX
-J l'
- y-
yxy
-z-
!'¥If<
y
Since the right-hand square is cartesian and 11 xz is flat, the morphism
( l)*(¥/)  w incl
induced by  l and  x is invertible (1.4). Since 0'6/'1 is an open
embedding, 0];X (w incl ) may be identified wi th /; the assertion follcws.
Section 4
Srrooth rrorphisms
4.1
Definition : Given a rrorphism of schemes !:    and a FOint
x of  , we say that f is smooth at x when the following conditions
are satisfied:
a) there is an open neighbourhood U of x such that ilQ. is finitely
presented;
b)
c)
f is flat at x;
-1
LlV.'l (x) : K (x) ] os; dimxf (f (x) )
f is said to be smooth if it is smooth at each FOint of X.. If
il
y'
2:'
> r i
) Y
2"

136
ALGEBRAIC GEa>1ErRY
I,  4, no 4
is a cartesian square of schemes and x' is a FOint of X' such that
g' (x') = x , then we have
'/2:" (x') = /};'(x) KX) K(x') .
To see this, we may assume that ,};"Y' are affine; if A = J)() , B = 19 (!:) ,
A' = () (') and B' = (Y') , then we have
::;.
'11' (x') = 0A'/B' :' K(X') ::;. A/B  B', K(X')
::;. A/B  K (x) x) K(x') = 1SI!: (x) KX) K (x')
(2.3). Hence
[, IY' (x')
K(X') ] = [/};'(x)
K (x) ]
and similarly
dim f-l(f(X)) = dim f,-l(f' (x'))
x- - x' - -
( 3, 6.2). It follows from  2, 3.2 that if f is srrooth a t x, then !'
is smooth at x' ; and conversel y, if f' is smooth at x' and <;I is flat
at f' (x') , then f is smooth at x
Given k E£:!. ' we say that a k-scheme X is k-smooth (resp. k-smooth at the
FOint x E) if the structural projection p: '}; ->-  k is smooth (resp.
smooth at x). When there is no risk of confusion, we shall also say "smooth"
instead of "k-srrooth". SUPFOse in particular that k is a field and k' is
an extension of k. The residue fields of FOints
onto x are then the residue fields of FOints of
and 5.7). It follows that, if k' contains K (x)
x'EX@ k' projected
- k
Sp K (x)@k k' (1, 5.2
, then there is a rational
FOint x' E @kk' which is projected onto x. By what we have already said,
it follows that, to verify that  is k- smooth at x we need only verify
that  @k k ' is k' - smooth at the rational FOint x' .
4.2 Let B be a rrodel, A = B[T l ,... ,Tr] the algebra of FOly-
nanials in r indeterminates and X =  A . We know that A/B is a free
A-ITOdule with base dT l ,... ,dTr (2.6) ; accordingly for each x E ' the
images dT i (x) of the dT i . form a base for /B (x) = A/B@A K (x) . If pEA,
write dP (x) for the image of dP in X/B (x) . Then we have
ap
dP(x) =  aT . (x)dT i (x)
l l

I,  4, no 4
SMXmI MJRPHISMS
137
where
EA
aT,
l
ap
and aT. (x) E K (x)
l
Srroothness theorem: Let f:X ->-Y be a locally finitely presented rrorphism of
schemes , x a FOint of  and y = i (x) Consider the fOllowinq assertions:
(i)
f is srrooth at x.
(ii) There is an open neighbourhood U of x, a natural number n
and a rrorphism g:Q->-";!;:X gn which is etale at x and satisfies ilQ = 12!' 1 og .
(iii)
For each local rinq CE1:1 ' each ideal I of C of vanishinq
square and each cormnutative square

I
r2y
) CjI
]=
1/J
> C
where  and 1/J are local haranorphisms , there is a local haranorphism
X: OX ->-C such that 1/J =Xfx and can 0 X =  .
(iv) There are affine open subschemes Q,y of )5.,X such that
f (Q) C Y' , FOlynanials P l ,...,P s E L9(y) [T l , . . . ,T r J and an open embeddinq
h: 11. ->- Y(Pl,...,P s ) such that the matrix (ap/aT j (!2(X)) is of rank s and
! I Q = pr 0 b ( pr being the canonical proj ection of the subscheme y (P l' . . . , P s)
of Xy. Qr defined by P l ,...,P s onto L.
(v)
-
(9 and () are noetherian and LJ x
x - y _
fonnal FOWer series () [[ T l ' . . . , T ]]
Y n
is isorrorphic to an al -
gebra of
Then we have (i) <=> (ii) <=>,(iii) <=> (iv) <= (v)
holds when lJ is noetherian and K (x) = K (y)
Y
The proof of the smoothness thecrem is deferred until Section 5 . In order to
reduce it to manageable proFOrtions we have left out many details: we advise
the reader to approach it only when he is feeling particularly industrious:
The implication (i) =>(v)
4.3
Corollary: The set of points of X at which a rrorphism of
schemes i: ->-X is smooth is open .

138
ALGEBRAIC GEOMEI'RY
I,  4, no 4
Proof: By the equivalence (i)<=> (ii) of 4.2.
4.4 Corollary: Given rrorphisms of schemes f:X -+ Y and c.E -+  ,
if i is smooth at x and -'1 is smooth at !. (x) , then .9f is smooth at
x .
Proof: Apply criterion (ii) of 4.2 and the obvious fact that the composition
of  etale rrorphisms is etale.
4.5
Corollary: Given a locally finitely presented rrorphism of
schemes !: -+Y , the followinq assertions are equivalent:
(i)
(ii)
f is smooth.
For each ITOdel C, each ideal I of C of vanishinq square,
each wE YJC) and each v E JC/I) , g f (v) = wc/l  (C/I) , then there is
uE(C) such that UC/I=V and .f(u) =w.
Proof: Assertion (ii) means that, given a cCXImUtative diagram
#
v

1 i
X
w#
there is a u"4/; C -+ K such that u# = v and !u#=w# . If C is a
local ring and u#,w# send the closed IXJints of Sp (C/I) and  C respec-
tivelyonto x EK and y Ey , then assertion (ii) is equivalent to assertion
(iii) of the SlTOOthness theorem. According to this theorem, then,! is
Sp (C/I)
c
C
smooth iff (ii) holds for each local ring C. In particular (ii => (i) .
Conversely, if f is smooth, the existence of u# is equivalent to the
existence of a section s of i sp C : (C) x f -+ Sp C such that the compo-
sition 20 can : (C/I) -+ (Sp C);; f has canponents  and v# . Re-
placing  by the C-scheme :!' with structural projection i sp C ' it is
enough to show that: if T is a smooth C-scherne and I is an ideal of C
of vanishing square, then the Canonical map T (C) -+! (C/I) is surjective.
To prove this last assertion, let Tl E'E (C/I) . For each prime ideal p of
C , let Tl be the ima g e of Tl in T(C /1 ) . By (iii) of the SlTOOthness
p - p p
,

I,  4, no 4
SMOOTH MORPHISMS
139
theorem, there is a t,pE !(C p ) which is projected onto Ylp. By  3, 4.1,
there is an f EC, f p and a t,fE !(C f ) which is projected onto the image
Yl f of Yl in !(Cf/I f ) . From these f's one can choose a sequence
f l ,...,f such that the (Sp C) f . cover Sp C . The existence of a t, E _ T(C)
n - l _
which is projected onto Yl E!(C/I) now follows from 2.8.
4.6
Corollary: Given a model k and a locally alqebraic k -scheme
 , the fOllowing assertions are equivalent :
(i)  is k- smoo th.
(ii) For each CE  and each ideal I of C of vanishing square,
the canonical map (C) -+  (C/I) is surjective .
4.7
Corollary: Let t:: -+1 be a locally finitely presented rror-
The following assertions are equivalent :
is etale .
is smooth and dim f- l (f (x)) = 0 for each x EX .
x- -
satisfies assertion (ii) of 4.5 and the element u of that
assertion is unique.
phism of schemes.
(i) f
(ii) f
(iii) f
(iv)
For each scheme  , each subscheme ' of  defined by a
nilIXJtent ideal , and each pair (s;r,g') consistinq of mo:r::phisms ..9:: -+1 and
2':'-+ such that .%9:' =21' , there is a unique b:-+2{ such that .2=
and bl '=.2-' .
Proof:
(i) => (ii): This follows immediately fran the definitions.
(ii) => (i): If dim f- l (f (x)) = 0 , then n = 0 in assertion (ii) of the
x- -
smoothness theorem.
(i) =>(iii): The existence of u follows from 4.5; uniqueness follows from
assertion (iii) of the cleanness theorem (3.1): for if u and u' satisfy
the conditions of 4.5(ii), then u#=u,# by 3.1, whence u=u'
(iii) => (iv): If J is the ideal defining Z' , we must show that, for each
cCXImUtative square
y(t)
incl t
y.(1 r + l )
g'
- r
l
2"r+l
7 '.£

140
ALGEBRAIC GID1El'RY
I,  4, no 4
tfr+ 1 r
there is a unique 'r+l: Y(q ) -+  such that 2 = 'r+lIY(t )
fg+l= 'J r +l . We may assume without loss of generality that 1 2 = 0
case, condition (iii) implies that, for each affine open subscheme
and
In this
U of
:9. , there is a unique rrorphism 11;= Q -+  such that .tbu = 21 Q and
b:Y 1g n' =' I n' . If ycg , then ty=.!1gIY by virtue of the uniqueness
of bv. Hence there is a unique £1: E -+  such that 1.\;= b I Q for each U.
This h is the required rrorphism.
(iv) =>(i): (iv) implies (iii), so f is srrooth (4.2) . By 3.1 , we also
have 0,£>/1.= 0 , and (i) follows.
4.8 Corollary: Let k be a ITOdel , k' k -ITOdel which is finitely
generated and projective as a k -ITOdule, and  a locally algebraic k scheme
each finite subset of which is contained in an affine open subset . If  is
smooth over k' , then T\ X i s smooth over k.
- k'/k-
Proof: By the construction of  1, 6.6, !TX is locally algebraic over
- k'/k
k . We must show that, for each C E.t\: and each ideal I of C of vani-
shing square, the map ( I k , A X) (c) -+ (X) (C/I) is surjective (4.6).
But by the definition of the functor Ik'l this map may be identified with
X (C Q9k k') -+ X ( (C/I)0 k k') , which, by 4.6, is surjective.
4.9
Corollary: Let x be a jX)int of a scheme  which is
locally algebraic over a field k . If  is smooth over k at x, then
the local ring l!J is an integral demain.
x
Prcof: By 4.1, there is a rational IXJint x E K Q9k K (x) which is proj ected
onto x. Since (J- is faithfully flat over r!J , the haranorphism iJ -+ tJ-
x /\ X A X x
is injective, and similarly for r!J- -+  . Since - + K(X)[[T l ,....,T n ]]
x x x
is an integral demain, so is &
x
Remark: A similar argument shows that, if  is smooth at x, then the
ring r!J x is regular , Le. is of finite hanological dimension. For since
8- is flat over i!J , we have
x x
"
A l?x 19- '" '"
t!J x - Q9 Tor, (M,tJ) + Tor. x(M <81 i!J_, N Q9 i!J_)
l?x l l JJ x x  x
for all r!J x -ITOdules M,N . If we know that K (x) [[T l' . . . , Tn ] ] has hanological

I,  4, no 4
SMCXJrH MJRPHISMS
141
dimension n, we have
" Ox
rJ- Q9 Tor. (M,N) 0
x t9 l
X
for i >n , whence
Ox 0
Tor. (M,N)
l
for i >n .
4.10 Corollary: Given a reduced locally algebraic scheme over a
perfect field k, the set U of points x EX at which X is smooth is
open and dense in X.
Proof: By 4.3, IT is open. To show that Q is dense in , we may obviously
assume that  = A where A is a finitely generated k-algebra. We now
show that IT is dense, that is, contains the generic IXJints of the irreducible
canponents of . For such a IXJint I; , the transcendence degree of the
residue field K (1;) over k is n =diml;X . Since K (1;) is separable over
k, K (1;) (K(I;) /k' K (1;)) is of dimension n over K (1;) , so that
K (1;) /k is of dimension n over K (1;) (Alg. V,  9, tho 2). In view of the
fact that X is reduced, we have K (1;) = VI; , so that K (1;) /k = (/k); by
Section 2. Hence E is srrooth at 1;.
4.11 Corollary: The prcxluct of tv.D irreducible schemes over an
algebraically closed field is irreducible.
Proof: Assume we have proved the assertion for the prcxluct of tv.D affine
irreducible schemes. Let X and X be tv.D arbitrary irreducible k-schemes;
we have to show that if  and ' are open and non-empty in  xX , then
their intersection is also non-empty. Now obviously there are non-empty
affine open subschemes Q, IT' , IT" of  and y, Y' , V" of Y such
that  meets QX'Y, hI.' meets 1['''-Y' and Q"cQOt and Y"cY0Y..'
Since 11.x Y is irreducible, Tf! 0 (IT.x y) , and hence , meets .Q".x y" ;
similarly Yi' meets 1[" x'::!" . Since 1[" x '{" is irreducible, it follows that
hi. O(!J".x y")n Ii' n (IT")< '{") is non-empty, and similarly for !! O'
We now turn to the affine case. Since we may assume that  and Y are re-
duced, it is enough to show that, if A and B are tv.D k-algebras which
are both integral danains, then the prcxluct AQ9kB is an integral domain.

142
ALGEBRAIC GEOMEI'RY
I,  4, no 4
For this purJ:Ose we may obviously assume that A is finitely generated. By
4.2 (v) and 4.10 there is then a maximal ideal m of A and an isanorphism
A
:A +k[[T l ,...,T]].
m n
/\
If i :A-+-A m is the canonical map (which is injective:), (i) Q\I is an
injection of A@kB into k[[Tl,...,TnJ]@kB, which, as a subring of
B[[T l ,...,T n J] is an integral danain. Hence A@kB is an integral danain.
4.12
COrollary: Let k be a field of characteristic 0 and let
!:  -+-  be a dominant rrorphism of k- schemes which are algebraic, irreducible
and reduced . Then the set of FOints of X at which f is smooth is open
and dense in X.
Proof: Recall that f is said to be daninant if % QP is dense in I  I .
If x and y are the generic FOints of  and X , then i (x) = y . Since
 and X. are reduced, the local rings of x and y coincide with the
residue fields K(X) and K(y) . It follows fran the results of Section 2
that
/x(x) = (/X)x = K(X)/K(Y) .
Since K (x) is a separable finitely generated extension of K (y)
we have
[£C/X= K (x) ] = [K (x) /K (y) : K (x) ] = tr K(y) deg K (x)
(Alg. V,
we have
 9, tho 2). Since K(X) is also the local ring of
. -1
tr ( ) deg K (x) = dim f (y) (3, 6.1), whence
K y x-
] -1
[YX (x) K (x) = dimxf (y) .
f-l( )
- Y
at x,
Since f is flat at x, f is smooth at x . The result now follows fran
4.3.
4.13 Corollary: . Given a rrorphism of schemes .i: -+-X , if f is
smooth at x E , then (/X) x is a free t9 x -ITOdule of rank
dim f- l (f(x))
x- -

I,  4, no 4
SMOOTH MJRPHISMS
143
Proof: With the notation of 4.2(ii), if we set Z = YXOn , then ( / ) )
- - - - :z, X g(x
is a free {) ( ) -ITOdule of rank n (2.6a)) . Since g* ( z/ ) :;. ( / ) -
c;rx - _ ¥ XYx
by 3.4, we see that (X/Y) x is free of rank n. Setting y = ! (x) 
corollary  3, 6.3 applied to the rrorphism g': f -1 (y) -+ On ( ) induced by g
- - -K Y
implies that
dim f -1 ( y ) = dim ' ( ) On ( ) = n ,
x- g X -K Y
and the assertion follows.
4.14 COrollary: Given a ITOdel k, a rrorphism of algebraic
k- schemes i:  -+ X , a JX)int x of  , and y = i (x) , if K- is k -smooth at
x and Y k- smooth at y, then the following assertions are equivalent :
(i)
(ii)
f is smooth at x.
'J'he Iffip X/k (y)@K(y) K (x) -+ 1S/k (x) induced by f is in-
jective.
Proof: (i) => (ii): By 2.9 we have an exact sequence
(X/k) Y y 0J x -+ ("f>./k) x -+ (V.Y) x -+ 0
the functor ? @ i!J K (x) transforms this sequence into the sequence
x
X/k(Y) @ K(X) -+ /k(x) -+ /X(x) -+ 0
K(y) -
The first arrow will then be injective if
[/k (x) : K (x) ] = [/X (x) : K (x) J + ['y/k (y) : K (y) ] .
But if z denotes the canonical image of y in Sp k , the three terms of
this equation are identical with dimX((@kK(Z))) dim x f-l(y) and
dimy(¥'@kK(z)) respectively. The desired equality now follows fran  3, 6.3.
(ii) =>(i): Since
(Vk) Y @ r!J y () x
and
(X/k)x are free x-rrodules of

I
144
ALGEBRAIC GEOMEI'RY
I,  4, no 5
finite rank (4.13), assertion (ii) means that (Y/k)y<3l,j GJ x -+ (Uk)x is
an isanorphism of (Y / k) <31 () {J x onto a direct factor or (X/k) x (Alg.
- Y Y -_
6). Since ( Y/k ) ::;  {l /k and (X/k)x -+ .2 /k (2.6),
_ Y Y - x
.,J into an D. -ITOdule M may be extended to a k-deri-
y x
into M (2.5). This enables us to verify assertion (iii) of
comm. II,  3, prop.
each k-derivation of
vation of ,j
x
the smoothness theorem (4.2): for since  is k-smooth at x, there is a
k-algebra haranorphism X' : () -+C such that canoX' =  (using the notation
x
of 4.2 (iii)). The k-linear map 1j; -X'f is then a k-derivation of
x
I . If D: r!J -+ I is a k-deri vation extending 1j; - X ' f , the map
x x
is a haranorphism satisfying 1j; = Xfx and can.x =  .
() into
Y
X=X'+D
4.15
Remark: The vector space over K (x)
t
/k(x) = K(xf/k(x), K(X))
is sanetimes called the (Zariski) tangent space of  at x. We may rephrase
cor. 4.14 in the case in which K (x) = K (y) by saying that f is smooth at
x iff f induces a surjection of the tangent space at x onto the tangent
space at y.
Section 5
Proof of the smoothness theorem
.A
5.1 /\ (v) => (i): We know that J)x is !lat over 0y if L9x is flat
over cJ (Alg. canm. III,  5, prop. 4). If lJ= rJ /J.m is the local ring
y x x xy
of f- l (y) at x and ill is its maximal ideal, we have
- x
( ) _ _ / _2 _  / 2
 -1 x -+ m x m x -+ m x m x
i (y) /K (y)
(2.2). These isanorphisms and the isanorphism
.A
o ::; K(y)[[T l ,...,T ]]
X n
imply

I,  4, no 5
SMJOTH MORPHISMS
145
[ -1 (x) : K (x) ] = n .
 (y) /K (y)
Likewise,

dimxCl (y) = Kdim 0Ix = Kdim J1 x = n
( 3, 5.14).
5.2 (ii) =>(i): Since g is flat at x and pr l is flat at
g (x) , f is flat at x . 'Ib show that
[ -1 (x) : K (x) ] s; n ,
 (y) /K (y)
-1
we may clearly replace X by ! (y) and .[ by the induced rrorphism
-1
! (y) -+ Sp K (y) . The proof is thus reduced to the case in which .x = Sp K (y)
Setting
X' = Sp K(Y) [Tl,...,Tn] =.x x Qn ,
we deduce fran the canonical isanorphism
2*(X'/K(Y))  /K(Y)
that
[K-l (y) /K (y) (x) : K (x) ] = [I /K (y) : K  (x) )] = n
(3.4). On the other hand, by  3, 6.3 we have
-1
dim X = dim X' + dim g ( g (x) )
x- <J(x)- x--
By 2.10, we have
-1 ]
dimx ((x)) s; [/I (x) :K(X) = 0

146
ALGEBRAIC GEOMEI'RY
I,  4, no 5
whence
dim X = dim X' = n
x- 2(x)-
( 3, 6.1), and the implication follows.
5.3 We na.v lead up to the imj:)lication (i) => (ii) by proving the
folla.ving result: given a field k and a point x of an affine algebraic
scheme , if [X/k (x) : K (x) J ::; dimx , then there is a rrorphism of
k- schemes <1: ->-- which is etale at x. This proves (i) => (ii) in the
particular case in which X = Sp k .
Consider functions f l , . . . , f n E t9 k () such that the canonical images df i (x)
of the differentials df i E X/k () form a base for X/k (x) over K (x) . We
claim that the rrorphism 2:{->-Q with comp:ments f l -:-... ,f n satisfies the
required condition. For let T, be the i th canonical projection of 0 k n
l -
onto O k and let g* (dT,) be the canonical image of dT. E  rYl (O k n ) in
- - l l ""k/k-
g* ( ) (X) . The canonical rrorphism g* (n ) ->-  X/k sends
- 9k/ k - - Qk/k -
g* (dT,) onto df. , so the map g* W ) ->-  /k ) is surjective. Replacing
- l l - o!1 / k x X x
-k -
 by a smaller affine open subscheme, we may assume that 2* Wo!1/k) ->- /k
-k
is an epirrorphism. By 2.9, we then have  =0 and it remains to sha.v
lC/9k
that g
is flat at x.
Tb prove this last assertion, consider an extension K of the field k and
set -g =Q9kK, 2: =2. <3IkK . For sufficiently large extension' K , there is a
rational FOint x EX which is projected onto x EX (l, 5.2 and 5.7). If
9 is flat at x, then g is flat at x , because rl- is flat over r.P
x x
and J) (x) . Since [X / K (x) : K (x) J = [X /k (x) : dx) J by 2.3, and
g - -
dim-X ;; dim X (3, 6.2), we may assume that x is rational. Setting
   
z=g(x) , we then have tJ /m () = k (3.1) . It follows that {} z ->- {) is
_ x z x x
surjective (assign g and @ the;;; -adic filtrations and apply Alg.
z x z
ccmn. III,  2, no 8, cor. 2 of tho 1). Now if z Ek n is of the form
A
(b l ,...,b ) , cY is the ccxrpletion of ,the local ring of k[T l ,...,T J at
n z n
(T l -b l ,...,T -b) , and is hence isamrphic to a ring of fonnal FOWer series.
n n", A /I
In particular, () is an integral danain. If cJ ->- r!J were not bijective,
z z x
then we would have
/' 1\
dim X = Kdim 0' = Kdim oj < Kdim 6' = n ,
x- x x z

I,  4, no 5
SMCDI'H MJRPHISMS 147
contradicting the hYFOthesis
that &x is flat over &z
/\ .II
n ::; dim X . This shows that t2 -+./) and hence
x- z V x
5.4 (i) => (ii): We Iffiy clearly assume that X and Y are affine.
Set A = t? qp , B = t.? () .
First of all suppose that B is noetherian . As in 5.3, choose fl,...,fnE A
in such a way that the images df i (x) of the differentials df i E X/Y ("f>.)
form a base for LC/2: (x) . We claim that the rrorphism 5)':  ->-  >< if' wi ili
canponents f, f l , .. . , f is etale at x. The equation  n (x) = 0 is
- n X/Y x 0 -1
established as in 5.3. Also, by 5.3 applied to the K (y) -schEme -! (y) , the
rrorphism f-l(y) ->- on ( ) induced by g is flat at x Ef-l(y) . In other
- KY _ _
words, if z =g(x) , CJ 1m () is flat over CJ 1m CJ . By Alg. ccmn. III,
- Xyx zyz
 5, no 4, prop. 3, 19 is flat over CJ .
x z
If B is not noetherian , we apply the lerrma of  3, 3.14, setting B =k
and x =p . Using the notation of that lerrma, if f: Sp A ->- Sp k is the
-0 - 0 -0
structural projection and if we set Yo=lo(po) , then we have
f-l(y) -+ f-l( y )@ ( ) K(Y) , whence
- -0 0 K Yo
f-l(Y)/K(Y) (p) -+ !l(Yo)/K(Yo) (PO)KPO)K(P)
d ;m f -l ( ) -1 ) ( R 6 ) -1 )
and 'p_ y = dimp!o (Yo ':f 3, .2. It follows that 10 (Yo is
o
smooth at po. By the remarks above, 10 is smooth at Po; hence there is
an open !J o in  Ao such that Po EQo and an etale rrorphism
o: lJ o ->- ( ko) x Qn such that o l!:T o = pr l oo . We finally set
g=g s: k k.
- -0 -1? 0
5.5 (ii) => (iii): Clearly we Iffiy assume that K and 2: are
affine and that 11 =£[ . If 2:. is etale at x, 51 is etale in a neighbour-
hood of x, so that we may assume that we are given an etale rrorphism
n
2:2:C ->-'i x Q such that ! =EElo . With this sUPfXJsition in force, we prove
the following assertion which obviously implies (iii), namely: for each C E '
each ideal I of C of vanishing square , each w E.?f(C) and each v E(C/I)
such that f(v) =w C / I ' there is sane u E(C) such that u C / I =v and
!:(u) =w (to see that this implies (iii), set w = (Ey( Sp 1/J))/? '
v = (E (Sp "'" and u = (E (Sp X )) b ) .
x v x-

148
ALGEBRAIC y
I,  4, no 5
For the proof of this assertion, supFOse that  (v) = (f(v) ,6 1 ,." ,cn) ,
with C. E C/I . Let C l '...'c be representatives of C l '...,c in C .
1 n n
Replacing f by  and w by (w,c l '... ,c n ) , we see that it is enough to
prove the assertion in the case in which t is etale. In this case, set
y' =  C, y" = Sp (C/I) ,]S' = Y' X0 ' " = X" x and consider the in-
duced diagram
'if
X" > X' w A ) X
i y " 1 fy. 1 # 1 1
Y" > y' w ) y
The rrorphism t: y " has a section ":x" -+X"x 0- with canponents I" and
# -
v . By 3.3 and  2, 7 .6b), there is an open and closed subscheme Z" of "
such that f X " induces an isorrorphism of " onto y" . Now we can identi-
fy the underlying toFOlogical spaces of K" and ' (since we have
t9 (") -+ \. ) /1 J) (') ). So let Z' be the open and closed subscheme of X'
which has the same underlying space as " . Then ly. induces a homeanor-
phism of . onto '!..,' . MJreover, if x'E' and y'= !Y' (x') , the map
,9 ,/I,!} , -+ J . /1,9. is bijective. By Nakayama' s lEmTJa, cJ-. -+ If. is then
y y x x y x
surjective. Since 0. is flat over JJ, , it follows that rJ . -+ rJ. , so
x y Y x
that fy. , induces an isorrorphism of . onto y' . Hence .t y . has a sec-
tion s" and we need onl y set u = (w# s · )
X- P
5.6 Before proving the equivalence of (iii) with (iv), which we
leave until 5.7, we make some prefatory remarks. SUPFOse that K and y are
affine and set A = ,9() , B =JCp . Since A is a finitely presented algebra
over B, we may assume that K is the closed subscheme of X x. (f defined
r
by a finitely generated ideal P , and that .f is induced by pr l :X "Q -+'!...
Let R be the local ring of '£ )( 9 r at x , and R that of K at x .
Setting Q=P x ' we have R=R/Q. With the notation of (iii), the honanor-
phisms f and \ji turn & and C into B-algebras, and the equality
-x x
\ji = X of implies that X is a B-algebra honanorphism. Thus let t l '...' t
x r
be the images of the indeterminates T l ,... ,T r under the CCITI]X>sition
can <p
B[Tl,...,Tr]----'" R/Q C/I

I,  4, no 5
SMOOTH MORPHISMS
149
if tl,...,t r are representatives of tl,...,t r in C, we evidently obtain
a CCXImUtative square of 
R .\ ) C
can 1 <t lean
R/Q ) C/I
such that .\ (T,)= t, i =l,...,r
1 1
Given this .\ , the other rrorphisms '\':R-.)C of  such that cano.\' = .CaIi ,
are of the form .\' = .\ - D , where D is B-linear and satisfies
D(xy) = '\(x)D(y) + .\(y)D(x) .
Denoting the equivalence classes rrod Q of x and y by x and y, we
have
'\(x)D(y) + .\(y)D(x) = (x)D(y) + (y)D(x) .
We can then assign I the (R/Q) -ITOdule structure derived from  and the
canonical (C/I) -ITOdule structure. The conditions imFOsed on D then mean
sirrply that D is a B-derivation of R into the (R/Q) -rrodule I. We now
see that the existence of a rrorphism X:R/Q-+C of  such that canoX = <P
is equivalent to the existence of a derivation D such that D I Q = .\ I Q . We
now reformulate this condition in rrore erudite terms:
1
Let 0: R -+ rJR/B<SiR(R/Q) be the derivation x -+dx<Sil . It follows fran 2.5
that each B-derivation D of R into an (R/C!-rrodule M may be uniquely
expressed in the form D = £0 , where £: /B <SiR (R/Q) -+ M is an (R/Q) _
linear map. M:)reover, since 0 (Q2) = 0, 0 induces an (R/Q)-linear map
j : Q/Q2 -+ RB <SiR (R/Q)
- 2 2
If .\ : Q/Q -+ I denotes the (R/Q) -linear map induced by .\ (.\ (Q ) = 0:) , the
existence of X is then equivalent to the existence of an (R/Q) -linear map
1\ such that I\j =.\ . We deduce fran this that assertion (iii) of the smooth -
ness theorem means that j is an isanorphism of Q/Q2 onto a direct factor
1
of the (R/Q) -ITOdule R/B <SiR (R/Q)
To prove this assertion, notice that, if j is such an isanorphism, there
is obviously an extension 1\ of .\ . The necessity of the condition is proved
by setting C = R/Q2, I = Q/Q2,  = Id and taking for .\ the canonical map
of R onto R/Q2 . Under these conditions, .\ is in fact the identity

"
150
ALGEBRAIC GEOMETRY
I,  4, no 5
2
map of Q/Q
5.7 (iii) <=> (iv): We may assurre that  and  are affine. With
the notation of 5.6, /B @R (R/Q) is a free (R/Q) -ITOdule with base
oTl,...,oT r (2.6). I>breover, if pEPCB[Tl,...,Tr] has image p in Q/Q2,
we have
'r.=, ) - \' 2E.
J IP - L aT, aT i
1
If K(X) is the residue field of R, assertion (iv) of the srroothness
theorem simply JreanS that j@R /Q K (x) sends the generators P l' . . . , P of
2 1 s
Q/Q onto the elements of a base for (R/B @RR!Q) @R/QK (x) . By Alg. carm.
II,  3, prop. 6 (it is unnecessary to assume that M is free in this pro-
FOsi tion), this implies that j is an isanorphism onto a direct factor, and
assertion (iii) follows by 5.6.
Conversely, if j is such an isarorphism, it is enough to chcose P l ,...P EP
s2
in such a way that 1'1'... ,Ps form a minimal system of generators for Q/Q
Then the matrix ((ap/aT j ) (x)) has a rank s. I>brecver, Pl,...,P s forma
minimal system of generators for Q, so that Y(P l ,...,P s ) and V(p) coin-
cide on a neighbourhood of x (3, 4.2). To obtain (iv) we take U to be
this neighbourhood and h to be the inclusion morphism of U in
y (P l' . . . , P s)
5.8 (iii) =>(i): Since (iii)<=> (iv) , we may assume that  = Sp B ,
where B EM , and that  is the closed subscheme of Y '" Qr defined by FOly-
nanials P l ,...,P E B[T l ,...,T ] such that the matrix ((ap./aT,) (x)) has
s n 1 J
rank s. Under these conditions, let Bo be the subring of B generated
Xo = Sp Bo '
Pi ; let
X = Sp B [T l ,...,T ]/(P l '...'P )
-0 - 0 r s
by the coefficients of the
and let x be the projection of x in X . Then ( (ap, faT,) (x )) has
o --Q 1 J 0
rank s and b y 4.1 it suffices to show that the mo rp hism X + Y is smooth
-0 -0
at x . We may accordinqly assume that B is noetherian.
If ( (ap ,/aT ,) (x)) has rank s, some square suhnatrix of order s is in-
1 J .
vertible. Therefore this holds throughout a neighbourhood  of x. Let x'
be a closed FOint belonging to the closure of x in  11!-1 (y) . Since the
set of FOints of  at 'Which ! is smooth is open (by (i) <=> (ii) and 3.2) ,

".,.
I,  4, no 5
SM:DI'H IDRPHISMS
151
it is enough to show that i is smooth at x' . Thus we may assume that x
-1
is closed in ! (y) .
Under these conditions, K (x) is a finite algebraic extension of K (y) . By
lemma 5.10 below, there is then an {) -algebra B' which is noetherian, flat
y
local and such that the residue field B' /n' coincides with K (x) . Applying
 1, 5.2 and 5.7 to f and the canr::osition
E:
Sp B'----? Sp CJ  y
- - y -
we see that there is an x' E  Xy ( B') which is projected onto x and
n' and satisfies K(X') =B'/n' =K(X) . By 4.1 it is enough to show that
f is smooth at x' . This we reduce to the case in which K (x) = K ( y )
- Sp B'
By 5.9 below, we thus have (iii) =>(v) =>(i) .
We now prove that (iii) => (v) 'When f) is noetherian and
- -  Y
Let t l '... , t be a base for m / (m + m ) and let t. be a
n x x y X 1
representative of t, in m . Setting S = 8. [[T l ,...,T ]] , we claim that
1 x /' y n
the continuous haranorphism of tf}-algebras :S+@' such that (T,)=t. ,
Y x 1 1
i =l,...,n , is bijective. For if s is the maximal ideal of S, we have
5.9
K (x)= K (y)
2
S/(s +nyS) '+ K(y)EbK(y)T l <:t...w K (y)T n
and
tfl ; (m 2 + m ) '+ K (y) <B K (y) t i E!:! . . .E!:! K (y)t
x x y x n
hence there is an !J -algebra haranorphism 1/J : J + S/ (s2 +m S) such that
y 0 x y
1/J (to) = T, . By (iii), there is a factoring of 1/J of the form
o 1 1 0
1J!1 2 can 2
t!J ---S/s > sirs +m S)
x y
similarly, 1J!1 factors through a morphism 1/J 2 :  + s/s3 . Continuing in this
way we build a caTlIT!Utative diagram of iJ -algebras
y
(!Jx
02
432 2
S/s  S/s ___ S/s ___S/(s -+ill S)
can can can y

152
ALGEBRAIC GECMErRY
I,4,n05
1
I
.
haranorphism
2
sirs +m S)
y
rrorphism of
By passing to the inverse limit we see that the Ij; induce an rJ -algebra
Any
Ij;: & .... S . By construction 1j;<jJ induces the identity map on
x
since Ij;. is an tJ y -algebra hom:::morphism, 1j;<jJ induces an auto-
/ 2  I 2 / 2
s s .... m m (:bs (s + m S)
y Y
Accordingly, if we assign S the s-adic filtration, Ij; induces an auto-
rrorphism of the graded algebra associated with S . By Alg. carrn. III,  2,
cor. 3 of tho 1, Ij; is an automorphism of S Using the exact sequence
2 2 2n
m 1m .... m /m .... m / (m + m lZ) .... 0
yy xx xxyx
we verify similarly that <jJ1j;
is an automorphism of
,/\
t.O . The claim follows.
x
5.10 Lenma : Let A
L bea field extension of K
residue field is isorrorphic to
be a local ring with residue field K and let
Then there is a flat local A- algebra B whose
L . If A is noetherian and L is a finite
algebraic extension of K, we may take B to be noetherian.
Proof: By well-ordering a set of generators of Lover K, we may confine
our attention to the case in which the extension L of K is generated by
a single element. If L is the field K (T) of rational fractions in one
variable T, set B = A[T ] , where p is the prime ideal of A[TJ con-
p
sisting of all polynanials whose coefficients belong to the maximal ideal m
of A. It therefore remains to consider the case in which L = K [t ] , where
t is algebraic over k and has a minimal polynanial of the form
P = a l + a 2 T+ ... +anr- l +  . If a l ,... ,an are representatives of
a l ,... ,an in A, it is sufficient to set P = a l + a 2 T+...+ an-l+ and
B = A[T ]/PA[T ] . For B is obviously a free A-ITOdule; rroreover, mB is
contained in the radical of B (Alg. ccmn. V,  2, cor. 3 of tho 1) and
B/mB = L. Hence B is local and the lenma is proved.
5.11
Remark: The proof given in 5.9 shows that in assertion (iii)
of the smoothness theorem we may impose further conditions on the ring C .
For example, if k is a field and  is a locally algebraic k -scheme , the
above arguments imply the following result: X is k- smooth iff , for each

I,4,n06
SM:OI'H ffiRPHISMS
153
local k- algebra C such that [C: k ] < + 00 and for each ideal I of C of
vanishing square , the canonical map (C) +(C/I) is surjective (ef. 4.6).
We leave the proof of the following result as an exercise for the reader: if
k is an infinite field and  is a locally algebraic k-scheme, then  is
k-smooth at each of its rational IX'ints iff for each integer n;;: 1 , the
canonical map
(k[TJ/(+l)) + (k[TJ/())
is surjective.
Section 6
Etale schemes over a field
Throughout this section, k denotes a field belonging to M, k denotes
... s
a separable closure of k such that k EM and IT denotes the tOlX'logical
s-
Galois group of the extension k /k .
s
6.1 PrOFOsi tion: A k-scherre  is etale iff the space of FOints
of X is discrete and the local rings of X are all separable finite ex-
tensions of k.
Proof: Clearly any scherre satisfying the latter conditions is etale over k.
Conversely, if  is etale over k, each IX'int x EX is closed (3.1 and
 3, 6.6), so trat each irreducible ccmponent of X reduces to a single IX'int.
Since each affine open subset contains only a finite number of irreducible
canponents, it follows that the underlying space of  is discrete. M::Jre-
over, the local rings r!) are separable finite extensions of k (to see
x
this, set m =0 in 3.1 (v)).
y
6.2
Corollary:
A k-scherre X is etale iff X 0 k k
- s
is a constant
k -scherre.
s-
Proof: Clearly X is etale over k iff  0 kks is etale over k, and a
k -scherre T is etale iff T is constant.
s
6.3
Corollary: If f:X+Y is a smooth morphism of schemes and

154
AlGEBRAIC GECMErRY
I, '4, no 6
y is reduced, then so is X
Proof: By 4.2 (ii) we rray assume that  =  B, x: = Sp A and f is etale.
There is a prcxluct of fields A' E.£1, and an injective horrorrorphism A -+ A I
since Sp (BI8i A A ' ) :;  18ip A' is etale over Sp A' , BI8i A A' is reduced
(6.1). As B is flat over A, B-+BI8i A A' is injective and B is reduced.
6.4 A II -set is by definition a small set E on which II acts
such that, for each x E E , the centralizer {y E II iYx = x} of x in IT is
an open subgroup of II . For instance, if X is a locally algebraic k-scheme,
II acts on X(k) via k If x EX(k) and if x#: Sn k -+ X is the
-s s -s :::Ls-
morphism associated with x, let w be the image under x # of the unique
FOint of Sp k ; then a honanorphism K (w) -+ k is associated with x#
- s s
which enables us to identify the residue field K(W) with a finitely gener-
ated subextension of k . This shows that X (k) is a II-set.
s - s
Corollary: The functor I-+  (k s ) is an equivalence of the full subcategory
of  fonned by the etale k- schemes with the cateqory of II -sets .
Proof: Modulo the characterization of etale schemes formulated in prop. 6.1,
this corollary is nothing more than a variant of Galois theory. Simply observe
that, if K is a finite subextension of ks' ( Sp K) (k s ) is IT/IT' , where
II' is the Galois group of ks over K. Since each II-set E is the direct
sum of II-sets of the form II/IT' , and since the functor K K(k s ) preserves
direct sums, we see that E is of the form X (k) , where X is etale over
- s -
k . We leave the rest of the proof to the reader.
6.5 ProFOsition: Let  be a locally algebraic scheme over a
field k. Then there is an etale k -scheme '11 0 () and a rrorphism
9i -+'11 0 () with the following universal property : for each rrorphism
i:  -+ ;I;;. of  into an etale k- scheme ;I;;., there is a unique 5l: '11 0 () -+].
such that i = 9Si
MJreover , qx is faithfully flat and its filires are the connected comp::m ents
of X (Le. the open subschemes of  whose spaces of FOints are the connect -
ed canponents of II)
Proof: First consider the case in which  = Sp A , where A is a finitely

I,  4, no 6
SMCXYI'H MJRPHISMS
155
generated k-algebra. If we can show that A contains a maximal separable
k-subalgebra As of finite rank, then the rrorphism SPkA -+1\As induced by
the inclusion rrap of As into A has the required universal property. To
prove the existence of A , consider the connected canponents X l '...,X of
s n
the underlying noetherian space of X. Clearly we have A::;' A i /. .. . A ,
- n
where A, = 12 x (X, ) , and the algebras A, cannot be further decoIt1p:)sed.
1 1 1
Accordingly the unit of Ai is its sole non-zero ideITljX)tent, so that each
finitely generated separable subalgebra K of A, is a field. If m, is a
1 1
maximal ideal of A. , it follows that [K:k] s; [A,/m, :k] . This shows that
111
the upward directed system formed by these K has a largest element A,
lS
Now set As = A ls J(...X Ans .
Now sUPFOse that  is arbitrary and consider the diagram of  formed
by the affine open subschemes Q of  and inclusion morphisms
:y -+Q between them. Clearly ,2S rray be identified with the direct limit of
this diagram. N:Jw set 'TTo(Q) = SPk JJ(\})s ; let 9u: Q -+'TTo(g) be the rrorphism
associated with the inclusion rrap of GJ(W s into C7(Q) ; if '{ cQ. , the
restriction (!) (g) -+ tj ('1) sends t!J (t:}) s onto 12 ('1) s and induces the unique
rrorphism ]:'TTo ('1) -+'11 0 (9) such that j0S1y =gy It follows imnediately
fran the construction of direct limits in  ( 1, no. 1) that the direct
limi t of the diagram ( 1'110 (g) I, I j0 1 ) is the geometric realization of an
etale k-scheme. By the ccxrparison theorem ( 1, no. 4 and 6.8), the diagram
('11 0 (Q) ,j) then has a direct limit '110 () in \:.; evidently therrorphism
g: -+'11 0 (,2S) derived fran the rrorphisms 9 11 by passage to e direct limit
has the required universal property ("the left adjoint functor '11 0 commutes
with direct limits").
To prove the final assertion of the pror::osi tion, we observe that the functor
'11 caTlIT!Utes with direct sums, which reduces the problem to the case in which
o
 is connected. Under these conditions the image of 9 is connected and
therefore contains only a single FOint w of '110 () . If Y is the open sub-
scheme of '110 ("f>.) whose only FOint is w and if 'il: -+X denotes the rrorphism
induced by 9x' then clearly (¥ ,q) is a solution of the universal problem
in qu estion, so that y = '11 (X) . Hence q -x is surjective and flat (1J is
- 0 - w
a field!) .
6.6
Definition:
With the ass ump tions of 6.5, we call '11 (X)
- 0-
"the"

156
ALGEBRAIC GID1ErRY
I,  4, no 6
k- scheme of connected canponents of  and g the canonical projection .
6.7
ProFOsition: Given a locally algebraic k-scheme X and a
field extension K/k with KE ' then the unique rrorphism
L : '11 0 (X OK) -r 'TT o (X)0 K
- - k k
such that SIX 0 k K = ;!xSIX@ K is invertible.
- - - k
Proof: Just as before we reduce the problem to the case in which X is
affine. We must then prove the following assertion: if A is a finitely pre-
sented k-algebra and if A k is the largest etale k-subalgebra of A, then
s
A k 0 k K = (A0 k K)K . To prove this, consider the set  of field extensions
s s k L
LE!:&. such that As 0 k L = (A 0 k L) s for each finitely presented k-algebra A
We show that
a) If L is a Galois extension of k, then LE  .
Galois group, then r acts on A0kK and normalizes
VIII,  4, prop. 7, it follows that (A Ok K)  = V 0 k K
Since V is etale over k iff V0 k K is etale over
For if r denotes the
(A0kK) . By Alg.
where % cV cA .
K , we have V=As
b) If ks is a separable closure of k and k an algebraic closure of
ks ' then kE Ek . For in order to prove that
s
k
A s0k
s k
s
(A ° k:)k
k s
s
we may assurre that Spec A is connected. If p is the characteristic ex-
FOnent of k, then each aE A0 k k has sane FOWer a rf1 in A. It follows
s
that the projection
Spec (A ° k) -r Spec A
k
s
is a haneanorphism, so that Spec A Ok k is connected. In other words
ks - k - s
A =k implies (A O k k) = k .
s s s
s
c) If T is an indeterminate , then k (T) E E1< . To prove this, apply the
argument of (a) to the group of autarorphisms r of k (T) of the form

I,  4, no 6
SM::DTH ffiRPHISMS
157
T aT+b
/----;> cT+d
with a,b,c,dE k and ad-be '" 0
d) If KE  and LE E K ' then LE  . This is clear.
e) If k C:K C:L and LE  ' then KE  . This is also clear.
f) If K is the union of an upward directed system of extensions K i E  '
then KE  . Again clear.
It now follows fran a) ,b) ,c) ,d) and e) that each finitely generated extension
belongs to . So by f), every extension has this property.
6.8 Corollary : The following assertions are equivalent for a
locally algebraic k- scheme :
(i)  is geometrically connected ( that is to say , .!0kK is connected
for each extension K of k)
(ii) if ks is a separable closure of k, 0kks is connected ;
(iii) '11 o() is isorrorphic to k k .
6.9 Corollary: If X is a connected locally algebraic k- scheme
which contains a rational FOint , then  is geometrically connected .
Proof: If X is connected, '11 0 (20 is of the form  K for some separable
finite extension K of k. If, in addition,  contains a rational FOint,
there is a ITOrphism kk -+  ' hence a rrorphism Sl?kk -+ kK , so that k = K
6.10 Corollary: Let  and .!': be locally algebraic k -schemes .
Then the canonical rrorphism '11 0 ( X X) -+ '11 0 () X '11 0 (y) is invertible.
Proof: In virtue of 6.5 and 6.7 we may confine our attention to the case in
which k is algebraically closed and  and I are connected. We must show
that X)< Y is connected. Since each open subscheme of  x r contains a
rational FOint ( 3, no 6.), it suffices to show that any t'MJ rational FOints
(x,y) and (x' ,y') belong to the same connected canponent. Now this is
certainly true for (x,y) and (x,y') (which both belong to the connected
subset ( Sp K (x)) X y ::; y) , it is also true for (x,y') and (x' ,y')

158
ALGEBRAIC GEOMEI'RY
I,  4, no 6
(which belong to  x( K (y')) -+ X , and the corollary follows.
6.11 Corollary: . With the assumptions of 6.10, if r is connected
and X is geanetrically connected , then xI is connected .
Proof: By 6.10 and 6.8, we have
'TTo(xI) -+ 'TTo()X'TToCO -+ kkX'TTo(I) -+ 'TTo(I) .

 5
PROPER MJRPHISMS
Section 1
Integral rrorphisms
1.1
Definition: Let !: +I be an affine rrorphism of scherres.
! is said to be integral (resp. finite, finite locally free, of rank n) if ,
-1
for each affine open subscherre ':{  1:, t)C (y)) is an integral algebra
(resp. a finite algebra , a finitely generated projective rrodule , a prolective
rrodule of rank n) over (y) .
If i is finite locally free and if yE X , f* (U) Y is a free V y -rrodule
(Alg. canm. II,S. tho 1). The rank n(y) of this rrodule is locally constant
by Alg. canm. II,S, cor. 2 of prop. 2. Accordingly X can be covered by
closed and open subscherres Y nE IN I such that the ITOrphism
-1 -n
f:f (Y) +Y induced b y f is of rank n for each nE/N
- n - -n -n
When kE 1:1 and 2 is a rrorphism of 1Jc' we say that g is integral
(resp. finite, finite locally free, of rank n) provided 'lP has the same
property.
1.2
ProFOsition: The following assertions are equivalent for an
affine rrorphism of scherres f:X +Y :
(i) .f is inteqral (resp. finite , finite locally free , of rank n) .
(ii) Each point yE I has an affine open neighbourhood V such that
t9(f-l (':{)) is an integral algebra (resp. a finite algebra , a finitely generated
projective rrodule , a projective ITOdule of rank n) over d(':{)
 Clearly (i) => (ii) . Conversely, supFOse that for each yE X we have
a y.. such that O(!-l (y)) is, for instance, an integral algebra over L?(y)
If V' is affine and open in y, we then have
U(!-l(':{,)) :;. LO(f-l(V)) 121 t!J(V')
- - t!J (y) -
so that (J C( 1 (Y' ) ) is also an integral algebra over U ('{') . If :Q is
aff ine and open in y, 11 may then be covered by open subschemes
f . E t!J(U) lin , such that the algebra
1
14:, '
1
(}(f-l CQf.)) :;. 0C(1 Cm) f.
1 1

160
ALGEBRAIC GE:a1!ITRY
I,'  5, no 1
is integral over l?(gf,)::;" 19(!J} f, for each i . If A = tD(W and
,n -1 1 1
xEv(f ()) , it follows that ALx] f , is a finitely generated Af,-ITOdule
1 1
for each i. By. Alg. ccmn. II,  5, cor. to prop. 3, A[x] is finitely
generated over A, hence 19 (! -1 ("(}) ) is integral over l1(W
1.3 Let us say that a rrorphism of schemes .t:....:£ is universally
closed if, for each rrorphism of schemes 2:!:..... I , the canonical proj ection
f : X x Y' .... Y' is closed.
-X' - y- -
proFOsition: An affine rrorphism of schemes is universally closed iff it is
integraL
Proof: This inmediately reduces to the case in which the schemes are affine.
The contention then follows fran:
Lerm1a : For each haranorphism of ITOdels <p :A.... B , the following assertions
are equivalent :
(i) B is an integral algebra over A.
(ii) Sp  : Sp B .... Sp A is universally closed.
(iii) For any indeterminate T, the map Spec  [T ] Spec B[T] .... Spec A[T] .
is closed.
Proof: (i) => (ii). If B is integral over A B0 A A' is integral over
A' . It is then enough to observe that Spec <p is a closed map whenever B
is integral over A (Alg. ccmn. V,  2, no 1, remark 2) .
(ii) =>(iii). This is clear.
(iii) => (i). Let bE B and consider the CamIUtative square
[T]
)B [T]
1 6
> B
A[T]
a l
'
A'
where B' is the localization of B at b, A' is the subring of B'
generated by lib and the image of A, ' is the inclusion map, and a
6 map T onto l/b. Since a and 6 are surjective and Spec  [T ] is
a closed map, so is Spec <P' . Since <p' is injective, the image of Spec B
is dense in Spec A' (l, 2.4), so that Spec <P' is surjective. Since no

I,  5, no 1
PROPER MORPHISMS
161
prime ideal of B' contains l/b, the same holds in A' . It follows that
lib is invertible in A' , that is, b/1EA' . Thus we have the equation
b  (a O ) <p (a l ) <p (an)
-+-+...+-
1 1 b b n
whence
s s-l s-n-l
b =  (aO)b +...+  (an)b
for sufficiently large s.
1.4
Proposition:
If f:X->-Y
is an integral and surjective rror -
phism of scherres, then dim X = dim Y
Proof: We have
dim Y = sup dim Y and dim  = sup dim i -1 (Y)
y. Y..
as y.. runs through the affine open subscherres of y. We may then assume
that K and Y are affine and set A = 19() , B = L?1() ,  = (/(!) . Factoring
A and B by their nilradicals, we may also assurre that A and B have no
non-zero nilFOtent elerrents. Under these conditions  is injective (l,
2.4). We may therefore assurre that B is an integral extension of A. Now
apply  3, 5.2 to Complete the proof.
1.5
ProFOsition:
If a rronanorphism of scherres
f:X->-Y
is a finite
rrorphism , it is a closed embedding.
Proof: By covering  by affine open subscherres i ' and replacing i by
the induced rrorphisms !-l(¥i) ->-¥i ' we reduce this to the case in which
 = Sp A and  = Sp B . Since the diagonal rrorphism oX/X: X ->-  X0 is
an isanorphism, the canonical map A0 B A->-A is invertible. The same then
holds for the canonical map (A/nA)0 B / n (A/nA) ->- A/nA for each maximal ideal
n of B. We then have CA/nA:B/n]2=CA/nA:B/n], so that CA/nA:B/n]=O,l
and so the map B/n ->-A/nA is surjective. By Alg. Ccmtl. II,  3, prop. 11,
B ->- A is also surj ecti ve.
1.6
ProFOsition: Any finite locally free rrorphism is finitely

162
ALGEBRAIC GEOMETRY
I,  5, no 2
presented .
Proof: Since such a rrorphism is affine, the problem reduces to proving the
following assertion: a B-algebra is finitely presented whenever the under-
lying B-ITOdule of A is projective and finitely generated. By  3, 1.4 and
Alg. carm. II,  5, tho 1, this reduces to the case in which A is a free
B-ITOdule with base a l ,...a . SUPFOse then that we have a,a ,= Lb ,an , with
.Q. n l J lv lJ lv
b. .EB . Clearly the kernel of the hom:::morphism : B[Tl,...,T ] -+ A such
lJ n .Q.
that <p (T, ) = a, is the ideal generated by the elements T, T ,- L n b, ,Tn. The
l l l J lv lJ lv
reader will verify that, rrore generally, a B-algebra A is finitely presented
whenever the underlying B-ITOdule of A is finitely presented.
Corollary: A finite locally free rrorphism is closed and open.
Proof: This follows from 1.3, 1.6, and  3, 3.11.
Section 2
The valuation criterion for properness
If kEl1, a k -model V is said to be discretely valued if its underlying
ring is a discrete valuation ring, that is to say, a ring which is principal,
local, and not identical with its field of fractions. If V is discretely
valued, Sl\.V then has exactly two FOints, one open, the other closed.
2.1
Definition: A rrorphism of schemes f:X-+Y is said to be
proper if it is separated, finitely generated and universally closed .
If k E Rl ' a rrorphism 2 of  is said to be proper if 'J;g is proper.
A k-scheme  is called ccmplete if the structural projection P!c:O -+ Sp k
is proper. Notice that any closed embedding is proper.
If f:!c -+:X is a proper morphism of schemes, t:?{:;::; X0 -+ ;::;
is proper for each rrorphism 2:  -+:X . Conversely, if  can be covered by
-1
O pe n subschemes Y. such that the induced ITO rp hisms f.: f (Y.) -+ Y . are
-l -l - -l -l
proper for each i , then !. is proper. This follows easily from  3, 1. 9
and the fact that the rrorphism f.? above is closed if the fiE are closed.
2.2
reover:

I,  5, no 2
PROPER M:JRPHISMS
163
Proposition: Let !:-+!' and 9::-+ be two ITOrphisms of schemes . 
(a) If f and .9: are pr o pe r, so is 9:0£
(b) If 'l<> is p r oper and 5l is se par ated, then f is p r oper .
-
(c) If .20 is proper, f is surjective, and '1 is se par ated and finitely
generated , then <:L is proper .
Proof: (a) follows fran the corresFOnding properties of separated and
finitely generated rrorphisms. Assertion (b) may be proved in the same way
as (c) of  3, 1.10. Finally, (c) becanes clear when one observes that the
assunptions rEfl1ain true after a "change of base" b:T -+  .
2.3 Corollary: Let k be an algebraically closed field and let
X be a ccrnplete, connected and reduced k -scheme . Then, for each k -ITOdel A
&A (XQ9k A ) may be identified with A.
Proof: By the lerrma of  2, 1.8, we have A (X@k:')-+ {(X)@kA . It is then
enough to show that we have k -+ tl k () . Let b: -+Qk be a function on 
and let 2":Qk -+ Sp k be the structural rrorphism; then .2 is separated and
:J0.}J is proper. Were h surjective, then 2. would be proper (prop. 2.2 (c)),
which is false by 1. 3. Since £1 QO is closed (prop. 2.2 (b)) and connected,
b- () is a FOint of Qk . Since  is reduced, 12 then factors through
SP kk , which shows that the map k -+ tOGS) is surjective.
2.4 Lerrma: Let A be a noetherian local integral danain of
dimension  1, m its maximal ideal , K its field of fractions , and L
a finitely generated extension of K. Then there is a discrete valuation v
of L such tha t v(x);, 0 if xE A and v(x) > 0 if xEm .
Proof: Let xo,x l '... ,x n be a set of generators of m. Since Kdim A ;, 1 ,
, n n+l
the graded nng gr (a) = E!:i m /m is not of finite length. Accordingly
2 n
the residue class ITOd m of one of the xi' Xo for instance, is not nil-
FOtent in gr (A) . Hence no relation holds of the form
x-l = P (x O ,xl' . . . ,x n ) , where P is a hanogeneous FOlynanial of degree
r l with coefficients in A . If C is the subring of K generated by A
and xl/x O ,... ,xn/x O ' we therefore have rrC =xOC t-C . It follows that, if
P is a minimal prime ideal of C containing Xo ' we have Kdim C p = 1
and r:C n A = m . If D is the integral closure of C in K , and n
p p

164
ALGEBRAIC GEOMETRY
I,  5, no 2
a maximal ideal of D, then D is a discrete valuation ring of K with
n
maximal ideal nD , such that nD nA =m (Alg. ccmn. VII,  2, prop. 5).
n n
The valuation w associated with Dn is FOsitive on A. Thus one may take
for v any extension of w to L (Alg. Caml. VII,  8, prop. 6 and  10,
prop. 2).
2.5
Lanrna: Let k be a noetherian ITOdel , !:!->-..?{ a ITOrphism of
algebraic k- schemes , x a FOint of  , and Y = i (x) . If y' E{ Y } is a dis-
tinct fran y, then there is a discretely valued k -ITOdel V with field of
fractions L and rrorphisms 2.: SP k V ->-  and b.: SI2J<.L ->-  such that  =.91 SP k L
and 1:1 (kL) = {x} , and that !l maps the closed FOint of SP k V onto y' .
Proof: Let Xl be an affine open neighbourhood of y' in the reduced sub-
schemef '£ carried by {y} . Replacing f by the induced rrorphism
-1
 (Xl) ->-X l ' we may assume that X='£l . Now set A=q,. ; since X is
assumed to be irreducible and reduced, and y is its generic FOint, the field
of fractions K of A is precisely 19. = K (y) . Setting L = K (x) , consider
y
the valuation v of 2.4 and the k-ITOdel V consisting of tE L for which
v(t):;; 0 . It is then sufficient to set h=E(x) (1, 5.2) and to take for
';!: the canposition of Ey' :k . ->- y. (l, 5.7) with the morphism
SP kV->-t9y' induced by the inclusion map of rJ y ' into V .
2.6
Properness theorem: For each noetherian ring k and each
morphism f: X ->- Y of algebraic k- schemes, the following assertions are equi-
valent:
(i) f is universally closed.
(ii) For each discretely valued k -ITOdel V with field of fractions L, the
map  (V) ->- '£ (V) Y(L) 25 (L) with components f (V) and  (incl) is surjective .
Proof: In virtue of the canonical isarorphisms !\!;( SPk L,25)-+ (L) and
!\!£(V,.¥)-+ I(V) , assertion (ii) means that, for each cCXImUtative square
h
(*)
SpkL
can 1
Spkv
5[
) X
l f
> Y

r
I
.
I,  5, no 2 PROPER MJRPHISMS
165
there is an !: SPkV ->-  such that J =! o! and b = o can .
(i) =>(ii): Set  =kV' E' = SP kL and consider the diagram
z " X 2X >- X
- Y - jf
:( If g
z' can ) z j Y
where g and f are the canonical projections and the components of ill
are can and h. The required rrorphisms  are of the form 2"6 0  , where
 is a section of lz such that  = o We now show that, since i z is
closed, such an E exists.
To prove this last assertion, let y be the unique FOint of Z' and set
x =!D(Y) . Since i z is closed, there is an x'E{x} such that f (')= ';{. ,
where y' is the unique closed FOint of  . We then have the following
caTlIT!Utative diagram for the local rings of x, x', y, y':
i!J ( y
x
l incl
:D ('
IX 's
v
where i!J is the local ring of V. at the prime ideal Ker(ay) . If m'
x x
is the maximal ideal of i!J ., 0- 1 (m' ) is the maximal ideal of V; since
x
0- 1 (Ker (ay)) = 0 , we therefore have m'fKer (ay)
(), in L contains V and is distinct fran L Since
x
proper subring of L, ay factors through a retraction
Accordingly the image of
V is a maximal
0: tJx'->-V of 0
The canposi tion
 k O E: .
,n x x
 ;> fu?k v x ' -----'"  ¥ X
yields the required section s.
(ii) => (i): We must show that, for each rrorphism f: ->-X , the canonical pro-
jection f :  Xx  ->-  is closed. To achieve this we assume first that Z
is algebraic over k. By observing that f also satisfies (ii) mutatis

166
ALGEBRAIC GEOMEI'RY
I,  5, no 2
mutandis, we reduce the problem to showing that a rrorphism f: ->-r is closed
whenever it satisfies (ii). Now if xE  , Y =! (x) and y' E{y} , we may
choose the square (*) above in such a way that b-(kL) = {x} and that 2-
sends the closed FOint of Sp V onto y' (2.5). With the above notation, if
-k
x' denotes the inage of the closed FOint of k V under .! , we have
£ (x') = x . This shows that the image of an irreducible closed subset is closed
( 1, 2.10). Since each closed subset is a finite union of irreducilile closed
subsets, the assertion is proved.
Now let  be arbitrary . We must show that, for each closed sub scheme .f of
f! x ' f _ z (E) is a closed subset of 9. By replacing y by the members of
-1
an affine open covering Ci) , and "f>. by the open subschemes .f (r i ) ,
we first reduce the problem to the case in which y is affine with algebra
B . If we now replace  by affine open sub schemes , we further reduce the
problem to the case in which 1- is affine with algebra C. Thus let Co be
a finitely generated B-subalgebra of C, Zo the k-scheme SP kCO '
f'o : .9 x;S ->- f!0 7< yX the ITOrphism induced by the inclusion map of Co into
C , and EO the losed image of PO I£'o . If Q is affine and open in X
and if .P: f! XyhI ->- EOXyQ is the rrorphism induced by Po ' the closed
image f6 of plin(9XyQ) is precisely fol (ox.¥W (2, prop. 4.14).
Since we obviously have
FnU =D( Q)-l(p!})
- - C Po - 0
o
for each U, we see that
n -1
£' = C Po (f 0)
o
Now we have
-1 -1
i z (PO (£'0)) = go (i z Cf O ))
-0
( 1, 5.4), where gO: 9 ->- 90 is induced by the inclusion map of Co into C
-1
Accordingly if! (po (EO)) is a closed subset of E and it is enough to show
that
n -1 -n -1
i z (C f'o (£'0)) - C i z (E'o ('O))
- 0 . 0-
-1
But this follows fran the fact that fz (z) is a noetherian space for each

I,  5, no 2
PROPER MJRPHISMS
167
zEZ , so that
-1 -1 -1-1
n(f (z)npO (£'0)) = t z (z)npO (f O )
for sufficiently large subalgebras Co of C .
2.7 Corollary: If l?: ->-  is a rrorphism of algebraic schemes
over a noetherian rrodel k, the fOllowing assertions are equivalent :
(i) p is a separated rrorphism .
(ii) If V and L are chosen as in theorem 1 , the map
(V) ->-Mi(L)(L) with canponents 12M and (incl) is injective .
Proof: Since the diagonal ITOrphism 0l(/: l( ->-   l( is an embedding, 0l(/ is
a closed embedding iff o/ is proper. Now apply theorem 2.6 to 0l(/.
2.8
Corollary: With the assumptions of theorem 2.6, the following
assertions are equivalent:
(i) .f is proper .
(ii) If V and L are chosen as in theorem 2.6, the map  (V) ->- X (V) Y(L) (L)
is bijective .
2.9
Corollary: If  is an algebraic scheme over a noetherian
ring k, the folling assertions are equivalent :
(i) X is a canplete k- scheme .
(ii) For each discretely valued k -ITOdel V with field of fractions L, the
map (incl):X (V) ->-  (L) is bijective .
Proof: I t is enough to apply cor. 2.8 to the structural rrorphism
.q: il- ->- SPkk , observing that (I\k) (A) is reduced to a single FOint for
each A E 1:\
2.10
Corollary :
The Grassmann functor G is a canplete scheme
-n,r -
QYgf if.
Proof: Apply corollary 2.9; if P is a direct factor of L n + l , pn+l
is a dir ect factor of + 1 (Alg. VII, 4, cor. theorem 1) :
It follows fran cor. 2.10 that G Q9 "k is a crvnnlete k-scheme for each kE'ti.
-n,r " ..""

168
ALGEBRAIC GEDME:I'RY
I,  5, no 3
Section 3
Algebraic curves
Throughout this section, k denotes a field belonging to .tl.
3.1
Definition: An algebraic curve over k is a k- scheme which
is algebraic , irreducible , separated and of dimension 1 . An algebraic curve
over k is said to be regular if the local rings at closed points are dis-
crete valuation rings.
3.2
ProFOsition: Each srrooth algebraic curve over k is regular .
The converse holds if the field k is perfect .
Proof: If an algebraic curve X is srrooth over k, the local ring rj) at
- - x
each closed FOint xE  is an integral danain and has harological dimension
1 ( 4, 4.9). It is therefore a discrete valuation ring (for the ideals of
(j are projective ITOdules, hence free of rank 1). Ccnversely, if & is a
x x
discrete valuation ring, let x be a rational FOint of K<8>kK(x) which is
projected onto x and let t be a uniformizing element of c!J.. . '!hen .J-
x x
is the local ring of t9 x <8>kK(x) at a ITBXimal ideal m and we have
iJ-/tJ- .:; ( <8>K(x))/t(!J <8>K(X)) .:; (K(X)<8>K(X)) .
x x xk x k m k m
If k is perfect, K (x) K (x)
Hence m_ = t D_ and
x x
is semisimple so that {J /t JJ- is a field.
x x
[X <8>k K (x) /K (x) (x) : K (x) ] .:; Cmx/m: K (x) ] :> 1 =d K (x)
It follows that  <8>k K (x) is smooth at x and X is smooth at x.
Remark: Using the "same methcxl" one can show that an algebraic scheme over
a perfect field is srrooth iff its local rings are "regular".
3.3
Given an algebraic curve  over k, we write w () for its
generic FOint and K () for the residue field of w (J . If 1=: -+ X is a
daninant rrorphism of algebraic es, we have f(w()) =wCD . We then write
K(f) : K (Y) -+ K () for the haranorphism induced by f and Qg] (, Y) for the
set of daninant rrorphisms of  into X.

r
I,  5, no 3
PROPER MORPHISMS
169
ProFOsition: Let  and  be algebraic curves over k . If  is reqular
and  is ccrnplete (2.1), the map ! -+K (:f) is a bijection of (,.!;) onto
(K(.!;)),K()) .
Proof: We first show that K (f) = K ('iI) implies! = 'I . If K (%) = K (g) , then
we have
fOE(W()) =E(w(Y))okK() =E(w(.!;))o SP kK(g) =<;1:0E(WW)
( 1, 5.2). Hence kK (w ()) is contained in Ker (!,5!) , which is closed in
X (2, 5.6). Since  is regular, hence reduced, we have Ker (%,<I) = , so
that :f ='I .
Now sUPFOse we are given a haromorphism v: K (y) -+ K () : we construct a g
such that K (';0 = v . For each closed FOint xE  , J7 x is a discrete valu-
ation ring whose field of fractions is K() . Since Y is canplete, by 2.9
there is a rrorphism r: sPk J)x -+:i such that
IK(X) = E(W(X))o (v) .
This rrorphism r has an extension 'Ix: 9x -+  to an affine open neighbour-
hood Qx of x (if r sends the closed FOint onto yE y , apply  3, 4.1
to thE harano rp hism J) -+  induced b y r) . If x X I are distinct
y x - ,
closed FOints of , we have
x x'
'iI llixngx') = <;! I (Qxngx')
in view of the uniqueness property proved a.J::Dve. We then obtain the required
rrorphism <I by "matching together" the rrorphisms { (l, 4.13) .
Remark: By the same methcxls, we can prove the following result: Let X be
a regular algebraic curve and X a canplete scheme over k . Then, for any
non-empty closed subscheme !J of X, the canonical map
(,X) -+ (g,X) is bijective.
3.4 Corollary: g  is a regular canplete algebraic curve over
k , then the autarorphism group of X is antiisorrorphic to the k- autorrorphism
group of K (
3.5
Corollary: If X and Y are regular canplete algebraic

170
ALGEBRAIC GEOMETRY
I,  5, no 3
curves over k,  is isarorphic to X. iff K () is isarorphic to K (:0
3.6 Corollary: If X is a regular ccmplete algebraic curve over
k, X is isarorphic to the projective line £'10 2 k iff K () is a pure
transcendental extension of k
Proof: Setting n = r = 1 in  1, 3.9, we see that the open subschemes !2{l}
and 9{2} defined there are isorrorphic to  [T] ; accordingly !'10 z k con-
tains an open sub scheme !J{l}0:ck isarorphic to kk[T]; it follows that
K() is the field of fractions k(T) of k[T] .
3.7
Theorem on the classification of curves: The functor  ->-K lli)
is an anti-equivalence of the full subcategory of  formed by regular
complete algebraic curves and dominant rrorphisms with the full subcategory of
 formed by finitely generated field extensions of k of transcendence
deqree 1 .
Proof: Since the functor r-+K() is fully faithful (3.3), it is sufficient
to construct, for each finitely generated field extension Klk of transcen-
dence degree 1, a regular complete algebraic curve  such that K () = K .
To this end, let T denote the set of valuation rings V such that kCVCK
and k (V) = K
The members of this set T then consist of the field K
together with sane discrete valuation rings (Alg. ccmn. VI,  10, cor. 1 to
th. 1). We endow T wi th a tOFOlogy by calling a subset open if it is either
empty or it contains K and its complement in T is finite. If U satis-
fies these conditions, we set 0T (U) = V V ; by taking inclusion maps as
restrictions, we thus define a sheaf of k-algebras <'?T. We show that (T,OT)
is the geometric realization of a curve satisfying the required conditions.
Let t be a uniformizing element of a ring VE T different fran K. By Alg.
ccmn. VI,  1, tho 3, lit is transcendental over k . Thus K is algebraic
over k(t) . Let A be the integral closure of k[t] in K. By Alg. carro.
VI,  1, cor. 2 to prop. 3, the rings V' E T such that t,i: V' daninate the
-1 - -1_ [ -1
local ring of k[t ] at the ideal t K t ]. By Alg. ccmn. VI,  8, prop.
2(b), there are only finitely 'many such V' . In other WJrds, the rings VliET
such that tEV" form an open subset U of T. By Alg. carro. VII,  2,
cor. 2 of prop. 5 and tho 1, U is the set of local rings of A. Since for

I,  5, no 3
PROPER MJRPHISMS
171
each sE A , we have A = n (A) = n A , where p runs through the prime
s s Ps p
ideals of A not containing s, we see on the one hand that (U, t)T I U) is
canonically isorrorphic to Spec A , and on the other that the local ring of
U at a FOint V" is precisely VI! . By varying V, we see that (T, t) )
T
is a spectral space. Since the k-algebra A defined above is a finitely
generated k[t]-ITOdule (Alg. ccmn. V,  3, tho 2), (T, c9 T ) is the geometric
realization of an algebraic k-scheme X. We claim that K is the required
curve.
To prove this, observe that since T is irreducible and of dimension 1 , 
has the same properties; we have already seen that K coincides with the
local ring of  at its generic point. It then remains to verify assertion
(ii) of 2.9. With the notation of 2.9, if the image of a rrorphism 2:fu:>kL->-
is a closed FOint x <;Ix ( ()) is isc:xrorphic to K (x) , and is therefore
a subextension of L of finite degree over k (3, 6.5). By Alg. ccmn. VI,
 1, tho 3, g factors through
-x
9Pk V . On the other hand, if x
haranorphism V' ->- V induced by
of CJ to E12kV.
V , so that g has a (unique) extension to
is the generic FOi11t, set V ' = g -1 (V) . The
-x
g defines the (unique) required extension
-x
3.8
Remarks: a): if  is a regular and complete algebraic curve
over k, the proof of the classification theorem yields, with the help of
K() , a description of the geanetric realization of  .
b): Let kE
extension of
sP= aE k and
be a field of characteristic p > 0 and let k ' = k(s,t) be an
2
k of degree p generated by elements sandt such that
t P = bE k . Let K be a field extension of k of transcendence
degree one generated by twJ elements SandT such that
P (S,T)= if - sP+l+aS -b = 0 The kernel of the rrorphism <P :k[S,T] ->-k' of
 such that <P (S)= s and  (T)= t is then the ideal m of k[S,T] gene-
rated by sP- a . Accordingly the local ring V=k[S,T is a discrete
valuation ring and its residue field is k On the other hand, by  4,
2
4.2 (iv), 9E k k[S, T] is srrooth at each FOint xl Qk such that
dP (x) = - (SP (x) - a) dS (x) 'I 0
Le. at each FOint other than m . Thus we see that a regular cat1plete alge-
braic curve X such that K() = K cannot be srrooth. If we set

172
AIGEBRAIC GE'.a-1ETRY
I,  5, no 3
V'=V0 k k' , we also have V' /mV' -+k'0 k k' ; since k'0 k k' is local, so is
V' ; since the maximal ideal of k'0 k k' is not generated by a single element,
V' is not a discrete valuation ring. Accordingly the algebraic curve
X 0 k' over k I is not regular.
- k

"
CHAPTER II
ALGEBRAIC GROOPS
Throughout this chapter k denotes a ITOde1. If AE and XE.t\Ei. ' we
write respectively pl2A and 0(1S) for .fu?kA and I9 k () . We write  for
the k-functor which assigns to each RE the set {e} whose only member
is e.
 1
GROUP SCHEMES
A rronoid is a set together with an associative law of canposition which has
a (necessarily unique) unit elEment which we denote by 0, 1 or e as the
case may be. Given two rronoids M and N, we say that a map f:M-+N is a
haranorphism if it ccmnutes with the laws of canposition and sends the unit
element of M onto the unit element of N (this second condition follows
fran the first if N is a group). A subset N of a rronoid M is said to be
a sulm:moid of M"" (resp. a subgroup of M) if it is stable under the law of
ca:nFOsition and contains the unit of M (resp. if N is a sul::rronoid of M
and if for each xE N , x -1 exists and belongs to N). Each rronoid contains
a largest subgroup, narrely, the set of invertible elements.
For each (not necessarily ccmnutative) ring A, we write A+ for the addi-
tive group of A, AX for the multiplicative rronoid of A, and A* for
the group of invertible elerrents of A
Section 1
Group-functors and group schemes: definitions
1.1 Let X be a k-functor. A law of canFOsi tion on X is a rror-
phism of functors
'11 : XxX ->- X
"f>. - - -
173

174
ALGEBRAIC GROUPS
II, . 1, no 1
It arrounts to the same thing to be given for each RE a law of canposition
'TTX(R) on (R) , such that the maps (): (R)+(S) are haranorphisms.
We say that 'TT is associative if, for each RE  ' 'TT (R) is associative,
that is to say, if the following condition holds:
The diagram
(Ass)
'TTXXI
(X x X) x X - "} X x X
- - - - -
can j X
- Ix'TTX <-
X x ( x ) - -,  >( X / '11 X
is caTlIT!Utative.
We say that 'TTX is a rronoid law if, in addition, each X(R) has a unit ele-
ment E (R) which depends functiorally on R. The family of E (R) 's defines
a rrorphism of functors E: +X which we call the unit section . It follows
that 'TTX is a monoid law iff in addition to (Ass) it satisfies the following
condition:
There is a rrorphism E"f>.: + K such that th
following diagram is ccmnutative:
(Un)
I x EX EX X I
Z'  I V z
K
'TTX is called a group law if each  (R) is a group. The syrrmetrizing opera-
- -1
tion x t-+ x of X (R) into K (R) then depends functoriall y on R and
defines a rrorphism a K : K +X Hence 'TTX is a group law iff it satisfies
(Ass), (Un) and the following condition (Sym):

1:W 1 '
II,  1, no 1
GROUP SCHEMES
175
There is a rrorphism a: + £l such that
the following diagram is ccmnutative:
(Sym)
x
can 1

(Id X ' ax)
- -) )<'
1'TT
> 
E:
A k-rronoid functor (resp. a k-group-functor) is a pair (,'TTx) where  is
a k-functor and 'TT a rronoid (resp. group) law on X. We frequently ccmnit
an abuse of notation by abbreviating this pair sirrply to . We say that
X is caTlIT!Utative if "f>. (R) is corrmutative for each RE t\ ' that is, if the
following axian holds (:  x  +  X  denotes the rrorphism with canFOnents
2 and pr l ):
The diagram
(Can)
XxX
l
- >(
is cCXImUtative
Given a k-rronoid-functor X, we write X or "f>.0PP for the k-rronoid func-
- -opp
tor which assigns to each RE t\ the oPFOsite rronoid of  (R) .
Given two k-rronoid-functors (, 'TT) and (y , '11 y) , we call a haranorphism of
X into x: each rrorphism :g of  into x: for which ! (R) is a rronoid
haranorphism for each RE  i that is, f satisfies the following two con-
iitions (the second being a consequence of the first if x: is a k-group-
:unctor) ;

176
ALGEBRAIC GROUPS
II,  1, no 1
The diagram
fxf
x ) KxX:
(H)
'TT"f>.
'TTX

t
) 1:
is ccmnutative
The diagram
(Han 2 )

c{ f T x
 y
is ccrrrnutative
The k-rronoid-functors form a category which we denote by  . The k-group-
functors form a full subcategory of , which is denoted by S?E k .
If the underlying k-functor of a k-rronoid functor is a k-scheme, we also say
that  is a k-rronoid-scheme or a k-rronoid . The expressions k-group-scheme ,
k-group are defined similarly.
1.2 Given a k-functor  equipped with a law of canposition 'TTX
and a k-functor X, the set (¥,) naturally carries a law of canposition
defined by (t",?!) I->- 'TTx o h where h:¥ +x:  is the rrorphism with canponents
f and <i!. The relation 'TT =P.Il o P.I2 holds in (Xx,) . If 'TTX is a
rronoid (resp. group) law, then E(¥' is a rronoid (resp. group) for each
k-functor ¥; in particular E: is the unit element of the m:moid
(,) (resp. CJ x is the ,inverse of 1% in the group (,) ) .
Similarly, if  and l' are two k-rronoid-functors, and if %:¥ + is a
rrorphism of k-functors, then ! is a hOITOITOrphism iff the following two

II,  1, no 1
GROUP SCHEMES
177
conditions are satisfied:
(HanJ.) In the rronoid  (X x X ,) , we have !O'TTy = (gopX l ). (t: PX2)
(Han;) gOE y is the unit element of t-he rronoid (,9 .
1. 3 The category of k-rronoid functors (resp. k-group-functors) ob-
viously admits inverse limits, and the functors  t-+ (R), RE , ccmnute
with inverse limits. Let us give sane examples.
Given two k-rronoid-functors (resp. k-group-functors) "f>. and X, the functor
 x ¥ is naturally equipped with a rronoid law (resp. group law), namely, the
prcxluct k-rronoid-functor (resp. k-group-functor) ;< X which assigns to
each RE the prcxluct rronoid (resp. group) (R)XX(R) . If we assign 
its unique law of canposition, we obtain a group-functor, sanetimes denoted
by 0 or 1, which is a final object in the categories  and 9Ek.
A subfunctor X of a k-rronoid-functor  is said to be a sub-monoid-functor
(resp. sub-group-functor ) of  if, for each RE  ' X (R) is a suhronoid
(resp. subgroup) of (R) . There is then a unique law of ccxrposition on X
such that the canonical inclusion rrorphism is a haranorphism: it is a rronoid
(resp. group) law. If  is a k-rronoid-scheme (resp. k-group-scheme), we
apply the tenu suhronoid (resp. subgroup) to those subfunctors of  which
are at the same tine sulm:moid-functors (resp. sub-group-functors) and sub-
schemes.
Given a sub-group-functor X of the group-functor "f>., we say that X is
normal (resp. central ) in X if, for each R, ¥ (R) is nonnal (resp.
central) in "f>.(R) .
If f:+y is a hcm:rrorphism f k-rronoid-functors, the kernel of f is the
sub-rronoid-functor Ker ! of  such that
( Ker D (R) = Kerf(R) = {xE(R) : f(R) (x)=l}
for each RE  . If X is a k-group functor Ker f is a nonnal sub-group-
functor of X thus f is a rronc:xrorphism iff Ker g ::;.  .
1.4 A k-rronoid-functor  has a largest sub-group-functor * ;
for RE, * (R) is the set of invertible elements of ]:;(R) . rbreover
if  is a k -scheme , so is * . To prove this, let 1 be the pullback of

178
ALGEBRAIC GROUPS
II,  1, no 1
the diagram
'TTx EX
 x  --=-;.+-=- 
and let j,: 1->- "f>. (resp.;i : 1->- ) be the canposi tion of the canonical pro-
jection E':l->-X)(X with PX1:x->- (resp. with N2:1:X->-£;) ; we thus
obtain a Cartesian diagram
"f>.* y ) r
11 1 j
1 > 
-1 -1
such that p(x)=(x ,x) and (x)=(x,x ) for REJ:\. and xE* (R) , so
that X* is obtained fran K by a pullback construction. This also shows
that if  is affine ( resp. algebraic) so is * .
1. 5 Let k' be a ITOde1. Let .£ be a full subcategory of l,1J:
which is stable under finite prcxlucts, and let F:f ->-£\ be a functor which
conmutes with finite prcxlucts. (Then E £ and F ():::::  ; rroreover if
,  E,£ , then  x IE £, and the canonical rrorphism
i(,) : F(x x.) ->- F() x F(X) , with canponents F(PX 1 ) and F(2) is an
isarorphism). If E £: and if 'TT is a law of canposition on £;, then the
canposite rrorphism
'(XX)-l F('TT X )
F () x F () l -, - ) F (X') - ) F ()
is a law of ccxrposition 'TTF()
(resp. a k-group-functor), then
a k' -group-functor) . If t::  ->- I
on F () . If (2:;,'TT x ) is a k-rronoid-functor
(F(),'TTF(X)) is a k'-rronoid-functor (resp.
is a horocm:>rphism of k-monoid-functors,
and if , E S ' then F (f) : F () ->- F (I) is a haranorphism of k-rronoid-functors.
Let F':£ ->- be a second functor which ccmnutes with finite prcxlucts, and
let h:F ->-F' be a functor rrorphism . Given a k-monoid-functor E f '
h () :F ()->- F' () is then a haranorphism of k-rronoid-functors . We now consider
sane examples of the above construction.
a) Let k I E  . The base-change functor ]E. ->-  ccrrmutes with finite
prcxlucts; it follows that, for each k-rronoid-functor , the k'-functor
0kk' canonically carries the structure of a k'-monoid functor (this may

II,  1, no 1
GROUP SCHEMES
179
also be verified directly fran the formula (@kk') (R) = 1S (kR) ).
b) In the above situation, the functor g:-;>:g: also ccmnutes with
finite prcxlucts (I,  1,6.6); hence, for each k'-rronoid-functor , the
k-functor ;f.Xx has a natural k-rronoid-functor structure.
c) Given a field k, take for .£ the category of algebraic k-schemes and
for F the functor "f>. 1--->- Sp c9() For each algebraic k-group-scheme ,
L9(g) is a k-group-scheme (I,  2, 3.3); by the above, the canonical mor-
phism \)!G: -+  C9(g) (I,  1, 4.3) is a haranorphism.
d) Again supI.XJse that k is a field, take for f the category of locally
algebraic k-schemes, and for F the functor '110 of connected comI.XJnents
(I,  4, 6.6). For each locally algebraic k-group-scheme g, '110 () is a
k-group-scheme and the canonical rrorphism %: -+'11 0 () is a group haranor-
phism. (This example will be treated rrore fully in  2, Sect. 2).
e) Of course, the above constructions are not confined to categories of the
form  . For example, they may be applied to the functor £;.-- which
assigns to each set E the constant k-scheme ; recall that we have
 (,) "'( II ,E) and Sc (,) "'.rE,(k)) for each k-scheme X.
If E is a rronoid, the natural k-rronoid structure on  arises fran the
first bijection: it is a group law whenever E is a group. For each k-rronoid
 , the second bijection induces a bijection of the set of k-rronoid hananor-
phisms  -+  onto the set of rronoid haranorphisms E -+  (k) .
We say that the k-rronoid g is constant if there is a rronoid E and an iso-
rrorphism  '" g . If Spec k is connected, this is equivalent to the canoni-
ca1 rrorphism YG:s(k)k-+ Q (I,  1, 6.10) being an isorrorphism Le. the k-
functor  being a constant scheme.
1.6
Affine rronoids and bialgebras
Let AE.t\. Specifying a law of canposition on Sp A is equivalent to
specifying a k-algebra haranorphism
I':, :A-+A@A.
A k
Accordingly, the axians of 1.1 may be rephrased as follows:

180
(Coass)
(Coun)
(Cosym)
(Cocorn)
ALGEBRAIC GROUPS
The diagram
Id 0to
A 0 A A A) A 0 (A ° A)
Yk k k
A 1 1
 to0Id
L\A A 0 A A A:> (A0A)0A
k k k
is ccmnutative.
There is a k-algebra haranorphism
EA:A->-k such that each of the following
canpositions is the identity :
to A
AA0kA
to A
A--;;>A0 k A
Id 0E
A A> A@k-A
k ->-
E A 0 IdA
> k0 k A-+A
There is a k-algebra haranorphism
:A-A such that the following
diagram is ccmnutative :
to A
> A0A
k
1 Id A ",0 A
A0A
k
yA
k
 A c prcxluct
The diagram
A0A
Y k
A A I j s

A0A
k
in which s (a0b) = b0 a , is ccrnmutative.
II,  1, no 1

II,  1, no 1
GROUP SCHEMES
181
Definition:  k-bialgebra is a pair (A,to A ) , where A is a k -model and
to A :A-+A0 k A is an algebra haranorphism , called the coprcxluct of A, 
satisfies the axians ( Coass ) and ( Coun ). The unique hom:.xrorphism EA:A -+ k
which makes the diagram ( Coun ) cCXImUtative is called the augmentation (or
counit) of A.
A haranorphism of the bialgebra
k-algebra haranorphism f:A -+ B
(f Of) oto A = to B o f and EBof = E A
(A,to A ) into the bialgebra (B,to B )
satisfying the two conditions
is a
In view of the above arguments, we imnediately obtain the following:
ProFOsition: The functor A f->- ePA is an anti-equivalence between the cate-
gory of k -bialgebras and the category of affine k -rronoids . Under this anti -
equivalence the k- bialgebras satisfying (Cosym) (resp. (Cocan), (Cosym) and
(Cocorn)), are associated with the affine k -groups (resp. the cCXImUtative
affine k -rronoids , the ccmnutative affine k -groups ) 
1. 7 Let A be a k-bialgebra,  = Sp A the associated k-ITOnoid,
!! a k-rronoid-functor and fE.!] (A) . By 1.2, a necessary and sufficient con-
dition for the rrorphism f*:t;! -+ which is canonically associated with f to
be a rronoid haranorphism is that the following two requirements be met.
(Han l ) : Consider the three maps to, i l ' i 2 :.!] (A) -+ tl (A 0 k A) induced by the
coprcxluct of A and the inje:tion: il:>-+a01, i 2 :a1-+10a . Then, in the
rronoid B (A 0 k A) , we have to (f) =il (f) . i 2 (f) .
(Han 2 )
E(f)
Consider the map E: !:HA) -+ !i (k)
is the unit element of H(k) .
induced by the augmentation. Then
We immediately deduce the following
LelT11'a: Let  =  A be' an affine k -rronoid . Let i l , i 2 be the maps of A
into A 0 k A defined by i l (x) = x01 and i 2 (x) =10 x . Then:
(i) in the rronoi d (A 0 k A) =.ill\. (A,A0 k A) , we have to=i l ,i 2
(ii) the unit element of the rronoid  (k) =  (A,k) is the auqrnentation
of A
(iii) if G is a group , the involution a A of A is the inverse of
IdA in the group t;! (A) =  (A,A) .

182
ALGEBRAIC GROUPS
II,  1, no 2
1. 8 Given an affine k-rronoid g , we can describe the bialgebra
structure of t!J() = (g,) in the following way:
a) the coprcxluct
to : (g) ->- Lf7()J(g)", (Qx,)
is defined by (M) (x, y) = f (xy) , for fE J!(g) , x, yE g (R) , RE  .
b) the augmentation
E : JJ(Q) ->- k
is defined by E:f = f (e) , where e is the unit element of g (k)
c) if Q is a group, the involution
a : J(Q) ->- J! (g)
is defined by (af) (x) = f (x -1) , for fEJ!(Q) , xE  (R) , REi:\. .
Let H be a closed subscheme of Q. defined by an ideal I of JJ (g) if
RE.t\. and xEQ(R) , we then have the equivalence
xEH(R)<=>{f(x)=O for all fEI}
We irmlediately infer that H is a suJ::m:moid of .Q. iff the following two
conditions are satisfied:
(i) E(I)= 0
(ii) to TIE.pS I into the image of JJ(Q)@kI +I@k(Q) in cO(g)@k J7 (Q) .
(iii) a (I) = I
The bialgebra structure of J!(H)"'(Q)/I is then the quotient of the bialgebra
structure of 19(G)
Section 2 Examples of group schemes
2.1 Groups defined by a k-ITOdule . Let V be a k-ITOdule. Define ThD
conmutative k-group-functor as follows: for each RE:  ' set

II,  1, no 2
GROUP SCHEMES
183
I's (V) (R) = k (V,R)
Ve. (R) = V  R
Of course, V..-.-D (V) is a contravariant functor and V>--"rV is a covariant
a g
functor. They both transform finite direct sums of k-ITOdules into prcxlucts of
k-group-functors.
If k' E  ' we have canonical isanorphisms
12s(Vkl) '" Pa(V)k'
(V@k') '" V @ k'
k a .sk
If RE  ' we have 12g (V) (R) =  (V ,R) '"  (S (V) ,R) where S (V) is the
symnetric algebra of the k-ITOdule V; if V is small, this shows that 12a (V)
is an affine k-scheme isanorphic to Sp S (V) The bialgebra structure of
S (V) is given by lemna 1. 7; the coprcxluct
to : S (V) -+ S (V)@ S (V)
is induced by the diagonal map V -+ V X V , taking account of the canonical
isanorphism S (V x V) '" S (V)@kS (V) ; the involution a is given by the auto-
rrorphism x -+ -x of V, and the augrrentation is the hanorrorphism
E:: S (V) -+ k = S ( 0) associated with the rrorphism V -+ 0 .
Let f:V-+V' be a k-ITOdule haranorphism, and let Qs(f):.s(V')-+Qs.(V) be the
induced haranorphism of k-group-schemes. Then the following conditions are
equivalent: f is surjective, a (f) is a rronanorphism, Qa (f) is a closed
embedding.
If V is projective and finitely generated , then we have a canonical iso-
rro rp hism V '" D () , so that V is an affine algebraic scheme isarorphic
a -a a -
to  S () -. If V I is a suhrod;:;le which is a direct factor of V, then
 -+ I is surjective, and so V' -+V is a closed embedding.
a a
2.2
+
by a (R) = R
The additive group . Write a for the -group-functor defined
for RE M . We then have canonical isaro rp hisms a k '" D (k) '" k
- -a a
- -
and the underying k-functor of a k is the affine line 9x.. We call 
the additive k-group. If T:a k -+Qk is the identity function, the bialgebra

184
ALGEBRAIC GROUPS
II,  1, no 2
of the affine algebraic k-group  is the free cmmutative k-algebra k[T]
we have toT = T @ 1 + 1 @ T, ET = 0, aT = -T. If Q is an affine k-rronoid
with bialgebra t9(Q) = A , the horocm:>rphisms of Q into  are the primitive
elements of A, i.e. the functions xEJ7(Q) such that topf =x@l + l@x .
Now supFOse that k is an algebra over the
We then define an endanorphism F of a k
FE!\ and each xE a (R) = R write rak
t.J,.en have, for RE  ' P
field IF , where p is a prime.
p
by setting Fx = xI' , for each
for the kernel of r :  + a k ; we
rak(R)
p
r
{xE R: x P = 0 }
The k-group-functor rak is an affine algebraic k-group with bialgebra
r p r
k[T] / (-rP) , where we identify T ITOd  with the inclusion rrorphism t
of rak into Qk . We have tot = t@ 1 + 1 @t, Et = 0 ,
p
an affine k-rronoid, the horocm:>rphisms of  into
CJt = -t . If G is
rak are then in one-one
Pr
correspondence with the primitive elements of zero p -th FOWer in the bial- <
gebra t9 (Q)
2.3 The multiplicative group of an algebra . Let A be a k-algebra
(associative, equipped with a unit element, but not necessarily cOIffilutative) .
We define a k-rronoid-functor by assigning to each RE l\ the rronoid
(A@kR))( ; we write I1A for the largest sub-group-functor of this rronoid-
functor. We then have
A
11 (R) = (A@R) * for RE M .
k .....1<
A
If A is a finitely generated projective k-ITOdule , then 11 is an affine
algebraic k-scheme . To prove this, define an element d of
J'(Ai'!) = k\!I(A,) by setting, for each RE,t\ and xEA@kR d(x) = deter-
minant of the R-endanorphism a>-+ ax of A@kR ( Alg. COI1ID. II,  5, exer-
cise 9). Then xEI1 A (R) iff d(x) is invertible, so that I1A is the affine
open subset (Ai'!)d of Aa defined by the function d (ef. also 1.4).
We give sane exarrples of this construction below.

II,  1, no 2
GROUP SCHEMES
185
2.4 The linear group . Let V be a k-ITOdule. For each RE ' let
4(V@kR) be the rronoid of all endc:xrorphisms of the R-mcxlule V@kR Define
a k-monoid-functor 1 (V) by setting
 (V) (R) =.t'R (V R) , RE 
We then get a canonical bijection
J'R(V@R) = (V@R,V@R) '" (V,V@R) .
If we carry over the rronoid law of  (V) (R) to  (V, V @R) we obtain the
following law: if f,gE (V,V@ R) , the prcxluct gf is the canposite of
the diagram
VV@R g@R> V@R@R VI/9m;> V@R
where m is the multiplication in R.
Suppose that V is finitely generated and projective over k then we have
the canonical bijections
(V,V@R) '" (tv@V,R) '" (S("\rQ9V) ,R) ;
it follows from this that (V) is an affine algebraic k-scheme. The iso-
rrorphism S ("\r@V) '" c.0( (V)) obtained above may be explicitly defined in the
following way: for RE ' fE(V) (R) and w@vE"\r@v, the value of the
function w @v at the FOint f is
(w@v) (f).= <wR,f(v R )>
Rerrarks : The preceding argument shows more generally that, if V is a finite-
ly generated projective k-ITOdule and W is a small k-ITOdule, then the k-func-
tor Mod (W,V) such that
Mod (W,V) (R) =  (W@R,V@R) , RE
is an affine k-scheme which is isorrorphic to Sp S ("\r@W)
On the other hand, if V is finitely generated and projective, the canonical
bijections .L'k (V)@R "'LR (V@kR) show that 1(V) is isc:xrorphic to the
k-monoid-functor associated with the k-algebra (V) (2.3) .

186
ALGEBRAIC GROUPS
II, '1, no 2
Now let us return to the general case. The linear group of V, denoted by
 (V) , is the largest sub-group-functor of l;.(V) ; we then have, for REl\ '
GL (V) (R) = (VR)
It follows immediately fran 1.4 and 2.3 that if V is finitely generated and
projective , then GL (V) is an affine algebraic scheme . We can also prove this
directly: define an element f of JJ ( (V) ) by setting f (x) = det (x) for
xE(V) (R) , RE ; we see inmediately that GL (V) is the affine open sub-
functor 1. (V) f of 1. (V) defined by the function f . In particular, we set
n
GL nk = GL (k) and call GL nk the linear k-group of order n; for each
RE ' we have accordingly
nk (R) = GL (n, R)
For each finitely generated projective k-ITOdule, the determinant defines a
group haocrnorphism
det : GL (V) -+ GL lk
whose kernel is denoted by SL (V) and called the special linear k-group of
n
V We set Lnk = SL (k) and call  the special linear k-group of order
n we then have for each R E 
SL nk (R) = SL (n,R)
2.5 S?k -ITOdules . An Ok -ITOdule is a k-functor  with a law of
canposition together with a rrorphism of functors 9kx -+  such that, for
each R E  ' the set  (R) , taken with its law of comp:>si tion and the map
R'>((R)-+(R) , is an R-ITOdule. For example. given a k-ITOdule V, the k-grol1p-
functors V@: and Qi'! (V) of 2.1 are naturally endCJlt.ed with the structure of
an  -ITOdule.
Given two Qk -ITOdules  and , a hono:rorphism of  into  is a rror-
phism of functors  -+!! which induces, for each R E  ' an R-ITOdule horro-
rrorphism of  (R) into  (R) . The set of these hono:rorphisms will be written
9k (,) For example, if V and W are k-ITOdules, we have the evident
maps

II,  1, no 2
GROUP SCHEMES
187
 (V,W) -+ Mod o (V ,W )
;":;;"j( ........ -k 9- i"l.
 (V, W) -+ O (Qa (W) ,Qa (V) )
-k - -
Proposition : The above maps are both bijective .
Prcof : Consider first a diagram of Qk -ITOdules
we have a ccrrmutative diagram
u:V -+ W . For each RE M. ,
- a a 
11 (k)
,> W
1
) wl8i R
V
1
VI8iR
1J (R)
where 1d (k) is k-linear and 1d (R) R-linear. We thus have 1d (R) = 1d (k) 18i k R ,
that is,  = 1,! (k) a . The map v 0--+ va is thus surj ecti ve; it is obviously
injective.
Now consider the second!T'ap: the isanorphism 12a (W):+ Sp S (W) of 2.1 induces
a bijection
 (V,S (W)) = 1;'9: (V) (S (W)):+ !:\m9: (W) ,1;' (V))
which nay be described explicitly in the following way: to y E  (V, S (W) )
is assigned the rrorphism of functors Et(1;'i"l. (W) '!::>i"l. (V)) such that_
U(A)=A*Y where RE ' AE(W,R)=Pa(W) (R) , and A*E(S(W),R) is
n -
defined by A* I W = A . Let A = A* I S (W) and let Y (v) be the canponent
n n
of d egr ee n of y(v), vEV . Clearly (U) =tnA for each tER ; by re-
. n n
placing R by R[T] , it follows that
u (TA) (v) = L TnA Y (v)
nEIN n n
in R[TJ . But if  is a haranorphism of Qk -ITOdules, we also have
u(TA) (V)=T'A Y (v)
- L n n
n
whence A Y (v) = 0 for n of 1 . Taking A to be the inclusion nap of W
nn
into S (W) , we infer that Y n = 0 for n of 1 , hence that u = Qi"l. (Y 1) , and
the proof is ccmplete.

188
ALGEBRAIC GROUPS
II,  i, no 2
Given Qk -ITOdules  and  , if we intrcxluce the functor Mod (M,N) such
that
Mod (M,N) (R) = Modo (M <2>R,N <2>R) ,
''''''''-R k k
we infer fran the pror::osi tion on canoniCal iSOlTOrphisms (2.4, renarks) that
Mod (V,W) -+ Mod (V9'W)
Mod (V,W) -+ Mod(D (W),D (V))
- - "9:
Analogously, if we write  (M) = Mod (M,M) and GL (M) = 1 (M) * for each 0k-ITOd-
ule M, we get canonical isorrorphisms
(Va) -+ 1.(V) , GL (V a ) -+ GL (V)
and
L (D (V)) '" L (V) , GL (D (V)) '" GL (V)
- - - opp - ""3. - opp
The definitions custanarily arployed in the theory of ITOdules extend to 2 k -ITOd-:-
ules. For instance, an 2k -algebra is an 2k -ITOdule  together with a rror-
phism M >< M -+ M which induces, for each R E 1:\ an algebra structure on the
R-ITOdule (R) . In virtue of the proFOsition, the algebra structure on the
<2 k -ITOdule V are in one-one corresr::ondence with the k-algebra structures

on V...
2.6 Autanorphism groups of algebraic structures . 2.4 may be gener-
alized to the case in which one is interested in the endOlTOrphisms of a
k-ITOdule V which carries additional algebraic structure (e.g., the struc-
ture of a not necessarily associative algebra, quadratic form, involution,
etc.). Consider, for example, the case of an algebra. Thus let A be a (not
necessarily associative) k-algebra. Define the sub-rronoid-functor End (A) of
1. (A) by assigning to RE the rronoid of all R-algebra-endOlTOrphisms of
A <2>R also define Aut (A) by Aut (A) = End (A) n GL (A) . If A is a finitely
k
generated projective k-ITOdule, End (A) is a closed subscheme of 1. (A) ,
hence an affine k-scheme. To see this, observe that End (A) is the inverse
image of the zero section of Mod (A <2>A,A) under the rrorphism
1. (A) -+ Mod (A <2> A,A) which assigns to E L (A) (R) the map

II,  1, no 2
GROUP SCHEMES
189
X 0Yt--+ f (xy) -f (x) f (y)
of
A0A0R -+ (A0R)0(A0R)
k k k R k
into A0kR .
Similar arguments apply in the case of a unital algebra. For example, the
k-functor of autarorphisms of the unital k-algebra M (k) is an affine k-
n
group-scheme which we denote by PGL nk and call the projective k-group of
order n.
2.7 The endorrorphism group of a scheme . Given a k-functor , write
EI1'\ () for the rronoid of endorrorphisms of  and AU ("f>.) for the group
of invertible elements of this monoid. Define the k-rronoid-functor End (
and the k-group-functor Aut ( by
End () (R) = En (  R)
Aut () (R) = AU(R) , RE
If 1:; =  A with A E  ' we then have End () = End (A) opp and
Aut(X)= Aut (A) . In par ticular, it follows from 2.6 that End (X) and
---opp --
Aut () are affine algebraic schemes whenever X is a finite locally free
k-scheme.
Observe that the underlying k-functor of End () is precisely H9!!k Q<,2:;)
(I, 2, Sect. 9). If X is a finite locally free k-scheme, by using I,  2,
9.3 we see anew that End () is an affine k-scheme. M:Jreover, it follows
from 1.4 that Aut (2:;) is a scheme (resp. an affine scheme) if End () is a
scheme (resp. an affine scheme) .
For example, set K = Qk . We have
 (Ok) '"  (k[T],k[T]) '" k[T]
the FOlynomial P (T) co=esFOnds to the endorrorphism of Qk which assigns
to RE!:\ and xE R the element P (x) of R. These renarks remain valid
in the rrore general situation in which k is replaced by an arbitrary k-
ITOdel; accordingly we get an iSOlTOrphism of k-functors

190
ALGEBRAIC GROUPS
II,  1, no 2
End (Qk)  (k[T])a
(of course, this iSOlTOrphism does not preserve the rronoid structure:)
ProFOsition: Each autarorphism of the k-scheme .Qk is of the form
xJ-+aO+alx+a2x2+...+anxn , where aO,al,a2,...,anEk,alEk* , and a=O
for i 2: 2 and sufficiently large r
Proof: If P (T) = a O + a l T +. . . + an r def ines an autonorphism of Qk' we
show that a l is invertible and a= 0 for i 2:. 2 and sufficiently large r.
It is enough to show that for each field KE!:J. and each horocmorphism :k+K,
 (a l ) is invertible and  (a i ) = 0 for i 2: 2 . Since  (a O ) +. . .+  (an) r
defines an autarorphism of Qk' we need only consider the case in which k
is a field. If Q (T) is a FOlynanial such that P (Q (T) ) = T , by examining the
terms of P and Q of highest degree we see imnediately that P and Q are
of degree 1 . Conversely, if a O '... ,an satisfy the conditions of the proFO-
sition, then, since the map xf-+-aOal-l+allx is bijective, we may replace
P by -aoa 1+ a lp . We may thus confine our attention to the case in which
a O = 0 and a l = 1 . In this case, we know that there is a fonnal FOWer series
2 3
Q(T)=T+b 2 T +b 3 T +... such that P(Q(T))=Q(P(T))=T. We also have
b.=p, (a 2 ,a 3 ,...,a.) where P. is a FOlynomial with integral coefficients.
1 1 1 1 r r.
If r 2 +2r 3 +...+(i-l)r i is the weight of a rronanial a/...a/, it is easy
to show that P. is isobaric and has weight i-l . It follows that b,= 0
1 1
for sufficiently large i. This ccxrpletes the proof.
Corollary 1 : For a reduced k -scheme K, the rrorphisms %:+ Aut () are of
the following form : there is an aE ur;:n * and a bEe? () such tha t, for
each RE£\ ' each xEX(R) and each yER , we have f(x) (y)=a(x)y+b(x)
Corollary 2: If k is a reduced ring , each autOlTOrphism of the k -group 
is hOlTOthetic, Le. of the form Xl--7-ax for aEk* .
2.8 Diagonalizable groups . Let r be a ccmnutative rronoid. Define a
Z-rronoid-functor by Q(r) (R) = (r,R) j for a group r , 12(r) is a 'If-
group-functor and we have 12 (r) (R)   (r ,R*) .
For each RE!1 we have:

II,  1, no 2
GROUP SCHEMES
191
D(r) (R) = Mon(r,R ) '" An(z[r],R) ,
where z[r] is the algebra of the rronoid r with coefficients in Z . If r
is small, the Z-functor D(r) is thus an affine Z-scheme isanorphic to
Sp z[r] .
Lemma 1.7 enables us to determine :imnediately the bialgebra structure of
Z[rJ.. The maps to: z[rJ->-z[r]@z[r], czlr]->-z and (when r is a group)
a:z[r]->-Z'[r] are defined by to(y)=y@y , E(y)=l , a(y)=y-l for yEr .
Evidently 12 (r X r') '" lJ (r) X 12 (r' ) and we write g< = D qN) , 11 = 12 (2') ,
11 =D (Z/nZ) j we then have, by definition
n -
gX (R) = R
g'" '"  Z[T]
11 (R) = R* ,
11 (R) ={xER:x n =l}
n
11 '" Sp z[T]/(r-l)
n -
-1
11 "'!2E Z[T, T ]
x
Qk "'(k) ,
Ilk '" GL lk .
We call 11 the standard multiplicative group and nil
roots of unity .
the group of
th
n
k
With the notation of 2.3, we have Ilk'" 11 and, if AE '
I1A '" lX I1 A
Notice that the orlly haranorphism of 12 (r) k into a k
phism. For each harrorrorphism corresFOnds to an element
such that tox=x@l+l@x (2.2), hence such that
is the zero haranor-
x = La y of kCr]
y
La y@y = La (y @1 + 1 @y)
y y
which irrmediately irrplies
a =0 , whence x=O .
y
2.9
Characters . Let G be a k-rronoid-functor. An element
f E,o (g) =  (, 9 k ) is called a character of  if f (e) = 1 and
f(xy)= f(x)f(y) for x,yEg(R) and RE . In other words, a character of
 is a haranorphism Q ->- g ; the set of characters of  is the rronoid
 (g,Qk) . It is custanary to write the law of canposition of this rronoid
additively; accordingly we write xf for the value f (x) of the character
f at the element xE (R) , RE ' so that we have the formulas

192
ALGEBRAIC GROUPS
II , . 1, no 2
f f f
(xy) = x y ,
xf+g= xfx g ,
for x,yE.rR) , RE and f,gEr:1£..(') . If  is a group, each char-
acter of  factors through Ilk ' and the monoid of characters of G is a
group isarorphic to k (,l1k) .
If g is affine , the rronoid of characters of  is the multiplicative
rronoid consisting of xE J7() which satisfy tox = x 0x, EX = 1. For example,
set  =  ' so that 19(G) = k [T ] . The characters of a k are then the FOl y-
nanials P (T) which satisfy the formulas P (0) = 1 , P (T+T') = P (T) P (T' )
If k is reduced, it follows that P = 1. In the general case, on the other
hand, a k does have non-trivial characters (cf.  2, 2.6 below).
ProFOsition: For a field k, distinct characters of a k -m:moid-functor are
linearly independent over k
Proof: Let fO,f l ,... ,f n be characters of the k-rronoid-functor . SupFOse
that f l ,.. .,f n are linearly independent, and let
fO = a l f l +...+a n f n ' aiEk ,
be a non-trivial linear relation. Let R,SEJ:\ ' xEg(R) and yE(S) . Then
in R0S we have
fO (x)0 fO (y)
f l (x)0alfO(y)+...+fn(x)0anfO(y) ,
and
fO (x)0 fO (y)
fO(x'y') = a l f l (x'y')+...+anfn(x'y')
f l (x)0a l f l (y)+...+f n (x)0 a n f n (y) ,
where x' ,y' are the :linages of x and Y in (R0S) . If fO is distinct
fran all the f, , it follows that there exist SE  and b, E S , not all
l .....1<. l
zero, such that for each RE  and each xE g (R) we have
f l (x)0b l +.. .+fn (x)0b n = 0
If u: S ->- k is a linear form on S such that u (b. ) are not all zero, it
l
follows that there is a non-trivial linear relation,
u(b l )f l +...+ u(b n ) f n = 0 ,

II,  1, no 2
GROUP SCHEMES
193
a contradiction.
2.10
Cartier duality for finite locally free carmutative groups
a) Let g be a k-rronoid-functor. Define the k-rronoid-functor !? () , the
Cartier dual of , by setting
12 () (R) =!:!e.. (,QR) for RE.t\ .
Define the biduali ty hcm:::irorphism
a G :  ->- 12 (12 () )
as follows: for RE and gE(R), aG(g) is the character of 12()R
- f
which sends fE!?() (S) , SEl% onto f(gs)=gsES. For each k'EA ' we
have Q(G@kk') =.!2(G)@kk' and a G @ k' =aG@kk'
- k
b) Let g be an affine k-rronoid and set A = 0(Q) . Assign the k-ITOdule
t A =  (A,k) the structure of an associative k-algebra by means of the for-
mulas (f.g,a) = (f @g,toAa) , where f,gEtA and aEA j the augmentation
E A of A is the unit element of t A . We call t A the Cartier algebra of
A .
This algebra is related to JJ (!? () ) via the horranorphism
SG: tJ1() ->-J'(Q())
defined as follows: if RE.t\ ' by 2.9 12(G) (R) is the set of
x = LX, @ r, EA@R.+c9(G@R)
l k l k -k
such that
toA@RX = xx
and
EA@R(X) = LEA (xi)r i = 1 j
if yEtt9() , RE.t\ and xE!?( (R) , by definition we set
S G (Y) (x) = (y,x) = h(x,)r.
l l


194
ALGEBRAIC GROUPS
II,  1, no 2
with the above notation. The fomulas
S(Y.O) (x) = (y.o,x)=(y@o,lIx)=(y@o,x@x)=S.(Y) (x).SG(o) (x)
and SG (E A ) (x) =(EA,X)= 1 show that SG is a horrorrorphism of unital algebras .
This horrorrorphism is related to the biduality horrorrorphism defined in a):
given gE(k) , let g:J7(g)-+k be the haranorphism f>-->-f(g) ; for each
RE  and each xE 12 (g) (R) , we have
ag(g) (x)=x(gR)=(g,x)=Sg(g) (x)
whence aG(g)= SG(g) .
c) If g is ccmnutative and affine , the Cartier algebra tLD(g) of Q is
corrmutative. Accordingly, given a hom::xrorphism SG:\?()-+L9(.!2(S;)) , we have
the canonically associated rrorphism
t
Y G : P(g) -+ Sp lQ(g)
which is defined explicitly as follows: if RE and xE.!2(S;) (R) ,
t
YG(x): cP(g)-+R is the hom:::m::>rphism yl-r(Y,x) (I,  1, 4.3).
If, in addition, 9 is fini te locally free , each k-linear map t £!7(g) -+ R is
of the form y -+(y,x) , where xE A@kR :+LD(@kR) . Such a linear map is a
horrorrorphism of unital k-algebras iff tox=x0x and Ex=l (ef. the formulas
of b) which show that SG is a haranorphism). This means that YG is an
isarorphism .
d) If 9 is ccmnutative and finite locally free, we have a canonical iso-
rrorphism tA@t A :+ t(A@A) ; write t m for the canposite map
t A -+ t(A@A) :+ tA@
obtained by transFOsition fran the multiplication m:A0A-+A. Similarly,
t 11: t A -+ k denotes the map Y I--+y (1) . fvbreover, since 12 (g) is affine by b) ,
l!)(.!2 (g) ) is a bialgebra whose coprcxluct and augmentation we denote sirrply by
to and E . For yE\!J(g), RE.t\ and x,yE!2()(R) , we have
to (S (y)) (x,y)
SG (y) (x,y)
- t t
(y,x.y) = ( m(y) ,x@y) = (Sg0 S g ) ( m(y) (x,y))

II,  1, no 2
GROUP SCHEMES
195
so that /',(BG(Y))=(BG@B G ) (tm(y)) ; similarly
t
E(B(Y))= B(Y) (l)=y(l)= ll(Y) .
It follows that t m and
bra structure over t L9(g)
t ll are the coprcxluct and augmentation of a bialge-
with respect to this structure B: t L9 (g)-+.Q (r.O(5,?))
is a bialgebra isomorphism.
Fran this we deduce the
ProFOsition: If G is a k -rronoid-scheme which is cCXImUtative and finite
locally free , so is Q() and the biduality haranorphism aG.:<:i-+.Q(Q()) is
an isomorphism .
Prcof: Since a G @ k,=aG@kk' for each k'E ' it remains to show that
aG. (k) is bijective Now-if gE  (k) , we have 0. 9 (k) (g) =B @ by b) i thus
it is enough to show that BG induces a bijection between the haranorphic
maps yEtJ!() and the oEt.O(()) such that M=o@o and Eo=l . Nowy
is a hcm:::xrorphism provided (y,l)=t ll (y)= 1 and (y,x.y)=(y,x)(y,y) for all
x,yE LD() this last equation is equivalent to
hence to
(tmy,x@y) = (y@y,x@y)
tmy =y@y . The assertion now follows fran the fact that B
is a
bialgebra isomorphi'kn.
Remark: If  is a k-rronoid which is corrmutative and finite locally free,
we have the conmutative square
tt!i() BDG ,;>Ji(Q () )
t so 1 1 11 (OG '
tt(J(g) can ) 0':G) ,
which yields a relation between the biduality hcm:::xrorphism a G and the
canonical isomorphism 19(G.)-+ttl9() . To show that this square utes, we
observe that all the relevant maps are algebra hcm:::xrorphisms. By applying an
extension of scalars, we reduce the problem to showing that for each

196
ALGEBRAIC GROUPS
II,  1, no 2
rrorphism : JI() -+ k of 1:\ we have
n -1 t
LY(ag) SIJQ =  (can ) Sg
Assuming the notation of b), if we have  =g with gE  (k) , this last
assertion is a consequence of the fact that the following diagram conmutes:
can
t a
) ttcQ(G) -< G
- 
\()
Sw
  (P (gG) )
r.J(a )
-3 :1 ()
L9(G)
g
g
where, for each k-ITOdule M and each mE M , m' denotes the canonical image
of m in t. (To prove that S(g) '=Vg)SW ' observe that SG(g)=G(g)
and 0.'= aS tl for each affine k-group tl and each 0< E l? (tI) (k) C t9(£!)) -
2 .11 Duality for diagonalizable groups . Let r and r ' be caTIIT!U-
tative rronoids. We shall determine the hom::m:Jrphisms 12. (r ' ) k -+ Q (f) k . First
of all observe that the rrorphisms of functors !:p(r'\ -+.!2(f)k corresFOnd
x
to the rronoid haranorphisms g:r -+k[r' ] . For f to be a hcmanorphism, it is
necessary and sufficient that g give rise to a bialgebra haranorphism
k[r] -+k[r'] , Le. that the following two conditions be satisfied:
tog(y) = g(y)0g(y) }
for y E r
Eg(y) = 1
If g (y) = L a (n) n , these conditions may be written:
nEr ·
a(n)a(n')= 0 for n1n' ,
2
a(n) =a(n) ,
La(n)= 1
These conditions determine a continuous map Speck -+ r ' (which takes the
value n on the closed and.open subset (Speck) a(n)) , Le. an element of
r ' k (k) It follows that we have canonical rronoid isorrorphisms
(l?(r')k,Q(r)k)  (r,rk(k))  (rk,rk) .

II,  1, no 2
GROUP SCHEMES
197
In particular, we get a canonical isarorphism
x
((f)k,gk)  fk(k)
and so the elements of f may be identified with characters of 12 (r) k for
RE ' yEf , gE(f) (R)=(f,R)() , we have gY=g(y)
If Speck is connected , we deduce the existence of an isorrorphism (I,  1,
6.10) :
(Q(f')k,(f)k)  (f,f') .
In particular, the rronoid of characters of 12 (r) k may be naturally identified
with f . Calling a k-rronoid which is isarorphic to sane 12 (r) k a diagonaliz-
able k-rronoid, we infer the:
ProFOsition: Suppose that Speck is connected . Then the functors
f r-+ 12 (r) k and g >--+ 12 (G) (k)
are quasi-inverse antiequivalences between the category of small corrmutative
rronoids and the cateqory of diagonalizable k -rronoids . These antiequivalences
associate finitely generated ccmnutative rronoids with diagonalizable algebra-
ic k -rronoids , and small cCXImUtative groups with diagonalizable k -groups.
In the general case, for RE£:\ we have isanorphisms
!2(f k ) (R)=( fR,g;)  (f ,g(R) )=12(f) (R)
accordingly there is a canonical isanorphism 12. (f k)  12. (f) k .
Fran the above results we infer the
ProFOsition: The functor G>---+D(G) is an antiequivalence between the cate-
gory of diagonalizable k -rronoids and the category of constant k -rronoids ; and
t->-12(G) is a quasi-inverse functor .
2.12 Boolean groups . Let f be a small Boolean toFOlogical group,
Le. a small topological group with a base of ccxrpact open sets. We know
(I,  1, 6.9) that the k-functor f k defined by f k (R)= :E5?J?(SpecR,r) is a
scheme. Fran this description it is clear that f k naturally carries the

'ill
198
ALGEBRAIC GROUPS
II,  1, no 3
structure of a k-group functor.
For instance, if r is a profinite toFOlogical group, the inverse limit of
finite discrete groups r i ' we have
r k (R) = Top (SpeeR, f) = lim Ton (SpecR, r .)
........ = 1
=(r i)k (R) ,
so that the k-group-functor r k is the inverse limit of the constant k-groups
(r i)k .
Section 3
Action of a k-group on a k-scherre
3.1 Definition: Given a k -rronoid-functor g and a k-functor X,
a (left) operation of g on X ( or simply a (left ) G-operation on ) is
a rrorphism of functors
1d:9)<->-K
such that , for each RE ' we have 1d(g,1d(g',x))=1d(gg',x) and 1d(e,x)=x
for g, g' E g (R), x E: K (R)
Under these conditions we shall say that G acts on X . We shall write gx
for 1d(g,x) .
Each rrorphism of functors 1d:g)( K ->-K canonically induces (I,  2, 9.1) a
rrorphism p:g->- End (X) . To say that  is a g-operation is equivalent to
saying that p is a horrorrorphism of rronoid-functors. The g-operations on K
are accordingly in one-one corresFOndence with the horrorrorphisms Q. ->- End (X) .
Notice also that if  is a group-functor, any haranorphism  ->- End () fac-
tors through Aut (K) , so that the g-operations on "f>. are in one-one corre-
sr::ondence with the k-group-functor haranorphisms  ->- Aut () .
If we express these conditions in diagranmatic form, we obtain the following
pair ofaxians :

\1'>"'"
II,  1, no 3
GROUP SCHEMES
199
The diagram
(Opass)
X)(
Id{  1
GXK
'TTgXI%
) Q)(
I
\!
) 
is ccm:nutative.
The diaqram
(Opun)
G xX  / X
,xr%\/-
x
is cCXImUtative.
Rel1'arks: a) in a similar fashion we define the riqht -opera-
3.2
tions 1J:  x 9 ->-  on . These are in one-one corresr::ondence with the haro-
rrorphisms of the oPFOsite rronoid of Q into End () .
b) SupFOse that G is a k-group-functor. Let f:gx->-gx be the rrorphism
such that f(g,x) (g,gx) for gEG.(R) , xE:(R) , REt!x: . This rrorphism
is an isorrorphism and we obtain a cCXImUtative diagram:
T  ) Gx
1% £2
 ) 
It follows that the rrorphism 1J:Qx->- is isorrorphic to the projection
g x  ->- . For example, if G. is a flat (hence faithfully flat) k-group, and
 is a k-scheme,  is faithfully flat.
3.3 Examples:
a) Let Et\ and rE . If yE r , {Y}k x  is open in r k x  .

200
ALGEBRAIC GROUPS
II,  1, no 3
If X is local , we have
(fk x ,p '" TTt\£;({Y}k x ,) '" !\£;(,,r
Y
(I,  1, 6.10). Thus each rrorphism 1,!:f k x ?;:->- corresr::onds to a family of
rrorphisms !(y):->-. Then 1,!, is a fk-operationon  iff Yf-+!(Y) is
a hc:m::m:Jrphism of f into the rronoid En () . The f k -operations on  are
thus in one-one corresr::ondence with the operations of the rronoid f on .
b) At this FOint we could reprcxluce the remarks of 1.5 concerning functors
conmuting with finite prcxlucts. However, we confine ourselves to only one
example of this type: let  be a Boolean space and f a Boolean group ac-
ting continuously on 'i. Let v:fx X->-X be the map (y,y) ->-y.y, and 1,!,
the canposi te rrorphism
i v k
r k x Y k ..::;) (f >< Y) k ----';'>Y k '
where i is the canonical isorrorphism. clearly 1,!, is a f k -operation on
Y k . If Spec k is connected, each f k -operation on 'Ik is of this type (for
in this case the functor Xt-+X k of I,  1, 6.9 is fully faithful).
c) The multiplication 'TTg:X->- in the k-rronoid-functor g satisfies
the axians (Opass) and (Opun) (cf. axians (Ass) and (Un) of 1.1) . Thus it is
an operation of g on itself, called the left translation operation. It is
associated with the horocmorphism Y:G->- End () such that y(g)x = gx . If G
is a k-group-functor, then y factors through Aut () ; if RE and
gE  (R) , the left translation y (g) is thus an autarorphism of the R-func-
tor g@k R . For example, if k is a field , G is a k -group-scheme and
gE (k) , the translation y (g) is an autarorphism of  which sends e
onto g From this it follows, for example, that the lccal rings of g at
its rational points are all isorrorphic.
The right translation operation is defined similarly: this is the right opera-
tion associated with the haroJro rp hism o:G ->-End(G) such that 0 (g)x = xg
-opp - -
d) Given a k-group-functor , define a left operation of g on itself
by (g,X)J-+ Int (g)x=gxg-l;. this is the inner autarorphism operation . This
operation preserves the group structure of , and accordingly induces a
......"'1'Jtroc:mo hism
. (,"'" '>, rp
( " " ' ,. . , ' .. ' . r :. : " " ' ",, ' ,, . . . . . , . , ' .: . , . . r-. 'i ;..<, :'<1:':
", _ _'; 7,':j;:,' ':?'iFf. -,
>
...i::"\ ,/

II,  1, no 3
GROUP SCHEMES
201
Int :  ->- Aut Gr (g)
where AutGr () is the k-group-functor which assigns to each RE the
autorrOrphism group of the R-group-functo r  0k R .
3.4
Definition: Let X be a k- functor , G a k -rronoid-functor
be the associated hom::mJrphism.
of 1; , the transporter of '£. into Y
acting on X and let p:  ->- End (29
a) Given two subfunctors 'i, '£'
is the subfunctor TransP G(Y''!.') of .9 defined as follows : for each RE: '
Transp CC,'!.) (R) is the et of gE(R) such that the canposite rrorphism
y' 0 R ;illQ1 ,:> X 0 R
- k - k
p (g) " X 0 R
-k
factors throuqh Y  R
b) Given a k- group-functor  and a subfunctor '£   ' the nonnalizer
of 'i is the subfunctor Norm G CO   defined as follows : for each
RE: ' Norm ('!.) (R) is the set of gE  (R) such that the autorrOrphism
p ( g ) of X 0R induces an autcmorphism of Y 0R .
- - k --k
c) If l is a subfunctor of 2:; , the centralizer of Y is the subfunctor
cent /O of  defined as follows : for each RE:  ' cent G) (R) is the
set of gEe (R) such that the endorrorphism p (g) of  0k R induces the
identity on Y 0k R
d) We write i 2 for the subfunctor of  defined as follows: for each
RE ' (R) is the set of xE:(R) such that for each sE and each
gEe  (S) we have p (g) Xs = Xs
If  is a group-functor, Norm G (X) is the largest sub-group-functor of
Transp C , '!.) and cent G ('£) is a nonnal sub-group-functor of G ('£) .
3.5 , proposition: Assuming the above notation, let RE . Then
we have
Trans 2 G (X' ,X) (R)
= {gE:(R) : P(g)R0S(yIR0S)E:X(R0S),\lSE:,VY'E:Y'(S)}
NofTIG (y) (R) = {gE: G (R) :

202
AU;EBRAIC GROUPS
II,  1, no 3
centG. (Y) (R) = {gE  (R) : p (g) R 0S Y R 0S =YR 0S ,'r/yE '1 (S) , VSE}
G
-(R) = {xE"f>.(R) : P(g)R0S0x=0s ,gEG.(S), SEl\}.
Prcof:
We have a cartesian square
(1)
) H ('1' ,)
1
TransPG (X' ,O-----7 H ('i' ,'D ,
a

J
in which a is the rrorphism formed by canposing p with the obvious rror-
phism End () = H (,) ->- H (T ,) ; for this is precisely the definition
of Tr ansp G. (1' , '1') . By I,  2. 9 . 1 and 9. 2, we have canonical isorrorphisms
H (Y',) (R) :+ ((R)x1",):+ (X', H (R,2S)) ,
for RE£:\ . By I,  2, 9.3, H9!Ik (X ' ,) (R) may then be identified with the
set of families of maps '1' (S)->- (R0S) which are functorial with respect to
S . We have a similar identification for H (1" ,'i) . Modulo these identi-
fications, the cartesian square
"r > H""" l' ,)(R:
TransP G ('1' , '1) (R)---? H ('1' , Y) (R)
then yields the first formula.
If  is a group, we infer without difficulty the formula for NormG. (1') (R)
The last two formulas are proved by means of the cartesian squares
(2)
B
f  f H"",, (X,o) """k T: (¥'6 X :
cent (X) ) H9!Ik (Y,   (X,)
where B(g) (y) = (p(g)y,y) , 8= (:@x,:L)

II,  1, no 3
GROUP SCHEMES
203
(3)
X H (G,) H (G,"f>.) r-' >H (g, X)
t diaq . ) ""'k (G, 'Xl
X- ? H (g,X) r-' ------J>. HS!!k (,
where y (x) = (11 (x) ,A (x)) with 11 (x) (g) = p (g) (x) , A (x) (g) = x
3.6
Theorem: Let  be a k -rronoid-functor which acts on a k-func-
tor  , and let I and T be subfunctors of X.
a) If y' is a locally free k- scheme (I,  2, 9.5) and Y is a closed
subfunctor of  ' then TranS P g (. ,) is a closed subfunctor of G.
b) If  is a qroup , and '1 is a closed subfunctor of  which is also
a locally free k -scheme , then Norm G () is a closed sub-qroup-functor of Q
c) If Y is a locally free k- scheme and the k- functor  i s separated , then
cent  () is a closed sub-rronoid-functor of G.
d) g  is a locally free k- scheme and the k- functor X
then xc;j is a closed subfunctor of X.
is separated ,
Prcof:
a) follows fran I,  2, 9.7, whereas b) follows fran the fact that
Norm ('i) = TranSPg (¥, '1 ) II a ( Transp (, I) )
To prove c) , observe that by definition Ox is a closed embedding and
apply I,  2, 9.5 to diagram (2). d) follows similarly, using diagram (3).
3.7 Corollary: SUPFOse that k is a field . Let  be a k -group-
scheme acting on itself by conjugation . If -'i is a closed subscheme of G
and Y' is a subscheme of , then TranS P G ('1' ,'i) , cent G ('1') and
Norm G ('1) are all closed subfunctors of G
Prcof: Since all schemes over a field are locally free, it is enough to show
that  is separated, and this follows fran the
3.8
Lemna :
Let  be a k -group-functor and let EG: ->- 9 be its
G is separated iff EG ts a closed embedding . M:Jreover.
unit section. Then

204
ALGEBRAIC GROUPS
II,  1, no 3
if k is a field and G is a scheme , then  is separated.
Proof: If  is separated, then EG is a closed embedding (I,  2, 7.6b)).
Conversely, we have a cartesian diagram
so is
'r; T' --'-. r'r;
.G. ------J> 
-1
f(x,y)=xy for x,yEs(R), RE . If EG is a closed embedding,
0G (I,  2,6.4). Finally, if k is a field and g a k-scheme,
is a rational, hence closed, FOint of g.
where
EG()
3.9 Let  be a k-group functor acting on itself by conjugation.
We set Ci G = Cent () and call Cent (G) the centre of . A subfunctor tl
of g is central iff it is contained in Cent (Ci) . If li is a sub-group-
functor of g, it is clear that H. is central (resp. nonnal) in  iff
cent (W = g (resp. Norm Ci (W = Ci) ; we say that tl is characteristic in G if .
ut (G) (W = Aut G r () ,
- Gr - -
Le. if for each RE!:1x: ' each autarorphism of the R-group-functor s:;.@kR
nonnalizes !!@kR
If !i is a nonnal (resp. characteristic) sub-group-functor of g and if K
is a characteristic sub-group-functor of H, then K is nonnal (resp. char-
acteristic) in Ci.
3 .10 Semi-direct prcxlucts . Let .G. and !i be k-group-functors, and
let : Cix!! ->-Ei be a Ci-Operation on !! which preserves the group structure
of H, Le. such that (g,hh')=g(g,h)1,!(g,h') for h,h'EB(R), gEG(R),
RE (or, in other rds, the haranorphism p:->- Aut (H) associated with
g maps Ci into Aut Gr (H) )
The semidirect product of Q by .!i with respect to the given operation is
the k-functor !! x' g with the following group structure: for R E  '
g,g' Eg(R) , set
(h,g) (h' ,g')
(h.p (g) h', gg')

II,  1, no 3
GROUP SCHEMES
205
This k-group-functor is written H><!G, or simply HX1 G . We have group hano-
p
rrorphi sms
tl
i
S
r- HXf G  --=--.. 
p p
in which i(h)=(h,e) , 12(h,g)=g , (g)=(e,g) for gE(R) , hEg(R) ,
RE£:\ . Note also the formulas 12o=G' (h,g)=:!:(h)(g) and
(g).:!:(h)(g)-l=i(p(g)h) .
The sections of p, Le. the horrorrorphisms a:G->-B)(J such that JZoQ=Id G ,
are the hOID:)ffi()rphisms of the form g>-+ (t:(g) ,g) =l'(g)s(g) , where l':->-!i
is a morphism of functors which satisfies the following condition:
!(gg') = !(g'). (p(g)!(g')), g,g'E(R), RE .
( " crossed horrorrorphism "). Sections of the form g r->-  (h)  (g) :!: (h) -1 with
hE!i(k) corresFOnd to the " trivial" crossed horrorrorphisms:
-1
gl--->- hop(g) (h ) , gE(R), RE .
We imnediately infer the following
ProFOsition: Let  and  be k -group-functors and le t q:£:: ->- and
.t:->-:g; be haroJrorphisms such that got=l.d G . For gEg(R) , RE ' write
p (g) for the autarorphism of ( Ker g)@kR induced by Int (t.(g)) . Then there
is an isorrorphism of k -group-functors u: (Ker g)XlGE which makes the follow-
- - - p- -
ing diagram ccmnute:
1 9( :r "LI
Ker _q -, E .9- 7' 
Prcof: Simply set '=!(h,g)=h.:t(g) if RE, hE Ker g(R) and gE<:;(R)
Let E be a k-group-functor, H a nonnal sub-group-functor of g: and G
a sub-group-functor of ;g: 0 SupFOse the rrorphism of k-functors H x  ->- g: de-
fined by (h,g)l->-hg for gEs:?(R), hEH(R), RE
sider the inner autarorphism operation of  on
is an isorrorphism. Con-
H i if we assign £IX'  the
corresFOnding semidirect prcxluct group structure, the isanorphism above is a

206
ALGEBRAIC GROUPS
II,  1, no 3
group isanorphism. We also call (imprecisely:) ];;. the semidirect prcxluct of
 by .
3.11 Exarr"ple: the triagonal group . Define the sub-k-functors
!2 nk ' :!'nk and 1J nk of GL nk as follows. If (a ij ) E GL n (R), REJ:\:. ' set
(a, ,) E Dnk (R) <=> a, , =0 for i fj
1J - 1J
(a, ,) ETnk(R)<=>a" =0 for i > j
1J - 1J
(a i ,) E.!J nk (R) <=>t ij = 0 for i > j
J a, ,= 1 for i =j
1J
These are closed sub-group-schemes of nk: for example, 1'nk is the closed
subscheme of GL nk defined by annihilating the functions (a, ,)>-+a, , ,
-  
i > j . We call !nk the (upper) triagonal group, !J nk the strict (upper)
triagonal group , and .!2 nk the diagonal group .
If a l ,... ,an E R* , RE  ' we write diag (a 1 ,... ,an) for the diagonal
matrix (a,o, ,) . Clearly
1 1J
diag : (Ilk) n .... !!nk
is a group isanorphism.
The group 1'nk is the semidirect prcxluct of 12nk by its normal subgroup
!lruc' and we have the formula
Int ( diag (a, )) (a, ,)
- 1 1J
-1
(a,a, ,a, )
1 1J J
for (a,)E(R*)n, (a,,)ET nk (R)
1 1J -
REl\;.
Let r>-+ (i(r),j (r)) be a bijection of the interval [1 ,1;2n(n-l) J onto the
set of pairs (i,j) for which 1::: i < j  n, such that the map r>-+ j (r) -i(r)
o 5 r $: 1;2n (n-l) , we define the closed subschemes
is non-decreasing . For
U(r) and A of
-r
1J nk by
U(r)(R) = {(a,,)E1L,.(R):a, ( ) ' ( ) =0 for s:2:r
- 1J l 1 S ,J s
A (R) = {(a,,)E1L,.(R):a, ( ) ' ( ) =0 for sfr}
-r 1J l 1 S , J s

II,  1, no 3
GROUP SCHEMES
207
for each RE  .
The U (r) are nonnal subgroups of 1'nk ' and we have
U = u (O) U (l) :::) u (k(n-l))=
-nk - :::) - :::). .. - 
F -J. 1 ( 1) A . ubgr f U (r) h ' h .. hi
or rr211 n+ , -r+l 1S a soup 0 _ W 1C 1S lSCITIOrp C to O.k.
fvbre precisely, the rrorphism Jr:g (r) -+ Qk such that !r ( (a ij ) ) = a i (r+ 1) , j (r+ 1)
is a group hcmcrnorphism whose kernel is !2 (r+1) , and it induces an isorror-
phism of r+ 1 onto O.k. It follows :i1m1ediately that .!I (r) is the semidi-
rect prcxluct of ;'+1 by .!I (r+1) Finally, if RE, (a ij ) E 4 (R) and
xE !2 (r-l) (R) , we have the formula
-1
f 1 (Int (a, ,) x) = a. ( ) ' ( ) a ' ( ) . ( ) f 1 (x)
-r- - 1J 1 r ,1 r J r , J r -r-
in particular x-ly-lxyEg(r) (R) for yEQnk(R), xEg(r-l) (R)

 2
LINEAR REPRESENTATIONS
Definitions
Section 1
1.1
Let  be a k-rronoid-functor and V a k-ITOdule. A linear re-
presentation of  in V is a hanomorphism of rronoid-functors
p:Q->-.J;;(V)
i.e., for each RE ' we are given a representation of Q(R) in the R-
ITOdule VQ9kR which depends functoriallyon R. We also call (V,p) a
k -G- ITOdule and define the category of k-G-ITOdules in the obvious way.
If Q is a k-group-functor, then p factors through GL (V) i the linear
representations of  in V are thus in one-one corresFOndence with the
group haroJrorphisms  ->- GL (V)
Let  be a k-rronoid-functor and let V be a k-rrodule. By  1, 2.5, this is
equivalent to being given the three following structures:
(i) a representation of  in V ;
(ii) a left G-operation on V such that for each REM and each
- a ""K
gEG(R) , the endarorphism of VQ9kR induced by g is R-lineari
(iii) a right peration on Q?J(V) such that, for each RE ' the action
of Q(R) on !:J 3 (V) (R) preserves the R-ITOdule structure of Qa (V) (R) .
1.2 Examples . a) Take for G the constant k-rronoid r k associated
with the abstract rronoid r . One verifies irtmediately that the linear re-
presentations of r k in V are in one-one corresFOndence with the linear
representations of r in V . Accordingly, the category of k-r k -ITOdules is
isom:::>rphic to the category of k [r J-ITOdules
b) If V is projective of rank 1, V Q9kR is a projective R-ITOdule of rank
1 for each R E  ' and 1. (V) '" Q ' so that the linear representations of 
in V are in one-one corresFOndence with the characters of G. With the
character x:->- is associated the representation P:G->-(V) such that,
for gE(R) , vEV0:9R , RE we have p(g) (v)=X(g)v .
c) SupJ:X>se that the k-rronoid-functor  acts on the right on the affine

II,  2, no 1
LINEAR REPRESENTATIONS
209
k-scheme £:;. Let RE: ' fEe J1()Q9kR '" JJ(£:; @kR) and gE:(R)
p (g)fE: t9(£:;)Q9kR by
define
(p(g)f) (x) = f(xg) for xE:£:;(S), SE: .
In this way we construct a rrorphism p: ->- J:. (cO (£:;) ) which is a haranorphism of
k-rronoid-functors, hence a linear representation of  in the k-ITOdule J) (£:;)
For gE:(R) , RE:l\: ' p(g) is an algebra endarorphism of J(£:;)Q9kR.
COnversely, a linear representation p of  in the k-ITOdule cO) such
that p (g) is an endonorphism of the algebra J7(£:;)@kR for each RE: and
each gE:(R) , yields a right sroperation on  .
d) SUPFOse in particular that  is an affine k-monoid-scheme; the right
translation operation of  on itself ( 1, 3.3 c)) induces a linear re-
presentation of  in cO (<2) called the (right) regular representation of
, written g ->-0 (g) ; accordingly we have (o(g)f) (x)=f(xg)
e) Returning to the situation in c), sUPFOse for simplicity that k is a
field. Let Y be a closed sub scheme of  which is stable under , i.e.
such that Transp C', X) =  . Let J be the ideal of J1(!5) defining X;
then J is a sub -k- Q-ITOdule of <2() . For if RE:£:\ ' J Q9kR may be identi-
fied with the ideal of J?()Q9kR defining XQ9kR ; if fE:JQ9kR and yE:(S),
SE:,tk , we have
(p(g)f) (y) = f(yg) = 0
so that p (g) fE: J Q9 R . The sequence 0 ->- J ->- cO () ->- J7() ->- 0 is therefore an
exact sequence of k-G-ITOdules.
1.3
Let p:->-(V) be a linear representation of the k-m:moid-
functor  in the k-ITOdule, V . A sub-k-ITOdule W of V is said to be pure
if, for each RE: ' the canonical rrap WQ9kR->- VQ9kR is injective, i.e. if
W rray be identified with a subfunctor of V . This is the case, for exam-
 
ple, if W is a direct factor of V. Let W' and W be pure sul:modules of
V such that W' C W . Define the subfunctor 9w', W of G by setting
',W(R) = {gE:(R): p(g)x-xE:W'Q9R, xE:W@R}
Clearly, ',w is a sub-rronoid-functor of . In particular, we have

210
ALGEBRAIC GROUPS
II,  2, no 1
go,W= cent (W) , ,w= TransPG (W'W) .
For simplicity we write centG. (W) , TranspG. (W'W) and NormQ. (W) for
centG (W) , TransPg (W,W) and NormQ (W) . (Recall that when we defined
the last object we assumed that  was a group.)
Lemma: Suppose tha t  is a group , and that W' is finitely generated . Then
, , W is a sub-group-functor of g . In particular , NOI111 G (W. )
Transp (w' ,W') .
Proof: Let RE: and gE Q.(R) ; the endanorphism p (g) of V@R is an
autanorphism. In particular, if gE Yw' ,w(R), p (g) maps W'@R into W'@R
and induces an injection W'@ R/mR ->- W'@R/mR for each maximal ideal m of
R . By Alg. ccmn. II,  3, prop. 11, p (g) induces an autarorphism of W'@R
and we have p (g) -1 (W'@ R) =W'@R . Thus, if xE W@R , we have
-1 -1 -1
p (g) x-x = p (g) (x-p (g) x) E p (g) (W'@ R) =\'I"@ R
-1
so that g E %' ,W(R) , as was to be shown.
1.4
ProFOsition: Let P:->-1:.(V) be a linear representation of the
k-rronoid G in the finitely generated pro-jective k -ITOdule V . Let W and
W' be tWJ sub -k -ITOdules of V which are both direct factors of V and satis-
fy W'CW . Then Gv,r',W' cent (W) and TransPG (W,W) are closed sub-
rronoids of Q. It G. is a group , then .Yw',W' cent (W) and Norm (W)
are closed subgroups o f .
proof: In virtue of Lema 1.3, it is enough to prove that Gw',w is a closed
subscheme of G. If (a. ) generates W and if (b.) generates the sub-
- l J
spa ce of  orthogonal to W' , then gEG_, W (R) iff b.(p(g)a.-a.)=O
=w, J l l
for all pairs (i,j) . It follows that Gv,r',w is the closed subscheme of G.
defined by the functions g!--T b , (p (g) a. -a ,) .
J l l
Rerrark: We could also have observed that W' is a closed subfunctor of W
9 
and that ',w is a transFOrter, and then applied I,  2, 9.7.
1.5 A pure sub-k-mcxlule W of the k-G-ITOdule V is said to be
stable under G if TranspG. (W,W)=G.

II,  2, no 1
LINEAR REPRESENTATIONS
211
Supr::ose that k is a field. We say that the representation p: ->(V) is
 or irreducible if V = 0 and 0 and V are the only subspaces of V
which are stable under . The representation p is said to be semisimple
or completely reducible if (V,p) is a direct sum of simple k-ITOdules.
A direct sum of semisimple representations is semisimple. A subrepresentation,
a quotient representation of a semisimple representation is semisimple.
If k is a field, the category of k-G:-ITOdules is always an abelian cateqory .
If V is a finite dimensional k-rrodule, it contains Jordan-Holder series;
the quotients of a Jordan-Holder series of V are simple k-ITOdules which
we call simple factors of V. V is said to be isotypical if all its simple
factors are isorrorphic.
1.6
Let .!.!): ->Q be a character of . Set
V!I! = {vE V: p (g) (viS! lR)=v@.!)](g), VgE g(R), RE},
(Va)m(R) = {xEVa(R):P(g)x,,=m(g)x s , \fgE(S), SE}
In particular, if we denote the unit character of G by 0, we have
G G
(VC}.) 0 = (V) - , and we set V-=V o
We irrmediately get V = (V) (k), (V) (R) = (V@ k R)6<1 R .But(V)and
!!]  !!!. C}. !I! !!] -k ill C}.
(V) are not necessarily identical, in other words, the subfunctor (V)
9 ill 9!!!.
of V may not be defined by a suJ::m:x1ule of V . However, this situation
ia.
cannot arise if k is a field.
ProFOsition: SUPFOse i;ha.t k is a field , and let p: ->1: (V) be a linear
representation of the k -rronoid-functor G in the k- vector space V . Then,
for each character  of  , we have
(V) = (V) .
ill s sill
Proof: Let RE and x E(V@kR)!!:lCSkR ; we show that x EVm@kR . Let (a i )
be a base for the k-vector space R, write x=Lv.@a, with v,EV . Let
l l l
SE and gE(S)
P(g)S@RxS@R = m(g)S@RxS@R
we have
which in V@S @R may be written
LP (g) (v,@ l)@a, = Lv.@m(g)@a,
l l l - l

212
ALGEBRAIC GROUPS
II,  2, no 2
l
But since (a i ) is a base for
p ( g ) v ,01 = v ,Om ( g ) for each
l l -
as required.
R over k, the last formula implies that
i . It follows that v, E V , hence xE V 0 k R
l ill m
1. 7 ProFOsi tion: SUPFOse that ' k is a field , and let p :  -+ h (V)
be a linear representation of the k -rronoid-functor g, in the k -vector space
V . Then the sum of the V ill as ill. ranges through  (,Qk) is direct .
Proof: Let m..,...,m be distinct characters of
-.1 -n
i = l,...,n with vl+...+v n =0 . If RE
follows that in V ° R
G , and let v. E V ,
- l IDi
and gE  (R) , it imnediately
o = p (g) (v l +.. .+vn)R = V10!!!1 (g)+.. .+v r 0g)n (g)
If the v. are not all zero, there is uE  such that the u (v , ) are not
l l
all zero. We then have u(v 1 )m.. (g)+. ..+u(v )m (g) = 0 , which contradicts
-.1 n-n
Proposition  1, 2.9.
Now assume that k is a field. The representation p:-+:&(V) is said to be
diagonalizable if V is the sum of the V or, in other words, if we can
ill
select a basis vi of V such that each subspace kv i is stable under Q
(the endorrorphisms p (g) then being represented over this basis by diagonal
matrices). A direct sum of diagonalizable representations is diagonalizable.
A subrepresentation, a quotient representation of a diagonal:i,zable representa-
tion is diagonalizable.
We define the tensor prcxluct of tw:J representations in the obvious way; the
tensor prcxluct of two diagonalizable representations is diagonalizable.
Remark: For each character ill: -+Qk ' let kill be the k-g-rrodule such that
kill.(R)=R and 9.X=ill(g)X for RE:, xE:R and gE:(R) . The above pro-
FOsi tion also follows from the facts that k m is a simple k-Q-rrodule for
each ill and that k m is not isorrorphic to k_ if ill f!l
Section 2
Linear representation of affine groups
Throughout this section G denotes an affine k-rronoid with bialgebra

II,  2, no 2
LINEAR REPRESENTATIONS
213
cJ(g) = A . Its coprcxluct is denoted by 6 and its augmentation by E
A " A
( 1, 1.6).
2.1
Let V be a k-ITOdule and let p: -+ 1. (V) be a rrorphism of k-
functors. We have canonical isorrorphisms
1:1Jd(,g(V)) "b(V) (A) "(V,VQ9kA)
The rrorphism p accordingly induces a k-linear map
: V -+ VA
 may be defined as follows: if go E (A) corresFOnds to the identity map
of A, we have
(v) = P (go) vA E VA .
For p to be a rronoid haranorphism, it is necessary and sufficient that the
conditions (Han l ) and (HOffi 2 ) of  1, 1. 7 be satisfied for f = . Adop-
ting the notation employed there, the prcxluct 1 1 ().12() is by defini-
tion (cf.  1, 2.4) the canposition of the morphisms corresFOnding to the
unbroken arrows of the diagram:
VV @A
I
V@i 2
) V@A@A
l @AQ9A
 VQ9A 19 i 2 ' VQ9i 19A19A. V 19m
V Q9A 19A - - - - - '7 V 19A Q9A 19A 1 "7 V (yA (yA (yA Q9A V (yA 19A
I Q9A
where m (a @b @c @d) = ac@ bd . Since the canposition of the arrows in the
second line is the identity map, condition (H) is equivalent to the
following condition:


214
1lli3EBRAIC GROUPS
II,  2, no 2
The diaqram
v
"vj
 > V@A
k
1 @A
v@to
A > VAA
(Modass)
V@A
k
is corrrnutative .
COndition (HGn 2 ) is equivalent to:
The diaqram

V V@A
I"v \ /:@'A
V'" V@k
k
(r-bdun)
is ccmnutative
Definition: Let A be a bialgebra with coprcxluct to A and augmentation f:'A
A (right) cancxlule is a pair (V,) where V is a k -ITOdule and
fly : V -+ V@ kA is a linear map satisfying (Modass) and (Mcd.un) . A rrorphism
of the carodule (V,) into the carodule (W, toJ is a linear map  : V -+ W
such that   = (@A) fly .
By the above remarks, a k--ITOdule structure on a k-ITOdule V is equivalent
to an A-carodule structure on V (Le. the category of k-G-rrodules and the
category of A-cancxlules are iscrrorphic) .
Given , we irrmediately construct the hcm:.JIrOrphism
p(R): (R) -+ (V,VR)'" 1 (V) (R) .
If gE(R) corresFOnds to the rrorphism f:A-+R, we have g=(f)gO) ,
hence
p (g)v = (I@ f) ((v)) .

II,  2, no 2
LINEAR REPRESENTATIONS
215
The endanorphism of V@kR associated with g is thus the canposition
VR
@R
) V A R
V@f
) VR
where f(a@r)=f(a)r , aEA , rER .
Remark: Left A-cancxlule structures are defined similarly; these corresFOnd
to linear representations of the opFOsi te rronoid of  A .
2.2
The axians (Coass) and (Coun) of 1. 6 show that the coproduct
toA:A->-A@k A endows A with an A-carodule structure. This structure is
associated with the regular representation of G in A (1.2 d)) . For each
k-rrodule W, write w O for the k-Q-ITOdule obtained by assigning W the
trivial operation. For each k-Q--ITOdule V, the axian (Modass) signifies that
o
:V ->-V @kA is a k-G:-ITOdule haranorphism. By (Mcdun)  has a k-linear
retraction.
If V is finitely generated and projective , there is a k-linear map vi k n
with a retraction; hence
VVO@A i &JA " kI1c9Ao: An
is a haranorphisrn of k-!£-ITOdules with a k-linear retraction. In particular,
if k is a field , each finite dimensional linear representation of G can
be embedded in a FOWer of the regular representation.
2.3 Let P:Q ->-(V) be a linear representation of  in a finitely
generated projective k-ITOdule V. By  1, 2.4, (V) is an affine k-scheme
which is isarorphic to S,(\r@kV}, and the k-algebra haranorphisrn
S (\r @kV) ->- A induces the k-linear map
c : \r @V ->- A
k
paired by duality with :V->-V@kA . If RE and wE\r&JR, vEV@R , we
write
c E A@R 0: r!) (G@R)
w,v k R-k

216
ALGEBRAIC GROUPS
II,  2, no 2
for the image of w il.9v under c iSikR , and we call this the coefficient of
p associated with v and w. By  1, 2.4, it is defined by setting
cw,v(g) = \Ws,p(g)v s )' gE(S), s .
It follows that c: (O)iSikV ->-A is a -ITOdule honurorphism, Le. for RE,k\,
gE (R) , w EiSiR and v EViSiR we have
o(g)c = c .
w,V w,p(g)v
SUPFOse that k is a field . The coefficient space of p , denoted by C (p) ,
is defined to be the image of c, in other words, the vector subspace of A
generated by the coefficients c , w E  , v EV . It is a subspace of A
w,v 0
which is stable under ; we have a rronarorphism of k-ITOdules V->-V iSiC(p) ,
and an epirrorphism of k-Q-ITOdules () °iSiV->- C (p) . In particular we infer
the
ProFOsition: SUPFOse that k is a field. Let p:->-!-(V) be a finite dimen-
sional linear representation of an affine k -rronoid  , and let C (p) be its
coefficient space. 'l'hen V and C (p) have the same simple factors . M:>re-
over , p is semisimple , isotypical or diagonalizable iff C (p) has the same
property.
Finally observe that p is a closed embedding iff S(iSiV)->-A is surjec-
tive, i.e. if the coefficients of p generate A as a k-algebra.
2.4
If (V,) is an A-carodule and mE A is a character of ,
we have
V m = {vEV : (v)=viSim}
To prove this, notice that if goEQ(A) is associated with the identity map
of A, by definition we have p(go)v A =(v) , hence (v)=viSim if vEV m
Conversely, if P (go)v A =viSim , we have p (g)v R =m(g)v R for each gE (R) ,
RE (ef. 2.1).
Similarly, a pure sub-k-ITOdule W of V is stable iff tow c. WiSikA .

II,  2, no 2
LINEAR REPRESENTATIONS
217
2.5
Exarrple 1: Linear representations of diagonalizable groups
Let r be a small caTlIT!Utative rronoid; let  =Q (r)k ' so that A =k[r ]
( 1, 2.8). If (V,) is an A-carodule, the map
 : V ->- ViSik[r]
sends vE V onto I yE r Py (v)iSi y where (p) yE r is a family of endanor-
phisms of V such that, for each vE V, p (v) vanishes for alrrost all y .
y
In this situation axian (Modass) becanes
(iSiA)(V) = I p (p ,v)iSiyiSiy' = (ViSitoA)(v) = I P (v)iSiyiSiy ,
y,y'Er y y yEr y
Le.
Py Py.= 0
for y f-y'
and Py Py =Py
Axiom (Modun) becomes
v = (ViSiE:A)(v)
I p (v)
yE r y
Le. Iyq Py = I .
These two conditions are accordingly equivalent to asserting that the p be
y
the proj ections of a grading of type r on V. By 2.1 we have for R c '
gE!? (r\ (R) and v E ViSikR ,
p(g)v = I gy (p iSiR) (v) ,
yE r y
where we have identified elements of r with characters of g(r)k' as ex-
plained in  1, 2.11: This formula also shows that p (V) is the k-rrodule
y
V intrcxluced in 1.6 and our remarks above imply that V is the direct Sillll
y
of the V We s\.1!ffi1arize all this in the
y
ProFOsition: Let P:!?(r)k->-(V) be a linear representation of a diagonal -
izable k -rronoid . If we identify r with a set of characters of .Q (r) k ' then
V is the direct Sillll of the V for yE r . The functor V>+ (V) E r is an
y - y y
equivalence of the category of k-Q (n k -m::xiules with that of graded k -ITOdules
of type r .
In particular, if k is the spectrum of a field, each linear representation
of Q(r)k is diagonalizable.

218
AlGEBRAIC GROUPS
II,  2, no 2
Remark: If D(r) k acts on the affine k-scheme X, then the J}rX) (see
- -  y
1.2 c)) form an algebra grading of cJJ("f>.) . Conversely, each grading of type
r of the k-algebra t!J(J induces a  (r) k -operation on  .
2.6 Example 2: Linear representations of the additive group . The
bialgebra of the group a k is A =klT] with toAT =T iSil+ll8iT and EAT  0 .
If (V,) is an A-coITOdule,  :V....ViSik[T] maps v onto I:=op(v)iSiT l ,
where P, E"f (V) and for each vE V, p, (v) = 0 for alrrost all i. The co-
l K l
rrodule axioms may be written:
(iSiA)(V) = ,'i ,Pj (Pi (v) )iSiTjiSiT i = (ViSitoA)(v) = Pi (v) (TiSil+l iSiT) i ,
l,J l
or P ,op, = ((i,j)) P, , , where ((i,j)) = (i+j) :/i:j: , and
J l HJ
v= (ViSiEA)(v) = Po (v) ,
or Po = I . A k[T ]-carodule structure is
(Pi) iEJN of endanorphisms of V such that
each vE V , P, (v) vanishes for alrrost all
l
thus defined by giving a sequence
P ,0 P, = ( (i, j ) ) P , + ' and, for
J l l J
i . For REl\: and tEa(R) = R , .
we have
co
p(t) = 'i ti(p.iSiR) .
i=O l k
We now analyse these equations in two particular cases:
a) Characteristic 0 : In this case the ring k is a ()-algebra.
Setting Pl = X , we have
for sufficiently large i
p. =/i: and, for each vEV , (v)
l
(!IX is locally nilFOtent") . We have
vanishes
\ i i
(v) = L X (v )iSiT = exp(XiSiT)v .
i=O i:
For RE1:\: and tE a (R)=R , we have p (t)= expt(XiSikR) . Accordingly the
linear representations of a k in V are in one-one correspondence with the
locally nilFOtent endanorphisms of V.
co
b)
Characteristic p ,,0 :
in this case the ring k
is an F -algebra with
p
p prime.
The calculati':>n is elementary although a trifle technicaL We get the

II,  2, no 2
LINEAR REPRESENTATIONS
219
following result: for iEIN , set s. =p i the si are horrarorphisms of
l P
V which satisfy the following conditions:
s,s. =s.s, , s=O and s. (v)= 0
lJ Jl l l
for each vE V and sufficiently large i
r
If n = nO+ nlp+...+ nrP ,
we have
o ::;; n, ::;; p-l , is the p-adic expansion of n E IN ,
l
no nl n r
So sl ... s
p = r
n n O I n l l...n I
. . r.
Setting
exp(s,X)
l
p-l
si p-l
l+si X +".+ (p_l) X
we get
co i
(v) = LPn (v)iSi r = (IJexp(SiiSi ))v
For RE and tEaCR)=R, we have
co i
p(t) = TTexpt P (s,iSiR)
i=O lk
Conversely, a family (si) iEIN of endorrorphisms of V satisfying the above
conditions defines, via the above formulas, a linear representation of a k
in V. For instance, taking V = k, GL (V) = Ilk ' the preceding remarks en-
able us to determine the characters of a k .
ProFOsition: Let p: a k .... (V) be a linear representation of a k . If V"I 0 ,
then vak=O
Proof: Let vE V , v"l 0 and let i E IN be such that
for j > i . Then p. (v) E V ak . For if n > 0 ,
l
p, (v) "I 0 , p, (v) "I 0
l l
p (P. (v)) = ((n, i) ) p . + (v) = 0 ,
n l l n
so that "'-_(p, (v))=p, (v) and p, (v) is invariant (2.4).
v l l l

220
2.7
that k is
ALGEBRAIC GROUPS
II,  2, no 2
Example 3 : Linear representation of the group . SupFOse
pr
an al g ebra over IF and set G = O. k . A calculation
p - r
similar to the one above shows that the p linear representations p of
 corresFOnd to the caTlIT!Uting families (si) 0::; i ::;r-l of endanorphisms of
th
V whose p FOWers vanish, via the fonnula
r-l i
p (t) = IT exp(t P s,) .
i=O 1
The same argument as above shows that V 'f 0 implies vCi 'f 0 .
In particular, the linear representations of p in V are in one-one
corresFOndence with the endanorphisms of V whose p th FOWers vanish. More
particularly, the characters of a k corresFOnd to the elements of k whose
th ' hh p ' 1 ' hi
p FOWers vams; ence we get a canonlca lSanorp sm
k(ak,l1k) '" pak(k)
which yields a canonical isanorphism
D ( a k ) '" a.. .
- p P K
By the above arguments, this isanorphism is given by the pairing
 : p ><. p ->- Ilk
such that (x,y)= exp(xy) for x,yEpa k (R) , RE
2.8 Example 4 : Linear representations of (U). Suppose that k
is a field and U is a finite dimensional k-vector space. Identify Qk with
a sub-k-rronoid of 1< (U) by assigning to each REA and each xE R the
endanorphism u .- ux of U 0kR . For each linear representation
p : 1«U)->-1.(V) , V accordingly carries a natural k-Qk-ITOdule structure and
we have V =EI) E fN v , where V denotes the subspace of V formed by the
n n n
v such that p (x) (v 0 1) = v 0 x n for all RE  and all x E R . Clearly V n
is a sub-k-1. (U) -ITOdule of V. We call V the horrogeneous canponent of V
n
of degree n. If V = V n ' V is said to be hanogeneous of degree n. For
example, if V =18Pu and p satisfies
p (g) (v l QSi .. .0v n ) = g(v l )0.. .0g(v n )

II,  2, no 2
LINEAR REPRESENTATIONS
221
then V is horroeJeneous of degree n
Identify the algebra of functions of b.(U) with the synmetric algebra
S(0U) as in  1, 2.4. With the notation of 2.3, if f.E and u,EU
l l
(i=l,... ,n) , we have
c Q9 0 f 0 0 (g) = (f 1 0.. .0f n , g (u l )0. . .0g (U n ) )
fl." n'u l ... un
(fl,g(u l ) )...(fn,g(u n ))
c f u (g).. .c f u (g)
1, 1 n, n
The rrap
c : t(U)0(£U) -+ S(0U)
accordingly coincides with the canonical map of
t n n n _
(0U)0(0 U) -+ 0( LJ0U)
into S (0U) . The image of this rrap is the space of harogeneous FOlynanials
of S ( 0U) of degree n. This space is therefore the hanogeneous canpo-
nent of S(0U) of degree n. Thus it follows fran ProFOsition 2.3 that
the simple factors of a horrocreneous k-.1 (U) -ITOdule of degree n already appear
as simple factors of  .
MJre generally, 2.3' shows that if  is a closed subronoid of .!:.(U) , the
simple factors of any k-Q-ITOdule appear as simple factors of the k-ITOdules
0 for nEJN .
2.9
Let k be a field and let V be a finite dimensional vector
space. For
g E GL (V) (R)  (VR)
and
Y E \r0R '" (V0R,R)
g (y) E0R by (g (y) ,x) = (y,g -1 (x) )
n m_
0 m v = (0V)0(0-v)
n
, xEV0R . If m,nEIN , we
we define
assign
a kyITOdule structure in such a way that

,
222
ALGEBRAIC GROUPS
II,  2, no 3
g (x l Q9 . . .181 x n 181 Y1Q9. . .181 Ym) = g (xl) 181. . .181 g (x n )Q9g (Y l ) 181.. .181 g (Y m ) .
Pror::osition: Let  be a closed subgroup ofGL (V) . Then the simple factors
of each k -Q-ITOdule appear as simple factors of the k-g -ITOdules  n,mEIN .
Proof: Set U =vxt.v and consider the haranorphism p: % (V) + b (U) defined
by p (g) (x, y) = (g (x) , g (y)) . We claim that this is a closed embedding . For
it can be split into the obvious closed embedding .1 (V) X b ( t.v) + b (U) and the
rronanorphism Pl :(V)+!:o(V)Xb(t.v) such that P l (g) = (g,g) . Now if
uE1(V) (R) and vE1(t.v) (R) (u,v) assumes the form P l (g) for some
gE GL (V) (R) iff (v(yQ9R),u(xQ9R)/ = (y,x) for all xEV , yEt.v, so that
P l induces an isanorphism of GL (V) onto the closed subscheme of b(V).x b(t.v)
defined by the functions
(u,v) + (v(yQ9R) ,u(x Q9R) - (y,x) ,
as (x,y) ranges through V.xt.v. In virtue of this, it is enough to apply
2.8 to the closed embedding  + GL (V) g b (U) , noting that the k -GL (V) -ITOdules
I8Pu are direct sums of ITOdules isarorphic to the 181 PV
q
Section 3
Existence of linear representations (in the case of a base
field)
Throughout this section k is a field .
3.1 Given a linear representation p:  + 1 (V) of the k-rronoid g,
the intersection of any family of stable subspaces of V is a stable sub-
space of V. In particular, for each subset of V there is a smallest stable
subspace of V which contains it: we call this the stable subspace generated
by the subset.
Lerrma: Let p:  + 1 (V) be a linear representation of the affine k -iIDnoid
 . Then each finite dimensional vector subspace of V generates a finite
dimensional stable subspace .
Proof: Let A be the bialgebra of g, and let : V + V Q9k A be the cano-
dule law of V. It is sufficient to show that each element x of V

II,  2, no 3
LINEAR REPRESENTATIONS
223
belongs to a finite dimensional stable subspace. Let (a i ) iE I be a base for
the k-vector space A . Set
!lx = I x,0a,
v . 1 1
1
Axiom (Modass) yields
I(y.)0a. = Ix.06 A a.= I x.0b, .0a.
ill ill i,j 1 1J J
in which we have set 6 a. =I,b. ,0a, ; this gives
A 1 J 1J J
(Mcdun) gives x = I ,x, E:(a,) . The vector
111
Xi contains x and satisfies W c W0 k A , and accordingly meets the re-
quirements.
!lx, =I.x,0b.. . Axian
v 1 J J J1
subspace W of A generated by the
3.2 Lerrma: Let H be a closed subronoid of the affine k""iTIOnoid G
--
Let I be the ideal of A. = 19(G) defining H Then!i = TransP G (1,1) in the
regular representation of Q. (1. 2 d)) .
Proof: Let REt\ . We show that !i(R) = Transp (1,1) (R) . If hE H(R) and
fEI0R , we claim that o(h)fEI0R. For each SE and each xEJ:!(S) , we
have (0 (h) f) (x) = f (xh) = 0, hence 0 (h) f E 10 R as claimed.
Conversely, let gEG(R) satisfy 0(g)fEI0R for each fEr0R . Then for
each fE I we have, f (g) = (0 (g) f) (e) = 0 , whence gE H (R)
3.3
Existence theorem for linear representations. Let Q. be an
affine algebraic k ""iTIOnoid and let .!i be a closed su1:m::moid o f g . Then there
is a finite dimensional linear representation p : -+!> (V) and a vector sub -
 U of V such that p is a closed embedding and .H = Transp.G (U,U) .
Proof: Let A be the bialgebra of  and I the ideal defining !i. Allow
 to act on A by means of the regular representation. By 3.1 there is a
finite dimensional stable subspace V of A such that V generates A as
a k-algebra and I nV generates the ideal I (since  is algebraic, the
k-algebra A is finitely generated). Set U = I nV . We claim that the pair
(U,V) satisfies the conditions of the theorem.
a) We have Transp (U,U)=.H . Since U generates the ideal I, and since
the action of  on A preserves the algebra structure of A, we have

'1
224
ALGEBRAIC GROUPS
II,  2, no 3
Transp (I,I)= TransPG. (U,I) . Since V is stable under G and UCV , we have
TransP.G. (U,I) = TransPG. (U,InV) = TranspG. (U'U)
Finally, by the lemna., we have TranSPQ (I,I)=.H: .
b) p:  ->-!: (V) is a closed embedding. For the haranorphism S (iSiV) ->- A sends
wiSivEiSiV onto the coefficient c (2.3). If we take for w the restric-
w,v
tion of sA :A->-k to V, we have for RE!:\ and gEQ(R),
cw,v(g) = (wR,o(g)v R ) = (0 (g)v R ) (e) = v(g) ,
so that c = v . It follows that the coefficients c generate A, and
w,v w,v
accordingly S(iSiV)->-A is surjective.
3.4 Corollary: An algebraic k -rronoid G is affine iff it is iso-
rrorphic to a closed suhronoid of an !:. (k n ) .
Proof: The condition is obviously sufficient; theorem 3.3 shows that it is
necessary (take H =) .
3.5 Corollary: Let  be an affine algebraic k -group and let H
be a closed suhronoid of G Then:
a) !i is a subgroup of G
b) there is a finite dimensional linear representation  ->- GL (V') and a
line (Le.  l-d imensional vector subspace ) D of V' such that .!i = Norm (D)
Proof: a) follows immediately from 3.3 and lemma 1.3. For b), take a linear
representation  ->-!:: (V) and a subspace U of V satisfying the conditions
of 3.3. Suppose U has dimension n. Set V' = 1\ , D = II  and con-
sider the representation Q.->- GL (V') such that, for RE ' gEQ(R) and
xl' . . . ,x n E ViSi R , we have
g (xl A. . . A x n )
g(Xl)!\...Ag(x n ) .
We have, for each RE l:1J<
UiSiR= {vEViSiR:vll(DiSiR)=O}
k k k

II,  2, no 3
LINEAR REPRESENTATIONS
225
Clearly HC NOrm (D) ; conversely, if gEG(R) leaves DR stable, we have,
for uEUiSiS , SE,
-1
g(u) ADS = g(u fig (D S )) = g(uJ\ DS) 0,
so that g(u)EUiSiS and gE TranSp( U,U)(R)=.Ii(R) .
3.6
Corollary: Let G be an affine algebraic k -m:moid . Then the
largest sub-group functo r G* of  is an affine open subgroup of g of the
form 9f' where fE t.?(C;;) is a character of G.
Prcof: By 3.4, we may assume that  is a closed sul:m::Jnoid of 1< (k n ) . Then
 n GL nk is a group by 3.5, and is a=ordingly equal to g* . Hence we may
take for f the restriction to G of the determinant function.
3.7 ProFOsition: Let  be an affine k ....m::moid (resp. k).
Then there is an inverse system () iE I of affine algebraic k -m:moids
(resp. k- groups ), with IE 11. , and a coherent system of haranorphism  ->-C;;i
such that
a)
the maps
c9(G.) ->- t.9(G)
l
and J\G.)->- t\G,)
- -l 
are all injective ;
b) t.?(C;;)
is the union of the images of the
J1(G, )
-l
Proof: Let (Vi) iE I be a directed family of finite dimensional vtor sub-
spaces of A = c9(C;;) which are stable under C;; and such that U i Vi generates
A (3.1). As we have seen in the proof of theorem 3.3, the image A. of
l
S (,iSiV, )->-A contains. V. . It follows that A is the directed union of
l l l
finitely generated k-bialgebras Ai . This proves the proFOsition in the case
where C;; is a k-rronoid; if  is a k-group, let d i be the image in A of
the determinant function det E  ( L (V, )) '" S ( . iSi V . ) . Then A is the di-
- l l l
rected union of the (Ai) d and  (Ai) di is a closed sul:m::Jnoid of GL (Vi) ,
hence a k-group (3.5).
Rermrk: Consider the case of groups. It can be shown that the morphisms
G ->-G, and G, ->-G. are faithfully flat and are e p irro rp hisms in the category
- -l -J 
of affine k-groups. It follows that each affine k-group is the inverse limit,
in this category, of a "strict" inverse system of affine algebraic k-groups.

 3
HOCHSCHIID COH(M)LOGY FOR GROUP SCHEMES
Throughout this paragraph  denotes a k""iTIOnoid-functor. A g-ITOdule is a
caTlIT!Utative k-group-functor N equipped with a -operation written
(g,m) -* gm , which preserves the group structure of N, Le. satisfies
g (m+m' ) = gm +gm' for R c  ' gE g (R) , m,m' EN (R) . A hOIIK:Jlilorphism of -
ITOdules is a hancm;)rphism of k-group functors which comnutes with the pera-
tions.
Section 1
The Hochschild complex and the exact cohorrology sequence
1.1
Let N be a Q.-ITOdule. For each n  0 assign the set
(G,) = l:.\£;(n,
the cCXImUtative group structure defined by the group law of M (1, 1.2).
An element f of C n (,) is called an n-cochain of  in M; this is
accordingly a system of n-cochains
i R E ((R) ,(R))
which is functorial in RE  .
Define a hananorphism (the boundary operator ) ()n :cn(,)-*cn+l(,) in the
n ,n+l i n
usual way: set () = Li=O (-1) ()i where, for each RE and
gl'''. ,gn+l E (R) , we have
n
()Of R ) (gl'.. ,gn+l)
gl
f R (g2'... ,gn+l)
n
(d i f R ) (gl' ..,gn+l) = f R (gl'." ,gigi+l'" . ,gn+l) , 1 $.i $.n
n
()n+lfR) (gl'... ,gn+l)
fR(gl'... ,gn) .
We verify imnediately that ()n()n-l= 0 for n > 0 we write C. (,) for the
ccmplex (((,)), ()n)) . Finally, set
Zn(S?,) = Ker()n , Bn(G,M)
Im ()n-l , B O (,)
o ,
_Jl n n
HO (,) = z (,N) /B (,) .

II,  3, no 1
HOCHSCHILD COH(M)LOGY
227
H(G,M) is called the nth Hochschild group of G in M
1.2
We now ccrnpute
o
HO(g,)
1
and HO (g,) . We have
o
C (g,) = £:.\£; ( k, ) '"  (k)
Cl(,) = (g,W .
000
By writing out d , we see irrmediately that HO (g,)= Z (g,) is the set of
mE(k) for which g = for all RE ' so that
H (,) '" G.(k)
By writing out dO and d l , we see that the l- cocvcles are the crossed hano -
rrorphisms of G into M, that is, the rrorphisms i:g -* satisfying
!(gg') = f(g)+gf(g') , g,g'Eg(R) , RE£:.\ '
and that the l- coboundaries are the trivial crossed haroJrorphisms of the form
!(g) = g- ' gEG.(R) , RE.£:\ '
where mE(k)
(cf.  1, 3.10).
1.3
Given a G.-ITOdule hom::Jmorphism f: -*  , we define in the obvious
way a hananorphism of ccxrplexes
C. (g,f) : C. (g,M) -* C. (g,!'!) ,
whence group hananorphisms
(,f) : (g,) -* (G.,W .
SUPJ:X>se that G. is an affine k-rronoid with bialgebra A . Then C n (,) may
be naturally identified with M(A <8h) where A@n is the k-algebra tensor
product of n copies of A. If
o -* M' -*  -* Mil -* 0
is an exact sequence of G.-ITOdules (Le. if for each RE.£:\ the sequence
o -* M' (R) -* M(R) -* " (R) -* 0

228
ALGEBRAIC GROUPS
II,  3, no 1
is exact), for each n, we then have an exact sequence
') ->- dl(,') ->- dl(, ->- dl(,") ->- 0
which yields an exact sequence of ccmplexes
o ->- C. (,') ->- C. (,) ->- C. (Q,") ->- 0
giving rise to the exact Hochschild cohorrology sequence
O->-H(G'')->-H(Q,) ->-H((2,II) ->-H(Q,') ->-...
->- H;(Q,II) ->-rS+l(Q,I) ->-rS+l(g,) ->-...
ProFOsition: If Q is an affine k -rronoid , the functor >-+ HO (G, , in
which  ranqes throuqh the catecrorv of G -ITOdules , is the derived functor of
G
the functor ...... - (k)
Proof: It is enough to prove that HO ((2,?) is effaceable , i.e. for each Q-
ITOdule  there is a 2-"ITOdule S; () and a rronorrorphism  ->-  () , such that
(Q,())= 0 for n > 0 .
Take  () = H (9, , where
H (, (R) = (QR') , RE .
Assign () the group-functor structure induced by that of  together with
the peration such that gf (h) = f (hg) : this turns  () into a 2-"ITOdule.
The rrorphism !1-() which assigns to mE!1(R) the rrorphism : %->-
such that (g)=gms for gEQ(S) , SEi%' isarronorrorphism (since
m(e)=m) , and is canpatible with the action of Q (since
g(f ) (h) = f (hg) = (hg)m = h(gm) = f (h) ,
-m -m -gm
for g,hE(2(S) , mE!!(R) , SE, RE)
Lerrma: For each ccmnutative k -group-functor M, we have
(G , H ((2,W) = 0
for n > 0 .

II,  3, no 1
HOCHSCHllD COHCMJ:UX;Y
229
Proof: We have canonical isarrorphisms
en( , H (Q,)) = !:(sf , H (Q,W) '" (Qx) = (+1, M)
consider the nap
n ....n+2 ....n+l
s : ( ,) ->- (n ,)
defined by
n
(s f) (g,g l ,...,g ) = f(e,g,g l ,..., g ) .
- n - n
If E en (Q , H (,M)) , n > 0 , we have successively
n n
s 3 f(g,g l ,...,g) = (g,gl,...,gn)-i(e,ggl,...,gn)+."
- n llil
+(-1) i(e,g,...,gn_l)
ln n
3 s f( g ,g l ,...g) = f(e,gg l ,...,g )-...+(-1) f(e,g,...,g 1 )
- n - n n-
whence
s n 3 n + 3 n - 1 s n - 1 = Id for n > 0
and so (s) is a horrotopy operator.
n
1. 4 As usual we can extend the definition of H (G,M) and H; (G,tlJ ,
as well as the initial stages of the cohorrology exact sequence, to rrore gener-
al situations. We will only need to do this in the simplest case, which
follows:
Let M, tl', tl" be k-group functors on which  operates by group endorror-
phisms and let
M'  M ¥ tl"
be hararorphisms of k-group-functors which are compatible with the action of
G and are further such that for each RE  ' the sequence
1 ->- M' (R) JtiR) M(R) ) Nil (R) ->- 1
is exact (i. e.  (R) induces an isorrorphism of 11' (R) onto the kernel of
y (R) , with Y (R) surjective). Suppose further that M' is carmutative , so
that Hi (G M') is defined.
o -'-
ProFOsition: Under the above conditions there is a map 3: "G (k)->- H; (Q,tl')
such that in the sequence of maps

230
ALGEBRAIC GROUPS
II,  3, no 2
1 -+ W (k) O G (k) ¥O "G (k) 1 H (,')
o
where !J o = HO (, y)
-1
3 (0)= ImO
o
and YO =HO (G,Y:) , we have Ker  = 1 , Ker YO = Im O '
Proof: The first two equalities are irrmed.iate. We now define 3 . If
G
m"E'-(k) , choose mE(k) such that y(m)=m" . For RE and gE(R) ,
we have y (gm) = gm"=m"= y (m) , hence gm. m -1 may be written in the form
 (R (g)) , with 1 R (g) EN.' (R) one verifies imnediately that the f R define
a rrorphism !:-+tl' which is a l-cocycle of  in' If mlE(k) is a
second element such that y()=m" , there is m'E' (k) such that
m 1 =(m')m , hence such that
g -1 g 9 -1 -1
m..m =u(m'). m.m .u(m' )
.L 1 - -
g -1
1J( m'. m' .1 H (g))
It follows that the class of f ITOdul0 B 1 (,')
is independent of the
choice of m; this we denote by 3 (mil) We now verify the last assertion.
If milE Im(yO) , we can choose mEG(k) and 3 (mil) = 0 ; conversely, if
g -1
3 (mil) = 0 , there is m' E M' (k) such that m'. m' . i R (g)= 0 for each
gE(R) , RE . Then u(m')m is invariant under 9. and is projected onto
ro" .
Remark: In general 3 is not a group hcrnarorphism. However, it is one if
!J (') is in the centre of M
Section 2
Extensions and cohanology of degree 2
Throughout this section we assume that G. is a k-group-functor.
2.1 Definition: Let  be a corrmutative k -group-functor . An H-
extension of G. .!2Y .!':l is a sequence of k- qroup-functor haroJrorphisms
M  E If G
- -
satisfying the following two conditions :
a) for each RE ' the sequence

II,  3, no 2
HOCHSCHILD COHCM:JI..CGY
231
1 -+ M(R) ;1 $) E.(R) 1<$) (R)
is exact ,
b) there is a rrorphism of k- functors E:9-+]; such that Qc=I9g .
The H-extension (g;,:j"p)
is a hom::Jrrorphism t:g; -+'
and (E.', 1-' , Q.' ) are said to be equivalent it there
such that to! = i' , 12.'0 f =12. .
The H-extension (,;h,Q) is said to be inessential if there is a k-group-
functor haroJrorphism :Q-+  such that Ec = I.9 Q .
One verifies without difficulty that equivalence of H-extensions is indeed an
equivalence relation. In the language vmich has just been formulated, propo-
sition 3.10 of  1 becomes: an H- extension of Q. l2Y .t1 is inessential iff
it is equivalent to an extension defined by the semidirect prcxluct of Q !2y
 with respect to a suitable operation of Q on .1'1.
Notice that condition b) implies that Q(R) is surjective for each REJ:\
since  (R) is a section of !2.(R) Conversely:
Lerrma: If G is an affine k -group , condition b) is equivalent to:
b') for each REl\ the map Q(R) is surjective;
and to
b") the nap p_((9()) is surjective .
Proof: Since b)=>b')=>b"), it is enough to prove b")=>b). But we have
a carmutative diagram
(r
l\ (Q,)
12. (J(G) ) -,. g(0' (G))
J
q
 (G,Q)
in which q(f)=I2.'f. If Q(<i7(Q)) is surjective, so is q and there is
g;Et\]i(Q,) such that !2..=:@Q.
2.2 Given an H-extension M k E. & G of Q by !':l, we define a
operation on !':l in the following way: Since !'i is nonnal in E.,  acts
by inner autarorphisms in ; since 11 is ccmnutative, !':l acts trivially

232
ALGEBRAIC GROUPS
II,  3, no 2
and the action of  factors through G . Accordingly we have
Int (x)!(m) = !(E(X)m)
for xE  (R) , mE!:1 (R) , RE  . This operation preserves the group struc-
ture of !:1 and depends only on the equivalence class of the given extension.
We call this operation the 9-O peration on  defined by the given class of
extensions .
We say that the H-extension M.t:g;  Q is central if i ( is central in
 , Le. if the i£"Operation on  defined by this extension is triviaL
2.3
ProFOsition:
Let M be a G-ITOdule. Then the set of classes
----
of H- extensions of Q. eY !:1 defining the given Q.-operation on M is canoni -
cally identifiable with H; (Q.,!:1) .
Proof: The proof proceeds as for ordinary groups.
a) If M    G. is an H-extension, and if s:G+E satisfies pO=Q ,
define a rrorphism :Gx G +M by
!?(g)(cj) = i(f (g,g'))s(g,g') , g,g'E(R)
- - -
RE .
If the above extension defines the given G.-operation on !:1, one verifies
inmediately that &.QEZ2(G.,M) . If g':+ is another secion of E, there
is h:G+!:1. such that g' (g)=:j'(h(g))(g) , and we obtain without difficulty
&s' = s + d' h , so that the class of i  in H (g ,M) depends only on the
extension in question; rroreover, it depends only on the class of this ex-
tension.
b) Given a 2-cocycle :gxQ+!:1. of Q in !:1, define an H-extension of Q.
as follows: on the prcxluct  =!1" Q. irrpose the group law
(m,g) (m',g') = (m+gm'+f(g,g'), gg')
for g,g'EG(R) , m,m'EM(R) , RE . Set .:ijm)=(m,e) and j2(m,g)=g .
If f' is a 2-cocycle which is coharologous to , one shows easily that
the H-extension associated with [' is equivalent to the H-extension associ-
ated with f .
c) It renains to verify that the two constructions above are mutually

II,  3, no 3
HOCHSCHILD COHCM)UX;Y
233
inverse, and this is inmediate.
2.4 Suppose Q acts trivially on M. Write z2 (G,M) for the set
- s --
of syrrrnetric 2-cocycles of  in M (Le. the 2-cocycles f:Qx Q+ such
that f(g,g')=f(g',g)) , and let
H 2 (G,M) = Z2(G,M)/B 2 (G,M)
s-- s-- --
The prcof above imnediately inplies the
ProFOsition: Let M be a caTlIT!Utative k -group-functor , and suppose that 
is caTlIT!Utative . Then the set of classes of H- extensions r:1.+!';+g such that
E is carrnutative is canonically identifiable with H 2 (G,M) , where G acts
s--
trivially on M.
2.5
Remarks:
1) As usual we can define directly the Baer sum of two H-extensions. This
corresponds to the addition given in H; (g,)
2) Here we have used a very restricted type of epirrorphism (those possessing
a section), and, accordingly, a very restricted type of extension.
3) The bijection of zl (G,) onto the set of sections of the semidirect pro-
duct MX1G described in  1, 3.10 nay be generalized as follows: let
(E) : M 1 f;  <2.
be an H-extension of  by  . Let us define an (E) -autarorphism to be an
autarorphism of E which induces the identity on  and Q. We obtain a
bijection of zl (G,M) onto the group of (E) -autarorphisms by assigning to the
cocycle f:Q +  the autarorphism  such that  (x) = J. (! (j2 (x) ) ) x for RE tl. k
and xE g; (R) .
Section 3
Coharology of a linear representation
3.1
We shall be concerned with the case in which M is of the form
v , where V is a k -ITOdule.
2.
ProFOsition:
Suppose that G is an affine k- rronoid and le t A = tJ(Q)
be its
-

234
ALGEBRAIC GROUPS
II,  3, no 3
bialgebra . Let P : Q +  (V) be a linear representation of Q and let
 : V +V0 k A be the correspondinq carodule law ( 2, 2.1). Let C. (,V) be
the canplex:
(Q,V) = V0A0A0.. .0A (n factors A)
n+l .
d n = I (-l)ld
i=O l
n tv. on tv. ° n+l
where d V'<YA + V'<>'A is defined by
i -
n
do(v0ai0...0an)= (v)0a10...Q5Ian '
n
d i (v0ai0.. .0 an) = v0a10...0 toa i 0.. .0 a n 1  i n ,
n
d 1 (v0a 1 0...0a) = v0a 1 0...0a 01 .
n+ n n
Then we have a canonical isorrorphism of ccrnplexes
c. (Q'V?J) ::: C. (Q,V)
Proof: By definition, we have
J1 On on
c (G,V ) ::: V (A ) = V0A
-?J a
which gives canonical isorrorphisms
A : (G,V )  Cn(G,V)
n - i'! -
We must nCM ccrnpare the boundary operators. For this purr::ose let g, ,
. On On. l
l = 1, .. .,n , be the elements of G (A ) :::  (A,A ) which corresFOnd to the
n injections a..--+10...010a01...Q5ll (a in the i th place). If fECn(Q,Va),
we have by definition
f(g l ,...g) = A (f)
- n n -
n
Let us show, for example, that An transforms the operator dO of
into the operator d of C.(,V) . Take x=vl8ia10...0anECn(Q,V)
-1 n n
show that if f =A (x) , we have A l d O f =d O X . But we have
- n n+ -
n n
An+l dof = (doD (gl'''. ,gn+l) = gl! (g2'''. ,gn+l) = gi (v010 a 1 0...0 an)
c. (G,V )
- a
; we must

II,  3, no 3
HOCHSCHILD COHOMJLCGY
235
(since gi+1E G(A0 (n+l) is the image of giE GCA0 n ) under the map associ-
ated with the haroJrorphism
a 1 0...0 a ......10 a,0.. .0a
n n
° n ° n+l
of A into A ). The latter prcxluct is the image of
v010a 1 0.. .0 an
under the canposi te map
V0A0(n+l)
0Id
" V0A0A0(n+l)
Id0110Id n-. ( 1)
'> V0AVY n+
where l1(a0b)=a.b . Accordingly, this latter prcxluct is
n
(v)0a10...0an' Le. dO x
3.2
If P:G +1 (V)
 , we write (G,V)
is a linear representation of the k-roonoid-func-
for the group (G,V) Thus we have, for
tor
example,
o G G
H (G,V) = V-(k) = V- .
- 
If s is affine, proposition 3.1 supplies an isomorphism
(G,V)  (C' (,V)) ;
in particular, we regain the description of vG. given in  2, 2.4.
Corollary: Suppose that Q is an affine k -monoid , and let
o +V' +vl'v" +0 be an exact sequence of k-Q -ITOdules such that p:V +V" has
a k- linear section . Then we have an exact sequence of ccmnutative qroups
G G G 1
o + V'- + V- + V"- + H (,V') + ...
... + (G,V,') + (G.,V) + (Q,V") + +l(,V)
-+- ...
Proof: For each R  £:\ ' the sequence
o +V'0R + VC9R + V"0R + 0
is exact; apply 1.3.
3.3 Proposition: Suppose k is a field and G is an affine k-
rronoid . Then the functor V t-+H. (G,V) , where V ranges through the category

236
ALGEBRAIC GROUPS
II,  3, no 3
of k -g-ITOdules , is the derived functor of the functor V>-->-V Ci .
 By 3.2, it is enough to show that H. (,?) is effaceable , Le. for
each kyITOdule V there is a kyITOdule E (V) and a rronorrorphism V -+ E (V)
such that  (,E (V)) = 0 for n > O. Take E (V) to be the k-vector space
V 0A , where  acts trivially on V and on A through its regular re-
presentation. We know ( 2, 2.2) that :V-+E(V) is a rronanorphism of k-
ITOdules; all that remains is to verify that  (,E (V)) = 0 for n > 0 .
3.4  Suppose tha t  is an affine k -rronoid ; let A be the
bialgebra of  and let V be a k -ITOdule . Then  (, V Ok A) = 0 for
n> 0 .
Prcof: Let RE ; we have canonical bijections
(V0A) (R) = V0A0R '" (V OR) ° (A OR) '" E(G0R, (V ) R ) ,
k  k k k k k ;w.., - k 
whence an isorrorphism of ITOdules
(V 0A) '" Hem (G, V )
k S. -1< - S
Now apply lerrma 1. 3.
Remark: By carrying over to the ccrnplex C. (,V0A) the horrotopy operator
of the corrplex C. ( , (,y)) , we obtain the operator:
sn : (V0A)0A0(n+l) -+ (V0A)0A0 n
sn((v0a)0a10...0an+l) = EA(a) (v0al)0a20...0an+l
3.6 corollary: With the assumption of proposition 3.3, let 
be the category of A- carodules (A = t!J ()) . Then we have canonical isorror -
phisms
(,V) '" (k,V) .
Prcof:
G .
We have the obvious isorrorphism V-'"  (k,V)
3.6
Suppose that  is affine and let k'El:\; . We have a

II,  3, no 3
HOCHSCHILD COHOM:)IJ:X;y
237
canonical isomorphism
C. (G V)@k' '" C. (G@k' V@k') .
-, k -k' k
If k' is flat over k, we accordingly obtain canonical isomorphisms
(G@k' , V@k') '" (G,V)@k' .
-k k - k
3.7 ProFOsition: SUPFOse that k is a field and tha t G is an
affine k Then the following conditions are equivalent :
(i) For each linear representation  -+ GL (V) , we have  (, V) = 0
for n > 0 .
(ii) For each finite dimensional linear representation  -+ GL (V)
we have  (G,V) = 0 .
(iii)
(iv)
simple.
(v)
Each linear representation of G is semisimple.
Each finite dimensional linear representation of G is semi-
The regular representation of G is semisimple.
Prcof:
(i) => (ii) : Trivial.
(iv) => (iii): By  2, 3.1.
(iii) => (v): Trivi!11.
(v) => (iii): By  2, 2.3.
(iii) => (i): By 3.3.
(ii) => (iv): Given two k-Q-ITOdules U, V which are finite dimensional over
k , assign  (U , V) a k-G-ITOdule structure as follows: if R E  ' identi-
fy (U,V)@kR with
McYJ (U @R , V @R)
M.V,.. k k,
-1
by means of the canonical bijection; then set (gf) (u) = gf (g u) for gE Q (R) ,
fE(u,V)@kR and uEU@kR. Now let O-+V' -+V-+V"-+O be an exact se-
quence of k-Q-m::dules of finite dimension over k. We have an exact sequence
of k-Q-ITOdules
o -+ (V",V') -+ (V",V) -+  (V",V") -+ 0 ,
hence a cohanology exact sequence

238
AU;EBRAIC GROUPS
II,  :3, no 4
(V"'V)->- (V",V")->-Hl (,  (V'I,V'))
It follows that the identity map on V" lifts to be a k-linear map V"->- V
which is Q-invariant, which means that the original sequence splits.
Section 4
Calculation of various cohorrology groups
4.1
ProFOsition: Let r be a rronoid and let  be a caTlIT!Utative
k-, on which the constant k -rronoid r k acts in a manner canpatible with
the qroup structure of M. Then we have canonical isorrorphisms
 (r J!-1) '" Hi (r ,tHk) )
(where the second member is the i th cohorrology group of the rronoid r in the
r-ITOdule M (k) )
proof: By  1, 1.5, we have
Cn(rk,M) = ((rk)n,M.) '" (rn,(k)) = (r,(k))
and the standard ccrnplexes c.(r k ,!1) and c.(r,(k)) are canonically iso-
rrorphic.
4. 2 ProFOsi tion: Let Q be a diagonalizable k -rronoid and
->-(V) a linear representation of G Then we have (,V)= 0 for
n> 0 .
Proof: Take =!2(r\; by 3.4, it is enough to show that L\r:V->-V0k[r]
has a retraction r which is G-invariant. Let p , yE r , be the P rojections
- y
associated with the grading of V (2, 2.5); set r(Lv Q9y)=LP (v) . Then
y y y
we have ro = I and L\r0r = (r 0A) 0 (V 06A) with A =k[r ] .
Remark: When  is a group and k is a field, it is sufficient to invoke
3.7 and  2, 2.5.
4.3
Corollary: Let G be an affine k -rronoid . Sup:[X)se there is a

II,  3, no 4
HOCHSCHILD COHOIDLC'GY
239
faithfully flat k' E such that 0kk' is diagonalizable . Let .Q.->-bM
be a linear representation of G Then  (, V) = 0 for n > 0 .
Proof: Irrmediate from 4.2 and 3.6.
4.4 We nCM proceed to the cohorrology of  acting trivially on the
k-ITOdule k. The affine algebra of  is kLx], which irrmediately yields
the standard COITg?lex. We have
(ak,k) = kLx 1 ,...,x n ]
if P(Xl,...,Xn)E(,\,k) , we have
(()) (Xl'... ,X n + l ) = P (X 2 '... ,X n + l )
n .
+ L (-1)l p (X 1 ,...,x,+x. l '...'X 1 )
i=l 1 1+ n+
n+l
+ (-1) P(X 1 ,...,X n )
Hence
o
H (,k)
J(ak,k)
1
z (ak,k)  k(ak,ak)
 k
ProFOsition: a) If k is an algebra over II), then the ring k (ak,a k )
is isorrorphic to k corresponding to A E k , we have the horrothetic map
x >-+AX, xEa(R)=R, RE
b) If k is an algebra over iFp , with p prime , the ring k (ak,a k ) is
isorrorphic to the non-corrmutative ring of polynanials kCF], where FA =APP
for A E k . CorresFOnding to A E k we have the horrothetic map x '->- AX , and
to F the Frobenius endorrorphism x I->- Y .
Proof: We knCM that
<1rk(')  {pEkLx]: p(X+y) =P(X)+P(y)}
By derivation, we obtain P' (X+y) = p' (X) . Hence P' is a constant a and
P =aX +Q , where Q'= 0 and Q(X+Y)= Q(X)+Q(Y) . If k is an algebra over
'l:l , we have Q=O, P=aX and the ring in question may be identified with

240
ALGEBRAIC GROUPS
II,  3, no 4
k . If k is an algebra over IF , we have Q (X) = R (X P ) and R (X+Y) =
P
R (X) + R (y) By induction on the degree, we nay assume that we have shown that
n-l
R = alx+a2xP+...+anXP
so that
n
P = aOx +a 1 x P +...+ anxP
By assigning aEk to the FOlynomial aX and F to the FOlynanial xP , we
obtain the required isorrorphism.
4.5
Corollary: There is a canonical isorrorphism of k -rronoids
End Gr (0.(2) '" Q
4.6 By definition, (ak,k) nay be identified with ({,ak)
and so naturally carries the structure of a left ITOdule over 'lr k (ak') .
If k is an algebra over IF , this structure behaves in such a way that, if
P
P(X 1 ,...,X n ) E (ak,k) ,
then
(FP) (Xl'... ,X n ) = P(X 1 ,... ,Xn)p
The boundary operators are accordingly kCFJ.-linear, so that Zn(ak,k) ,
Bn('l<,k) and rf(ak,k) are all kCFJ.-ITOdules. If P is a prime number and
0< i < P , let us agree to write {J?} for the integer (J?) /p . We also write
l l
WE :iCx, Y ] for the FOlynanial
p-l , . 1
W(X,Y) =) {l}xlyP-l = - p ((x+y)p-XP-Yp)
l=l
and W (X, y) for the image of W (X, Y)
in IF Lx,Y]
P
Theorem: SUPFOse that k is a field . Then
o , we have
2
H (ak,k) = 0
a)
if k
is of characteristic
b) if k is of characteristic p> 0 , the FOlynanials
r
W(X, Y) and xy P (r > 0)
2
f0 rm a basis for a kCFJ -ITOdule ccxrplement for B (ak,k)
2
in Z (ak,k) and

II,  3, no 4
HOCHSCHILD COHCM)UX;Y
241
the jX)lynomial W(X,y) is a basis for a k [F] -ITOdule ccmplement for
2 2
B (ak,k) in Zs(ak,k)
Proof: Since the boundary operator is harogeneous with respect to the total
degree, z2(a ki k) is a graded subspace of k[X,y] , so that it is sufficient
to consider the harogeneous canponents. Thus let
n _J1-1 __J1-1_J1
P = aOx +a 1 x Y+...+a n _ 1 xy +any,
n>O
be a FOlynomial such that
(*)
P(X,Y)+P(X+Y,Z) = P(X,Y+Z)+P(Y,Z)
By derivation with respect to X and setting X = 0 , we get
P(O,y)+ P(Y,Z) = P(O,Y +Z) ,
i.e.
P(X,Y) = an_l[(x+y)n-l_-l]
By derivation with respect to Z and setting Z = 0 , we get
P(X +Y,O) = P(X,Y)+ P(Y,O) ,
Le.
P(X,y) = a 1 [(x+y)n-l_ r -l]
If we nCM invoke Euler's formula, we see that
(**)
nP = XP' + yp' = a [(X + Y) n _ x n _ ynJ
X Y 1
+ (a n _ 1 -a 1 ) [X(x+y)n-l_]
We nCM distinguish several cases:
a) a 1 an_l . It follCMS frpm (**)
rence of a cocycle and a coboundary,
(*) and X by -Y yields
(Y+Z)n-l_r-l_zn-l = 0
n-l _J1
that Q = X(X +Y) -x
is the diffe-
hence a cocycle. Replacing P by Q in
If p = 0 , this last formula irrplies that n = 2 , which contradicts air! a n - 1 .
If p  0 , the formula irrplies that n = 1 + pr ; hence n is invertible
ITOdul0 p and (**) irrplies that P is cohanologous to (a l -a n _ 1 )Q/n

242
ALGEBRAIC GROUPS
II,3,n04
But
r r r
Q = X (X + Y) p - xP + 1 = xy P
b) a l =a n - l '10 and p=O or nO ITOdp . By (**) we then have
P = a l ((X +y) n_ i?-) /n and P is a coboundary.
c) a l = an-l 0, p '10 and n == 0 ITOd p . In this case the binanial coeffi-
cient
( n-l )
p-l
(n-l) (n-2) . . . (n-p+ 1)
1. 2... (p-l)
n-l _Jl-l
is congruent to :H ITOd p . If n 'I p , it follows that a 1 ( (X + y) - y )
contains the term :t:a l 2f- P yP-l , which is absurd since the forrrer expression
p-l p-l -
is P recisel y P y ' (X, y) . Hence n = p so that P' =a ((X+y) - Y ) = a W'
, y 1 lY
Similarly Px =a 1 w X ' so that the partial derivatives of p-aiw vanish.
Hence P = a 1 W + aoxP + apYP but this can only be a cocycle if a O = a p = 0
d) a = a = 0 P 'I 0 . In this case p' = P' = 0 so that P = P i (Xp , yp)
1 n-l ' X Y
where Pi is a cocycle of degree nip < n (if P '10) .
By induction on the degree of the FOlynanial, it follows from the preceding
2
discussion that the FOlynanials in question do indeed generate, ITOd B (ak,k)
2
the kCF J-ITOdule Z (ak,k) .
M:>reover, the F S (xy V ) (r > 0 , s  0) and the FSW (s  0) are all of diffe-
rent degrees. Finally, since the xypr are not syrrrnetric, they cannot be
coboundaries, and FSW is not a coboundary since the coboundary of
xpsE k[X] '" C 1 (ak,k) vanishes. This completes the proof.
4.7 ProFOsition: SUPFOse that k is a field , and let B- be a
closed subqroup of O.k. Then the canonical maps J(ak,k)->-Him,k) , iO
2 2
and Hs(ak,k)->-Hs(g,k) are surjective.
 Since the assertion is trivial for tl =a k ' assume that tlf;a k . Let
I be the ideal of k[T] =A defining g, and let n be the degree of a
FOlynomial P such that 1= (P) Let UCk[T] be the k-vector subspace of
k[T] generated by 1,T,... ,r- 1 . The canonicl map U ->-A ->-A/I is bijective
and LlUCU@U . By what we have'already proved, zi(ak,k) is the kernel of
"i . A @i A @(i+l) S . ' 1 1 Z i (H k) . th k 1 f the l ' ndUCed map
o.  . 1JtU ar y, _, lS e erne 0
a i : (A/I)@i ->- (A/I)@(i+l)

II,  3, no 4
HOCHSCHIW COHOMJUX;Y
243
Let :XE Zi(H,k)C(A/I)@i . Replacing x by xEU@i , we have aixEu@i+l
and aix=OITOdI, so that aix=o and XEZi(ak,k) . This proves the first
assertion. If i = 2 and x is syrrmetric, so is x, which proves the second
assertion.
4.8 Corollary: SUPFOse k is a field of characteristic p¥'O .
2
Then H (a k ,k) is a k- vector space of dimension 1 generated by the class
- p
of W.
2
Proof: By 4.6 and 4.7 H (p a k ,k) is generated by the class of W. This
class cannot be zero because the coboundaries belonging to
k[x]/(XP)@k[x]/(X p ) are all of degree < p
2
4.9 Remark: Clearly W(X,Y)Ez (a,a) Henceforth we shall write
s
!Y 2 for the Z'-group whose underlying Z'-scheme is Q2 and is such that
(x,y) + (x' ,y') = (x +x' , y +y' -W(x,x'))
for REl1 and x,y,x' ,y'E R . By 4.6 a), we have 21Q '" a

 4
DIFFERENTIAL CALCULUS ON GROUP SCHEMES
Section 1
Infinitesimal FOints of a group-functor
1.1 Let R be a ring; if R[T] is the algebra of FOlynanials in
2
T , write E for the class of T mcxl T and R(E) for the quotient algebra
R[T ]/ (T 2 ) , which is called the algebra of dual numbers over R. We have
a deccxrposi tion R (E) = REf) E R and ham::m:>rphisms i: R.... R (E) , p: R (E).... R ,
defined by i(l)=l , p(l)=l , p(E)=O , such that pi=I.
u (1)=1 , u (E)=aE ; we have pu =p
a a a
a 1-+ u a is a ham::m:>rphism of the rronoid
of the R-algebra R(E) .
there is an endanorphism u of R(E) such that
a
and uai = i . MJreover, the map
R X into the rronoid of endorrorphisms
Associated with each aE R
1. 2 Let  be a k-group-functor. For each RE  ' consider the
hcm:m:>rphisms G(i): G(R)....(R(E)) and (p): (R(E))....(R) . Let Lie (G) (R)
be the kernel of (p) . Since pi =I ' we have a split exact sequence
i
1 .... Lie () (R) .... G(R(E)) :: (R) .... 1
p
in which we have simply written :j, and E for (i) and G-(p) . We now de-
fine a (R) -operation on Lie (G) (R) by setting, for each gEG-(R) and
xE Lie () (R) ,
Ad (g)x = 1 (g)xi(g)-l .
Similarly, for each aE R , the haroIrorphism u : R(E)....R(E) induces an endo-
a
rrorphism (ua) of (R(E)) which is compatible with 1? and i, hence an
endorrorphism Lie () (u a ) of the group Lie () (R) . We abbreviate simply to
u the haranorphism a J-+ Lie (G) (u) of R into the endorrorphism rronoid of
-- a
the group Lie () (R) . For xE Lie (G) (R) we set ax = u (a) (x) .
The ThD operations defined above preserve the group structure of Lie () (R)
and are compatible with one another: if gE(R) , x,x'E Lie () (R) and
aE R , we have
Ad (g) . (xx')
( Ad (g). x) ( Ad (g). x')
a(xx')
(ax) (ax')
Ad (g) (ax) = a( Ad (g)x)

II,  4, no 1
DIFFERENTIAL CALCULUS
245
Since all of the above constructions are functorial in RE t\ ' we have in
fact defined a k- group-functor Lie () , as well as operations
gx Lie () .... Lie ()
Qkx Lie () .... Lie ()
which are ccxrpatible both with one another and with the group law of Lie ()
1. 3 Let  and Ii be k-group-functors and f: ....!i a horrorrorphism.
The harorrorphisms K(R(E:)):(R(E:))""(R(E:)) and f(R): (R)""!i(R) are canpat-
ible with the rrorphisms :i, and E relative to  and tl. It follows that
.t: (R (E) ) induces a horrarorphism
Lie (f) (R) : Lie () (R) .... Lie (B) (R)
and a haromorphism of split exact sequences:
1 .... Lie(G) (R) .... G(R(E))
Lie ( f) (R )[ f(Rc))l
1 .... Lie (B) (R) .... B(R(E))
i
,:f
1
:B(R)
p
.... 1 .
.... 1 '
In particular we obtain the formulas:
and
Lie (K) ( Ad (g) .x) = Ad (f(g)) . Lie (f) (x)
Lie (f) (ax) = a Lie (f) (x)
for gE(R) , xE Lie (g) (R) and aER
1.4 If k' E, we have
Lie ( _ G iSik') = Lie (G)iSik'
- -k
by 1.2 and I,  1, 6.5. We then verify that the operations Ad and u
relative to the group  iSikk' nay be obtained from those of the group  be
extension of scalars.
1. 5 Finally, we see inmediately that the functor Lie transforms
prcxlucts of k-group-functors into prccb lctS of k-group-functors. MJreover, if
l....HK

246
AIGEBRAIC GROUPS
II,  11, no 2
is an exact se:]Uence of k-group-functors (i.e. for each RE,\\ , the sequence
1 -;.  (R) iiR) tl (R) glR) !S (R)
is exact), then the sequence
1 -;. Lie (g)  (f) Lie (H)  (g) Lie (!S)
is exact.
Section 2
Examples
2.1 M:rlules . Let .!'1 be an Qk-ITOdule ( 1, 2.5). For each REl\
consider the map : x r-+E(E:) of l:l(R) into N(R(E)) . Clearly
E (E) E Lie (N) (R) . As R ranges through  ' we obtain a rrorphism of k-
group-functors
 : N -;. Lie (tl)
which is canpatible with the actions of Q on 11 and Lie () . This rror-
phism g is evidently an isarorphism if 11 = V  or N = 12 (V) , for a k-module
V.
2.2 The linear group. Let  be an Q k -module, 1(N) and GL ()
the k-functors defined in  1, 2.5. For each R E  ' let E R be the nap
f>->-Id+dR(E) of 1() (R) into 1() (R(E)) . Clearly Id+dR(E) is the
inverse of Id - d R (E:) and the latter belongs to the kernel of
GL (11) (p) : GL (M) (R(E:)) -;. GL () (R)
In this way we define a rrorphism of k-functors ];;:1 (N) -;. Lie ( GL (N)) . When M-
is of the form V or 12£ (V) , we deduce fran  1, 2.5 that the hararorphism
1 () (i) : 1 (N) (R) -;. 1 () (R (E:) )
induces an isarorphism
1(!1) (R)R(E) -+ 1(!:P (R(E)) .
This implies the last assertion of the following proposition; the other asser-
tions are triviaL
ProFOsition: L-=t M be an 9k -ITOdule . For RE ' x,x'E1() (R) ,

II,  4, no 2
DIFFERENTIAL CAICULUS
247
gE  (!':!) (R) and aE R , we have the formulas
E (x + x') = J:;; (x) 0 E (x I )
a (J:;;(x) ) = (ax)
Ad (g)E(x) = (g.x.g-l)
If V is a k -ITOdule and if l:1 is isanorphic to Va or to !2 a (V) , then
E;:f:. () ->- Lie ( GL () ) is an isorrorphism.
2.3 Autorrorphisms of an algebra . Let A be a (not necessarily
associative) k-algebra. Consider the sub-group-functor Aut (A) of GL (A)
( 1, 2.6) . By the above discussion, Lie ( Aut (A)) nay be identified with
. the subfunctor f of f:. (A) such that
xE f (A) (R) <=> E; (x) E Aut (A) (R (E:) )
If a,bE MSi R and xE :k(A) (R) , we have in A 0R (E)
g;(x) (a.b) = a.b+Ex(a.b) ,
(x) (a) .E;.(x) (b) = (a + EX (a)) . (b + EX (b))
= a.b+E:(a.x(b)+x(a) .b)
This irrplies the
ProFOsition: Let A be a ( not necessarily associative ) k -alqebra . Let
Der (A) be the subfunctor of (A) such that Der (A) (R) is the set of deri -
vations of the R- algebra A 0kR . Then the isanorphism E;.:  (A) ->- Lie ( GL (A) )
of 2.2 induces an isanorphism Der (A) '" Lie ( Aut (A) )
2.4
Autarorphisms of a scheme . Let K be a k-functor and let
 E Lie ( Aut (K)) (k)
For each k (E) -ITOdel R we thus have a permutation of "f>. (R) (also denoted
by ) which reduces to the identity when ER = 0 . If f:f': ->-Qk is a function,
SE and xEK(S) , then f(XS(E)) is of the form a+Eb with a,bES
Setting E = 0, we obtain a =! (x) . Setting b = (D) (x) , we have
accordingly
X
f(XS(E)) = (x)+ E(Dt:) (x) .
Since this formula is functorial with respect to S, we see that the maps

248
AIGEBRAIC GROUPS
II,  4, no 2
X ->- (Dt:) (x) define a new function D: K ->-  . Moreover, one verifies
easily that the operator :  1-+ Dt: is a k-derivation of the algebra iJ()
We now turn our attention to the action of  on the geanetric realization
I Q9k k (E:)j of the k (E) -functor  Q9 k k (E:) . Given a geanetric k-space T , let
T (E) be the geanetric k (E) -space which has the same underlying space as T
and satisfies T(E:)= 8 T (E)= c?TEbEdiT . If T= Ix I , it is easy to see that
there is a canonical isorrorphism 1"f>. Q9 k k(E) I .:;: II (E) : if AEl\ ' the in-
clusion map A ->- A (E) induces a homeom::>rphism i of Spec A (E) onto Spec A
(the prime ideals of A(E) = AEti E:A are of the form p EtiEA , where pE Spec A ) ,
and an isorrorphism tJ spec A (E)':;: i* (l!I spec A (E) ) . Since the functors
I-+ IKQ9 k k(E) I and  1-+ IKI (E) ccnmute with direct limits, the argument of
I,  1, 4.1 shows that there is a unique isorrorphism of functors
j () : IKQ9 k k(E) ->- IKI (E:) such that, for each AE ' j (Spec A) is the can-
FOsi te isorrorphism
\ SPkA Q9kk(E) I .:;: lk(E)A(E:) I .:;: SpecA(E)':;: (SpecA) (E:)
(I,  1, 6.5 and 4.1) .
MJre generally, let Q. be an open subfunctor of K. Since, by the displayed
formula irrmediatelyabove, QQ9 k k(E:) and Q obviously have the same space
of FOints,  induces the identity on the space of FOints P of Q Q9k k (E) .
Since, with the notation of  1, 4.10, we have QQ9 k k(E:) = (Q9kk(E))p' we
see that  induces an autarorphism  (Q) of QQ9 k k(E) such that
 (Q)Q9k (E) k = Id. This defines a k-derivation D¥ (Q) of J(Q) . By varying
Q , we obtain a k-derivation D<jJ of the structure sheaf cOx of X.
Pror::osition: For each k- functor  , each aE k , each pair
E Aut () (k) , we have
Xn -1 X 0
DAd (u)  = v(g) 0 D. "tw
,. E Lie ( Aut ) (k)
and each
Do' = D +D<jJ' ,
Da = aD '
If Z is a scheme (resp. an affine scheme) , the map <jJ J..->- D  (resp. <jJ  D;
is a bijection of Lie ( Aut ) (k) , onto the set of k -derivations of J)x
(resp. of cD() ).
Proof (sketch): We Jrerel y give the inverse of the map  r+ D  in the case
in which X is a scheme. To each k-derivation D of cf)x we assign the

II,  4, no 2
DIFFERENTIAL CALCULUS
249
autorrorphism 1jJ of rV x (E) =  + EJ1 X such that 1jJ (a + Eb) = a + E (b + Da) where
Q is an open subsc of - and - a,bE (? X(!J) . The required autanorphism
 of X@kk(E) induces the identity on the underlying toFOlogical space of
1@kk(E) I -+ 1"f>.1 (E) and the autorrorphism 1jJ of the structure sheaf c.9(E)
Let Der(t9 x ) be the k-ITOdule formed by the k-derivations of the sheaf L9x'
where  iascheme. If RE and dE Der(c.9 x ) , define a derivation 
of the sheaf of R-algebras L!7X@R as follows: if g is an affine open sub-
scheme of , then i.2 (Q @kR) ;;;- J!(Q)@kR and  (g @kR) is obtained from d (Q)
by an extension of scalars. We may therefore define a functor Der () such
that Der () (R) = Der(@R) . The proFOsition then implies the existence of
a canonical isorrorphism of k-functors
Lie ( Aut ()) " Der () ,
which is ccmpatible with the group laws and the action of .Q
2.5 Groups of invariants . Let  and tl be kroup-functors and
let f:tl->- AutGr (G) be a hanorrorphism. For each RE and each hE!i(R) ,
f (h) is an autorrorphism of the R-group  @k R , so that Lie (f (h) ) is an
autarorphism of Lie (@k R) " Lie () @k R . Fran this we derive a haranorphism
fi (R) ->- Au (Lie () @kR) and by varying R we get a haranorphism
!i ->- Aut ( Lie ()) . Since the actions of !i (R) preserve the group structure of
Lie ()@kR , this horrorrorphism factors through Aut Gr ( Lie (G)) .
ProFOsition: Let  and !i be k roup-functors and let !: ->- Aut G r () be
a haranorphism . Then gB is a sub-qroup-functor of  and Lie (gH ) = Lie (G) fi .
Proof: It is enough to show that Lie (#) (k) = Lie (9) B (k) since k is arbi-
trary. Now by  1, 3.5 we have
(k(E)) = {gEG(k(E:)):f(h)R(E)gR(E)
gR(E) , hE tl(R), RE}
This gives
Lie () (k) = Lie () (k) n (k(E))
= {xE Lie (G) (k) : f(h)= , hEfi(R), RE}
= Lie (G) fi (k)

250
ALGEBRAIC GROUPS
II,  4, no 3
Section 3
Infinitesimal FOints of a group scheme
3.1 Consider a k-scheme X and a yE X (k) . Let i be the section
of X associated with y, Le. the rrorphism y#:  ->-1' . This rrorphism is a
section of the canonical projection )2:1' ->- ' and is accordingly an embedding
(I,  2, 7.6 b)). write w for the k-rnodule w, (e.) formed b y . .the sections
y !o -k
of the ITOdule w. of the embedding i (I,  4, 1. 3). If A EM. and
l - MK
Y = Sp A , there is a canonical isano rp hism w '" 1/1 2 , where I is the kernel
- - y
of y:A ->- k . If k is a field, y may be identified with a rational FOint
of y (I,  3, 6.8) and w with m / m 2 .
y y
Returning nCM to the general case, let Xi be the first neighbourhood of i
ip X and let

:1: 1
->- y,
-J:
:h2
->- X
be the canonical factoring of ;!.
rrorphism of  onto the closed
vanishing squ are, Y. is affine
-b.
be identified with the algebra
multiplication such that
(I,  4, 1.1). Since 1 1 induces an iso-
subscheme of Xi defined by an ideal of
(I,  2, 8.1) .-By I,  4, 1.5, <J(y,) may
-l
k Cbw , if we assign the k-rnodule k Cbw the
y y
(A,s) (A',s') = (n' , As'+A's)
for A , A ' E k, s C" I E w . The rro rp hism i l : e. ->- y ,
,<" y - -k -!
associated with the haranorphisms (A,s) >->- A of
h->- (1..,0) of k in kEBw y .
and 12.1 2 : ¥:j, ->-  are
kE17 wink, and
Y
The construction of w is functorial with respect to the " FOinted scheme"
y #
(X,y) . For let X' beak-scheme, y'ET(k) i'=y' and ?I:1"->-X a
rrorphism such that g (y' ) = y . By I,  4, 1. 3, g induces a rrorphism
w. ->-w.. hence a map of w =W. (e.) into w ,=w" (e.) which we denote by
J, b. Y b. -k Y l k
W . Assumin g this notation, the hornorro rp hism c9(y,)->-c9(y) associated with
g -! -J,
the rrorphism Xi ->- X b induced by ?I sends (A, s) E k Cb w y onto
(A ,w g (s) ) E k <tJ w,.
- y
Furthennore, it is clear, that the maps w : w ->- w ( , ) and
EEl Y y,y
w : w , ->- w ( , ) induce an isanorphism w E!] w 1-+ w ( , ) , where
pr 2 y y,y y y y,y
(y , y') E X (k) x X' (k) = (X x X') (k) .

II,  4, no 3
DIFFERENTIAL CALCULUS
251
3.2 With the notation of 3.1, let J be an ideal of k of vanish-
ing square . To each k-linear map d: W ->- J we assign the haroIrorphism
y
kwy->-k which sends (A,i;) onto Hd(i;), next, the corresFOnding rrorphism
d':->-¥i of , and finally the element ay(d) of X(k) associated
with the canposi tion

g'
> Y.
-
i 2
> Y
The map
a : M-rl (w ,J) ->- Y(k)
y 1<. Y -
thus defined is functorial in (;{ , y)
comnutative square
with the notation of 3.1, we have a
f\h1_ (w "J)
;-'=""1<. Y
 (W1,k) \
f\h1 (w ,J)
;';;;"'1<. Y
a y .
> ¥. (k)
1 (k)
 ¥(k)
ProFOsition: Let '.i be a k -scheme , yE I(k) , J an ideal of k of vanish -
ing square and q: (k) ->-  (k /J) the map induced by the canonical projection
k ->- k/J . Then a : Mal (w ,J) ->- Y (k) induces an isomorphism of !>bd k (w ,J)
-1- Y k Y -  y
onto q (q(y)).
Proof: The problem is a special case of the situation discussed in I,  4,
1.5. We merely give the inverse map of q-l(q(y)) into (Wy,J) . Let
zE q -1 (q (y)) ; since  coincides with the first neighbourhood of
Sp (k/J) ->-  k in Sp k =  ' z #:  ->- '.i factors through ¥ i ' hence
zEY.(k)=M. (kEBw ,k) . Evide.ntly z(w )cJ , and so the required inverse map
- ""K Y Y
assigns to z the induced map of w into J .
Y
3.3 If RE£:\ ' we may apply the preceding prorosition to the
R(E)-scheme ¥ Q9kR (E) , the canonical image t=YR(E) of y in (R(E))
and the ideal ER of R(E) . Now w t may be identified with WyQ9kR(E)
(apply I,  4, 1.6 to the case in which f. is the canonical projection
 R(E)->- Sp k) . Accordingly (E) (Wt'ER) may be identified with
 (E) (W y Q9k R (E),ER) , hence with (wy,R) . Thus we obtain the

252
ALGEBRAIC GROUPS
II,  4, no 3
Corollary: Le! :£ be a k- scheme, REl\ '
ated with the horrorrorphism a+bu>-a of R(E:)
-1
there is a bijection  (Wy,R)':; q (q (y) )
(,y) .
q:X(R(E:))->- X(R) the map associ -
into R and yE Y (k) . Then
which is functorial in R and
Let us recall the definition of this bijection: let d:w ->-R be a k-linear
y
map and let d": kEbw ->-R(E:) be the haranorphism such that d"(;\.,s) =
a+d (s) E: . If DE q -1  (y) ) is the image of d under this bijection, D # is
the canposition
R(E:)
d"
, y,
-;h
, x
If R = k ,
-1
we call q (q (y) )
the tanqent space to X at y. When k is
a field, this space may, by the corollary, be identified with the Zariski
tangent space Mcit (w ,k) (I,  4, 4.15).
K Y
3.4 Now consider a k-group g. Let e be the unit element of g(k)
and let E: = E: =e# be the unit section of g. Set wg/k =w e =wE: () . Simi-
larly, if f:g->-.tr is a h::Jrrarorphism of k-groups, we write w Uk for w f
(3.1) .
ProFOsition: If  is a k -group , .!2: ->-  is the canonical projection , and
G/k is the sheaf of differentials of g over k, we have canonical iso -
rrorphisms
WG/k .:; E:g (g/k) and G/k':; Pg (W g / k ) ,
where
Wg/k is the quasicoherent ITOdule over  associated with w g / k .
Proof: Since wg/k =wE: () , w G / k may be identified with the ITOdule wE:
of the embedding E:. The first fornula then follows from I,  4, 2.2. The
second follows fran I,  4, 1.6 and fran the Cartesian square
GxG y > G
-IO/k Pg -f E:g
g > 
in which °G/k (g) = (g ,g) and -1 for REJ:\ and g ,g' E (R)
y(g,g')= g'g

II,  4, no 3
DIFFERENTIAL CALCULUS
253
3.5 Theorem on infinitesimal FOints: Let .Q. be a k -group and let
J be an ideal of k of vanishing square . Then the map a e :  (wGIk,J)-+ g(k)
of 3.2 is a group haranorphism and the sequence
a e
o -+  (wglk,J) -+ g(k) -+ g(k/J)
is exact.
Proof: The second assertion follows fran 3.2. To establish the first, notice
that, by 3.1, the functor (Y,y) r---->!Vhl (w ,k) CCXImUtes with products. We in-
- K Y
fer as usual (cf.  1, 1.5) that, if  is a k-group, the k-linear map
 (w'TT ,k) imposes a group structure on  (wG / k,k) in such a way that
G -
a is an isorrorphism. The following lemna shows that !Vhl ( w 'TT ,k) coincides
e --=-K G
with the natural addition in  (wg/k,k) : -
 Let M be a set equipped with a law of canposition possessing a unit
element and let m:MxM-+M be a harorrorphism . If M contains an elenent e
such that m(e,x)=m(x,e)=x for all xCM , then M i s COIm1Utative and
m(x,y)=xy for all x,ycM
Prcof: We have x =m(x,e)=m( (l,e) (x,l) )=m(l,e)m(x,l)= l.m(x,l)=m(x,l)
similarly m(l,y)= y , so that m(x,y)=m(x,l)m(l,y)=xy ; similarly
m(x,y)= m(l,y)m(x,l) =yx .
3.6 Corollary: Let .Q. be a k. Then for each RC '
Lie (g) (R) , equipped with the R-operation defined in 1.2 , is an R -ITOdule
canonically isanorphic to Qg (wG/k) (R) =  (Wg/k,R)
Prcof: Clearly the iscrrorphism of 3.3, where X= and y=e, is ccxrpatible
with the actions of R.
The k-functor Lie (g) , equipped with the -operation defined in 1.2 is
accordingly an -rrodule ( 1, 2.5) canonically iscrrorphic to Qa (w G / k ) .
If f: g -+ £! is a horranorphism of k-groups, Lie (f) may be identified with
IJ (w Yk )
3.7
We now introduce a notion which will facilitate our calculations
a great deaL Let us write the group law of Lie (g) additively . If RCl\ '
SC and a is an element of S of vanishing square, there is a unique

254
ALGEBRAIC GROUPS
II,  4, no 4
hcm:xrorphism of R-algebras R (E:) ->- S which sends E onto a . The image of
of xE Lie (g) (R) under the canposite hoIroJrorphism
Lie (g) (R) ->- g(R(E)) ->- (S)
will be written
ax
e
(in g (R (E:)) , we have x = e EX :).
For x,yE Lie (G) (R)
we thus have in g (S)
(1) eO. (x+y)
ax ay
= e e ,
while the definition of the external law of Lie (9) (R) may be written as
follows: for xE Lie (g) (R) and aER , we have in Q.(R(E)) ,
(2)
(Ea)x E(ax)
e = e
If !: G ->- H is a hcm:xrorphism of k-group-schemes, if x E Lie (g) (R) and if S
is an R-ITOdel with an element a of vanishing square, we have in l!(S)
(3)
!. (e ax ) = eO. ( Lie (:1=:) x)
Section 4
The Lie algebra of a group-scheme
Let  be a k-group-scheme. We shall assign to Lie (g) the structure of an
"Qk-Lie algebra" .
4.1 The adjoint representation . Let RE and gEg(R) . The auto-
rrorphism Ad (g) of Lie () (R) defined in 1. 2 preserves the R-ITOdule struc-
ture of Lie (g) (R) . If we write GL ( Lie (g) ) (cf.  1, 2.5) for the k-group-
functor of linear autorrorphisms of Lie (g) (which, by loco cit., is iso-
rrorphic to the group-functor @, (w 9 / k ) opp) , we derive a hananorphism
Ad : g ->- GL ( Lie (g) ) ,
which we call the adjoint representation of G.
If xE Lie (g) (R) and gE g (R) , we thus have in 9 (R (E:)) ,
(4)
EX -1 E:Ad( g )x
ge g = e -- .
4.2
The bracket . By 2.2 and  1, 2.5, Lie (( Lie (g)) may be

II,  4, no 4
DIFFERENTIAL CALCULUS
255
identified with ( Lie (g))
ad = Lie ( Ad )
hence we get a canonical Qk -ITOdule isorrorphism
Lie (g) ->- I: (Lie (g)) ,
thus a "bilinear" rrorphism Lie (Q)x Lie (Q) ->- Lie (Q) which sends (x, y) onto
(ad x) y for x,yE Lie () (R) and RE  . Set (ad x) y= lx,y] . In virtue of
the identification we made in 2.2, we get the formula
(5)
EX
Ad (e ) = Id +Ead(X)
Le. Ad (eEx)y=y+dx,y], where x,yE Lie (g) (R) and where Ad (e EX ) belongs
to the algebra
I: (Lie g) (R (E)) '" :I:: ( Lie g) (R) Eb EI: ( Lie ) (R) .
ProFOsition: Let REl\ and x,yE Lie (Q) (R) Let SE and let a,B be
two elements of S of vanishinq square . Then in (S) 'We have
(6)
ax By -ax -By aBlx,y].
e e e e =e
Prcof: It is enough to prove the contention when S = R (E, E ') is the R-algebra
generated by twJ elements of vanishing square. Noting that S '" R (E) (E') , 'We
get successively
EX
eEXeE'Ye-EX = eE' Ad (e )y
= eE' (Y+Elx,y])
E'y E'(dx,y])
= e .e
= eE'Y.eEE'lx,y]
= eEE'lx'Y].eE'y
((4) )
((5) )
((1) )
((2) )
which by (1) gives the required result.
-E'y EX E'y -EX
We note in passing that the above prcofs also gives e e e e =
eEE'lx,y] , which by (6) implies that lx,y]=l-y,x]=-ly,x]. To prove this
, . , EE'U EE'V, .
lt lS enough to observe that the equallty e =e lll1phes u =v .
For if cp: R(E:)->-R(E,E') is the rrorphism of  such that cp(E:) =EE' ,
then Spec cp irrluces a homeorrorphism of the underlying spaces of
Spec R(E,E') and Spec R(E), and a rronorrorphism of the structure sheaf of
Spec R (E:) into its direct image in Spec R (E, E '). It follows that Spec cp
is an epirrorphism of k' so that Sp  is an epirrorphism of . In
other \\Drds, for each k-scheme , "f>.(cp) : "f>.(R(E:)) ->-0-(R(E,E')) is an

256
ALGEBRAIC GROUPS
II,  4, no 4
injection.
For each RE we endow Lie () (R) with the R-algebra structure defined by
the Lie prcxluct. We show later (4.5) that this gives us a Lie algebra . Notice
that it follows fran the definition of the Lie prcxluct (or fran (6) and the
preceding remark) that if t:  ....!i is a hcmarrorphism of k-group-schemes, then
Lie () (R) : Lie (G) (R).... Lie (!i) (R) is an R-algebra haranorphism for each R E  .
4.3 Let P :  .... GL () be a linear representation of f, where M is
of the form V 9 or Q 9 (V) . By 2.2 we get a rrorphism
Lie (p) : Lie () .... ()
which satisfies the following condition: for RE: and xE: Lie () (R) the
equation
(7)
p (e EX ) = Id + E Lie (p) (x)
holds in  () (R (E:) )
have
Using (6) we see inlnediately that in l:() (R(E,E')) we
EE I Cx Y J EX E 'y -EX -E I Y
Id+EE' Lie (p) (Cx,yJ) = pre ') = pre )p(e )p(e )p(e )
= (Id+ E:Lie (p)x) (Id+E' Lie (p)y) (Id- ELie (p)x) (Id-E' Lie (p)y)
Id + EE' ( Lie (p) xLie (p) y - Lie (p) yLie (p) x)
which implies
(8)
Lie (p) [x,y J = ( Lie (p) x) ( Lie (p) y) - ( Lie (p) y) ( Lie ( p) x)
U,ie (p)x, Lie (p)yJ .
For each R E  ' we endow !: () (R) with the Lie algebra structure which un-
derlies its R-algebra structure. From 2.2 and the preceding discussion we
deduce the
ProFOsi tion: Let p :  .... GL () be a linear representation of  , where  is
of the form Va or Qa (V) . Then , for each RE ' Lie (p) (R) is a hcrno-
rrorphism of Lie () (R) - into the Lie algebra  () (R)
In particular, if we take p to be the adjoint representation of , we ob-
tain Cx, y J = [ad (x) , ad (y) J , which, in virtue of the antisynrnetry of the Lie
prcxluct, is pn.'cisely the Jacobi identity:

II,  4, no 4
DIFFERENTIAL CALCULUS
257
(9)
Ux,y],z]+ [[y,z],x]+ [[z,x],y]= 0,
for x,y,zE Lie () (R)
RE .
4.4 Now let  be' a k-functor and let  x  ....  be a Q.-operation
on  , which we write in the form (g ,x) 1-+ P (g) X , so that it is associated
with the hanorrorphism P:"" Aut () Let Der(J/x) be the set of derivations
of the sheaf of k-algebras J) X equipped with it natural k-Lie algebra struc-
ture ([D,E] = Do E-EoD for DEEDer(J7x)) In 2.4 we defined a map
Lie ( Auq p (k)....Der(c9 x ) by canposing this with Lie(p) we get a map
p' : Lie () (k) .... Der (tPK'
By definition, we have accordingly for each open subfunctor Q of , each
fEJ)(Q) , each RE ' each mEg(R) and each xE Lie (Q.) (k) , the following
relation in R(E:) (with the nonnal abuse of notation): f (p (eEX)m) =
f (m) + E (p' (x) f) (m) . If aE R satisfies 0. 2 = 0 , we rray apply to this formula
the haranorphism  :R(E:).... R such that  (a+bE:) = a+ ba where a,bE R . Thus
we obtain
(10) f (p (eax)m) = f (m) + a (p' (x) f) (m)
ProFOsi tion: Let  be a k- functor and let p :  .... Aut ( be a horranorphism.
Then the map p' ',.: Lie (Q.) (k)....Der(JJ x ) defined above is an anti-haranorphism
Qf Lie (<;:.) (k) into the k- Lie alqebra Der (cOX) .
Proof: In virtue of 2.4, it is enough to show that if x,yE Lie (<;:.) (k) , we
have [p' (x) ,pI (y)] =p' ([y,x]) . Let k(E,E') be the ring generated by tv..D
variables of vanishing square. Let 9 be an open subfunctor of "f>., fE cP(g) ,
RE  and mE 11 (R) . By (6) and (10) , we have
EX E'y -EX -E'y
EE' (p' ([x,y]f) (m) = f(p(e e e e )m)-f(m)
By means of a step-by-step calculation using (10) we see that the right-hand
side of this equation becomes
EE' ([p' (y)p' (x)- p' (x)p' (y) ]f) (m)
which implies the required relation, g,f,R having been chosen arbitrarily.

258
ALGEBRAIC GROUPS
II,  4, no 4
4.5 The preceding discussion may be applied to the particular case
in which  acts on itself by translations. Accordingly the haranorphism
y: ->- Aut () (1, 3.3 c)) gives rise to an algebra-antihananorphism
y' : Lie () (k) ->- Der (tP ) . Therefore we have by definition
(11)
f (e EX g ) = f (g) + E (y' (x)f) (g) , where
f(ge Ex ) = f(g)+ E(O' (x) f) (g) , where
y' (x)=D
y(x)
0' (x)= D o(x)
for each open subfunctor .Q of , RE  ' gE  (R) and xE Lie () (k)
Similarly
y' (x+y) = y' (x)+y' (y)
0' (x+y) = O' (x) + 0 I (y)
(12)
y' (ax) = ay' (x)
o I (ax) = ao' (x)
y' ([x,y])= [y' (y), y' (x)] , o' ([x,y])= [0' (x) ,0' (y)]
for x, yE Lie () (k) , aE k .
Since Y:->- Aut (Q) is a rronarorphism, y': Lie () (k)->-Der(J7 f ) is injective.
It follows that Lie () (k) is a k-Lie algebra (it remained to show that
[x,x] = 0 for xE Lie () (k) , while the above argurrent gives a new proof of
the Jacobi identity). Replacing k by a variable RE l\ ' we get the
ProFOsition: For each RE ' Lie () (R) is an R- Lie algebra .
By direct manipulation of (11) and (12) we derive the usual formulas:
(13)
o ' (x) y' (y) = y' (y) 0' (x) ,
(14)
(0' (x)f) (g)
(y' (x)f) (g)
(y' (Ad (g) x) f) (g)
(0' (Ad (g-l) x) f) (g)
4.6 Let dEDer(c9) and let cpE Lie ( Aut ) (k) satisfy d=Dcp
(2.4). For each gE Q(k) , define gd and dg by the formulas
gd=gD = D and dg=D g=D
cp y(g)cpy(g)-l cp o(g)CPo(g)-l
Fran the definit:ion of Dcp we :imnediately obtain the following equivalent

II,  4, no 4
DIFFERENTIAL CALCULUS
259
descriptions of gd and dg: if Q is open in  and fEe J/(Q) , let
fgE 0!(y (g)-\r) and gfEc.D(o(g)-l W be the functions which satisfy (fg) (x) =
f (gx) and (gf) (y) = f (yg) then we have (gd) f = (d (fg)) g -1 and (dg) f =
g -1 (d (gf)) .
Fran the definitions we .inmediately obtain the fonnulas
y' (x)g = y' (x),
go' (x) = 0' (x)
(15)
-1
0' (x)g = 0' (Ad (g )x) , gy' (x) = y' (Ad (g)x)
for gE  (k) and xE Lie (Q) (k) .
A derivation dE Der (cQ G) is said to be left (translation) invariant (resp.
right (translation) invariant ) if, for each RE and each gE (R) , we
have g = (resp. g =) (2.4).
ProFOsition: The map y' : Lie (Q) (k) ->-Der(J7 G ) (resp. 0' : Lie (Q) (k)->- Der(JJ g ) )
induces an antiisomorphism (resp. an isomorphism ) of Lie (Q) (k) onto the sub-
algebra of Der (LJd fonned by the right (resp. left ) translation invariant
derivations.
Proof: Obviously it is enough to prove the assertion for y' . Let  act
on Aut (.Q) as follows: if REt\: ' gE:(R) and uE Aut R(:.iZIkR) , define
the autarorphism gu of giZlR by gu(x)=u(xg)g-l for xEg(s) , SE '
Le. by gu = 0 (g) -lu'& (g) . We quickly infer that the harorrorphism
y :  ->- Aut (.Q) induces an isorrorphism of .Q onto Aut (g) G . For if u (xg -1) g =
u(x) , we have u(e)x=u(x) , so that u=y(u(e)) ; the converse is clear.
It follows fran 2.5 that Lie (y) : Lie (Q) .:; Lie ( Aut (Q) ) induces an isorrorphism
Lie( Q)->- Lie(Aut (Q))Q ; taking the values on k of the t.v..D members, we get the
required assertion.
4.7 Corollary: (a) The following assertions are equivalent for
xE Lie ( ) (k)
(i) y' (x) is right-and-left translation invariant .
(i') 0' (x) is right- and -left translation invariant .
(ii) y' (x) = o' (x)
(iii) xE ( Lie ) G.(k) .
(iv) xE Lie ( Cent (.Q) ) (k)

260
ALGEBRAIC GROUPS
II,  4, no 4
b)
t2

onto this algebra.
The Lie algebra of right-and-left translation invariant derivations of
is caTlIT!Utative and y and 0 induces the same bijection of ( Lie  (k)
Proof: a) follows from (14), (15) and 2.5. (b) follows from (a) and 4.6.
4.8
Lie () (k)
Lie (iZIR)
Let RE  and let SE  . Then the canonical map Lie (9) (R) -+ Lie () (S) is
canpatible with the algebra structure of the t:w,) canponents and with the ring
homorrorphism R -+ S . Fran this we derive a canonical S-algebra hcm:morphism
The Lie algebra of , written Lie() , is the k-algebra
Thus for each RE!:1x we have an algebra isorrorphism Lie () (R) '"
Lie () (R)iZI R S -+ Lie (9) (S)
This hcm:morphism is not bijective in generaL In particular, the canonical
homorrorphism (Lie (9) ) 9: -+ Lie (Q) is not always an isarorphism, the upshot of
which is that the k-algebra Lie () is not in general sufficient to determine
the Qk -algebra Lie ((0 . However, there is an important case in which it is
sufficient:
PrOFOsition: The following conditions are equivalent .
(i) The canonical hcm:morphism Lie ()  -+ Lie ( is an isorrorphism , Le.
the map Lie(G)iZIkR-+ Lie (9) (R) is biiective for each REJ\: .
(ii) The L9 9 -ITOdule 9/k is finitely qenerated and locally free.
(iii) The k-ITOdule w 9 / k is finitely generated and projective .
Proof: (ii)<=> (iii) by proposition 3.4.
(iii)<=> (i) : by proposition 3.6 and Alg. II,  5, prop. 8.
In particular, the above conditions hold when:
(a) k is a field and  is locally algebraic over k (by (iii))
(b) when  is srrooth over k (by (ii) and (I,  4, 4.13)) .
4.9
Let us sum up these results in one particular case:
Suppose that k is a field. To each locally algebraic k-group  we assign
functorially a finite dimensional k-Lie algebra Lie(9)=5L ' a linear

II,  4, no 4
DIFFERENTIAL CALCULUS
261
representation Ad :-+ GL (9:) and, for each RE ' a map x-+e EX of <pZiR
into Q(R(E)) . These assignments satisfy (1), (2), (5) and (6); rroreover,
for each REk\. ' the sequence 1 -+ 2<SiR -+ Q(R(E) -+ .G.(R) -+ 1 is exact.
4.10 Example 1: the additive group. Take G.=a k . By 2.1 Lie ()
may be identified wi th k, where e EX = EX for xE RE t\. ; the Lie algebra
of a k is the commutative Lie algebra k.
Let p :a k -+ g, (V) be a linear representation of a k in a k-ITOdule V. Let
us determine the horranorphism Lie ( p) of 4.3. By  2, 2.6 there are endo-
rrorphisms p. of V such that p(t)v=Itip. (v) for tEREM . Applying
l l 
this formula with t = e EX = EX , we get p (e EX ) = Id + EXPl . Identifying
Lie ( GL (V)) with b.(V) by means of formula (7) of 4.3 we get Lie (p)x=xPl
In particular, if k is a Q- algebra we obtain by  2, 2. 6a) ,
p(t) = exp Lie (p) (t) , tEREt\.
If k is an IF -algebra, similar arguments apply to the group a k : its Lie
p 
algebra may be identified with k, and if p is a linear representation of
k ' we have, for each tE RE such that t P = 0, P (t) = exp Lie (p) (t)
4.11 Example 2: diagonalizable groups . Take Q = Q (r)k ' where r
is a small caTlIT!Utative group ( 1, 2.8). For RE we have
*
R(E:) = {a + Eb : aE R* , bE R} ,
which immediately yields an exact sequence
0-+ @"'(r,R)  Q(r) (R(E:)) -+ 12(f) (R) -+ 1 ,
in which u(b)= l+Eb . It follows that Lie (Q) (R) is isanorphic to Gr(r ,R)
and e EX = 1 + EX . Accordingly we have wG/k '" r <Sizk in particular, the
Lie algebra of Ilk may be identified with k .
4.12 Example 3 : linear groups . Let V be a finitely generated pro-
jective k-ITOdule; take = GL (V) . By 2.2 and 4.3, Lie (Q) (R) may be natu-
EX
rally identified with the Lie algebra  (V) (R) , with e = Id + EX , the ad-
-1
joint representation being given by Ad (g) X = goxog .
If H is a sub-group-scheme of Q, Lie (H) (R) may be identified with the

262
ALGEBRAIC GROUPS
II,  4, no 5
Lie subalgebra of fo(V) (R) consisting of all x for which Id+ExEIHR(E:)).
For instance, if li = SL (V) , then xE Lie (!iJ (R) iff det (Id + EX) = 1 , which
is equivalent to Tr (x) = 0
Section 5
Differential operators
In this section, for each k-scheme '1,  denotes the category of sheaves
of k-ITOdules over I X I .
5.1 Let !: ->'1 be a rrorphism of k-schemes and let d: J7 y -> f* (cOx)
be a rrorphism of  . For each open subscheme 11 of x: and each <jJ E  @
denote by (ad<jJ)d the element of Ab(du,t*(J1x) Iw such that
((ad <jJ)d) (x)=d(x)-d(<jJx) for xEc9(y)  YC:Q (it being assumed that
!* (JJ x ) is assigned the structure of a ITOdule over t!J y defined by the horro-
rrorphism I! I! : t9 y -> !* (J7 (;) induced by f).
Definition: A k-deviation of order :£ n , with origin y and target X is
 (!,d) consisting of a rrorphism !:->X- of  and a
dE(JJy,;f*(J)x)) such that
- -
(ad<jJo) (adl) ... (ad<jJn)d = 0
for each open subscheme Q of Y and all sequences <jJ 0' .. . ,  n of J! (ill .
We call (t ,d) a k- deviation , or simply a deviation , if there is a natural
number n such that (;f,d) is a k-deviation of order :£ n . We also say that
d is a k-deviation of f, and write d for (;f,d) whenever there is no
FOssibility of confusion. Finally, for each rrorphism f of , fO de-
notes the k-deviation (!,Itl t ) (I,  1, 1.4). Like rrorphisms, deviations
will be represented by arrows: X (f ,dJ {; or (! ,d) : x: ->  .
Let (f,d) :X-> and (2,e): -> x: be deviatiOns of order :£n and :£p
respectively. Write de E IW (J) z.' () * (t!J ) for the rrorphism which assigns
to each open subscheme Q <?f Z. the canposi te map
-1
t!J(g) e(g) ;>t .9(<;!-1(g)) d(<;! (Q)) ) O(!-1(2- 1 (Q))

II,  4, no 5
DIFFERENTIAL CALCULUS
263
It is easily shown that de is a k-deviation of ! of order $; n+p: use
the formula
(ad<jJ) (de) = d((ad<jJ)e)+(ad(<jJg)d)e ,
.n n 
where <jJE \;I(lf) and where g is the image of <jJ in v( (Q)) . Set
Cf,d) (2,e) = (gf ,de) ; we call (<J,de) the canrosite deviation of (g ,e) and
(:f,d) . Canposition of deviations is associative and its unit elements are the
deviations of the form Id O
Notice that (<J) 0 = .lO
5.2 Example: SuProse that  =  = k . Then a rrorphism :f:  -> 'i
of  is called a section of X and a deviation d of f is called a
distribution of 'i. carried by 1 . If i factors through an open subscheme
Q of ¥. and if <jJE J1(lf) we say that d(<jJ) E t.9()=k is the value of the
distribution d at <jJ . Setting 11 = (:f ,d) , we also write
d(<jJ) = Jdl1 = f (y)dl1 (y) .
Notice in particular that  (:ft/) = f dfO , where :fp E y(k) is associated with
f
5.3 ExaInI?le: Let  be a k-scheme. A differential operator on 
of order $; n is a k-deviation of order $; n of the identity rrorphism of
 . The set of differential operators on  is written Dif Q[) ; the subset
consisting of those of order $; n is written Dif n () . By 5.1, Dif () is
a k-subalgebra of  (J!,J)) : the algebra of differential operators on X.
If DE DifO () , we have D (fg) = fD (g) for each open Q in  and all
f,gEJJ(Q). If g=l, we thus see that D(f)=fD(l) . Accordingly we may iden-
tify J1() with Dif 0 () by assigning to  E JJ() the differential opera-
tor f f-rf<jJ .
If DE Dif () , D (1) E cO() is called the constant term of D . We write
Dif+(X) (resp. Dif()) for the subset of Dif() (resp. Difn()) con-
sisting of all D for which D (1) = 0 . We then have Dif () = Der (J1 x ) , where
Der (J?x) denotes the set of k-derivations of the sheaf <.fl X . This is -proved
as follows: if DE Dif  () , Q is an open subscheme of  and <jJ E t1 (Q) ,

264
AU;EBRAIC GROUPS
II,  4, no 5
then (ad<jJ) DE DifO (Q) ; hence there is a function a() EJJ(Q) such that
D (<jJg) = <jJD (g) + a (<jJ)g for gE 0(U) . Setting g = 1 , it follows that a ()
D(<jJ) so that D is a derivation. The converse is obvious.
5.4 Let i:  ->- '£ be a rrorphism of k-schemes. We now formulate a
methcxl for calculating the deviations of f. If dE Ab¥-(tJ ,f. (<.9 )) , write
"""'k Y - X
d:f for;,e element of (t9Q9kf" (tfl X ) , 0') defined as follows:-for each
xE  , dx: sends
<jJ Q91jJ E (rj x Q9 f" (rP )) = JJ Q9 tJ ( )
-k- Xx xkf x
onto d(1jJ)
iJ .
x
MJreover, we write 1f for the sheaf of ideals of tP2):Q9kC (W X ) generated by
sections of the form lQ9<jJ-(H)Q91 , where <jJf denotes the section of J}x
over an open set U induced by the section <jJ of !:" (J) 'i) over U..
where d(1jJ) is the canonical image of d(1jJ)Ef*(tO)f(X) in
ProFOsition: The map d I-->-d t induces a bijection of the set of k- deviations
of i of order  n onto the set of elements of (J)Q9kf. (J)X) , J)"f>.)
. hi r1n+l
vanlS ng on J:f
Proof: Clearly the map d I->-d:f is a bijection of W? (lOX' f. (t9)) onto
(t9Q9kf. (JJ'i) , r!J?2 . It follows directly from the definitions that, under
this bijection, deviations of order s; n corresFOnd to rrorphisms which vanish
"1 n + l
on J f .
In future we shall write 11 for the X-ITOdule (J'XQ9k f . (J y )) /+l . The
k-deviations of f of order s; n are thus in one-one corresFOndence with the
elements of  ( ' J)"f>.) .
.f)n
5.5 The X-ITOdule J i may also be constructed as follows. First
consider an arbitrary embedding : 2:; ->- E. of k-schemes; let y be an open sub-
scheme of  such that i is the comFOsition of a closed embedding 1: ->- y
and the inclusion rrorphism of . Y in . If 'J is the kernel of the rrorphism
l :,j}v ->-1* (J)x) induced by. 1 ' it is clear that the closed subscheme
Y(1 n +1) (I,- 2, 6.8) of Y depends only on :h and not on the choice of
'! . We denote this closed subscheme by  and call it the nth neighlxJUrhood

II,  4, no 5
DIFFERENTIAL CALCULUS
265
of i in Z . We entrust to the reader the task of generalizing to nth neigh-
bourhoods the functorial properties of leading neighbourhoods described in
section 1
ProFOsition: Let 1::]:[ ->- ¥ be a rrorphism of k- schemes and let X be the
space of FOints of X. Then the geanetric realization of the nth neighbour -
hood of  with respect to the embedding y: ->- )( ¥. with canponents Id and
! is canonically isarorphic to (X, P) .
Prcof: Set Q =x. '!.. and i =y and apply the preceding remarks; clearly y
induces a homeomorphism of X onto the space of points of . It thus re-
mains to compare the structure sheaves. Let ;;::- and  be the presheaves of
k-algebras over  such that
:7(U) =  J7 y (T),  (U) = lim c.Ov()
%(Q)c'!' - y(1J) c lY -
Then the associated sheaves of T and  are precisely f. (Jl y ) and
r(J1 v ) =y.(tD XXY ). The maps cCycr)->-t9Xxy(Qx 'I:) induced by P..!"2:x'!..->-'!..,
detee a rrO"rph1.sm u: '.f"->- " hence'; oorphism u:f. (&J->-:( (tJ/y) of the
associated sheaves. Similarly, the maps JJ x C9)->-JJ xxy (Q ,,) , induced by
pr 1 :  x ¥ ->-  , induce a rrorphism v: e?x ->- 2. (lJ) , hence a rrorphism of
t.O<29kf. (Ji y ) into r (cY y ) : a<29bl-+v(a)u(b)  The canFOsition
J) @ f" (tJ/) ->- ,. (cY) c '. (J /+l)
 k- y 1 y. ] y
vanishes on + 1 . By comparing the stalks of the sheaves in question, we see
that the induced rrorphism
( cj <29 f" (J! ) ) / 12+1 ->- ,. (t.f) /+ 1)
 k- ¥ I 1 y
is an isorrorphism.
Remarks: Henceforth we shall identify II with (,P) by means of this
isanorphism. If d is a deviation of f: ->- ¥. of order s; n and if
d I :'P ->- cP x is the rrorphism associated with d (5.4) , we may reconstruct
d fran df by observing that d I is a deviation of the morphism
fn: X ->- Zn induced by _ f and d is the canposite deviation
-y - -y

266
ALGEBRAIC GROUPS
II,4,n05
d f
K -----'1 zn
-y
. 1 0 0
lIlC Rr
-- xIY
5.6
Corollary: For each rrorphism of k- schemes f:X-+Y
pn is a
i
quasicoherent ITOdule.
Proof: With the notation of 5.5, let  be the canposi te rrorphism
incl P!l
 -) "f>.xy ---) K
Identifying  with (,) by means of the canonical isorrorphism, we
have q (x) = x for xE i; : and the -ITOdule cr is the direct image under
q _ of the sheaf of functions of Zn. The proof is canpleted by invoking I,
-y
 2, 2.4.
(ad b )D = 0
n
Corollary: Let h: B -+ A be a rrorphism of  . Then the map
is a bijection of the set of k- deviations o f S h of order
set of k- linear maps D:B -+A such that (ad b O ) (adb l ) ...
for all b O '... ,b n E B .
5.7
d >->- d( B)
$; n onto the
Of course we set
( (ad b) D) (x) = h (b) D (x) -D (bx)
for b,xE B . A map D satisfying the conditions of the corollary will be
called a k-deviation of h of order $; n .
Proof of the corollary: First assume the notation of 5.5 for the case in
which  = SEjl . By 5.6 and I,  2, 1.10, we have accordingly
(;)n ,n  i7.)l1
(..Ti,V') -A(:fi(),A).
Moreover, by arguing as in 5.4 we see that the k-linear maps D:B-+A satis-
fying the condition of the corollary are in one-one correspondence with the
elements of A (A@kB/T+l , A) ., where J is the ideal of A@k B generated
by the elements 1 @b - h (b)@ 1: this follows by assigning to
Jl+l . Jl+l
A:A@kB/J -+A themap b......A(l@bITOdJ ). The corollary now follows
fran the fact that ()  A@kB/T+l (5.5) .

II,  4, no 5
DIFFERENTIAL CALCULUS
267
5. 8 Given AE  ' set P/k = (A 0 k A) / T+ 1 , where J is the ideal
n
of A 0k A generated by the elements a 0 1 - 10 a, aE A . Endow P A{k with
the A-algebra structure induced by the haranorphism a r-+ a0 1 ITOd In+ ; set
Jl+l
0(a)=10a ITOdJ for aEA, . By 5.6, there is a canonical bijection
n
A(PA/k,A) :;. Difn(A) ;
namely, if A EA (P/k ' A) , the differential operator associated with A
sends a E A onto (A (0 (a)) EA.
The construction of P/k is functorial in A; given a hcrnorrorphism h:A-+B
of !:\., we write P/k: P/k -+ P/k for the map induced by h0 k h: A0kA -+
-+ B0kB . If S is a multiplicatively closed subset of A and h is the
canonical map of A into A[S-l], P/k induces a bijection
[ -1 ] n _ n
A S  P A / k -+ PA[S-l]/k
Similarly, if a is an ideal of A and h:A -+ A/a
P/k induces a bijection of p// (aP/k + A 0 (a) )
is the canonical map,
onto pn
A/a,k
5.9
Example: If A=k[T] , we get an isanorphism
n _ n+ 1
Pk[T]{k -+ k[T ,h] / (h )
by sending T 0 1 ITOd T+ 1 and 10 T ITOd T+ 1 onto T ITOd h n + 1 and
T+hITOdhn+l , respectively. With the k[T]-linear map
A. :k[T,h]/(h n + l ) -+ k[T]
l
such that A. (h j ITOd h n + 1) = 0.. is associated the differential operator
l lJ
d
dT i
k[T] -+ k[T]
such that
P(T+S) = }}-, p)Si
i dT l
if PE k[T] , whence P(T+S)E k[T,S] . We have accordingly

268
ALGEBRAIC GROUPS
II,  4, no 5
 Tr = ()-i
aT1 1
It follows that
a
aT i
a
aT j
_ ( i+j ) a
- i aTi+j
so that in particular,
a
aT 1
n
a
- n'-
- .ar
(Taylor's formula) ,
and so Dif (Qk) is the free k[T] -ITOdule generated by
Id = ...L a
aT O ' aT 1
, ... ,
a
aT i
, ...
5.10
Example: By 5.8 and 5.9 , there is a k[T,T- 1 ]-algebra iso-
n [ -1 ] n+l ,
Pk[Trl]/k onto kT,T ,h/(h ) which sends o(T) onto
, J r n+l
, hence '1' onto (T+h) ITOd h =
rrorphism of
T+hITOdhn+l
T r + () -lh+ () T r - 2 h 2 +...+ () -n ITOdh n + 1 (nEZ)
-1
It follows that. Dif (Ilk) is the free k[ T , T ] -ITOdule generated by the
operators a /aT 1 , iE IN , such that
  = ()Tr-i, rE .
aT1 1
5.11
Let f: -+X be a rrorphism of .@S . Assume that we are given:
a) affine O pen subschemes U, , iEI, covering
-1
schemes V, , iEI, of Y such that f(U.)cv,
-1 - - -1 -1
U. ' 1 ' lEI., covering u,nU, . Let f.:U, -+V,
-lJ 1J -1 -J -1 -1 -1
f and let h. = 12 k (f,) . B y 5.7, each deviation
- 1-1
is associated with a deviation d. of f.
1 -1
to be a deviation d of E of order :;; n which induces
K b) affine open sub-
c) affine open subschemes
be the rrorphism induced by
D. of h. of order :;; n
1 1
of order :;; n
In order for there
d. for each

it is necessary and sufficient that, for each i,j,l, the deviations of
i ,
¥ -+Qijl inUuced by d i and d j coincide. In other words, the canFOsite

II,  4, no 6
DIFFERENTIAL CALCULUS
269
maps below coincide:
D. can
1 cJ ,fl
C!J. k (V , ) -----7 k (U , ) ----?v k (U, ' i )
-1 -1 -lJ
and
D, can
W k (V , ) (1 k (U ; ) ----;. V k (U, ' 1 )
-J -J -lJ
Now supFOse we are given a k-scheme . If W is an affine open subscheme
of Z, it is clear that the k-linear map
J1 k () D i : V k (xYi) -+ V k (xgi)
is a k-deviation of JJ k ()iSikhi or order :S: n . If the D i satisfy the above
"matching" conditions, the V k ()iSikDi satisfy the "matching" conditions with
respect to the covering of Z xX by the affine open subschemes W><U, .
-- --1
Accordingly there is a deviation  x d of "f of order :S: n which induces
JJ k ()iSikDi for each i and each l'!. This deviation Z-x d depends only on
.z; and (f,d) .
If e is a deviation of a rrorphism : -+1' , one defines ex K in a similar
fashion. Set ex d = (e x) (1' X'd) ; then we also have ex d = ()( d) (e") .
Finally, if R, we write diSikR
order :S: n which induces, for each
or  for the deviation of
i , the k-linear map
[iSikR
of
DiiSikR (}R (Y iiSikR) -+ Jl R (Qi iSikR)
Section 6
Invariant differential operators on a group scheme
6.1 Let G. be a k-rronoid-scheme and let EG.:  -+ G. be its unit
section. A distribution of Q. carried by EQ. (5.2) will be called a distri-
bution at the origin , or simply a distribution on G..-. We write Dist 
(resp. Dist G) for the k-ITOdule of distribution (resp. distributions of
n-
order :S: n) . If 11 E Dist G., 11 (1) = dl1 E k will be called the constant tenn
of 11 . We set
Dist+G = {I1EDistG: 11(1)= O}

270
ALGEBRAIC GROUPS
II,4,n06
and
Dist + G = (Dist + G) n Dist G
n- - n-
Let e be the unit element of _ G (k) . If k is a field, Dist G may be
n-
identified with the space of k-linear maps V e -+k vanishing on m+l (apply
5.4 to the case f =EG ) .
If 11 C Dist G and v C Dist G , define the convolution p rcxluct 11 *v E Dist G
m- n- - m+n-
to be the canposite deviation
o
 11 x v 'TT G
 +-- x  +-- x  +=- Q (5.1,5.10) .
The convolution prcxluct is obviously associative and induces an associative
algebra structure on Dist Q , the algebra of distributions on .
Given a haranorphism !:  -+ g of k-m::moid-schemes, and ailE Dist m  , we
write 1(11) or (Dist f) (11) for the canposite deviation
fO
   -<-=- E!. .
Clearly f (11) C Dist H and Dist f
-. m-
phism.
Dist G -+ Dist H is a k-algebra haranor-
Moreover, if RE ' the map I1I-+I1 R (5.11) of Dist into Dist(iSikR)
is canpatible with the convolution prcxluct and may accordingly be extended
to a haranorphism of algebras (Dist G)iSi k R -+ Dist (GiSikR) .
6.2
Example: SUPFOse that Q. is affine and has bialgebra A. Let
J be the kernel of the augmentation EA. By 5.4 and 5.6 , DistmQ. may be
identified with the k-ITOdule of k-linear maps I1:A -+ k such that 11 (.rn+ 1) = 0
If v :A -+ k is a second k-linear map satisfying v (+ 1) = 0 , 11 * v is the
canposite k-linear map
L 11 iSi v
A  AiSiA  kiSik --=--.. k .
k k
For ample, if g = a k ' let
E. (T J ) = 0" . Then Dist ex..
1 1J '+' K
We have E. *E . = (1. J) E, + ' .
1 J 1 1 J .
If G=l1 k ' let v.: k[T,T- 1 J-+k
- 1
k[T,T- 1 J/(T-l)i+l and satisfies
E . : k [T ] -+ k be the k-linear map satisfying
1
is'the free k-mxlule generated by E O ,E 1 ,E 2 ,...
be the linear map which vanishes on
v. ((T-l)j)=o,. . Then we have v. (Tj)=() ,
1 1J 1 1

II,  4, no 6
DIFFERENTIAL CALCULUS
271
where () denotes the coefficient of T i in the series development of
1
(1+T) j (jE Z , iEiN) . If k is an algebra over i[), then we have
( Vi ) 1 ,
vi = i = i: Vi (v l -l)... (Vi-Hi)
In this case, Dist Ilk is therefore the free ccmnutative k-algebra generated
by vi (this does not hold in the general case) .
6.3 Ccnsider a k-scheme  and a right operation :;-; x Q -+  of
the k-rronoid-scheme G on X. Let p: G -+ Aut X be the hancrno rp hism
-opp
associated with  If I1E DistmQ , we write p' (11) for the canposite devia-
tion (5.1):
X>< u O
"f>.":::- x  xQ 
Clearly p t (11) is a deviation of order :;; m of the canposite rrorphism
_ XE:G P
--+ x ---+ x Q-+;-;
Le. of the identity rrorphism of X Accordingly p' (11) is a differential
operator on X of order :;; m .
Proposition: For each k-scheme , each k -rronoid-scheme  and each hano-
rrorphism p: G -+ A'Ut X , the map p': Dist G -+ Dif X is a k- algebra hano-
-Dpp
rrorphism.
Proof: Since p' is obviously k-linear, it is enough to show that
p' (11 *v) = p' (l1)oP' (v) for l1,vEDist.
This follows fran the ccmnutative diagram below: in this diagram, the caTIIT!U-
tativity of the b-KJ base triangles follows fran the definition of p' ; also
the canposi te deviation of  x '11 g,  )< Q x V and  x 11 coincides with
K x (11 * v) .

272
ALGEBRAIC GROUPS
II,  4, no 6
o
 X'TT G
)(gx < xQ
x1\q
xl/:\
lCX ( p' (11) "f>.x < p' (v) 
o
1!
6.4 The definition of the convolution prcxluct (6.1) may be rephrased
as follows. Let 11 be an open subscheme of g such that EG factors through
 and let Er2(!l) . Then we have )d(I1*V) = 5('TTG)d(I1XV)-, where 'TTG is
the function induced by  on '11  1 (Q) . We write this last formula in the
form:
f d (I1*V) = )  (xy) d (11 (x) x v (y) )
Also, assuming the notation of 6.3, let y be an open subschEme of ,
1/JE0tv), I1EDist(G) and xEV(k) . By 5.2 (P'(I1)1/J) (x) isthevalueat
p' (11)1/J of the dis-tribution }o carried by x#: ->-'{ ; accordingly it is
also the value at 1/J of the canposite distribution
#0 0
X r--' XXI1 u
 --  +-- )( +--"Q +=--
which coincides (5.11) with
#0 0
11  x xG \J
 +'--9 __x Q ( - xG. · "f>.
Hence we have
(p' (11) 1/J) (x) = f 1/J (xg) dl1 (g) .
6.5
We nON apply the results of the preceding discussion to the
case in which G. is a k-group-schEme,  = and  ='TTG . In this case p
is the hcm::mo rp hism 0: G ->-Aut G of  1, 3.3c)
-opp --
If  is affine, with bialgebra A, we may regard a distribution I1E DistQ

II,  4, no 6
DIFFERENTIAL CALCULUS
273
as a linear form on A which vanishes on  1 (6.2), and 0 · (11) as the
canposi te map
A
tl A
A0kl1
A0A----+ A0k----+ A (s.6.).
k
For example , if ,=ak " we have 0' (E i )= :J/:JT i (6.2 and 5.9). If g=l1 k ,
we have 0' (V,)=T1:J/:JT 1 (6.2 and 5.10).
1
In general, if g Eg(k) and D is a differential operator on G, we write
gD for the composite deviation
o
G y(g) , G
D
-1 0
, g y (g ) , Q.
-1
Le. the differential operator such that (gD) () = (D(g))g for each sec-
tion  of cY Q. over an open U (cf. 4.6). We say that D is left invariant
if, for each R E!,\ and each g E g (R) we have gD R = DR .
Invariance Theorem . For each k -group-scheme  ,
o I : Dist Q ->- Dif Q.
is an isorrorphism of the algebra of distributions of G onto the subalgebra
of Dif G formed by the left invariant differential operators .
Proof: We first show that, if 11 EDist  , then o' (11)
Let U be an open subscheme of G E t!7(Q) , R E '
we must show that
is left invariant.
x EQ(R)
gE(R)
((go I (11) R) R) (x) = (0' (11) RR) (x) .
By change of base we reduce to the case in which R = k . In this case we have
((go'(I1)))(X) = (O'(I1)(g))(g-lx) = r(g)(g-lxt)dl1(t)
= f(gg-lxt)dl1(t) = (0' (11)) (x) :
Consider a differential operator D on G and the value D (e) of D at the
origin, Le. the ccmposite deviation
D EO
G
G
> G
> 
f dD (e) = (D) (e)
. Let U be an open subscheme
Then by definition we have

274
AffiEBRAIC GROUPS
II,  4, no 6
of , E t9()
x EQ(k)
If D is left invariant, we then have
(0' (D(e) )) (x)
= J  (xg) d (D (e) (g)) = D (e) (x)
-1
((D(X))X ) (xe)
(D) (x) .
Since this calculation may be repeated after an arbitrary change of base, we
see that D = 0' (D (e) ) if D is left invariant.
Finally let 11 EDist  . If we set x =e in the formula
we see that
(0' (11)) (x) = J(Xg)dl1(g)
(0' (11) (e)) () = J(eg)dl1(g) = 11() ,
0' (11) (e) = 11 and DI-+D (e) is the required inverse map of 0 I
so that
For example , if  = , D is the differential operator P (T) d /dT i , and
g E a k (k) = k , then we have gD = P (T-g) d /dT i ; this operator is invariant if
P(T) is constant. imilarly, i =l1k' D=P(T) d/dTi and g E Ilk (k) = * ,
then we have gD=glp(T/g) d/dTl ; this operator is invariant if P(T)=AT 1 ,
A Ek .
Remark: The invariance theorem may be generalized to k-rronoid-schemes. If
9 is a k-rronoid-scheme , let us say that a differential operator D on 
is left equivariant if, for each open subscheme Q of , each E c.O() ,
each R E.£:\ and each g E g(R) , we have DR (Rg) = (DRR)g . Then 0 I is an
isanorphism of Dist g onto the subalgebra of Dif G formed by the left
equivariant differential operators.
6.6 Let : 9 x  ->-  be a left operation of a k-monoid-scherre 
on a k-scheme  and let p: 9 ->- Aut !;. be the associated haranorphism. De-
fine a map p': Dist g ->- Dif  by assigning to 11 E Dist 9. the canr::osi te devi-
ation
o
 11>< X u
.2S. +-- x   x -<-=-
The map p , is then an antiharanorphism of k-algebras:
p , (11 * \!) = p' (\!) 0 p' (11) for 11 , \! E Dist g

II,  4, no 6
DIFFERENTIAL CALCULUS
275
When  = and  ='TT G , P is the haranorphism y of ( 1, 3.3c) . If,
rroreover,  is a group, y' is an anti-rrorphism of Dist  onto the alge -
bra of right invariant differential operators , provided one defines right in-
variance as follows: if DE Dif Q and g E  (k) , define Dg by the formula
(Dg) ()=g-l(D(g)) ,where . is a section of J7 over an open Q. (4.6);
then D is said to be right invariant if, for each R E  and each g E Q (R) ,
we have DRg = DR
6.7
(Ad g) 11
We then have
subscheme of
If  is a k-group-scheme, 11 E Dist Q
is by definition the ccxrposite deviation
11  tillt g) 0 G
and g E  (k) , then
 (
J d( Ad g) 11) = J  (gtg -1) dl1 (t)
g such that 1 EQ(k) .
 EV(y)
for
, where U is an open
This definition gives rise to the formulas
1 (0' (11)) (g) = (y' (( Ad g)I1)) (g) ,
-1
(y' (11)) (g) = (0' ( (Ad g )11)) (g)
-1
and 10'(I1)g=o'(( Ad g )11),
, igy' (11) = y' (( Ad g)l1) .
For example, the first of these is proved as follows:
(y' ( (Ad g)I1)) (g) = f(g)d((Adg)l1)
= J (g) (gtg-l)dl1(t) = J(gt)dl1(t)
Arguing as in 4.7, we infer the
(0' (11)) (g) :
ProFOsition: Let  be a k -group-scheme : a) if 11 E Dist  ' the following
conditions are €qUi valent :
(i)
(i' )
(ii)
(iii)
Y I (11) is left invariant.
0 ' (11) is right invariant.
y' (11) = 0' (11) .
11 E (Dist G) G. , Le. (Ad g) I1 R = I1 R for all R E  and g E g (R)
b) The algebra of left-and-right invariant differential operators on  is
CCXImUtative and y' and 0' induce the same isanorphism of Dist () g onto
this algebra.
Remark: If l1,vEDist, we have y'(v)o'(I1)= O'(I1)Y'(V). For if

276
ALGEBRAIC GROUPS
II,  4, no 6
: }C.Q x.Q+Q is the rrorphism (x,y,z)H-x.y.z, the caTlIT!Utative diagram be-
o
low reveals that y' (v)o' (11) = (vx Gx 11)m. . We see similarly that
o
o' (l1)y' (v) = (v x Gx 11).m .
",1\"
"'",//"\0\
G<ce XGxe. ( Gxe.  G
- -k - -1<: y' (v) - -1<: 0' (11)
It follows that Dist (9) 9 is contained in the centre of Dist Q . For if
y' (11) = 0' (11) , we have
y' (I1*V) = y' (v)y' (11) = y' (v)o' (11) = 0' (I1)Y' (v) = y' (11)Y' (v) = y' (V*I1)
so that l1*v = V*11 for each v E Dist Q. .
6.8
We now relate the above results to those of section 4. Let Q.
be a k-group-scherne, let QE be the first neighbourhood of  with respect
to E G :  +9 , and let
El E 2
-,%----Q
be the canonical factoring of Eg (I,  4, 1.1). By 5.4 and 5.5, the devia-
tions of EG of order :s; 1 , Le. the distributions 11 E Dist19 , are of the
o -
form VE 2 ' where V:.QE is a deviation of E 1 . Since QE is affine
with algebra cP(.QE)=kEBW G / k (3.1), the deviations v of E 1 are associ-
ated with linear forms k EB wC;;/k + k (5.7). Elements of Distt g corresj:Ond
to linear fonns which are zero on k. Accordingly we have a canonical iso-
rrorphism
DistQ + :WG/k,k)
Write
v Q Dist G + Lie Q

II,  4, no 6
DIFFERENTIAL CALCULUS
277
for the canposition of this canonical isanorphism with the isanorphism
Lie  '"  (w g / k ' k) of 3.6. The inverse isanorphism v 1 may be explici tl Y
described as follows: let T1: k(E:)+ be the deviation of the canonical
embedding  +  k (E:) such that T1 (l) = 0 and T1 (E) = 1 ; if
 ELiecQ.(k(E)) , Vl(O is then the canposite deviation
#0
k ---2l- Sp k (E) ..-L- 
ProFOsition: If 9 is a k- group-scheme and 11 E Dist , then we have
y' (v (11)) =y' (11)

and O'(V(I1)=O'(I1).
#0
Proof: Let  =v G (11) , so that 11 =n . If U is an open subscheme of
g, E JJ(U) and mE(k) , we have by 5.4 and 4.5
(0' (11)) (m) = f(m)dl1 = J((m)#-)dT1
= n( (m) ()) = T1 (((E)))
= (0' (O) (m) .
By applying a change of base, we see that the equation (0' (11)) (m) =
(0' ()) (m) remains true for all R and mEQ(R) . The equation y' (0
y' (11) is proved similarly.
6.9
ProFOsition:
,+
If 9 is a k -group-scheme and 11 , vEDist 1  , we
have
[V g (I1),V g (V)J = V.Q(I1*V-V*I1) E LieG
Proof: By (12) of 4.5, 6.5 and 6.8, we have
o' ([V g (l1) ,v(v)J) = 0' (V g (I1))O' (vg(V))-o' (Vg(V))o' (V g (I1))
= O' (11)0' (v)-o' (v)o' (11)
= 0' (I1*V-V*I1)
= 0 I (v (I1*V-V*I1))
Q

278
ALGEBRAIC GROUPS
II,  4, no 7
Section 7
Infinitesimal groups
7.1 Definition: !2 k -group-scherre G is said to be infinitesimal
if the canonical projection  + is finite locally free and its unit sec -
tion Eg:  + induces a hcmarorphism of II onto Ig.l
The first condition means that g is affine and that J7(Q) is a finitely
generated projective k-ITOdule. By I,  5, 1.6, J7(g) is thus a finitely
presented algebra over k, and the kernel I of J7(E G ): cY(Q) +k is a finite-
ly generated ideal (I,  3, 1.3). The second condition in the definition means
that (K) reduces to the unit element for each field KEk\ ' or that I is
a nil ideal. Since I is finitely generated, this last condition is equivalent
, to saying that I is nilFOtent. By 5.4 and 5.6, Dist  may then be identi-
fied with  (J(g) ,k) = tJ7«) and the multiplication in Dist  may be ob-
taed by transFOsing the coprcxluct of J7 (g) :
tl',
\!!(G)0 t V(G) .::;. t(J(G)0J{G)) - (J(G), \?(G)
-k - -k - -
Also, it follows fran axian (Coun) (1, 1.6) that the unit element of
Dist Q is the augmentation EcO (G) : &(Q) + k of J!(G)
Throughout the rest of this section, we write
I', : Dist (G) + Dist (G) 0 Dist (G)
- - k -
for the map derived by transFOsition of the multiplication miP() in J!()
I',
tL9(0
t
mJ!(G)
,
t(19(G)0J!(G)) .::;. \9(G)0\G)
k k
We then have, by definition,
(1',11,a0b) = (l1,a.b)
for I1EDist(Q) , a,tEJI(G) . Setting a =0. +x and b = S +y , where a,SEk
and x,yEI , we get
(1',11,a0b) = (l1,aS+ay+Sx+xy) .
Moreover we have

II,  4, no 7
DIFFERENTIAL CALCULUS
279
(111&>1 +11&>11 , a I&> b) = a\l1,b) +6(I1,a) = (11 ,20.6 +ay+ 6x) .
Accordingly we have
LlI1 = 111&>1 +11&>11
iff 11(1)=0 and 11(12)=0 ,Le. iff I1EDist.
Finally, we write E: Dist G + k for the map 111->- 11 (1)
7.2
ProFOsi tion:
Let  be an infinitesima l k -group and let K be
p H- P , of 6.3 induces a bijection of the set of
a k-scheme. Then the map
right - operations on K onto the set of alqebra haranorphisms
v: Dist +Dif such that v (11) (1)=E:(11) and
V(I1) (fg) = L(V(I1,) (f)) (v(v,) (g))
. 1 1
1
where I1E DistG, LlI1 =L.11 ,I&> v, and f,gEJi(Q) and U ran g es throu g h the
1 1 1
open sub schemes of X.
Proof: Let \,!:!;,x G + be a right peration on X. Since the canposite
rrorphism
Xi< E U
;.;x - G, ;';xG---+X
is the identity and I;.; xEGI is a hanearorphism,  induces a rrorphism of
Q x G into Q, hence a Q-operation on !}, for each open sub scheme _Q of X
This fact allows us to confine our attention to the case in which X is
affine.
In this case, we show that the map pH p' establishes a one-one corresFOnd-
ence between right operations }d on X and haranorphisms v:Dist G+ Dif 
such that V(I1) (l)=E(I1)
and
v (11) (fg)
L (v (11 .) (f)) (v (v.) (g))
,1 1
1
for f,gEc9() . If M,N,P are k-rrodules, where P is finitely generated and
projective, and if A :M+Nl&>kP is a k-linear map, write A' :Ml&>k+N for
the canposite map
M I&> t p A I&>t p , N I&>PI&> t p NI&> can, N I&> k  N
k

280
ALGEBRAIC GROUPS
II,  4, no 7
We know that ,\,\' is a bijection of (M,N@P) onto (M@,N) .
COnsider in particular a linear map
,\ : J? (X) -+ &(X)@ t.9 (G)
- -k-
and the canposi te maps
c9()@ J () mul t..,.  ()
a :
,\-+ rY(?:;)@ t9()
and
B : J(X)@J1(X) t9(?:;)@J1()@(?:;)@d() 
J1()@ ()@ c9()@ d(G) mult@mult+. J!()@ J?()
Then
a' : J\)@ J\) @ Dist  -+ cO(2P
sends f@g@11 onto ,\. (fg@l1) , while S' sends f@g@11 onto
L('\'(f@I1.)'\'(g@\J,) , where tol1=LI1.@\J. .
. l l ,l l
l l
Accordingly'\ is ccxrpatible with the multiplication (i.e. 0.= B ) iff
a' = S' , i.e. iff
,\'(fg@l1) = L(,\'(f@I1.)'\'(g@\J.))
, l l
l
for all 11 E Dist  and g, fE J} () .
One shows similarly that a)
,\' (1@11) = E: (11) for all 11 ;
the equality ,\ (1) = 1 is eqUivalent to
b) the cCXImUtativity of the diagrams
T ,\ ) 6()@6(G) 
6(x) 6(X)@k
I &(){)01,<,,() - can. -
(*) et (**) ,\ / &(19@<&()
6()@6() '\@6()  )@6()@6( 6()@(?()
is equivalent to asserting that ,\': JI()@ Dist () -+c.9() turns J7 qp into a
(unital) left rrodule over Dist() .
Now if ,\ is induced b y a harano rp hism p:G -+Aut (X) , ,\. is P recisel y
-opp --
the map f@W-+ p' (11) (f) . Our assertion follows from this and the fact that

II,  4, no 7
DIFFERENTIAL CAICULuS
281
,\ is induced by a hcm:::m:>rphism p
CCXImUte, and is such that ,\ (1)= 1
iff ,\ makes the diagrams
and a=B .
(*) and (**)
7.3
Proposition:
Let = Sp A
be an infinitesimal k -group.
x is an algebra hararorphism iff
a) If x E Dist () '"  (A,k) , then
E: (x) = 1 and I', (x) = xi&> x .
b) If x,yE(k) =(A,k) CDist ,
volution prcxluct x*y.
x.y ='TT G (x,y)
coincides with the con-
Proof: x E Dist G is an algebra hcm:::m:>rphism provided that for f ,gEA we
have
(/',x,fi&>g) = (x,f.g) = (x,f)(x,gj = (xi&>x,fi&>g}
and E: (x) =(x,l) = 1, Le. I', (x) =xi&>x and E: (x) =1 .
Moreover, if x,y E Dist (G) are algebra haranorphisms, we have, for each
f E t9() ,
(x.y) (f)
(xi&>y) (1',t1()f) = (Xi&>y,I',c9()f) = (x *y,f)
(x *y) (f)
so that x.y = X*y .
Remark:
Let X be a k-scheme,
 an infinitesimal k-group and
If x E Dist  satisfies E: (x) = 1 and
p : G ->- Aut (X) a , hcm:::m:> .,. rphism.
-opp -- ,
/',x =xi&>x , it follows directly fran the definitions that the endanorphism
I p (x) I! of  X induced by p (x) coincides with the differential operator
p' (x) .

 5
IDCALLY ALGEBRAIC GROUPS OVER A FIELD
Troughout  5, we shall assume that k is a field. Let k be an algebraic
closure of k \\hich belongs to \ ' and let ks be the set of elements of k
\\hich are separable over k. We denote by II the Galois group. of k over k,
s
i.e. the profinite topological group of k-autOITOrphisms of k. Those members
of  \\hich are fields are called extensions of k.
Section 1 The neutral component, etale groups
1.1 Neutral component theorem : Let  be a locally algebraic
k -qroup , e the unit element of (k) and GO the open sub scheme of G car-
ried by the connected component of e. Then
(a) t is a characteristic s ubq rou p of G .
(b) For each extension K of k, we have GO @ K (G @k K)O .
k
(c) The connected components of G are irreducible, algebraic over k
and all have the same dimensions.
Proof: (b) Since Q is locally algebraic, Q is locally connected (Alg.
corom. II,  4, prop. 10). Accordingly o is closed and open in Q. For each
extension K of k, o @k K is closed and open and connected by I,  4 , 6.9,
and hence coincides with the connected component of e in Q @k K
(a) By I,  4 , 6.11, QO x o is connected, hence the =rphism
'TT G
QO x QO -7- G x G.  G.
factors through GO ; similarly, the =rphism
°G
GO -7- Q  
factors through \2. It follows that G. 0 is a sub-group-scheme of G. We show
that it is characteristic. If K is an extension of k, in virtue of (b),
o @k K is invariant under each autorrorphism of the K - scheme  @k K pre-
serving e. In particular, Q. @k K is invariant for each automorphism of

II, 5, no 1
I.CX::ALLY ALGEBRAIC GROUPS
283
the K -group  Q9k K . Now let R E l\ and let  be an autorrorphism of the
R - group G Q9 k R ; we must show that, if x is a point of GO Q9 R ' then
- - k
u (x) E o Q9k R . Let I be the prime ideal arising as the projection of x
into Spec R , and let K be the field of fractions of R/I; then x is the
image of a point x' of oQ9k K , and (x) is the image of (x') \\hich
by the preceding discussion belongs to o Q9k K , so that we have
(x) E G O Q9k R as required.
(c) We show first that G. 0 is irreducible. It is enough to prove that oQ9k k
is irreducible, and by (b) we may confine ourselves to the case in \\hich k
is algebraically closed. If st is not irreducible, then there are two closed
points x,y E G. 0 such that x belongs to exactly one irreducible component
of , and y belongs to at least two distinct irreducible components. The
ring i!J then has exactly one minimal prime ideal and the ring r2 at least
x y
two; but this is impcssible since these rings are isorrorphic (1, 3.3 c) ) .
Let 11 be an open sub scheme of o \\hich contains e and is algebraic over
k . By lemma 1. 2 below, the rrorphism g x 11 ->- <2,0
tive; hence SO is quasicompact and so algebraic
induced by 'TTs is surjec-
over k (01,3.8 and
I ,  3 , 2.1) .
Finally let .!:! be an arbitrary connected component of  and let x be a
closed point of !i. Let N be a finite quasi-Galois extension of k contai-
ning K (x) , and' consider the projection rrorphism p: k ((iQ9k N)  G .
This rrorphism is closed and open by I,  5 , 1.6. All the points of the stalk
p-l(x) are rational over N (for by I,  1, sect. 5 the residue fields of
these points are the residue fields of the local factors of K (x) Q9k N , and
are accordlligl y isorrorphic to N). If Y is such a point, and if u is the
-y
translation (left translation, for example) of  Q9k N \\hich sends e onto
y , the connected component of y is u ( G o Q9 k N) , and is therefore irredu-
-y -
cible. But then E(y(GoQ9k N) is irreducible, closed and open in .!:!, so
that it coincides with .!:!, \\hich implies that !i is irreducible. Finally,
.!:!Q9k N is the union of the \,ly (O Q9k N) as y ranges through 12 -l(x) , hence
is algebraic over N, and its dimension is the same as that of sOQ9k N ,
Le. that of o. It follows that H is algebraic over k and has the same
dimension as o (I,  3 , 6.2 ) .

284
ALGEBRAIC GROliPS
II, 5, no 1
1.2
It remains to prove the
Lemma: Let U be dense and open in the algebraic k -group G. Then the com-
posite rrorphism
UXQ.  <;!x<;!
'TTG.
) 
is faithful1v flat .
Proof : This rrorphism is flat by  1, 3.2b) , so it remains to show that it
is surjective, i.e. that Q(k) = Q(k) Q.(k) . Let gEQ(k) and let y(g) be
the left translation which it defines. Since IIk and y(g) aG(g are dense
and open in  ' they have non-empty intersection and ther are u, v E U (k)
with u = g v -1 , hence g = u v, which is what we are required to prove.
1.3
If G is a locally algebraic group, GO will be called the
neutral component of Q; it is an algebraic and irreducible open subgroup
of G. The theorem implies that the dimension of Q is finite, and that
dim G = dim GO = Kdim J}
- - e
Finally, since o is an open subgroup of , we have Lie <2 = Lie QO . Also
notice that
[Lie  : k]
[m 1m 2 : k] ;;: K dim t!J
e e e
dim G
1.4
ProFOsition : Let Q be a locally algebraic k- group . Then the
following conditions are equivalent :
(i) Q is etale.
(ii) Q Ok ks is constant.
(iii)
QO = 
L9 = k
e
(iv)
(v)
Lie G = 0
(i) = > (iv)
we have
cf. I, S 4 , 6.2 .
K(e) = k , hence <.!J = k if G is etale.
e
is equivalent to m 1m 2 = 0 , hence to
e e
m = 0
e
Proof: (i) <=> (ii)
(iv) <=> (v)
Lie Q = 0
(iv) => (iii)
apply I,  3 , 4.2 to the canonical projection <;!o ->- 

II, 5, no 1
LOCALLY ALGEBRAIC GROUPS
285
(iii) => (ii) : if 5t =  ' then (Q Ok k)O =  ; by translation we in-
fer that for each closed point g of Q Ok k the connected component of g
is iSOl1Drphic to -. This implies that each connected component of G ° k
-k - k
is isorrorphic to , so that  Ok k is constant.
1.5
Example : Let K be a finite extension of k and let .<:! = Aut K .
We know that  is an affine algebraic k-group (s 1 , 2.6) ; we also know that
Lie  is the space of k-derivations of K (s 4,2.3). It follows that Aut K
is etale iff the extension K I k is separable. Observe rroreover that if
n = [K:k]  3, Aut K is not constant, for ( Aut K)(k) has at rrost n elements;
-n
on the other hand, since K Ok k is isorrorphic to k , ( Aut K)(k) is iSOl1Dr-
phic to the symmetric group on n letters.
1. 6 Example : By 9 4 , 4.11, we have Lie (n l1 k)  {x E k : nx = O} . It
follows from 1. 4 that nl1 k is etale iff n 1k t- 0 . Suppose the latter holds;
then Il k is constant iff l1(k) = l1(k) , i.e. if k contains the nth roots of
n n n
unity. In this case l1(k) is a cyclic g=up of order n and each primitive
n
nth root of unity accordingly defines an isorrorphism of (/n onto nl1 k .
1.7 If k = k , each etale k-group  is constant. In genera 1, II
s
acts on the group G(k s ) .
Proposition: The functor GI-+G(k) is an equivalence between the category
- - s
of etale k- groups and the category of small discrete groups on which IT acts
continuously ("II - groups" ) .
Proof : Let C be the category of etale k-schemes and C' that of II - sets;
then the functor X 1-+ X (k) induces an equivalence between C and C'
s
( I , S 4,6.4), hence also an equivalence between the category of groups in C
onto that of groups in C' .
1. 8 ProFOsi tion : Let s:; be a locally algebraic k- group . Then
there is an etale k -group 'TTo() and a horrorrorphism %: Q ->- 'TTo(G) such
that ,  k -g=up !.I and each horrorrorphism i: Q->-]i there is a
unique hOl1Drrorphism : 'TTo(s:;) ->-!.I such that f = .9" % . Moreove r % is
faithfully flat and finitely presented ; its fibres are the irreducible com-
ponents of G and its kernel is GO

286
ALGEBRAIC GROUPS
II, 5, no 1
Proof : Let 'TTo(r;?) be the scheme of connected cOITIfOnents of  (I ,4, 6.6).
We know that the canonical rrorphism : r;? -->- 'TTo(G) is faithfully flat and
finitely presented and that its fibres are the connected canponents of ,
hence the irreducible cOITIfOnents of Q (1.1 c) ). By  1 , 5.1 d), there is on
'TTo(G) a unique k-group structure such that % is a group hom::morphism. Let
tl be an etale k-group and let t: Q -->-!! be a horrorrorphism. By I,  4 , 6.5
there is a unique rrorphism 2:: 'TTo() -->- Ei such that ! = 2: 'k; ; by the lemma
below,  is a homomorphism.
1.9
Lemma : Let R be a ITOdel , Q,  and !i R - group-schemes ,
12 : G -->- lS a faithfully flat quasicanpact horrorrorphism and 'I:  -+ Ei 
rrorphism of schemes . Then 3: is a homanorphism iff gp is a horrorrorphism.
Proof: One way round is obvious. Conversely, if 2"!: is a homomorphism, we
have
'TTH(FrJ (px!:) = 'TTH(:!t) x 212) = 2"P'TTQ = 'I 'TTrs.(r:: x !:) .
Since 12 x 12 is faithfully flat and quasiccxrpact, it is an epirrorphism of
schemes, so that 'TT H (g x 2) = g 'TT K and g is a horrorrorphism.,
1.10 As the solution of a universal problem, the pair ('TTo(G), %)
is evidently unique. We call '11 0 (9) the group of connected cOITIfOnents of Q.
If K is an extension of k, by I,  4 , 6.7, we have a canonical isorrorphism
'TTO(QQ/)k K)
 '11 (G) Q/) k K .
0-
If H is another locally algebraic k-group, we have a canonical isorrorphism
(I,4,6.ll)
'11 (G x H)  '11 (G) X'TT (H) .
0-- 0 - 0-
Since the connected cOITIfOnents of Q are algebraic over k (1.1), G is al -
gebraic over k iff the set of connected cOITIfOnents is finite, Le. if the
k-group
'11 (G)
o -
is finite.

II, 5, no 2
LOCALLY ALGEBRAIC GROUPS
287
Section 2
SnDoth groups
2.1
SnDothness theorem for groups over a field : Let Q be a local -
ly algebraic k -group . Then the following conditions are equivalent :
(i) Q is srrooth .
(ii) GO is smooth.
(iii) 9 is srrooth at e.
.A
(iv) The canpletion ring 01e is isorrorphic to an algebra of formal power
series k[[X 1 ,...,X n ]] .
(v) There is a perfect field K E L\ such that the ring G7 e Q9k K is re-
duced.
(vi) [Lie ( g) : K] = dim 9
(vii) For each R E  such that [R: k] < -tOO and each ideal 1 of R such
that 1 2 = 0 , the homorrorphism g(R) -r Q(R/1) is surjective .
Proof: (i) => (ii) => (iii) : triviaL
(iii) => (iv)
cf. 1 ,  4 , 4. 2 .
(iv) => (v) : triviaL
(v) => (i) : assume (v), and let K E  be an algebraic closure of K.
Since Elk is separable, the ring 0 e Q9k K = (Q9k K ) Q9K K is reduced.
Since it is enough to show that 9  K is smooth (I, 4 , 4.1), we may as-
sume that k is algebraically closed and 0 reduced. By translation, G is
e -
then reduced at all of its closed points (& 1, 3.3 c)), and so is reduced.
By 1 , & 4 , 4 .12, there is a dense open 1J in Q which is smooth over k. But
G is the union of the smooth open sub schemes y (g) Q for g E Q (k) , and is
therefore smooth over k.
(iii) <=> (vi) : this follows from the equalities
dim 9 = Kdim tJ
e
and
2
[Lie () : k] = [m 1m: k] ,
e e
from the definition in 1, & 4,4.1 and from 1,  4,2.2 .
(i) <=> (vii): cf. I,4,5.ll.

288
ALGEBRAIC GROUPS
II, 5, no 2
2.2
Corollary: If f: G-+H is a flat horrorrorphism of locally
algebraic k- groups , and if  is smooth , then so is B..
Proof : Since f Ok k is flat, we may assume that k is algebraically closed.
But J! ->- to is injective, so that J! is reduced and H is smooth
H,e Q,e .!:l,e
(by (v) of 2.1) .
2.3 Corollary : Let  be a locally algebraic k -group . If k is
perfect , then red is a smooth subqroup of G.
Proof: Since k is perfect, red x red is reduced (I ,  2 ,6.14), so that
the rrorphism
red x red -->-  x 
'TTg,
G
factors through G ed ; similarly, the comr::osite rrorphism
-r
a
G -->- G
-red -
Q
factors through ed. It follows that G red is a subgroup of Q; condition
(v) of the theorem iJrplies that it is smooth.
2.4 Corollary : Let G be a locally algebraic k- group of dimen -
sion O. If k is perfect ,  is the semidirect prcxluct of red !2Y QO .
Proof: Since GonG d = GO d is smooth, connected, and O-dimensional, it
- -re -re
is identical with e. by 1.4. The comr::osite rrorphism G d -+ G-+ '11 (G) is
-k -re - 0 -
accordingly a rronorrorphism. But, by 1.8, it is faithfully flat and quasicam-
pact; hence it is an isorrorphism, which completes the proof (1, 3.10) .
O-dimensional connected groups, i.e. those which are spectra
of local k-algebras of finite rank, are called infinitesimal (4, 6.1) .
Each O- dimensional locally algebraic group is accordingly the semidirect
prcxluct of an etale group by an infinitesimal group , provided the base field
is perfect.
2.5
Remark : We show in  6 , sect. 1 that if k is of characteris-
tic 0, each locally algebraic k-group is srrooth.

II, 5, no 2
LOCALLY ALGEBRAIC GROUPS
289
2.6 Examples : The group a k is srrooth: it is entertaining to veri-
fy this by means of condition (vii). If V is a finite dimensional k-vector
space, the group GL (V) is smooth; this may also be verified directly from
(vii). If r is a finitely generated commutative group, the diagonalizable
algebraic group 12 (f)k is smooth iff the torsion of r and the characteristic
of k are relatively prime (a vacuous condition if k is of characteristic 0) .
To see this, apply condition (vi): the dimension of 12 (f)k is the Krull di-
mension of k [f], Le. the rank of rover IQ, while the Lie algebra of
Q (f)k is isorrorphic to  (r ,k). In particular, Ilk is smooth.
2.7 We now give a partial generalization of theorem 2.1. Let AEl1,
and consider the A-group GL nA , which we identify with an open subset of
b(A n ) ""
SP A A [X, .] 1 <. '< .
- lJ  l, J  n
Proposi tion : Let G be a closed sub - group - scheme of GL nA let
P l ,...,P EA[X, ,] be such that , for each REl1" , 'jJe have
r lJ rt
C!(R) = {(x,.) EGL (R) : P l (x, ,) = ... = P (x..) = O}
lJ ----n lJ r lJ
Suppose that for each s E.e£A , the Lie alqebra of the K (s) -qroup C! I8i A Kts)
has rank n 2 - rover K(S). Then C!. is smooth over A.
Proof: In virtue of I,  4,4.2 it is enough to show that for each FOint x
of , the dPi(x) are linearly independent elements of A[X, ,]/A I8i A K(X) .
lJ
For this puq;ose we may assume that A is a field (replace A by K(S) )
and that x is rational over A. Since  G/A (x) is the quotient space of
the A-vector space with basis the dX,. by the subspace generated by the
lJ
dP i (x) (I,  4, def. 2.10 and 2.6 b) ), it is enough to show that G/A (x)
has rank n 2 - r . By translation (x being rational over A), it is enough
to verify that G/A(e) has rank n 2 _r. But G/A(e) is the dual of the
vector space Lie(G) which was assumed to have rank n 2 - r .
This proposition enables us to show that the classical groups
are srrooth over '3f. Take for example C!. = nZ' defined by the single equa-
tion det (X. ,) = 1; since for each field k the Lie algebra of SL nk has di-
lJ -
mension n 2 - 1 (4, 4 .12 ), SL n 2' is srrooth over :?? .

290
ALGEBRAIC GROUPS
II, 5, no 2
2.8
Srroothness of centralizers theorem : Let  be a smooth l=ally
algebraic k -group ,   k- group and !: B-+ Aut Grg a horrorrorphism ; there
is a natural linear representation p:  -+ GL (L ie c:; ) If H l (!:1 ,Lie c:;) = 0 ,
then g5 is a smooth l=ally alqebraic k- qroup.
Proof : By \:'i 1 , 3.6 and 3.8, c:;H is a c lased subgroup of g, and hence a
locally algebraic k-group. To establish its smoothness, by 2.1 it is enough
to show that if AE£:\ has finite rank over k and if I is an ideal of A
of vanishing square, then the horrorrorphism (A) -+ gtl( A/I) is surjective.
Define the k-group-functors C:;1 and C:;2 by
gl(R)=g(A0 k R ), g2(R) = Im(g(A0 k R )-+g((A/I)0 k R)).
By \:'i 4 , 3. 5, there is an exact sequence
(*)
-->- E (R)
1J (R) ) G (R)
-1
y(R) ) G (R) -->- 1 ,
-2
VJhere
E(R) A0R(wG /A0R,I0R)='" (wG/k,I0R)
-A0R -
='" Lie (c:;) 0 I 0 R ( Lie ( g) 0 I )  (R)
The map
(R) :  (W g / k , 10 R) -->- gl (R)
has a description similar to that of the isorrorphism
 (Wc:;/k ' R) -->- Lie (c:;) (R)
of \:'i 4 , 3.3. If h E H _ (R) , f (h) induces autorrorphisms of G 1 (R) and
- A0R -
C:;2 (R) and f (h) induces an autorrorphism of Lie(c:;) 0 R , hence an autorror-
phism of Lie(c:;) 010 R . It follows immediately from the definitions that
the horrorrorphisms g (R) and y (R) of the sequence (*) are ccxrpatible with
the actions of B.(R) . Accordingly we get an exact sequence of k-group-func-
tors acted uFOn by B.:
1 -->- (Lie (£;)0I)  9 1  g2
-->-
1 .
Now apply \:'i 3 , 1.4 to this situation; we get an "exact sequence" of ]X)inted
sets:

II,  5, no 3
LOCALLY ALGEBRAIC GROUPS
291
o 1!0 0 :{O 0 d 1
1-r H O (tl,(Lie(g) 0I)a) -->- H O (tI,gl) -->- H O (H&2) -->- HoOj,(Lie()0I)a) .
What are
o
HO (B,G. 2 )
of G(A)
so that
o H
these various terms? By  1,3.5, H O (l1,gl) = g- (A) , VJhile
is the set of elements of d-i(A/I) VJhich are images of elements
. But, since g is smoo th, g(A)-+-G.(A/I) is surjective (2.1),
H(£I,Q2) = gtl(A/I) . Moreover, we have
Hq,(Lie(Q)<2II)a) "" H(£I,Lie(g)a) <211
[ for if n is the rank of I over k, we have
Hence we get the "exact sequence":
(Lie (G)18i I) "" (Lie (G) )n ].
- a -a
1 -+- H O (H,Lie (g) ) I8i I -r #(A) -r Jj(A/I) -r H 1 (Ei,Lie (G) ) I8i I .
So if H 1 0.J,Lie(g)) = 0, gH(A) -+- g9A/I) is surjective, and this is VJhat
we had to prove.
Remark: The above proof still works when H is a k-rronoid-
functor, provided one already knows that gH is a l=ally algebraic
k-scheme.
Section 3
Orbits
3.1 ProFOsition : Let g be a k- group-functor , let  and 1: be
algebraic k- schemes acted upon by , and let f: X -->- Y be a rrorphlsm
VJhich is compatible with the actions of .
(a) If X is non-empty , 1: is reduced and g(k) acts transitively on X(k) ,
then f is faithfully flat .
(b) If g(k) acts transitively on X(k) then fC9 is a l=ally closed
subset of Y.
(c) ,Lf  is reduced and g (k) acts transitively on  (k) , then i factors
into "f>. -->- t: ()red -->- X , VJhere the first rrorphism is faithfully flat and the
second is an embeddinq . Furthermore f ()red is stable under G..
Proof: (a) Since 2:(k) f- 0 (I,  3,6.9) and (k) acts transitively on 1:(k) ,
f(k) is surjective and so therefore is f (I,  3,6.11). By I,  3,3.6,

292
ALGEBRAIC GROUPS
II,  5', no 3
there is a non-empty open subscheme .!! of r such that i induces a flat rror-
phism of f-l(Q) into ¥. To show that ! is flat, we may replace , ¥ ,.., Q
by X <:Ok k , Y <:Ok k ,..., Q<:Ok k , and hence assume that k is algebraically
closed. 'rhen g(k) acts transitively on y(k) and Q(k) t- Z; the translates
gQ for g EG(k) accordingly cover r. It follows that f is flat over
-1 -1 -1
U g f (Q) = U f (gm = f (U gQ) = I  I
g g g
(b) In virtue of I,  3 , 3.11, the proj ection rrorphism k ( ¥ <:Ok k) ->- Y is
open and surjective. To show that 1 (lli is locally closed, it is enough to
prove that its inverse image in I ¥ <:Ok k I, in other words (1 <:Ok k) (1S <:Ok k) ,
is locally closed. Accordingly we may assume k to be algebraically closed.
By I,  3 , 3.9, 1 Q9 is a constructible subset of y; hence there is a sub-
set U of ! () which is open and dense in 1 () . 'rhus f-l(U) is open in
-1 -1
II . Since the translates gf (U)=f (gU), gEg(k), cover II (U being
non-empty because 1S is non-empty), i (lli is the union of the gQ, hence
open in its closure.
(c) Let l' = Im f be the closed image of i and let  = f()red be the open
subscheme of l' carried by f Q) . By (a) it is enough to show that, for each
R E  ' the sub scheme R of ¥R is stable under g (R) . For this, let
P (R) = i R ("f>.R) be the set of points of R. Using the notation of I,  1,4.10
and I,  2 , 6.11, we then have
R = (T R ) P (R) = (( Im i )R)P (R) = ( Im () ) P (R)
(I ,  2, 6.15) and it is clear that the last expression is stable under g (R) .
3.2 Proposition : Let g be a smooth locally algebraic k -group
actinq on a locally alqebraic k -scheme .
(a) Let Y be a reduced closed sub scheme of X. 'rhen Y is stable under G
iff r(k) is stable under (k) .
(b) If Y is a stable subscheme of , then 1¥l red and (II- Irl)red are
stable under G.
(c) Each non-empty stable subscheme of X of minimum dimension is closed .

II,5,no4
LOCALLY AIbEBRAIC GROUPS
293
Proof: (a) By I,  4,6.3 applied to the projection gx x: -+ x: , gx X is re-
duced. A rrorphism .12: Q.x X -+  thus factors through Y iff 12(k) factors
through  (k) .
(b) Identifying X(k) with the set of clos:d points_ of 1S.<Si k k (I ,3, 6.6
and 6.8), I i! red (k) is the closure of  (k) in  (k) . Since each g E g (k)
leaves X(k) fixed and induces a continuous autorrorphism of (k) , it leaves
!! red (k) fixed; (a) now applies. Similar reasoning works for (IYI-!Y ! ) .
- - red
(c) If  is a non-empty stable sub scheme of  of min.imum dimension, we
have dim(II-!I)red < dim, whence by (b) Ixl-lxl=  and Y is closed.
3.3
Let g be a reduced algebraic k-group and let X be an alge-
braic scheme acted upon by g. Given x E X(k) consider the rrorphism
i : g -+  defined by i (g) = gx for g E g(R) , RE  . We may now apply
3.1 (c) to this rrorphism: this explains why we shall call the subscheme
!:(g)red of X the orbit of X.
Proposition : Suppose that k is algebraically closed , and let G be a srrooth
algebraic k- group acting on a non-empty algebraic k- scheme X . 'Ihen there is
a point x E 1S. (k) with a closed orbit.
Proof: Let Y be a non-empty stable sub scheme of X of minimum dimension.
By 3.2 this subscheme is closed. Moreover, if x E X(k) , the orbit of x is
a stable subscheme of Y; accordingly it coincides with Y red (3.2 (b)) ,
and is therefore closed.
Section 4
'Ihe group of rational points over an algebraically closed field
4.1 SUppose that k is algebraically closed. For each locally al-
gebraic k-scheme "f>. identify  (k) with the set of closed points of  (I,
 3,6.6 and 6.8). We know (I,  3,6.9) that A 1-+ An(k) is a bijection of
the family of closed (resp. open, locally closed, irreducible, constructible)
subsets of  onto the corresponding family of subsets of X(k) . If G. is a
k-group-functor acting on X and Y is a reduced sub scheme of , we imne-
diately obtain
Norm G ('£) (k) = Norm (k) (X (k)) , cen t (Y) (k) = cent g (k) (X (k) )

294
ALGEBRAIC GROUPS
II, '5, no 4
In particular, if r is closed (resp. if 1:f. is separated) , we know that
Norm.{ O (resp. cent (y)) is a closed subfunctor of . If, in addition, 
is a reduced k-group, it follows that it nonnalizes x: (resp. centralizes y)
iff  (k) normalizes 2:: (k) (resp. centralizes x: (k) ).
Now if k is arbitrary  is a smooth k-group, and H a smooth
closed subgroup of , we may apply the above results to the reduced
k-groups  @ k and H (9k k '_ thus obtaining Norm (B)(k) = Norm<2 0<) nHk) )
and cent G(tl)(k) = cent (k)Qi(k)) . In prticular H is normal (resp. central )
iff G is normal (resp. central) in (k) . For example,  is camnutative
iff (k) is cornnutative.
4.2 If  and x: are l=all y algebraic k-schemes, the prcxluct topo-
logy of  (k) x x: (k) is not in general the topology of O:f x x: )(k) (for in-
stance, if G is a l=ally algebraic k-group, (k) is not in general a topo-
logical group). However, if A (resp. B) is a subset of 1:f.(k) , we have
A x B = A x B : To prove this, notice that, if a E A , the rrorphism of r into
 x x: which sends y onto (a, y) for y E Y (R) , R E  induces a continuous
- -
map of y (k) into (ii x X) (k) . It follows that a x B C a x B . Hence
A x 13 C U a x B C A x B , so that A x 13 C A x B C A x B . The reverse inclusion
a
is obvious . Accor dingly, for each rrorphism .f:  x y ->-  , we have
.f(AX B) C i(Ax B) .
4.3 Proposition : Suppose that k is alqebraically closed and let
G be a l=ally alqebraic k -group.
(a) The map H >-+ H(k) is a bijection of the set of open (resp. closed re-
duced ) subqroups of  onto the set of open (resp. closed ) su1xJroups of (k)
(b) If A and B are constructible (resp. irreducible , resp. dense construc -
tible ) subsets of (k) , then A.B is constructible (resp." A.B is irredu-
cible , resp. A.B = ,G(k) ).
(c) The closure of a subgroup of  (k) is a subgroup of  (k) . Moreover ,
each constructible subgroup of (k) is closed .
Proof: (a) Clearly, if H is an open (resp. closed reduced) subgroup of ,
then tl(k) is an open (resp. closed) subgroup of G(k) . Conversely, if L is
an open (resp. closed) subgroup of <;!(k) , let tl be the open (resp. closed

II,  5, no 4
LOCALLY ALGEBRAIC GROUPS
295
reduced) subscheme of  whose space of points is the open (resp. closed)
subset L I of  such that L' n (k) = L. It is immediate that the rrorphism
'TT G
-;-GxGG
H x H
factors through B, as does the rrorphism
a G
H

G
and so H is indeed a subgroup of .
(b) Let A and B be constructible subsets of (k) and let AI and B' be
the corresponding constructible subsets of . The subset C' of  x  de-
fined by C' = J2!" 1-1 (A ') n .E!" 2- 1 (B') is constructible, hence also the subset
'TTG(C') of G . But obviously we have 'TT(C') n (k) = A.B. The irreducibility
assertion is proved similarly (4.2 and I,  4,4.11). Finally, if A and B
are constructible and dense, A' and B' are constructible and dense in II
and hence contain dense open subsets V and W of I G I (I,  3 , 3.2). But if
U = Vn W , then U(k) U(k) = G(k) (1.2) so that A.B = (k) .
(c) Let H be a subgroup of (k) ; since a G is a scheme-autorrorphism of ,
-1 - - -1 -
x f-+ x is an autorrorphism of (k) and so (H) H. Since H.H C H.H = H
(4.2), Ii is a subgroup of (k) . Finally, if H is a constructible subgroup
of (k) , let H' be the closed reduced subgroup of  such that fl' (k) = H
(H' exists by (a) ). Applying (b) to B I , we get B'(k) = H = H.H = B, so
that H is closed.
4.4
Lemma: Let R and S be ITOdels and let p: R ->- S be a rror-
phism which makes S a faithfully flat R -ITOdule . Let  be an P- group-scheme
and let H be a subscheme of G. Then B is a subgroup of  iff B @R S is
a subq"roup of  @R S .
Proof: Consider the cOlT1]X)site rrorphism
f:HxH -;- GxG
'TT G
G.
-1
Then f factors through H iff the canonical embedding f (lI) ->- tl x [
isorrorphism, and by I,  2 , 3.5 this holds iff f -inn (9R S ->- (B x BJ @R S
an isorrorphism. A similar argument applies to the cOlT1]X)site rrorphism
a
is an
is
H
G


296
ALGEBRAIC GROUPS
II,  5, no 4
!
!
4.5 ProFOsi tion : Let g be an algebraic k- group and let x E g (k) .
Let H be the smallest closed subgroup of g for which x E ji(k) . Then B is
commutative , smooth , and B (k) is the closure of the subgroup of g (k) gene-
rated by x .
Proof : Let : Z'k ->- G.. be the horrorrorphism such that  (1) = x . Let B. be
the closed image of ! (I,2,6.ll); by I,2,6.16, .!i0kk is the closed
image of f Ok k and it is carried by the closure E of the image of I 0k k I .
But En G (k) is the closure of the image of f(k): Z -+ G.(k) , hence is the
closure of the subgroup of (k) generated by x. By 4.3 (a) and (b), Eng(k)
is then of the form H' (k) , where g' is a closed reduced subgroup of
 Ok k . Hence H Ok k = H' . It follows that .!i is a subgroup of  (4.4);
by 2.1 it is srrooth and by 4.1 corrmutative since g(k) is commutative.
We call H the closed subgroup of g generated by x. If K is
an extension of k, the subgroup of  Ok K generated by the canonical image
of x in (0k K) (K) =  (K) is g Ok K .
Example : If x E G. (k) has order n < + 00, g is isorrorphic to
the constant group (Z!nZ)k
4.6 Proposition : Suppose that k is algebraically closed and let
G be an algebraic k -group . Let (Ai)iE I be a family of constructible and
irreducible subsets of (k) containing the unit element . Let H be the sub-
group of G(k) generated by the A, . Then H is closed and irreducible.
- l -
Moreover , there is a sequence B 1 ,... ,B n , n:S 2 dim G, of subsets of g (k) ,
-1
chosen from the A. and the A. , such that
l l
H = B 1 B 2 ... Bn .
-1
Proof: Clearly we may assume that each A. is a member of the family (AJ.
l l
Consider the collection of subsets of G.(k) of the form Ail A i2 ... A.ip , as
(ii' . . . , i p) ranges through the set of finite sequences of elements of I.
Each subset of th is form is an irreducible closed subset of G.(k) (4.3 (b) ).
If Ajl2... A:iq is maximal (q:Sn), then the inclusion A.13 C A.B (4.2)
implies
Ajl." Ajq.Ail... A = Ail... A.Ajl... Ajq
Ajl. .. A
for all
i 1 :...,i p

II,  5, no 4
LOCALLY ALGEBRAIC GROUPS
297
In particu lar, A j1 . .. Ajq is a subgroup of <2 (k) containing H. Thus we
have H C A j1 . .. Ajq , so that, by 4.3 (b) applied to the closed reduced sub-
group of  defined by H (4.3 (a) ), we get
-
H = Aj 1 ... Aj
. q
Aj 1 ... Ajq
-
This implies that H H, and also yields the final assertion.
4.7 Corollary : Suppose that k is algebraically closed and let G
be a locally algebraic k- group . Let A and B be closed subgroups of (k) ,
where B is irreducible . Then the group of conmutators (A , B) is closed and
irreducible . Moreover , if n = dim  , each element o f (A, B) is the pro-
duct of at rrost 2n conmutators.
Proof: For each a E A let B be the image of B under the map
- -1 -1 a
b  a b ab . Since Ba is the set of rational points of the image of a
rrorphism  -  , where  is the closed reduced subgroup of Q such that
B (k) = B , B is a constructible and irreducible subset of <;t(k) which
- a
contains the unit element. The proof is completed by applying 4.6 to the
family (B a ) a EA.
4.8 Existence theorem for the derived group : Let  be a smooth
algebraic k -group . Let 0() be the derived group of g, Le. the subfunc-
tor of G defined as follows : for each REl\, .9()(R) is the set of
gEG(R) for which there is SE, faithfully flat and finitely presented
over R, such that gs belongs to the group of commutators of (S) . Then
(a) j) (<;;;) is a smooth closed subgroup of g, which is connected if G is
connected ;
(b) for each algebraically closed extension K of k, 2 (<;;;)(K) is the group
of conmutators of  (k) .
Proof : First consider the group Q(k) and its group of conmutators H : we
show that H! (G(k) ,QO(k)) is finite. Let K = 9(k)! ((k) ,<;t(k)) ; then since
the image of G.-°(k) in K is central, the centre of K is of finite index
in K. By a classical result in group theory*, it follows that the group of
*
See for example B. HUPPERT, Endliche Gruppen I, chap. IV, S 2 , Satz 2.3,
Springer - Verlag, 1967.

298
ALGEBRAIC GROUPS
II,5,n04
commutators of K is finite. But this group of commutators is precisely
H / ( G (k) ,(:t(i)) . By 4.7, (  (k) , t(k)) and hence Hare closed subgroups of
 (k) ; rroreover, if each element of H / (  (k) ,O(k)) is the prcxluct of at
rrost q commutators of K, and if n is the dimension of , then each ele-
ment of H is the prcxluct of at rrost N= (2n+ q) commutators.
Now we consider the rrorphism 11: g2N -+  such that for each REl\: we
-1 -1 -1 -1
have (xl'Yl'...''YN) = xl Yl xlY l ...  Y N YN ' and let V be
its closed image (I,\;'i2,6.11). By I,\;'i2,6.15, V<:Okk is the closed image
of 11 <:Ok k ; it is carried by the closure E of the image of I <:Ok kl. Since
En g (k) = H by the preceding remarks, V Ok k is the closed reduced sub-
group of <:Okk whose set of rational FOints is H (4.3 (a)) . By 4.4 and
2.1, V is therefore a srrooth subgroup of , which is connected if G is
connected.
We now show that I1 = V (g) . To do this, notice first that the rrorphism
f : 2N -+ Q induced by Q is dominant. Hence, by I, \;'i 3,3.6, there is a
2N
dense open subfunctor of 9 such that flQ is flat and i(!,J) is dense in
D Accordingly, the coTf1!xJs i te rrorphism
u x U
fill x flu) Q x D
'TTJd
D
is flat (\;'il, 3.2) and surjective (1.2). If gE Q(R) , consider the induced
Cartesian square
ll x Q
D
r
Il
v
Sp R
If (V.) is a finite affine open covering of y, and if S = II G!(V,) , then
-l l
S is faithfully flat and finitely presented over R and we have a commuta-
tive diagram
g2N x g2N 'TT G ( fx f) G
I Il
S SpR
so that gs belongs to the group of commutators of g(S) and gE V()(R)

II,  5, no 5
LOCALLY ALGEBRAIC GROUPS
299
Conversely, since the rrorphism of g x s:; into s:; which sends (x,y) onto
-1 -1
x Y xy factors through ]2, it is plain that the group of conmutators of
G(R) is contained in ]2(R) . If gE (R) and if gs belongs to the group of
corrrnutators of G.(S) , hence to 12(S) , S being faithfully flat over R, then
g E ]2 (R) by I, 5 2 , 3 . 6 applied to the corrrnuta ti ve diagram
D
(gs)# I
Sp S
G.
Il
) Sp R
This proves (a). To prove (b), remark that, for each S E  finitely gene-
rated over K, by the Nullstellensatz there is tp E (S ,K) ; with the nota-
tion of the theorem, g = tp(gs) then belongs to the group of corrrnutators
of tp((S)) c g(K)
4.9
One proves similarly the
Proposition : Let  be an algebraic k -group , !! and  two srrooth closed
subgroups of , where .!i is connected . Then the sub functor C!i,IS) of -.G.
such that for each RE, (!i,IS)(R) is the set of gE (R) for which
there is S E £% ' faithfully flat and finitely presented over R, such that
gs E (!i (S), (S)) , " is a srrooth , connected and closed sub:Jroup of .
Section 5
Horrorrorphisms of algebraic groups
5.1
ProFOsition : Let f: G -+ H be a horrorrorphism of algebraic
k- groups .
(a) The image f (s:;) of 'f is a closed subset of 11, and we have
dim f(S:;) = dimS:; - dim Ker f
(b) If f is a rronorrorphism (Le. if Ker f =  ), then f is a closed em-
bedding.
(c) If R is reduced and if f is surjective (resp. and if fO: GO ->- HO is
surjective ) then f is faithfully flat (resp. flat).
(d) If G is reduced (resp. srrooth ), then f (G)red is a closed reduced

300
ALGEBRAIC GROUPS
II,5,no5
(resp. smooth ) subqroup of H, and f factors into -- f (G)red -- B where
the first morphism is faithfully flat and the second is a closed embeddinq.
Proof: (a) Let  act on.  and H by translations ( g E  (R) sends g' E Q (R)
onto g g' and h EH (R) onto f(g) h ). By 3.1 (b), f (Q) is locally closed.
Since t:((k)) is a subgroup of tl(k) , it is closed (4.3 (c)), and accor-
dingl y i () is closed by I, 9 3 , 6.11. We defer the proof of the second part
of (a) until after we have proved (d).
(b) We may assume that k is algebraically closed (1,92,7.3). By I I 93,4.7,
-1
there is dense open subscheme y of tl such that t: I f (Y) is an embedding.
By translation f is an embedding and by (a) it is closed.
(c) If f is surjective, it is faithfully flat by 3.1 (a) . If t: QO -+!:L 0
is surjective, it is flat by the preceding remark, hence, by translation, i
is flat (by 1,92,3.2 rem. we may assume that k is algebraically closed) .
(d) Apply 3.1 (c) ; we simply show that f(G)red is a subgroup of Q. Since
the former is obviously stable under a, it is sufficient to show that it
is stable under the prcxluct. Now we have the commutative diagram
fJXE i xi
:r , f () red x f ()red ) Hx H
- j
E i
 i (Q)red , H
where fJxp is faithfully flat and where i is an embedding. Applying I,
9 2 ,3.6, we obtain the required conclusion. If G. is smooth, so is f(Q)red
by 2.2.
Finally, we take up the last assertion of (a). We may replace k by k (I,
93,6.2) and hence assume that k is algebraically closed. Then Qred and
£!:red are subgroups of Q and £!: (2.3) and by (d) we have a faithfully flat
morphism G red ---;. f (G.)red whose kernel is ( Ker f) n Qred . By I, 9 3, 6.3,
we have dim f (g) = dim gred - dim ( Ker f) n gred = dim G - dim Ker i .
5.2
Corollary : Let G be an algebraic k- group. Then the following
conditions are equivalent :
(i) G is affine ;

II,  5, no 5
LOCALLY ALGEBRAIC GROUPS
301
(ii) there is a faithful linear representation p:  -r GL (V) of  in a
finite dimensional k- vector space . (Recall that p is faithful if it is a
lIDnorrorphism. )
Proof: By 5.1 (b) and 92,3.-4.
5.3 Proposi tion : Let i: g -r li be a hOITOITDrphism of algebraic
k- groups. Suppose that g is smooth over k. Then the following conditions
are equivalent , and they iJrply that !i is smooth :
(i) Lie (f): Lie (G) -+ Lie (H) is surjective;
(ii) Ker f is smooth and ! (G)red is open in H;
(iii) i is smooth.
Proof: Let .K = Ker f, 2: = Lie (G) , h = Lie (tl! , Js = Lie (:[S) , cp = Lie (f) . By
5.1, we have dimG=dim:[S+ dimf(G) . Moreover, evidently [J:;:k]+[cp():k] =
[g:k] ; finally, since G is smooth, we have dim'G = [g:k] . It follows imme-
 - 
diately that
[h:k] - [cp (g) :k] = ([h:k] - dim H) + (dim H - dim f (G)) + ([k:k] - dim K) .
vv ........ /""'" - - __............ _
If (i) holds, the left-hand side of the above equation is zero, hence so
also are the three expressions on the right-hand side. This implies that K
and H are smooth 'and that f(G)red is open in g, and (ii) follows.
Now assume (ii); we derive (iii). Since f (G) d is O pen in H _ and smooth
- - re
(5.1), Ii is smooth (2.1) and f is flat (5.1). To show that ! is smooth,
by extension of the base field we may confine ourselves to the case in which
k is algebraically closed (I, S 4 , 4.1). Since the set of points of Gat
which! is smooth is open (1,94,4.3), it is enough to show that i is
smooth at each rational point g E G.(k) . For this purpose we need only veri-
-1
fy that f (f (g)) is smooth, and this is the case because the left trans-
lation y(g):G-+ induces an isanorphism of Ker f onto f-l(f(g))
5.4
PGL nk = Aut (H n (k) )
gEn(R) , RE,
of t1 n (R) .
Finally, (iii) implies (i) by 1,94,4.14 or 4.15.

302
AlliEBRAIC GROUPS
II,  5, no 5
of GL nk may be identified with M n (k) (4, 2.2) and that of PGL nk with
the algebra Der (M (k)) of derivations of M (k) (4, 2.3). Let us compute
n n
the map Lie CO . If x E Lie (Q) , we have
e EXme-EX = (1+ EX) m (1- EX) = m+ E (xm - mx) ;
it follows that Lie (f) sends X E M (k) onto the inner derivation
n
m t-+ xm - mx . By a classical result (cf. Cartan and Eilenberg, Horrological
Algebra, chap. IX, 5.1 and 7.8), Lie(f} is then surjective. Applying 5.3,
we infer that PGL nk is srroo th. Moreover, we know (Alg. VIII,  10, no.l,
cor. to th. 1) that the k-au tomorphisms of M (k) are of the form m t-+ gm; -1.
n
It follows from 5.1 that f is faithfully flat. Finally, Ker f is a srrooth
subgroup of  by 5.3; since the centre of M n (k) consists of scalars
only, ( Ker f)(k) consists of horrothetic maps. It follows that Ker f is the
subgroup 12 of g,nk ' isorrorphic to llk ' such that, for each R E  ' 12. (R)
is the set of homothetic maps.
5.5
Corollary : Let !: Q -->- H be a horromorphism of algebraic
k -groups . SUPFOse that .Q is srrooth over k. Then
(a) Lie cn is bijective iff Ker f is etale and ! (Q)red is open in ;
(b) f is an open embedding iff f is a rronorrorphism and Lie (f) is bi-
jective.
Proof: (a) By 5.3, Lie (f) is bijective iff i(Q)red is open and Ker f is
srrooth and has zero Lie algebra, i. e . is etale (1. 4) .
(b) One way round is obvious. Conversely, any monorrorphism i is a closed
embedding (5.1 (b)) ; if Lie(t) is bijective, f()red is an open subgroup
of H. The induced morphism Q -->- f (Q)red is a rronorrorphism which is also
faithfully flat (5.1 (d)), hence a strict epirrorphism (I,  2,3.4). It
follows that Q ->- i (Q)red is an isomorphism and i is an open embedding.
5.6 Corollary : Let 1.1- be a smooth subqroup of an algebraic
k -group G. Then !i. is an open subgroup of  iff Lie (!.!l = Lie (0
Proof: This is just 5.5 (b) applied to the embedding B -->- Q. .

II,  5, no 5
LOCALLY ALGEBRAIC GROUPS
303
5.7
Lemma : Let g be a l=ally algebraic k -qroup.
(a) Let H be a subgroup o f g. Let l! act on Lie () via the adjoint re-
presentation of g. Let  = NormG (!i) ,  = Cent G(!.n (cf.  1 , 3.7). Then we
have Lie () = Lie (g)tl., Lie @) /Lie (!i) = (Lie (Q) /Lie (!i)) .
(b) If p: g -.... GL (V) is a finite dimensional linear representation of .Q.,
and if W'c W are two vector subspaces of V, then the Lie algebra of the
subgroup ',w of Q (2,1.3) is the sub-Lie algebra Lie(Q\v"w of Lie(g)
consisting of all x ELie(g) such that Lie(p)(x) E L(V) maps W into W' .
Proof: (a) The assertion about  follows from  4 , 2.5 applied to the ho-
rrorrorphism
H. --+ g
JJ:1t
) Aut Gr (g) .
Let x ELie()
Then x E Lie (!'J) iff for each M E!\ and each h E 1.1 (M) ,
we have
EX -EX
(e k(E:) hM(E) (e )M(E) E.!.!(M(E)) and
-EX EX
(e k(E) \i(E) (e k(E) E B(M(E)) ,
by  1,3.5. But this latter condition may be written in the form
E -E-l
e \i(E)e \i(E) E H(M(E))
and
-E E-l
e \i(E) e \i(E:) E B(M(E)) ,
hence in the fOTI11:,
:!: E(- Ad(h))
e E B(M(E)) ,
Le. - Ad (h)1E Lie(!i)0M , which implies the required result.
(b) Let X E Lie () . Then X belongs to Lie (Sw " W) iff e EX - Id maps
W0k(E) into W'0k(E:) , Le. iff Lie(p)(x) maps W into W' .
5.8
Corollary : Let !! be a srrooth subgroup of an algebraic k -group .
(a) If (Lie () /Lie q:j))H = 0 , then B is an open subgroup o f NormG (B) .
(b) If Lie(H) is self-normalizing in Lie (g) , then !i is an open sub-
group of NODnG (B) and Norm G (B) is an open subgroup of Norm G ( Lie (!i)) .
Proof: Let Q and D,' be the Lie algebras of NormG (1.1) and Norm G ( Lie (!i) ) .
By 5.7 (a) and (b), we have

304
ALGEBRAIC GROUPS
II,  5, no 5
I
U/Lie (g) = (Lie (G.) /Lie (g) Ii c (L ie (g) /Lie (!i) fie (g) = E,' /Lie (B)
Now apply 5.6 to complete the proof.
5.9 Corollary : t  be an algebraic k -group , Q a conmutati ve
subgroup of  and B a srrooth subgroup of  containing CentG (Q) If
Hi (g , Lie (g) ) = 0 , then B is an open subgroup of Norm (g) . -
Proof: We have the exact sequence (3, 1.4 or  3,3.2 if Q is affine)
0-+ Lie(g)Q -+ Lie (G)Q -+ (Lie(g)/Lie(g))Q -+ Hl(g,Lie(H)) .
Since B::::> Cent G(g) , Lie(H)::::> Lie(gQ (5.7 (a)) , hence Liemf2.= Lie( .
It follows that - (Lie (g) /Lie (En,'2 = 0 i also QC; now apply 5.8 (a) .
5.10 Corollary : Let G be a smooth algebraic k- group and Q a
closed conmutative Subgrou;of -. If Hl(g,Lie()) = 0 and k(g)= 0,
then cent (g), Norm (9) and NormG ( Cent (g)) are smooth , and each is an
open subgroup of its successor .
Proof: We already know that Cent G(g) is smooth (2.8). Moreover, we have
H\g , Lie ( CentG (Q)) = Hi (12, Lie ()9)
by 5.7 (a). Since 9. acts trivially on Lie (G.,9., we have accordingly
H\Q,Lie( CentG. (Q))  H 1 (Q,k)@Lie(g,';{ "'" (Q,)@Lie(g)Q = 0 .
Applying corollary 5.9 to the pair (g, Cent G(Q)) we infer that cent G(Q)
is an open subgroup of NormG ( CentG. (Q)) . But NormG. (Q) is contained be-
tween these two, and this completes the proof.
5.11 Remark : By  3 , 4 .2 and 4.3, the above corollary applies in
particular when 9 is diagonalizable (or, more generally, when 9 @k k is
diagonalizable) .

 6
THE CHARACTERISTIC 0 CASE
Throughout  6 I we assume that k is a field of characteristic o.
Section 1
The enveloping algebra and invariant differential operators
1.1 Let  be a locally algebraic k-group. The canonical isomor-
'U +
phism Lie  _ Dist 1  of  4 , 6. 8 may be extended to an algebra homomor-.
phism
c : U(Lic) -+ Dist 
where U (Lie l2) denotes the enveloping algebra of Lie  . (Gr. et. alg. de
Lie, I I  2 , prop. 1). This homomorphism is compatible with the filtrations,
i.e. we have c (1U (Lie G) c Dist G for each n E!l'J (Gr. et. alg. de Lie, I ,
n - n-
5 2 , no. 6 ) .
Cartier's theorem : Let G be a locally algebraic k -group . Then
a) Q is srrooth ,
b) the canonical isomorphism c : U(Lie) _ Dist  is bi-jective.
Proof: Let e be the unit element of (k) I regarded as a point of . By
 4 15.4 I we may identify Dist n  with the space of k-linear fonns on V e
which vanish on m n+ 1 I hence also as the space of fonns on:9 which vanish
A n+ 1 e e 2 -
on m . Let I be the ideal of the synmetric algebra S (m 1m ) generated
e 2  2 e e
by m 1m , S (m 1m ) the completion of S (m 1m 2) in the I -adic topology I
e e e e e e
D the space of k-linear fonns on S (m 1m 2) which vanish on i n + 1 and
nee
DUD
n EJN n
Each section s: m 1m 2 --+ m of the canonical projection extends to a con-
e e e
tinuous homomorphism S (m 1m 2) --+ JJ ; the transpose map sends
e e e
D , and hence induces maps h : Dist G --+ D
n n n- n
below that the composition
and
Dist G into
n-
h : Dist G --+ D . h'e show
( . ) c . h
U Lle  -->- Dlst  -- .... D
 2 
is bijective. Since h is injective (for the hOI!DI!Drphism S (m 1m ) --+ t9
e e e
induces a surjection of the graded algebras associated with the 1- adic and

306
ALGEBRAIC GROUPS
II, S 6, no 1
ill - adic filtrations; by Alg. COmTI. III ,  2 , cor. 2 to tho 1 it is therefore
surjective); it follows that h is bijective, and so also are c and
A 2 A
S (m 1m ) -+ [) (for if this last map is not injective, by Alg. COmTI. III ,
e e e
An
 2, cor. to prop. 5, its kernel is not contained in I for sufficiently
large n, which contradicts the surjectivity of h). Since S (m 1m 2) is iso-
e e
rrorphic to the algebra of formal power series in [m 1m 2 : k] variables,
e e
a) and b) follow.
To show that hoc is bijective, we prove that, for each n EN , the indu-
ced map
1.1 :u/u i -DID 1
-n n n- n n-
(vvhere Un = Un(Lie G) ) is bijective. To see this, choose a basis wi'... ,w d
f I 2 d ba . + 'V ( I 2
or m m and a ual sis £; 1 '.'" £; d for Dlst 1 G - Hcxl k m m , k) .
e e - e e
By the Poincare - Birkhoff - Witt theorem (Gr. et. alg. de Lie I,  2 , no. 7) ,
un/un_1 then has a basis consisting of the residue classes mod u n _ 1 of
0.1 ad _ " .
the quantities £;1 ... £;d ' vvhere 0. 1 + ... + ad - n. S1111l1arly, the resldue
B 1 Bd. _
classes of wi ... w d wlth B 1 + ... + Bd - n constitute a basis for
I n /I n + 1 . The bijectivity of hoc now follows from the canonical isomor-
phism D ID 1 --->- tiocl (I n /I n + 1 , k) and from 1.2 below.
n n-  -k
1.2
Lemna: \'i'ith the notation of 1.1, we have
0. 1 ad B 1 Bd
( h ( c ( £;1 . .. £;d )), wi . . . w d ) 0
provided
a. t- B,
l l
for at least one i, and
0. 1 ad 0. 1 ad
( h ( c ( £;1 . .. £;d )), wi . . . w d )
0. 1 : 0. 2 : ... ad:
Proof: It is enough to show that, rrore generally, we have
(*)
(h(c(x 1 ...x n )),a 1 ...a n ) = aJs (x 1 ,b a (l) ... (xn,ba(n)
II
2
a i ' . . . , a E m 1m and b = s (a,) E m
nee i l e
vvhere X l '.'.' x E Lie G ,
n -
Now we have, by definition
(h(c(x 1 ... x n )), a 1 ... an)
f b 1 ... b n d (xl * . . . * x n )
If
'11
n
G x ... x G --+ G satisfies
'11 (0 1 ' ..., g )
n  n
g 1 . .. gn
for R E 

II,6,nol
THE CHARACTERISTIC 0 CASE
307
and giEg(R), then by 4, 6.1, xl *...* x n is the composite deviation
'V xl x . . . x x n
 +--.E\x ...x (
'11 0
n
x...xg+--g.
Moreover, if SE is the leading neighbourhcx::x:J. of  with respect to the unit
section E<;!: k -->- S and i:.<;: E -->-  is the inclusion rrorphism, then each
xi is a composition of the form
Yi
.E\ -+--- SE
jO
-<---"'- G
\'i'e then have
f b 1 ...b d(x l *".*x) = f((b l ...b) '11) d(x 1 x ...xx)
n n n n n
= f((b 1 ...b n ) 'TT n (2 x...x j)) d(Y 1 x ...xYn) .
Now, by 1.3 below, we have
(b 1 . ..b ) '11 (j X ... x j )
n n - -
II (a.@...@l+ ... +1@ ...@a, +c,)
ill l
where
2
c , E M and t1 is the maximal ideal of
l
L (a (1) @ . .. @ a ( ) )
aE S a a n
-n
eJ/m @ ...@ c9 1m
e e e e
This and the equality
f aa(l)@ ...@aa(n) d(Y 1 x...xYn)
(Yi,aa(i)
II ( x. , b ( . ) )
i l a l
now give fOrTIRlla (*) .
1.3
Le.rrrua : Wi th the notation of 1.1 and 1.2, the homomorphism
induced by
o : 0; ---+ G' 1m 2 @ ... @ tJ 1m 2
e e ek k e e
'11 (jx...xj) Gx
n - - '-eo;
x G --->- G
--eo:
satisfies
o(b) == b@l@...@l+l@b@...@l+...+l@l@...@b mcxl M 2 ,
for
- 2
bE me ' and b = b mcxl me

308
ALGEBRAIC GROUPS
II,  0, no 1
Proof: Let 8 : 19 -+ () Q9 k () 1m 2 I5Q ... 6Q [) 1m 2 be the horrorrorphism induced
nee e e K: -k e e
by '11 (G x j x ... x j) : G x G x ... x G -+ G . It is enough to show that
n - - - - -E: -s-
8 (b):: bQ910...01+1@biSi...01+... rrodN 2 where N is the maximal
n 2 2
ideal of {) 0 rJ 1m 0...0 <2 1m . Now, when n = 2, we have 8 2 (b) = b'0 1+
e e e e e 2
liSib" +b'" , where b"'Em 0m 1m . Since the comr::osition G  Gx e. --+
e e e - --k
G ' '112
xJ..' QxCi--+0
is the identity, we see that b = J:!. Noting that
GxG
- -s
the composition of G  e x G --+ G x G wi th
-E: -k -E: --E:
!!Drphism, we similarly obtain b" = b . The general
'TT2(x i) is the inclusion
case may be inferred from
this by induction, using the formula
o ( 0 2 iSi () 1m 2 iSi . . . iSi t!J 1m 2) 0 0 ,
nee e e n-l
which follows from the fact that
'11
n
'11 0
n-l
('11 X G n - 1 )
2 -
1.4
Recall that with each element x E Lie G. we have associated
(  4,4.5) a derivation 0 '(x) (resp. y'(x)) on G which is left (resp. right)
translation invariant. This map extends to a hO!!D!!Drphism (resp. antihomo-
!!Drphism) of algebras 8': 1lJ (Lie Q) -+ Dif G (resp. y': U (Lie ) -+ Dif Q
where Dif G is the algebra of differential operators on 0 (4, 5.3) .
Corollary : The map 0': U (Lie G) -+ Dif G (resp. y': V (Lie G) -+ Dif G
induces an isorrorphism (resp. an anti-iso!!Drphism ) of the alqebra 1JJ(Lie C;Z)
onto the algebra of differential operators on .G. which are left (resp. riqht )
translation invariant.
Proof: Immediate from 1.1 and  4 , 6.5 .
1.5 Since the isomorphism c of 1.1 is compatible with the adjoint
representation of Ci, it induces an isomorphism U(Lie G)G --+ (Dist Q)Ci .
Supr::ose now that  is connected. Applying 2.1 c) , which is proved below, to
V(Lie G) (which is the union of the 1JJ (Lie G) , all of which are finite
- n -
dimensional and stable under G ), we get U (Lie G) G = U (Lie Ci) Lie G. . But
since Lie & generates the algebra U(Lie Ci) , the right-hand side is pre-
cisely the center of U(Lie Q) . Applying  4 ,6.7 , we obtain the
Corollary :  se that G is connected : a) Let x E U (Lie G)
following conditions are equivalent :
Then the
--

II,  6, no 2
THE CHARACTERISTIC 0 CASE
309
(i) y'(x) is right-and-left translation invariant.
(i') o'(x) is ri g ht-and-left translation invariant.
(ii) y'(x) = o' (x) .
(iii) x is in the tf:<:)t U(Lie C?.) .
b) The maps y' _ and o' induce the same isorrorphism of the c eTl -t:.re of QJ (Lie )
onto the algebra of right-and-left - translation invariant differential ope-
rators on G.
Section 2
Relationships between groups and Lie algebras
2.1
Pror::osi tion : Let G be a locally algebraic k
(a) Let H and !:; be subgroups of Gi g B is connected , then BClS; iff
Lie tl C Lie !5 . If H and !:; are connected , H = 1S , iff Lie!i. = Lie K .
(b) If t 1 and %2 are hO!!D!!Drphisms of  into a k -group G' and if G is
connected , then %.1 = t 2 iff Lie 1 1 = Lie:h.2 .
(c)
If
p : G -+ GL (V)
- -
is a finite dimensional linear representation of ,
and if  is connected , then a vector subspace of V is stable under  iff
G Lie G . .
it is stable under Lie G . We have V- = V - ; !!Dreover , p lS sllllple
(resp. semisimple ) iff Lie p is .
(d) Let H be a connected subgroup of G. Then we have *
Lie(Cent H)
-(2-
Lie ( Norm G H)
Cent . G (Lie H)
Lle _ -
NOrTIL. G (Lie H)
Lle _ -
f, in addition , G is connected , then B is normal (resp. central ) in 
iff Lie H is an ideal of Lie G (resp. is in the centre of Lie  ).
(e) If G is connected , then  is corrmutative iff Lie G is corrmutative ,
and the centre of G is finite iff the centre of Lie  is zero .
* If g is a Lie algebra and h is a subalgebra of g, we set
Cent h = {x E g : [x,h] = O} , Nonn h = {x E g : [x,h] C h}
g g

310
ALGEBRAIC GROUPS
II, . 6, no 2
Proof: (a) li c 1$. is equivalent to J:j:nK=li. Since BnK is srrooth by 1.1,
this last condition is equivalent by  5,5.6 to (Lie H)n(Lie.IS) = Lie B ,
i.e. to LieBc Lie.IS . The second assertion follows immediately.
(b) Let.IS be a subgroup of Q such that K(R) = {g E g (R):  (g) = i 2 (g) }
Then
Lie 1S = {x E Lie  : (Lie i 1 ) (x) = (Lie 2)(x)} ,
and i 1 = i 2 is equivalent to .IS = 53, hence to Lie lS Lie  by (a), hence
finally to Lie!l = Lie f 2 .
(c) Let W be a vector subspace of V. Then we have Lie ( Norm G W) = (Lie )W W
- ,
by  5 , 5.7. Now W is stable under  iff Norm W =  , hence by (a) iff
(Lie 9)W,H = Lie  , Le. if W is stable under Lie  .
By the same argument, WC V is equivalent to Cent W = G , hence to
Lie G -g-
( Lieg) 0, W = Lie  , and so to W C V -. The second assertion follows
from the first.
(d) Writing C = centg .!i, t! = Norm .!i , we have by  5,5.7
Lie C = (Lie ).!i, (Lie tl) / (Lie En = ((Lie ) / (Lie HJ,.tl
( ) th " . ( ' ) Lie H t (1 ' H)
B y c, lS g lves Lle G = Lle  - Cen . G ,le
- -Lle _ -
argument, Lie t! = NO ie  (Lie H) .
The last assertion thus follows from (a).
by the same
(e) This follows immediately from (d).
2.2 Corollary : Let  be a connected alqebraic k-. Then the
followinq conditions are equivalent :
(i) The Lie alqebra Lie  is semisimple .
(ii) Each normal connected corrmutative subgroup of G is zero.
(iii)  has finite centre , and all finite dimensional linear representa-
tions of g are semisimple .
(iv) G has finite centre , and the adjoint representation of G in Lie G
is semisimple .
(v) G has finite centre , and 9. has a finite dimensional semisimple li-
near representation whose kernel is finite.

II,  6, no 2
THE CHARACTERISTIC 0 CASE
311
Proof: (i) => (ii): If !i is a normal connected commutative subgroup of Q.,
then Lie!i is a commutative ideal of Lie G , and is therefore 0 if
Lie  is semisimple.
(ii) => (i) : Let h be a commutative ideal of Lie G . Then K = (Cent h)o
- --
is a connected subgroup of  whose Lie algebra is Cent. (h) (5, 5.7) ;
Lle 
this latter is an ideal of Lie Q , so K is a normal connected commutative
subgroup of Q. whose Lie algebra contains h. If (ii) holds, we thus have
h = 0 .
(i) => (iii): By 2.1 (c) and (e).
(iii) => (iv) TriviaL
(iv) => (v) The Lie algebra of the kernel of the adjoint representation
of Q. is the kernel of the adjoint representation of Lie Q. If Q. has fi-
nite centre, the adjoint representation of Q. is therefore finite, and, if
(iv) holds, the adjoint representation is semisirnple.
(v) => (i) : If V is the space of the representation and 15. is its ker-
nel, then Lie K = 0 is the kernel of the associated representation of Lie 
in V. By 2.1 (c) and Gr. et alg. de Lie, I ,  6 , prop.5, Lie Q. is reduc-
tive. Since Q. has finite centre, the centre of Lie Q. is zero. Hence
Lie Q is semisimple.
2.3
Proposition : Let  be an algebraic k- group and let .v() be
its derived group (5, 4.8). Then
a) the Lie algebra of .0(G) is the smallest vector subspace d of Lie Q.
such tha t, for each
into d0R
R E Iv1 and each g E  (R) ,
-k
Ad(g) - Id maps (Lie ill 0R
b) if G is connected ,. we have Lie:iJ () = [Lie , Lie GJ .
Pr=f: a) By the pr=f of  5 ,4.8 we may choose a natural number N and
an open sub scheme U of G 2N in such a way that the morphism f of U into
9() satisfying
(gl,hl,...,gN')
-1 -1 -1 -1
gl hI gl hi ... gN  gN 
for (gl' .. .,hI") E Q (R) , R E t\: ' is faithfully flat. Since .Q and  () are
smooth (1.1), tl"e set of points at which f is srrooth is dense and open in U

312
ALGEBRAIC GROUPS
II, '6, no 2
(I ,  4 , 4.12). We may assume that k is algebraically closed, and choose a
rational r::oint u = (gl'... ,) of Q. at which f is smooth, hence at which
the tangent map to L is,surjective (I,  4,4.15) . By translation the tan-
gent space to 11 at this r::oint may be identified with (Lie )2N , similarly,
the tangent space to g (g) at i (u) may be identified with Lie S () . The
tangent map to f at the r::oint u thus corresPJnds to a map t of (Lie )2N
into Lie f) () c Lie  such that
f( EXl h EYl EYN ) = f( h ) Et(xl'Yl,...,xN'YN)
_gle , l e ,...,e -gl'."'-'N e
for xi' y i E Lie 5!. If d is the subspace of Lie g defined in the statement
of the theorem, we see irmnediately that e Ed is a normal subgroup of (k(E)).
1;Uso, if g E g (k) and x E Lie G , we obtain directly
EX EX -1 E Ad( g )x (.Ad( g ) x - x) EX
ge =ge g g=e- =e- e g
so tha-t- g and e EX corrmute rrodul0 e Ed . It follows at once that t maps
(Lie )2N into d , so that Lie S () cd. r1oreover, in C2.(R(E)) RE£:\
we have
-1 -EX EX
9 e ge
-1
e E:(x - Ad (g ) X )
for g E (R) , X E Lie  @ R , which proves that Lie $()  d .
b) With the notation of  2 , 1.3, Lie .0(g) is the smallest vector subspace
d of Lie  such that, in the adjoint representation of , we have
G = G . Applying 2.1(a) and using 95,5.7 b), we infer that if G is
- -=-a,Lie 
connected, d is the smallest vector subspace of Lie s. such that
[Lie g , Lie ] cd, which establishes b) .
2.4 Definition : Let G be a locally alqebraic k-. A subalge -
bra of Lie G i s said to be algebraic if it is the Lie alqebra of a sub-
group of .
By 2.1 (a) , the map !! 1-+ Lie!! is a bijection of the set of connected sub-
groups of  onto the set of algebraic subalgebras of Lie  .
Clearly the intersection of algebraic subalgebras is algebraic. In particu-
lar, for each subalgebra h of Lie s. , there is a smallest algebraic subal-
gebra A (h) of Lie s. containing h; this we call the algebraic hull of h.

II,  6, no 2
THE CHARACTERISTIC 0 CASE
313
2.5
Lemna: Let  be a locally algebraic k -group, h a Lie subal -
gebra of Lie G and W,W' two vector subspaces of Lie G such that
[h,W] C W' and W' C w. Then [A(h),w] C W' .
o 0
Proof: Consider the adjoint representation of!2. and the subgroup G-
-"W',W
of r;zO (2, 1.3). Its Lie algebra is the set of all x E Lie  SUC;l. that
[x,W] C W' (5, 5.7). Since it contains h, it also contains A(h) .
2.6 Pror::osi tion : Let  be a locally algebraic k -group .
(a) Let h be a subalgebra of Lie  . Then each ideal of h is an ideal of
A(h) ; we have [h,h] = [A(h) ,A(h)] and A(h) /h is corrmutative . Furthenrore
the algebra [h,h] is algebraic.
(b) The derived ideal , the radical , the nilpotent radical , and the Cartan
subalgebras of Lie  are all algebraic .
Proof: (a) Let k be an ideal of the subalgebra h; then we have [h.k] C k
so that [A (h) ,k] C k by 2.5 and k is an ideal of A(h) . Similarly
[A(h) ,h] C [h,h] ; applying 2.5 again, we get [A(h) ,A (h)] C [h,h] v.hich im-
plies that A(h) /h is corrmutative and [A(h) ,A (h) ] = [h,h] . Finally, let .!i
be the connected subgroup of  with Lie algebra A(h) . By 2.3 b), the Lie
algebra of qi) is [A (h) ,A (h) ] = [h,h] and so [h,h] is algebraic.
(b) We already know that the derived ideal of Lie  is algebraic. Let r
be the radical of Lie G . We have [Lie g,r] C r, so that [Lie  ,A(r)] C r
by 2.5. Hence A(r) is an ideal of Lie G ; by (a), it is solvable, and we
have A (r) = r . The nilpotent radical of Lie g is [Lie g ,Lie ] n r ; it is
therefore algebraic. Finally let h be a Cartan subalgebra of Lie G and
let !i = (Nom h) o . B y  5 , 5.7, Lie H = NOITIL, G (h) = h .
 - Lle_
2.7 Corollary : Let G be a locally algebrak k -group . Then each
subalgebra of Lie  which coincides with its derived algebra is algebraic .
In particular , each semisimple subalgebra of Lie  is algebraic.
2.8
Corollary : Each finite dimensional k- Lie algebra which coin-
cides with its derived algebra is the Lie algebra of an affine algebraic
group.

314
ALGEBRAIC GROUPS
II,  6, no 2
Proof : By Ado's theorem (Gr. et alg. de Lie J I,!:i 7, no. 3, th.l) , there is
a rronorrorphisrn of the given Lie algebra into an algebrR 4 (V) , hence into
the Lie algebra of a group GL (V) . Now apply 2.7 .
In particular, each semisimple k-Lie algebra is the Lie algebra of an affine
algebraic k-group, and 2.2 applies.
2.9
Proposi tion : Let  and li be connected alqebraic k- groups .
(a) Let f:  ->-!i be a horrorrorphism. Then f is faithfully flat and has
finite kernel iff Lie f is bi4ective.
(b) Let cp: Lie  ->- Lie H be a horrorrorphism of k- Lie algebras , and suppo -
se that  = E() . Then tllere is a faithfully flat horromorphism with finite
kernel p:' ->-, and a homorrorphism f: Q.' ->-fl. such that Lie f = cpo (Liel?).
Pr=f: (a) Irmnediate from 1.1 and !:i 5 , 5.1 and 5.5 .
(b) Let k C (Lie g) x (Lie!D be the graph of cp ; this is a subalgebra of
(Lie ) x (Lie !i) which is isorrorphic to Lie G and hence identical with its
derived algebra. By 2.7 there is a connected subgroup g' of G x H such
that Lie' = k. By (a), the projection p:' ->- is a faithfully flat
horrorrorphism whose kernel is finite. If f: G' ->-H is the second projection,
we have Lie i cp 0 (Lie p)
2.10 Corollary : Let G l and G 2 be connected alqebraic k- qroups,
both identical with their respective deived groups and let
cp : Lie 0 1 -+ Lie Q.2 be an isorrorphism. Then there is a connected algebraic
k -group  and fai thfull y flat horrorrorphisms with f ini te kernels i 1: Q. ->- 1
and i 2 :  ->-.Q2 such that Lie i 2 = cp 0 (Lie i l )
2.11
Corollary : Let  be a connected algebraic k- group which coin-
cides with its derived qroup and satisfies the following condition :
(SC) Each faithfully flat horromorphism with finite kernel G' ->- ,
where 0' is connected , is an isorrorphism.
Then for each l=ally algebraic k -group B , i 1-+ Lie f is a bi 4ection of
k(') onto the set of k-alebra horrorrorphisms Lie g ->- Lie!i .
Pr=f: The map in question is injective by 2.1 (b) and surjective by 2.9

II,  6, no 3
THE CHARACTERISTIC 0 CASE
315
Section 3
The exponential map
3.1 Let  be a k-group-functor and let R E  . We denote the ele-
ments of Q.(R[ [T]]) by function symbols such as f (T) . Given an R-algebra
S E  which is linearly tOpOlogized and complete, and a topologically nilpo-
tent element t of S, we write f (t) for the element of (S) which is the
image of f (T) under the continuous rrorphisrn of R[ [T]] into S which sends
T onto t . Thus we will have, for instance, the element fiE) of G(R(E)) ,
the element f(T+T') of G(R[ [T,T']]) , etc. .
Proposition : Let R E 1\ and let  be a k-. Then for each x E Lie (£2.0 R)
there is a unique elemen t exp(Tx) of G(R[ [T]]) such that
(a) exp(E x) = e E x in Q.(R(E)) .
(b) exp(T+T')x = exp(Tx) exp (T'x) in Q(R[ [T,T']]) .
Moreover , if x,yELie(Q0R) , and if [x,y] = 0 , we have
exp(T(x+y)) = exp(Tx) exp(T y) .
Proof: Let E 1 ,..., En be n variables of vanishing square and let
R = R(E 1 ,...,E) R 1 ( E) . Consider the element X of G(R) defined by
n n n- n n n
X
n
El x En X
e ... e ,
where xELie(0R) . By 94,4.2, the element X n is invariant under pennuta-
tions of the variables E, . Now consider the R-horrorrorphisrn a : R[T]/r+ 1 _R
l n n
such that an (T) = El + ... + En . A straightforward argument shows that, when
k has characteristic 0, a is a bijection of R[T]/r+ 1 onto the sub-
n
the invariants under
ring of R formed by
n
the E, . It follows that there is a unique element E
l n
that a (E ) = X : to see this, let U be an affine O pe n sub scheme
n n n -
taining the origin whose ring is A. Since  R and  R have
n
the group S of pennutations of
n n+l
of G(R[T]/T ) such
of  con-
the same
space of points and the corrp::>si tion
#
can X n
 R -+-  Rn -+- .c2.
factors through
E G ' we have X E U(R ) M (A, R) . Since we have
n - n MK n
X E U(R )Sn = U(R Sn)
n - n - n '

316
ALGEBRAIC GROUPS
II,  6, no 3
 belongs to g(Im an) , and is therefore of the form an (En) , where
E E U(R[T]/+l)  G(R[T]/+l) .
n - -
Now consider the commutative diagram
R[T] /+1
a
n
R
n
Pn j
R[T] /
a n - 1
l
Rn_l
where p is the canonical map and q sends E:, onto E:, for i 'f n and an-
n "n l l
nihilates E: . We have q (X ) = X 1 , so that P (E ) = E 1 . Hence there
n "n n n- n n n-
is a W1ique element E(T) EG(R[[T]]) such that E = E(T ITOd +1) for
n
each n. To prove this, take 11: and A as above; each En correspcnds to a
horrorrorphism A -+ R[T]/-l . Hence these form an inverse limit system,
which in turn yields a horrortDrphism A -+ R [ [T]] , associated with an ele-
ment E(T) of Q(R[ [T]])  (R[ [T]]) . Let E'(T) be another element of
G(R[ [T]]) such that E = E'(T ITOd +1) for each n. Since _ G is separated
- n
(l, lemna 3.8), Ker (E(TJ#',E'(TJ#') is closed in fu2.R[[T]] (I,2,7.6)
and is acccrdingly defined by an ideal I of R[ [T]] . By hypothesis we have
.J1+1
IS;;'l' R[ [T]] for each n, so that 1=0 and E(T) = E'(T) .
We now show that the element E(T) E (R[ [T]]) meets the conditions (a) and
(b). This is immediate in the case of (a), for E(E:) = X = eE:1X . As for (b),
1 1
we have for each n a conmutative diagram
R[T]/+l 0 R[T]/+l
R
v n !
R[T]/T 2n + 1
a 0 a
n n
R 0 R
n R n
! un
a 2n
R 2n
where u (E:,) = E:, 01 for 1;'; i;'; n and u (E:,) = 10 E:, for n+l;'; i;'; 2n ,
n l l n l l -n
while v n (T) = 10 T + T0 1. Since by construction we have un (X 2n ) = i 1 (XJ i 2 (X d
where i 1 and i 2 are the injections of R into R 0R , it follows that
n n n +1
V n (E 2n ) = jl (En) j2(E n ) , where \ and j2 are the injections of R[T]/T n
into its tensor square. Accordingly E(T+T' and E(T)E(T') have the sarre
image in G(R[[T,T'Jj/(T n + 1 ,T,n+l)) so that E(T+T') = E(T)E(T') by the sarre

II,  6, no 3
THE CHARAcrERISTIC 0 CASE
317
argument as above.
lEt us show finally that E(T) is the unique element of (R[ [T]]) which
meets the conditions in question. If F(T) = exp(Tx) satisfies (b), it fol-
lows immediately by induction that F ( L T,) = IT F (T,) in G (R[ [T ,... , T ] ] )
i l i l - 1 n
Now if F(T) satisfies (a), we infer that in G(R) we have
- n
E' X
F ( 4: E,) = I): F (E.) = II e l = X
l l l l i n
which implies F(T ITOd r+ 1 ) = E(T ITOd r+ 1 ) and finally F(T) = E(T) . Lastly
we show that if x,yELie(0R) and if [x,y] = 0, then exp(T(x+y)) =
exp(Tx) exp(Ty) . In virtue of our arguments above, it is enough to show that
in G(R) we have
- n
(e E1x ... eEnX) (eElY ... eEnY) = e E1 (x+y) ... eEn(x+y) .
But e Eix e EiY = eEi(x+y) and the e Eix and e EiY conmute by 94,4.2,
and the contention fo] 1 '.ls.
3.2 Remark: Under the conditions of 3.1, if f (T) E s (R [ [T] ]) and
if f(T+T') = f(T) f(T') in s(R[[T,T'J]), there is a unique xELie(0R)
such that f(T) = exp(Tx) . To see this, note that we have f(O) = f(O)f(O)
in G(R) , so that f(O) = 1 ; it follows that f(E)Es(R(E)) is projected on-
to 1 and is accordingly of the form e E x for a uniquely determined
x E Lie (00 R) . Thus we have f (T) = exp (Tx)
3.3 Example : Take  = GL (V) where V is a finite dimensional
k-vector space and xE.i'k(V)0 R ZR(V0R) . Then
Tixi
exp ( Tx )
L
i2:0
, I
l.
To verify this, notice that we have
EX
e
Id+ EX by 94,4.2, so that
X n = (Id+ E 1 x) ... (Id+ EnX)
n
Id + (E 1 + ... + En) x + . .. + (E 1 E 2 ... En) x
Id + tx + ... + (tnjn:)x n
where t a (T ITOd r+ 1) . By passing to the limit we obtain the required
n
formula.

318
ALGEBRAIC GROUPS
II,6,no3
3.4
Ccrollary : Let Q, R and x be as in 3.1 .
(a)
If aER, then exp(aT)x = expT(ax)
-1
If gEQ(R) , then gexp(Tx) g = exp(TAd(g)x)
(b)
(c)
If f : Q -+ 1i is a horrorrorphism of Q into a k -group !i, then
f(exp(Tx)) = exp(TLie(f)x) .
(d) If Q is locally algebraic , then in GL (Lie ()) (R[ [T]]) we have
T i ad (x) i
Ad exp(Tx) = L
i;;:O
. ,
l.
Proof: (a), (b) and (c) follow immediately from the uniqueness assertion of
3.1; (d) follows from (c) applied to the horrorrorphism Ad : Q ->- GL (Lie (Q) )
and from 3.3 .
3.5 Ccrollary : Let .Q be an affine algebraic k -group , let RE
and let x E Lie (Q)@ R. Then the following conditions are equivalen t.
(i) There is a faithful finite dimensional linear representation
p: Q -+ GL (V) such that Lie(p) (x) is nilpotent .
(ii) For each finite dimensional linear representation p J2f Q, Lie(p) (x)
is nilpotent .
(iii) exp (Tx) E Q(R[T])
(iv) There is a hOIIDrrorphism i: a R ->- Q R such tha t Lie (f) (1) = x .
Proof: (ii) => (i)
By 92,3.4.
(i) => (iii) : By 3.3, exp(T Lie(p)x)E GL (V) (R[T]) . By 3.4 (c) , we have
the commutative square
<..O(p)
· °jc2
 ; (VII
R[T]
can
R[[T]]
where PI and P2 correspond .to exp(T Lie(p)x) and exp(Tx) . Since J(p)
is surjective (95,5.1) and can is injective, P2 factors through R[T] .
(iii) => (iv)
Let S E  ; for each tE S = a (S) , consider the horrorrorphism

II,6,no3
THE CHARACTERISTIC 0 CASE
319
R [T] -r S which sends T onto t, and the image f (t) of exp (Tx) under this
horrorrorphism. We thus obtain a rrorphism f: a R - ; it is inmediately seen
to be a homorrorphism ((b) of 3.1). We then have Lie(f) (1) = x in virtue of
(a) of3.1.
(iv) = (ii)
By 92,2.6.
3.6
Suppose R = k . The horrorrorphism f whose existence is asserted
by (iv) is uniquely determined (2.1 (b)), and it is a rronorrorphism when
x  0 . To see this, notice that its kernel is a subgroup of  of dimen-
sion 0, hence etale (1.1), while a (k) has no non-zero subgroups.
3 . 7 When the conditions of 3.5 are met, we say that x is nilpo-
tent , and we write exp(x) for the element of (R) which is the image of
exp(Tx) under the horrorrorphism R[T] ->-R which sends T to 1. Accordingly,
if R= k , we have f(t) = exp(tx) for each tES, SE .
If x is nilpotent, we may replace T by 1 in corollary 3.4; in particular,
we obtain the formulas
-1
g exp(x) g
exp( Ad (g)x)
g E G(R)
Ad ( exp(x) )
Z
i2':O
i
ad (x)
, .
l.
Similarly, if x and yare two nilpotent elements of Lie () 0 R , and if
[x,y] = 0 , we have exp(x+y) = exp(x) exp(y) by 3.1.
3.8
Remark: It follows from 3.5 that the subalgebra of Lie ()
generated by a nilpotent element is algebraic.
3.9 Let k [ [T] ] be the sub ring of k [ [T]] consisting of fonnal
exp
power series arising as solutions of linear differential equations with con-
stant coefficients. If k = iC, these are linear combinations of formal power
series of the fonn P (T) exp (aT) where P E k [T] and a E k . If k = IR, they
are linear combinations of formal power series of the form T) exp(aT) sin(bT) ,
P(T)exp(aT)cos(bT) , where PEk[T] and a,bEk .

320
ALGEBRAIC GROUPS
II,  6, no 3
Pror::osition : Let Q be an affine k- group . Then , for x E Lie (), exp (Tx)
belongs to G(k[ [T]] ).
- exp
Proof : Let 0: k[ [T]] -+k[ [T,T']] be the horrorrorphism f(T)t-+f(T+T'). By
3.1 (b), we have (o)exp(Tx) = exp(T+T')x E (k[[T]] I8Ikk[[T']]); whence
exp(Tx) E G(o)-l(G(k[[T]] 181 k[[T']]))=(fl(k[[T]] 181 kilT']]))
- - k k
since G is affine. It is therefore enough to prove the following
3.10 LeIm1a : Let f E k [[T]] . Then f E k [ [T] ] iff
exp
f(T+T') Ek[[T]]\ kilT']] .
Proof: If
f(T+T') =  a,(T)b,(T'),
ill
by applying a derivation with respect to T n times and setting T' = 0 we
obtain
f(n) (T) =  bn)(O) a, (T) ,
ill
which shows that the f (n) (T) generate a finite dimensional vector space,
hence that f E k [ [T]] . Conversely, if f E k [ [T]] , there exist
exp exp
a 1 ,... ,a r E k[ [T]] such that for each n we have
f(n)(T) =  b, a,(T) .
i l,n l
Taylor's formula now applies, to give
f(T+T')  1 f (n) (T) T. n  a,(T) b,(T')
=
n n: i l l
where
b,(T')  1 b. T. n
l n n: l,n

 7
THE CHARACTERISTIC P -f 0 CASE
In s: 7, if qJ : k -+ .£ is a hOTIDrrorphism of IIDdels and if R E lj.£, cpR denotes
the k-rrodel obtained froD R by restriction of scalars. The external law of
qJR is then (A,X)>->-qJ(A)X, AEk, xER. Similarly, if SE1:1 k , we set
Sl8icp.£ = Sl8i k .£ ; an element of Sl8iqJ.t is then a linear combination of elements
which we denote by Sl8icpA (s E S,A E.£) satisfying Sl1l8icpA = s I8iqJCP (11) A for
11 E k .
Throughout s: 7,
p denotes a fixed prime number and
k an JF -rrodel.
p-
Section 1
The Frobenius rrorphism
1.1
If RE
for x E R
Let f be the endoTIDrphism of k such that f (A) = A P for A E k .
we write f:R -+ f R for the TIDrphism of r1 k such that f (x) = xP
R ( ) - R
For each k-functor , we write  P for the functor derived
If .£ E t 1 ] and X E ME, we have:
"""K - """"'k"""
Thus, if k =lFp , we have f = IC\
l8ik.£ (in general, of course, f -f 1,\ ) .
In the general case, if n 2; 0 , we define (pn) by (pn) (R) = (fnR) for
(rP+ l ) (rP) (D)
R E  . Accordingly we have "f>. = (K ) C . Similarly, we define
 :  -+ (pn) by the formula .f(R) =  (f) if R E and if f: R -+ fn R
snds x onto xrP . Then , ich we abbreviate to  r is the composition
from X by the extension of scalars f (I, 0 1 , 6.5) . Then we have
"f>. (I») (R) =  (fR) for R E.t\ . Similarly, if g: -+ X is a TIDrphism of l:\:E: '
u(p):x(p)-+y(p) is theTIDrphismof !k£: such that (P)(R)=\l(fR) for
R E!1 k . Pinally, the rrorphism of  into X (p) which assigns to R El\
the map (fR) : 'iR) -+X(fR)=(P)(R) is called the Frobenius morphism with
domain ; "'Ie denote it by fx or simply E .
(XI8i .£) (p) = X (p)18i .£
- k - k
, so that X(p)= 
and fx .£ = £X l8i k.£ .
and (X@) = X<p) I8i .£
- k - k
FX ( ) fx(p)
-=-p
(p2) -+... -+(rP-l) (rJ1)
1.2 Exarrple : Let T be a geometric k-space (I,  1 , 6 .8) and let
T (p) be the geonetric k-space which has the same underlying tor.;ological

322
ALGEBRAIC GROUPS
II,7,nol
space as T, and whose structure sheaf is the sheaf of k-algebras 19 T (p) =
c?T@fk (Le. the associated sheaf of the pre sheaf U I-r tiT(U) @rk) . Let
F T : T --+ T(P) be the morphism of Esqk asociated with the identity map of
T and the =rphism <!J @.ck --+,y induced by the maps s@- A --+ sP A .
T L T r
Given REt!k' considerthek-functor .2 k (T(P)) and a UE.2 k (T(P)), Le. a
=rphism (u,ut:) : Spec R --+ T(P) of k. Writing () for the structure
sheaf of k-algebras of Spec R , let u' be the composition
f
rfl  tJ 181 k  u; (t9) .
T T f
Clearly u' is a =rphism of the sheaf of k-algebras of rf) into the sheaf of
T
k-algebras derived from u (rJ) by the restriction of scalars f: k --+ k . As
R varies, we accordingly obtain maps (u,u:f) f-+ (u,u') which define an
isorrorphism 11 (T) :.2k (T (p)) -2:...e k (T) (p) . This isorrorphism satisfies
11 (T) 0 e k (F T) = Is T .
_k
1.3 Pror::osition : There exist =rphisms \!C9: 1(P)lk --+ II)
which are functorial in  E L\ ' satisfying \! () 0 Ix  I k = X IKI k for each
X and are such that \! (20 is invertible when X is a scheme .
Proof: Set T = I  I k in 1.2 . If CP:  --+ .e k I  I k is the canonical rrorphism
which arises f=m the adjointness of k to I? I k ' it is enough to take
\! () to be the rrorphism assigned to the composition
X(p)
cp(p) ) S (Ixl )(p) I1(T)-l , S (lxl(P))
-k - k -k - k
by the bijection
l.\:£;. ((p) ,ek( II)) )  k (1(p)lk' IKI) )
of I ,  1 , 6. 8 .
As an application of this pro]XJsition consider the case in which  =  A ,
A E  . For each R E  ' we then have a canonical bijection
K(P)(R) =(A'fR) (Ak,R) (kAk)(R) I
V.nence an isorrorphism
A (A) : (k A) (p)  k (A  k)

J'
II,  7, no 1
THE CHARACTERISTIC p7'0 CASE
323
If cp: A<2Ifk --+ A
A (A) 0 F K =  cp .
the same is true of
is the horroI1Drphism a<2l f A  a P A , one shows easily that
Since If"f>.1 is bijective in virtue of the projXJsition,
Spec cp Spec A -->- Spec !\. <21 f k .
1.4
If G
is a k-group-functor the fonnula G(P)(R) = G( R) shows
- - f
a k-group-functor and
that d p ) is naturally endowed with the structure of
that F : G ->- G(p)
-G - -
r : G --: G (p) . 'Ihis
is a horromorphism. We write .fIG for the kernel of
latter is a characteristic subqroup of G. To see this,
observe that, if R El\: and 1,!, is an autoI1Drphism of <2IkR, we have a
commutative diagram
r (Q.  R )(pn)
G<2I R
- k
1,!, 1 ( n) 1
JJP
n
G R .f (Q. R)(pn)
pnG  R
- 1
rG R
We say that g has heiqht ;:: n if Fn G G. For <"ach k-group-functor g
f1Q. obviously has height ;:: n
Observe that Lie (G) = Lie (FG) To see this, it is enough to verify that
Lie () (k) = Lie (fG) (k) , or that Lie (G) (k) S;.£'G (k (E:)) . But this follows
immediately from the fact that the hOI1Drrorphism fk(E:): k(E:) ->- (':(E:) anni-
hilates E: , and so factors th=ugh k .
a) If
1.5 Examples :
Q(p)
G =  I then
.....n G = n
.!:. p
a k
(1.1) ane: .£'(x)

for x ERE  .
Hence
b) If  = 12 (f) k ' R E l\: ' and x E 2E (r , R*)
F (x) = x P ,whence £ ('1) = x ('1f = x (p'1) for
be identified with 12(r/p0 r )k . For example,
, we have Q (p) = Q. and
'1 E r Consequently, pnQ
nl1k nl1 k .
lC n P
may
c) If  is constant , then G. (p) G and f = Id If G is etale , then F
is an isomorphism.
1.6
P=jXJsi tion : Let Q be a k- group-scheme and let n
 0 . Then the followinq conditions are equivalent :
G has height ;:: n .
be a na-
tural number
i)

324
ALGEBRAIC GROUPS
II,  7, no 2
ii) Q is affine ; if I () is the kernel of the augmentation of
th
(!l(G) , then the p power of each element of I (Q) vanishes.
Pr=f: (ii) => (i) : If G = Sp A , and if XEL\(A,R) = Q(R) , then !'(x)
is the composition
x f R R
A -r R ,
so that  (x) (a) = x (a pn ) , which implies that f (x) factors through the
augmentation of A.
(i) => (ii): The Cartesian square
pn
- 1
S\
'V
G
lrn)
G P
and the fact that Irl is bijective together imply that  ->- k is in-
jective, hence that the canonical projection P G : <::i ->- S\ is bijective, and
hence that the unit section E G : S\ ->- Q is a closed embedding. If (!li)
- -1
is an affine open cover of G., EG () is affine for each i. Since
-1 -1
!li = p (E G (1.J i )), it follows from 1,92, 5.6 and 5.2 that 9i is affine.
The remainder of the argument is immediate.
Section 2
th
The p -r::ower operation in Lie (G)
Throughout t.'1is sec'cion G denotes a k-group-scheme.
th
2.1 We now define a map of Lie () into itself called the p _po_
wer operation and written x I->- x[p] . Let x E Lie (Q) ; consider the algebra
k (E 1 , . . ., E p ) obtained by adjoining to k variables E 1 ,...,E all of square
p 2
zero. Set a = El + . .. + Ep and '11 = E 1 . . . Ep . Then we have a P = 0 , '11 = 0
and it is easily shown that the subalgebra of k (E 1 , . .., E p ) generated by a
and T is the algebra of elements invariant under all pennutations of the
E, . Consider the element eEl x eE2x ... eEpx of
l
This element is invariant under
Ker ( G (k (E 1 ' . . ., E )) ->- G (k) ) .
- p
all pennutations of the E . ( 4 , 4.2 (6) ) .
l
Arguing as in 9 6,3.1, we infer that it belongs to Ker ((k(a,'TT) ->- G(k)) .

II,  7, no 2
THE CHARACTERISTIC prfO CASE
325
If we apply to this element the horrorrorphism of k(a,'TT) onto k('TT) which
annihilates a, we obtain an element of Ker ( Q (k ('11)) ->- Q (k) ) , which is
thus of the form e'TTY , where y E Lie (Q) . Set y = x [p] . Modulo
Ker (Q(k(a,'TT)) ->- Q(k('TT))) , we then have
[p]
eE1x ... eEpX == e(El... E p)x
Given a k-group hOITorrorphism f: G ->- H and x E Lie () , we have
(Lie() (x)) [p] = Lie(!) (x[p])
2.2
Examples :
1) Set Q =  and identify k with Lie(Q) via the map x H-EX (4,4.11).
,Then we have
El x EpX _ _
e . .. e - (E 1 + ... + E p ) x - a x ,
[p] th " ( )
so that x = 0 , and the p - [Ower operatlon In Lie Q
is zero.
2) Take G = Q(r)k ' where r is a small comnutative group, and identify
(r,k) with Lie(Q) via the map x f-+ l+Ex (4, 4.11). T:1en we have
eEl x ... eEpX
a P - 1 p-l p
 (l+E i x) = l+ax + ... + (p_1) X- +'TTX-
so that
'TfX[pJ
e = 1+
and x[p] = £ .
3) Take Q = GL (V) , where V is a finitely generated projective k-ITOdule.
Identify (V) with Lie(Q) via the map x 1--+ Id+ EX (4, 4.12) . The same
computation as the one above then gives x[p] = x P .
2.3 Pro[Osition : Let xELie(Q) .
+ 'V
a) If v G : Dist 1 (9) ->- Lie () is the canonical isorrorphism (  4 , 6.8 ) and
if x E Dit ()  Dist(Q) , then x * ... * x (p terms identical with x)
+ .
belongs to Dist 1 (G) and we have
v (x) [p] = V G ( x * x * .. . * x )
Q
b) li 2f. is a k- scheme, : Q ->- Aut () a horrorrorphism , and u': Lie (Q) -+
Der qp c Dif () the corresponding antihorrorrorphism , we have u '(x) p = u' (x [p])
c) In the algebra Dif (9), we have
Y' (x[p]) = Y' (x)p , o' (x[p])
0' (x)p .

326
ALGEBRAIC GROUPS
II,  7, no 3
Proof: The notation is taken from S 4, sections 4 and 6. Assertion c) fol-
lows from b) a pp lied to the horrorrorphisms y: G -+ Aut (G) , 8: G -+ Aut (G)
- -- C)pp--
( S 1, 3.3) . By S 4 ,6.6, we have y' ( x * ... * x) = y' (x) P ; since y' is injec-
tive, a) follows from c) by S 4,6.8 . It is therefore enough to prove b) .
Let Y be open in X and let RE
we have
f E L9(Q) and mE Q (R) . By definition
EX
f (  (e ) m) = f (m) + E ( ::!,. (x) f) (m)
and it follows without difficulty that
2
f((eqx ... eE:pX)m) = !(m) +a(u'(x)f) (m) + g: (u'(x)2 f) (m) + ...
-1
(p-l): (u'(x)p-l f) (m) + 'TI( u '(x)p f) (m)
... +
Setting a 0, we get
'TTX [p]
f(!:! (e ) m)
i.e. u' (x)p = u'(x[p]) .
f (m) + 'TI( u'(x)p f) (m)
Section 3 Lie p-algebras
3.1 Def ini tion : Let .t be a k- Lie alqebra and let Xo ' xl E.t .
For 0 < r < p , set
1
sr(x O 'x 1 ) = -r: ad Xu (1) ad x u (2) ... ad xu(p_l) (xl) ,
where u ranqes throuqh the maps [l,p-l] -+ {O,l} which assume r times the
value o.
For instance, sl (x O ,x 1 ) coincides with [x O ,x 1 ] for p= 2 and with
[xl' [xl ,x O ]] for p = 3 .
3.2
ProfOsition : Let A be a k- algebra ( associative , but not ne-
cessarily commutative ). Given a,bEA, set (ad (a) b) = [a,b] = ab - ba .
Then we have the Jacobson formulas
a) ad (a)p = ad (a P )
aEA,
b)
(a+b)P= aP+b P + 1:: s (a,b), a,bEA.
O<r<p r

!If
II,  7, no 3
THE CHARACTERISTIC WO CASE
327
Proof: Setting La (b) =  (a) = ab , we have
(ad (a P ) ) (b) = (L P - R P ) (b) = (L - R )p (b)
a a a a
ad (a)p (b)
which gives a) . Also, if a 1 ,.. . , apE A , we have
(*)
 as(l) ... as(p) = t ad at(l)." ad a t (p_l) (a p )
-p -p-l
To see this, notice that the right hand side of (*) is
1::
i,j
p-l-r
1:: (-1) a t ( , ) ...at ( ' ) a at ( ' ) ...at ( ' )
tES II lr p Jp-l-r Jl
p-l
where (i 1 ,... ,i) ranges through the strictly increasing sec:ruences of na-
'tural numbers in the interval [l,p-l] and (j l '.. . , j 1 ) denotes the
p- -r
strictly increasing sequence whose members are the integers in [l,p-l] dis-
tinct from i 1 ,.. . , i r . This sum may be written
p-l-r p-l
1:: (-1) ( 1 ) L. a (1) ." a ( ) a a ( 1) ." a ( 1)
r p- -r vES v v r p v r+ v p-
p-l
p-l £-1 p-l p-2
But the fonnula (x-i) = _ 1 = 1< + 1< +... + 1 which holds in
x-
k[x] , implies that (_l)p-l-r ( p-l ) = 1, which gives (*) . Now for b)
p-l-r
if Xo ,xl E A we have
(x o + x 1 )p = {+ xi + 1:: 1:: xW(l ) ...xw(p)'
O<r<p wE F (r) -
where F (r) is the set of maps of [1, p] into {O , l} which assume r times
the value O. If we assign to each s E S the Ifap w E F (r) such that
-1 -1 -1 -p - s
w (0) = {s (l),...,s (r)}, we obtain a surjective map of S onto F(r) ,
s 
such that the inverse image of each wE F (r) has r (p-r)  elements. Setting
a 1 =...= a r = Xo ' a r + 1 =...= a p = xl ' we have xw s (l)". .xw s (p) = as (1)".. as (p)
and
L
wEF(r)
x w (l) ... xw(p)
1
. ( ) , L. a (1) ". a ( )
r. p-r . tE S ssp
-p
Similarly we obtain
1 1
sr(x O 'x 1 ) = r r (p-l-r)  ad at(l)... ad a t (p-l) (a p )
-p-l
Using (*), we obtain the required formula.

328
AIGEBRAIC GroUPS
II,  7, no 3
3.3 Definition : Let .c be a k -Lie alqebra .  pth- power operation
on .c is a map x<--+x[p] of .c into itself satisfying the following condi -
tions :
,
l'
(P-AL 2)
(Ax) [p]
ad(x[P])
(x+y) [p]
>P x[p]
AEk,xE.c
(p-AL 1)
(p-AL 3)
(ad(x))P xE.c;
x[p] + y[p] + 1: sr(x,y),
O<r<p
t,
!
x,yE.c .
A Lie p- alqebra over k is a pair consistinq of a k- Lie alqebra .c and a
th . 0
P - power operatlon on .c
The upshot of pror::osition 3.2 is that the Lie product [x,y] = xy - yx and
the p th - power operation x [p] =  endow each associative k-algebra A vlith
the structure of a Lie p-algebra over k. In particular, each Lie subalgebra
of A which is stable under the p th - povler operation is a Lie p-algebra.
(.
'I
!
Given two Lie p-algebras .c,.c' , a horromorphism rp:.c -+.c' is by definition
[ p ] [p]
a k-linear map satisfying rp([x,y]) = [rp(x) ,rp(y)] and rp(x ) = rp(x)
for all x,y E.c . The category of Lie p-a1sebras is denoted by k.
From 2.3 we ilmnediately infer the
<
\
3.4
th
Pror::osition : For each k- qroup-scheme , the p - power oper-
,
'\
,.
I
j
1
(
ation definied in 2.1 endows Lie() with the structure of a Lie p- algebra
over k. If X is a k- scheme and u: G ->- Aut (X) a homoITQrphism , then
- - - - - -opp
u ': Lie () ->- Der 0:0 is a horrorrorphism of Lie p -alqebras .
We have, naturally, assigned Der GO the p th - r::ower operation induced by the
p th _ power operation in the associative algebra Dif GP
3.5 Theorem : Let .c be a Lie p -algebra , which is also a finitely
generated projective k- ITOdule . Then there is a k- group-scheme (.c) and an
isorrorphism of Lie p- algebras a.c:.c q. Lie ((.c) ) such that the followinq
condition holds:
(*) for each k- qroup-scherre  and each horrorro=hism of Lie p- alqebras
rp : .c ->- Lie () , there is a unique horrorrorphism i:  (.c) ->- G. such that
rp = Lie () oa.c
t
Remark: Concbtion (*) rreans that the map !t-+Lie(!) 0 a.c is a bijection

II,  7, no 3
THE CHARACTERISTIC pr60 CASE
329
k (  (.t) ,)  k ( .t,Lie ()) . As the solution of a universal problem,
the pair ((.t) ,a.t) is "unique".
In proving this theorem we will make use of the results to be proved in
3 .6 - 3.10 .
3.6 First of all consider two arbitrary Lie p-algebras .t and .t I
over k and a Lie algebra horrorrorphism rp:.t ->- .t · (which does not necessa-
rily preserve the pth-power operation). Set c(x) = rp(x) [p] - rp(x[p]) ; let
D
A E k, x,y E.t . By (p-AL 1), we have c (Ax) = A c (x) ; by (p-AL 3), Vie have
c (x+y) = c (x) + c (y) ; by (p-AIJ 2), we have [c (x) ,rp (y)] = 0 . If we apply
these results to the case in which rp is the canonical map of .t into its
enveloping algebra U (.t) , we see that c (x) belon<]s to the center of U (.t)
for each x E.t . If (x,) is a system of generators of the k-rro::1ule .t, the
l
left ideal of U (.t) generated by the elements c (x) is tvlo-sided and coinci-
des with the ideal generated by the c(x,). Let u[p](.t) be the quotient alge-
l
bra of U(.t) by this ideal and let j be the corrposite map .t --+U(.t)-+u[p](.t) .
Clearly j is a horrorrorphism of Lie p-algebras (with respect to the Lie p-al-
gebra structure on u[p] (.t) derived from its associative algebra structure) .
Pror::osition : a) Let A be an associative unital ( not necessarily commuta-
tive) alqebra equipped with the structure of a Lie p-algebra defined in 3.3
Then for each horroxrorphism of Lie p- algebras rp:.t ->- A, there is a unique
. [ ]
horromorphism of unital alqebras g: U p (.t) ->- A such that rp = gO j .
b) If the k-rro::1ule .t is freely generated by (xi)iEI ' where I is totally
ordered , then the k-ITOdule u[p] (.t) has a basis consisting of 1 and the
,  , n 1 , 
products IT J (xi ) J (xi) . . . J (xi) such that i 1 < i 2 < .. . < i r and
s sir
o < n. < p for each j E [1, r ] .
J
Pr=f: a) follO'ds immediately from the universal property of the enveloping
algebra U(.t) of .t . l'.s for b), identify .t with its image in U(.t) and set
c, = c (x,) =  - x p] . Then the c, belong to the centre of V (.t) anc1 gene-
l l l l l o£ [ ]
rate the kernel J of the canonical map VlIJ (.t) into lIJ P (.t) . Let
m,
subrro::1ule of 1lJ (.t) generated by the IT X. l for  m, ;"; r . Since
ill l
rro::1 U p-l ' it follows from the PoincarE? - Birkhoff - vii tt
de Lie I , S 2 , no. 7, tho 1) that the rronornials
U be the
r
c, :: xI?
l l
theorem (Gr. et alg.

330
ALGEBRAIC GroUPS
II,7rn03
TIx,niTIc,mi, O;';n,<p, m.O,
I I I I
"
form a basis for the k-I1Ddule 11.1 (,£) ; and the result follows instantly.
3.7 Corollary : If the k -I1Ddule ,£ is finitely generated and pro-
jective , then u[p](,£) is a finitely generated projective k -ITOdule and the
canonical map j: ,£-+U[p](,£) is injective .
-1
Proof : For each multiplicatively closed subset S of k, we may endow S ,£
with the structure of a Lie p-algebra by setting
[p]
()[p] =  ,
s sP
for x E'£ and . s E S. In particular, consider a partition 1 = It, f. of uni-
I I
ty in k such that '£f. is a free ITOdule over kf. for each i. Clearly the
I I
pair w[p] ('£)f' ,jf') is a solution of the universal problem of 3.6 relative
I I [
to k f , and E f ' . It follows that U [p] (,£) f . is isorrorphic to U p] ('£ f ') and
I I I I
is therefore free, and that jfi is injective for each i. Accordingly
u[p](,£) is finitely generated and projective and j is injective.
3.8 The construction of U [p](,£) is functorial in '£. This fact
and the universal property of the pair (u[p](,£),j) yields the following
results:
a) There is a unique horrorrorphism of unital algebras E: u[p](,£) -+k such
that Eoj = 0
b) There is a unique horromorphism of unital algebras
D : u[p]W -+ u[p](,£)Q9u[p]W such that to(j(x)) = 1Q9j(x)+j(x)Q91 for xU.
c) There is a unique antihorrorrorphism of unital algebras Tl: u[p](,£) ->-u[p](,£)
such that Tl (j (x)) = - j (x)
If u E u[ p1 W and u =
for xE,£
I U,Q9V, , we have
I I
I u, 181 v. I v. 181 u.
I .L I I
I u, 181 DV. I DU. 181 v.
 I I I
I &(u.) v. = u
I I
I Tl(u.) v. E:(u)
I I

II,7,n03
THE CHARACTERISTIC Pia CASE
331
Tc prove the above fonnulas it is enough to verify them when u = 1 or
u=j(x) , xE.t, and this is imrnediate. Henceforth, if u=u[p](.t) and
R E,&\ , we simply denote the maps
lI0R co
UR- UUR--- (UR) (UR)
and U0R  k0R 2..,. R
k k
by II and E respectively.
3.9 Proposition : Let .t be a Lie p- algebra over k which is also
a finitely generated pro-jective k -rro::1ule . For each P. E let (.t) (B) be
the subrronoid of (U[P](f)R( formed bY 'x such that tox=x0x , Ex=l .
Then
a) 1:;.(.t) is a finite locally free k- group-scheme of height  1 .
b) We have a commutative square
u [p] (.t)
I
S.t
Dist ((.t)
I
.t
a.t
Lie ((.t) )
where the riqht hand vertical arrow is induced by the canonical isoJ11Orphism
, co +
Lle ( E (.t) ) -+ Dist 1 ( .& (.t)) of  4 , 6.8, a.t is an isorrorphism of Lie p- alqe-
bras and S.t is <in alqebra isorrorphisrn such that E = E 0 S.t ano II 0 S.t =
( Sf S.t) 0 to , with the notation of  4 , 7.1 .
Proof: Observe first that since x E  (.t) (R) implies 1l (x) x = E (x) = 1 , so
that x is invertible, E(.t) (R) is a group. Let A = MD<\: (1!J[p](.t) ,k) ; then,
equipped with the multiplication
t
AA  t( U U)  t u = A ,
where U = u[p](z), A is an associative comnutative k-algebra with unit
element E. Morecver, since u[p](.t) is a projective k-mcx'I.ule of finite rank
(3.7), we have a biduality isorrorphism
i : u[P](.t)R  (A,R)
As in S4 ,7.3, we see that, for xEU[P](f) 0kR
i (x) is a horrorrorphism

332
ALGEBRAIC GROUPS
II,  7, no 3
I
of unital k-algebras iff x E:£; (.£) (R) . Accordingly i induces an isomorphism II
'V
)2(.£) -->- Sp A , so that (.£) is a finite locally free k-scheme. One veri-
fies that the coproduct I':,A: A ->- A@A associated with the group structure
of :£; (.£) is derived by transposition of the multiplication U@ U ->- U (ap-
ply  1,1.8 a) ). We now prove b). By definition, Lie ( (.£)) may be identi-
fied with the set of elements of  (.£) ( k (E:)) of the form 1 + EX. If
.£' = {x EU[P](.£) : I':, (x) = x@l + l@x} ,
the map a: x r+ 1 + E x is a bijection of .£ I onto Lie ( :£; (.£)) . This map is
in fact an isorrorphism of Lie p-algebras, for if A E k and x, y E .£ I , we have
Aa(x) = 1+ (AE:)x = 1+ E(AX)
a(Ax)
a (x) + a (y) = (1+ EX) (1 + E y) 1 + E (x+y) = a (x+y) ;
-1 -1
(l+Ex)(l+E'y)(l+Ex) (1+ E'Y) = l+EE'(xy-yx),
whence [a (x) ,a(y)] = a([x,y]) ; and finally
(1+ E1x) ... (1+ EpX) == (1+ E 1 ... EpXP)
rrod E 1 + ... + Ep , so that
P [D]
a (x) = a (x L ) .
The canonical map j:.£-+ u[p](.£) is injective (3.7) and maps .£ into .£.
We claim that j (.£) = .£. . To prove this we may assume that the k-rrodule .£
is free with basis (cf. 3.7). Then u[p](.£) is free and has a ba-
(Xi)iEI
sis consisting of te
u
n
n.
II xi l
II ni:
where n = (n i ) i E I and 0  n i < p for each i . It is easily shown that
I':, (un) = r+=n u r @ Us '
where we have set (r+s).
l
r. + S, ; the claim follows immediately.
l l
We have thus constructed an isorrorphism 0..£:'£ ->- Lie( (.£)) . In virtue of
the universal property of U [p] (.£) and 2.3, this extends to an algebra horro-
rrorphism
B.£ V[p](.£) --+ Dist(J;;(.£))

II,  7, no 3
THE CHARACTERISTIC pfO CASE
333
If J is the kernel of the augmentation E A of A, we know that Dist (  (.£) )
may be identified with the set of 11: A ->- k which vanish on a power of J.
Accordingly we get a canonical injection
y : Dist (  (.£)) -->-  ( A,k) = OJ [p] (.£)
in view of the definition of the convolution product and the fact that multi-
plication in V[p](.£) and the coproduct of A correspond to one another by
transposition, y is a homorrorphism of unital algebras. One verifies that
y S.£j = j by writing out the definitions of y and S.£. It follows that y S.£
is the identity and that S.£ is bijective.
Finally we show that the height of  (.£) is ;<:; 1 . If f E A , we have
o'(x)(f P ) = 0 for each xELie((.£)), because o'(x) is a derivation. Since
Lie (  (.£)) generates Dist (  (.£) ), we therefore have 8 · (u) fP = 0 for each
u E Dist + ((.£)) . By duality, this gives fP = 0 if f (1) = 0 , and  (.£) has
height ;<:; 1 .
3.10 Proposition : Let  be a k -scheme . Then for each horrorrorphism
of Lie p- alqebras 1jJ:.£ --...;- Der ( ) , there is a horrorrorphism
p : E(.£) -->- Aut(X) ( thus correspondinq to a riqht E(.£) - operation on X)
-opp -- - - - - -
such that 1/J = p' 00..£ .
Pr=f : By 3.9 b) and the universal property of OJ[p](.£) , there is a unique
algebra homorrorphism v: Dist (  (.£)) ->- Dif  such that v S.£ j = 1/J. If v
meets the conditions of prop.  4,7.2, it is associated with a (unique)
horromorphism p: E (.£) ->- Aut (X) . Assuming the notation of this propo-
- opp --
sition, one shows easily that the set of 11 E Dist ((.£)) such that
V(I1) (1) = E(I1) and
V(I1) (f g ) = L: (V(I1.) (f)) (v(v.) (g)) , where toll = !: 11, @ v. ,
ill ill
( 5 4 , 7.2) is a subalgebra of Dist ( J;; (.£)) . Since this subalgebra contains
the image of .£, it coincides with Dist( .!2(.£)) , and the assertion follows.
3 .11 Theorem 3.5 is now an irmnediate consequence of 3.10. To see
this, given cp :.£ ->- Lie(Q) , let 1/J be the cOffi[Osition
8'
.£  Lie () -->- Der (  G)

334
ALGEBRAIC GROUPS
II,  7, no 4
and p : E (,£) -+ Aut (G) the corresponding homomorphism. Since 0' p (x) is
- opp --
left translation invariant for each x E,£ , so is p; hence we have
p(a) (gg') = gp(a) (g') for aE(,£) (k) , g,g' EG , and the property is pre-
served by changes of base. It follows that 1": a 1--+ p (a) (1) is the required
unique homorrorphism.
3.12
Corollary : Let ,£ and ,£' be two Lie p- algebras which are
also finitely generated projective k -rro::1ules . Then the map which assigns to
cp E k ('£,,£') the unique horromorphism £,: (cp) : £,: (,£) -+ £,: (,£ ') such that
'V
Lie (  (cp)) 0 0.,£ = 0.,£. 0 cp , is a bijection :!:dP k (,£ ,,£') ->- B;"k( £; (,£), £; (,£') )
Proof: Immediate from theorem 3.5 .
Section 4
Groups of height ;:: lover a field
We assume that k is a field (of characteristic p).
4.1
Proposition : The functor ,£ 1--+  (,£) is an equivalence between
the category of finite dimensional Lie p- algebras and the category of alge -
braic k- groups of height ;:: 1 .
Proof: In virtue of 3.9 and 3.12 it is sufficient to show that each alge-
braic k-group of height ;:: 1 is isomorphic to a group  (,£) : In fact more ge-
nerally we have the
4.2
Structure theorem for groups of height ;:: 1 : Let G be a
k -group-scheme . Then the following conditions are equivalent :
(i) Q is alqpbraic , (k) = e, and the canonical homomorphism
U[P](Lie(G))->- Dist(G) is bijective.
(ii) G is algebraic , (k) = e, and the unital algebra Dist()
is generated by Lie ( ) .
(iii) G is algebraic and of height ;:: 1
1l
(iv) There exists a finite dimensional Lie p- alqebrafsuch that 
is isomorphic to (,£)

II,  7, no 4
THE CHARACTERISTIC pfO
335
(v) For each k -group !i, the canonical map 2r k (Q,ill ->-k(Lie() ,Lie(tl))
is bijective.
(vi) G is affine , and U(Q) is isomorphic to the quotient of an alqebra
of polynomials k [Xl' . . . ,X n ] by the ideal generated by the xl.
Proof: (i) => (ii) : Trivial.
(ii) => (iii) This is proved in the same way as the final portion of 3.9 .
(vi) => (iii) By 1. 6 .
(i) => (v) cf. the proofs of 3.10 and 3.11 .
(v) => (iv) Set.£ = Lie (g) ; (v) then applies to give a homanorphism
!:  -+ (.£) such that LieCO = 0..£ . Apt?lying 3.5, we get a homomorphism
: :g:(.£) -+  such that Lie(5!:) 0 0..£ = Id . Applying (v) again and 3.5, we get
52 0 f = Id, f 0 5!: = Id
(iv) => (vi)
First let '2.. be an algebraic
k-group of height  1 , let m
2
n = [m/m : k] . If (m. ) is a
l
a surjective homomorphism
be the augrrentation ideal of G), and let
2
basis for the k-vector space m rrod m , there is
(1.6)
cn : k[X 1 ,...,x ]/(x?) ->- J1(G)
.,. n l -
such that cp (x.) = m, . If .£ is a finite dimensional Lie p-algebra and if
l l 2
Q=(.£) we have [.£: k] [m/m: k] . Hence, by 3.9 and 3.6, we have
[(Q) : k]
[Dist(g) : k] = [1JJ[p](.£) : k]
n
p
and cp is bijective.
(iii) => (i) Set.£ = Lie () a'1d consider the homomorphism f:  (.£) ->- 
such that Lie (f) 00..£ = Id.£ . Assume the notation of the preceding paragraph
and set mi = t9()(mi) E L9(!:'2 (.£)) ; the hanorrorphism d(f) 0 cp of
k [X 1 '...,X ] / (X?) into J}( E (.£)) sends X, onto m and is therefore bi-
n l - l l
jective by what we have already seen. Since cp is surjective, cp and J?(!)
are bijective, and [ is an isomorphism.
4.3
Corollary : Let Q be a locally algebraic k -qroup .
a) The map E  Lie (tI) induces a bijection of the set of subgroups of G
of height  1 onto the set of sub - Lie - p- algebras of Lie (g)
If !i and K

336
ALGEBRAIC GROUPS
II,  7, no 4
are two subqroups of 9 and if B has heiqht  1 , the inclusion B cj\ is
equivalent to Lie nn C Lie () .
b) If !l and .f 2 are homorrorphisrns of  into a k -qroup ' and if g has
heiqht  1 , then %1 = 1 2 iff Lie (Ii) = Lie (!2) .
c) If p: g->- GL (V) is a finite dimensional linear representation of Q.
and if g has heigh t  1 , then a vector subspace of V is stable unde r G
iff it is stable under Lie () . We have v G = ie (g); moreover p is simple
or semisimple iff Lie (p) has the same property .
d) If H is a subgroup of 9 of height  1 , we have
Lie ( cent G C!iJ )
Lie ( Norm g (H) )
Cent . ( G ) (Lie (H) )
Lle1..',2 -
N°rI1J,ie (G) (Lie (tl) )
e) If G has heiqht  1 the map B t-+ Lie (tI) is a bi iection of the set of
nonnal subgroups o f g onto the set of p- ideals of Lie (Q) (9- p- ideal beinq
a subspace I such that xELie(Q) and yEI  [x,y] EI and x[P]EI).
The proof of this result is similar to that of S 6 , 2.1 .

"FUNCTIORAL" DICTIONARY
In this dictionary we only want to fix the tenninology and the notations
used in this treatise.
abelian
A category C is abelian iff it satisfies the following conditions:
a) C is additive; b) for any rrorphism f of C , Ker f and Coker f
exist and the canonical rrorphism Coim f -+ Im f is invertible.
If a and b are twJ objects of C , we write a,bEC, and we denote
by C n (a,b) , n  1 , the group of Yoneda-extensions of order n of a
by b. In particular, CO(a,b)=C(a,b) is the group of rrorphisms fran
a to b.
additive,
A category C is additive iff it satisfies the following conditions:
a) C has a zero-object; b) if a,bEC, au.b and a'TIb exist and the
canonical rrorphism a.llb -+ a'TIb is invertible; c) if a,bEC, the "natural"
law of canposition of C (a,b) is a group law.
If f:a -+b is a rrorphism of the additive category C , we set Ker f=
Ker (f,O) , Coker f= Coker(f,O), Im f= Ker (b-+ Coker f) and Coim f
Coker (Ker f -+ a) . In case f is a rronanorphism, we also write b/a in-
stead of Coker f
A functor between twJ additive categories is additive iff it cCXImUtes
with finite products.
adjoint
Consider two functors l:C -+D and r:D -+C . An adjunction-isanorphism
fran 1 to r is by definition an isanorphism of functors D(lx,y) :::;.
C (x,ry) , xEC,yED. If such an isanorphism exists, 1 is said to be
left adjoint to r and r right adjoint to 1. In this case 1 com-
mutes with direct limits and d with inverse limits. Moreover, if C
and D are abelian, and if 1 (resp. r) is exact, then r (resp. 1
337

338
.FUNCI'ORIAL DICTIONARY
maps injective (resp. projective) objects onto injective (resp. projec-
tive) ones.
For instance, let S,T be hK:> tor::oligical spaces, f:T +S a continuous
ImP, C and D the categories of sheaves of sets over SandT. The
direct image functor r = f. (denoted by f* in GODEMENT, Ch II, 2.12)
is right adjoint to the pull back functor f. (denoted by f* in
GODEMENT, Ch II, 2.11).
antiequivalence
If C and D are hK:> categories, an antiequivalence fran C to D is
an equivalence fran c opp to D .
cartesian
A cartesian square is a COImn.1tative diagram of C of the form
[3
a--->b
y 1 1
c --)d
such that the rrorphism a-. b 'TIC with canponents [3 and y is invertible.
d
category
We denote by Ob C the class of objects of a category C , by Fl C the
class of  or rrorphisms . We simply write aEe instead of aEOb C )1
if a,bEC, we denote by C (a,b) the set of arrows fran the domain a
to the  b ; in case a = b Ida
phism of a. Finally, if fEe (a,b)
ccxrposed arrow ; we also say that gf
is the identity or identical rror-
and gEe (b,c) , gof =gf is the
factors through g,f or b
is denoted by C opp or cO
The opr::osite (or dual) category to C
cocartesian
Dual to cartesian.
cokernel
a
This is the direct limit of a diagram of the form ab it is denoted
[3
by COker (0.,[3)
COImn.1tative
A diagram is called caTlIT!Utative if comr::osed arrows with the same domain
and the same range always coincide.
camtUte
Let f :c-, D be a functor and c = ( (c.) , (" k ) ) a diagram of C . Set
1 1J

FUNCTORIAL DICTIONARY
339
fc = ( (fc.) , (f, ' k )) . We say that f cam1Utes with the inverse (resp.
l lJ
direct) limit of c if lim c (resp. lim c ) exists and if the canonical
 -
rrorphism f (lim c) ->- lim fc (resp. lim fc ->- f (limc) ) is invertible. We
  ----7--"
say that f cCXImUtes with inverse limits (resp. with small inverse limits,
resp. with filtered inverse limits ...) if f COlffiUltes with lim c for
--
any c (resp. any small c , resp. any small filtered inverse system
c ...)
contravariant
A contravariant functor fran C to D is a functor fran Copp to D .
derived
Let f:C ->-D be an additive functor between tv.KJ abelian categories.
Supr::ose that each cEC has a projective resolution
d n d
.. .Pn--->-Pn-l--->- .. . ---->- PO ---+c --+ 0
Then we set (LOf) (c) =Cokerf(d 1 ) and (Lnf) (c) = Kerf(d )/Imf(d 1 )
th n n+
if n:<: 1 ; we say that c r+ (L n f ) (c) is the n left-derived functor
of f. For instance, if A is a ring, C =A' D =1W, dEC and
n n
f(c) = dc , we set Tor A (d,c) = (L f) (c) .
The nth right-derived functor of f is defined similarly by means of
injective resolutions; it is denoted by Rnf .
diagram
A diagram (or system of objects and arrows) of a category C is a family
(c.) of objects of C together with a family (. ' k ) of arrows
l 
" k :c,->-c, .
lJ J l
For instance, we rray assign to any functor f:D ->-C the following diagram
(which we still denote by f): (c i ) = (fd) dED' (ijk) = (f) EF1D
When D is obtained from an ordered set I (Le.
Ob D = I, Card D (a, b) = 1 if a:;; b and Card D (a,b) = 0 otherwise) , the
diagram f is called a filtered inverse system (resp. a filtered direct
system ) if for any a,bEI there is a cEI such that c:;; a and c s b
(resp. a:;; c and b:;; c )
A diagram is called small (resp. finite ) if the indices of the objects
and arrows range through small (resp. finite) sets.

340
FUNCIDRIAL DICTIONARY
dimension
If C is an abelian category, the projective dimension of an object
aEC is the upper round pj(a) of the numbers n such that (a,b) O
for sane b . The injective dimension is defined in the dual way. The
global dimension of C is the upper round of the pj (a) , aEC.
effaceable
If C is an abelian category, a functor f:C ->- is called effaceable
if, for any cEC and any dEf(c) , there is a rronanorphism y:c ->-c'
such that f (y) (d) = 0 .
epirrorphism
A rrorphism p:c ->-d of C is an epirrorphism if up =vp always implies
u =v . We say that p is a strict epirrorphism if there is a family of
double-arrows a" S . : c ,=:;c with the following property: every morphism
111
q:c ->-e such that qa. =qS, for any i factors through p.
1 1
equivalence
A functor f:C ->-D is called an equivalence if there is a functor g:D->-C
such that gf and fg are isorrorphic to the identical functors of C
and D respectively. Such a functor g is said to be quasi-inverse to
f
A functor f:C ->-D is an equivalence iff it is fully faithful and every
dED is isorrorphic to sane image fc
exact
A functor is called left exact (resp. right exact , resp. exact) if it
CCXImUtes with finite inverse limits (resp. finite direct limits, resp.
finite direct and inverse limits) .
In an abelian category C, a sequence a r:; b .l} c is called exact if
So. = 0 and the canonical rrorphism Im a ->- Ker S is invertible.
extension
Let C be an abelian category and c,dEC . We denote by (c,d) the
group of Yoneda-extensions of order n of c by a, i. e. the group of
the classes of exact sequences of the form
./
O->-d->-x ->-x ->-...->-x ->-c->-O
1 2 n
faithful
A functor f:C ->-D is faithful (resp. fully faithful ) if for any a,bEc
the map f(a,b) : C(a,b) ->-D(fa,fb) is injective (resp. bijective).
j,
"1

FUNCTORIAL DICTIONARY
341
final
An object c of C is called final if Card C (a,c) =1 for any aEC .
full
A subcategory D of C is a full subcategory if the inclusion-functor
fran D to C is fully faithfuL
functor
If f:C->D is a functor and a,bEC, we denote by f(a,b) : C(a,b)->-D(fa,fb)
the map y>-->- f (y) attached to f. The category of functors fran C to
D is denoted by CD .
initial
An object a of C is called initial if Card C (a,x) =1 for any xEc .
injective
An object c of the abelian category C is called injective if the
functor cOP P ->-@, dH-C(d,c) is exact.
inverse
The inverse of
and f-lf =Id
a
fEC(a,b)
-1
. If f
-1 -1
is a morphism f EC (b, a) such that ff = I
exists, f is called invertible or an iso-
rrorphism .
isanorphic
'IWo objects a,bEC are called isanorphic if there is an isanorphism
f:ab
kernel
a
The inverse limit of a diagram of the form ab is called kernel and
is denoted by Ker(a,S) .
limit
Let c = ((c,), (. ' k )) be a diagram of the category C . Whenever this
l lJ
is possible, we choose once for all an object denoted by  c or
lim c. and rrorphisms
--r l
in, =in.. ' k for all
J l lJ
rrorphisms p,:c.->-d satisfying the equations p. =p.. ' k ' there exists
l l J l lJ
a unique p: limc->-d such that p. =p°in. for any i. We say that
--> l l
lim c is the direct limit of c, that in, is the canonical induction
--> th l
of index i and that Pi is the i canponent of p
in, : c .  lim c , satisfying the folla.ving conditions:
ll--yl
i, j , k ; rroreover, for any d Ec and any family of
For the dual notions with say inverse limit (denoted by  c or
lim c, and canonical p roj ection of index i (denoted by pr l , ) .
 l -
l

342
FUNCI'ORIAL DICTIONARY
rronanorphism
The notions of ITOnanorphisms and strict rronanorphisms are dual to those
of epirrorphisms and strict epirrorphisms.
prcxluct
Let (c i ) iEI be a family of objects considered as a diagram for which
the family of arrows is empty. The inverse limit is denoted by .TT E c.
1 I 1
and is called the cartesian (or direct) prcxluct of the family
(ci)iEI. If I={l,2} , we :et 'lci=cl'TTC2=clxc2. If c i =c for
any iEI, we write TJ c, = C
1 1
pull back
This is the inverse limit of a diagram of the form
a S
a ---->- C +--- b
It is denoted by a'TTb,
c
a 'TT b ->- a (resp. a 'TT b ->- b
c c
a b ) .
projective
An object p of an abelian category C is called projective if the
a x b or a 'TT S b . The canonical projection
c a,
is denoted by pr 1 or Sa (resp. pr 2 or
II
functor Cr+H?, x 1-+ C (p,x) is exact.
push out
This is the direct limit of a diagram of the form
a S
a+-c-->-b
f
c a,S .
It is denoted by a.\lb or a 11 b . We wrl te
c
in 2 : b ->- a l!. b for the canonical inductions.
representable
Let C be a category. A functor f:C ->- (resp. f:COP P ->-].) is called
representable if it is isorrorphic to the functor xr-+ C (d,x) (resp.
XI-->- C (x,d) ) for some dEC .
c
in l :a->-allb and

retraction
A retraction of a rrorphism f:a ->- b is a rrorphism r:b ->- a such that
rf = Id
a
resolution
If C is an abelian category, a projective resolution of cEc is an
infinite exact sequence
. .. ->- Pn ->- Pn-l ->- ... ->- PO ->- c ->- 0
'1
A

FUNCTORIAL DICTIONARY
343
such that Pn is projective for any nE IN . For the dual notion we say
injective resolution .
section
A section of a ITOrphism f: a + b is a ITOrphism s: b + a such that
fs = I .
sum
Let (c i ) iEI be a family of objects. The direct limit of (c i ) iEI '
considered as a diagram for which the family of arrows is Empty, is
denoted by i*Ici and is called the direct sum of the family (c i )
If I={l,2}, we set c 1 uc 2 =lfc i . If c.=c for all iEI, we
(I) _ il l
set c - iEIci
subcategory
We say that the category D is a subcategory of C if Ob DCObC ,
Fl DCF1C and if the canposition of D is induced by that of C .
universe
A universe U is a set subjected to the following axians: a) If
(Xi) iEI is a family of sets XiEU and if IEU, then N?i is an
element of U b) xEU implies {x}EU; c) xEX and XEU imply
xEU; d) if XEU, the set of all subsets of X belongs to U
e) a pair (x,y) belongs to U iff both x and y belong to U
zero
A zero-object 'bf a category C is an object which is both final and
initiaL

INDEX OF NOTATIONS
I Notations of N. Bourbaki
We use the following notations due to Bourbaki:
tC
IF
P
(V)
GL(n,R)
(V)
IN
N*
I!J
IR
SL(n,R)
z
field of complex numbers
z /pZ , p prime
linear group of the R-rncdule V
linear group of order n with coefficients in R
ring of endanorphisms of the R-mcxlule V
{0,1,2,...} = set of natural numbers
I
{1,2,...} =!N,{O}
field of rational numbers
field of real numbers
special linear group of order n with coefficients in R
ring of rational integers
If g is a Lie-algebra and if x,yEg , we set [x,yJ= (ad x) (y) . If g acts
linearly on V, V g is the subspace of V formed by the v such that
x(v) = 0 for all xEg.
II Categories and universes
M2 ccrrmutative groups GC
 sheaves of k-rncdules, over the k-scheme X II, 4 , 5 Intrcxl.
 (cCXImUtative unitfil) rings GC
 (associative cCXImUtative unital) k-algebras GC
344

A
E
.-

k
T
Gr
-
k
K
.....

tl
1%


k\E;
¥
W/E
A
S

O
-k
A


II



INDEX OF NOTATIONS
345
cancxlules over the bialgebra A II, 3, 3.5
sets GC
geanetric spaces I,  1, 1. 4
k-geometric spaces I, 1, 6.8
geanetric spaces over a geanetric space T I, 1, 6.8
groups GC
k-group-functors II,  1, 1.1
fields belonging to the universe .Q. I, 1, 4.5
p-Lie-algebras over k II, 7, 3.3
models GC
f-rrodels I, 1, 4.1
k-rrodels GC
Z'-functors I, 1, 3.1
k-functors I, 1, 6.2
.e.-functors I, 1, 6.2
Z'-functors over a Z'-functor S I, 1, 6.2
A-rrodules GC
.e.- rrod ules I, 2, 4.1
sheaves of ITOdules over the geometric space X I,  2, 1.1
9k-rrodules II, 1, 2.5
small A-modules I, 2, 1.7
rronoids GC
k-rronoid-functors II,  1, 1.1
quasi-coherent sheaves of modules over  I, 2, 1.7
quasi-coherent .e.-modules I, 2, 4.1
schemes I, 1, 3.11
kschemes I, 1, 6.4

346 INDEX OF NOTATIONS
12e tor::01ogical spaces GC
2,'::: fixed universes GC
.J
III Ord inary letters
Aut II, 1, 2.7
B n II, 3, 1.1
C n , cO II, 3, 1.1
Der II, 4, 2.4
Dif, Dif II, 4, 5.3
n
dim, dim x ' Kdim I, 3, 5.1
Dist, Dist n , Dist+, Dist+ II, 4, 6.1
n
e II, 1, Intrcxl. and  4, 3.7
End II, 1, 2.7
exp II, 6, 3.1
F II, 3, 4.4
Fract I, 3, 5.3
Hi II, 3, 3.2
i 2 II, 3, 1.1
HO ' Hs
Lie II, 4, 4.8
m I, 1, 1.1
x
Spec I, 1, 2.1
U() II, 6, 1.1
u[PJ () II, 7, 3.6
W II, 3, 4.6
Zn , z2 II, 3, 1.1 and 2.4
s

INDEX OF NOTATIONS 347
IV Underlined letters
Ad II, 4, 1.1
Aut II, 1, 2.6 and 2.7
 II, 1, 3.3d)
Cent II, 1, 3.4 and 3.9
D II, 1, 2.8 and 2.10
D II, 1, 2.1
-a
 II, 1, 3.11
Der II, 4, 2.3 and 2.4
det II, 1, 2.4
 II Intrcxl. and II, 4, 2.1
E II, 4, 2.2 and  7, 3.5
End II, 1, 2.6 and 2.7
, x II, 7, 1.1
F (r 1 ,...,r) I, 2, 4.5
-i'l s
GL , II, 1, 2.4
G I, 1, 3.4
-i'l,r
 I, 2, 7.1
Im I, 1, 4.2
Im I, 2, 4.10
Int II, 1, 3.3d)
Ker I, 1, 5.1 and II, 1, 1.3
L II, 1, 2.4
Lie II, 4, 1.1
Mod II, 1, 2.4
Norm II, 1, 3.4
0 I, 1, 3.3 and 6.1

348 INDEX OF NOTATIONS
12 I, 1, 6.3 and 6.5
PGL nk II, 1, 2.6
p I, 1, 3.4
-r
S I, 1, 3.5
S I, 1, 6.6
"""
T I, 1, 6.8
SL ,  II, 1, 2.4
EE I, 1, 3.2 and  2, 5.4
!nk II, 1, 3.11
Transp I, 2, 7.4 and II, 1, 3.4
 II, 1, 3.11
V I, 2, 6.2
b: 2 II, 3, 4.9
V Greek letters
a a II, 1, 2.2
r
p
y ,0 II, 1, 3.3c) and  2, 1.2d)
y' , 0' II, 4, 4.5 and  6, 1.4
0 II, 1, 3.5
°x/y I, 2, 2.1
to II, 1, 1.6 and  2, 2.1
E II, 1, 1.1,  1, 1.6 and  4, 1.1
E I, 1, 5.6
x
E(X) I, 1, 5.2
K(X) I, 1, 1.1
11, nil II, 1, 2.8
A II,
11 1, 2.3

INDEX OF NOTATIONS 349
'TT II, 1, 1.1
'TT (X) I, 4, 6.5 and II, 5, 1.10
0-
II I, 1, 6.6; I, 4, 6 Intrcxl.
II, 5 Intrcxl.
a II, 1, 1.1
1/J I, 1, 2.9 and 4.3
 I, 4, 2.1 and 2.5
w I, 4, 1.3; II, 4, 3.1 and 3.4
VI Scri p t letters
YJ II, 5, 4.8
() I, 1, 1.1, 2'.1, 3.3, 4.2 and 6.1
jJ II, 4, 5.4
VII Expo nents or indices
 II, 1, 2.1
,f I, 1, 1.4 and 4.2
F II, 7, 1.4
opp II, 1, 1.1
red I, 2, 4.10 and 5.1
VIII Miscellaneous Notations
A($)) II, 6, 2.4
A A = ring, sEA I, 1, 2.1
s
D(a) a = ideal I, 1, 2.1
S(a) a = ideal I, 1, 2.1

350
INDEX OF NOTATIONS
V(a)
a = ideal
5l(P)
f- l
9: = rrorphism of schemes; P = set
f = rrorphism of schemes
I p  = Z-functor; P = subset of II
 ' k! >  = schemes; k = ITOdel
1'S'k,1'k' T,S = schemes; k,k' = ITOdels
kR R = k-algebra
 X = set or boolean space ; k = ITOdel
"f>.G. "f>. = k-functor;  = k-group-functor
tl)<1 £I, = k-group-functors
G G = k-grou p- functor; W,W' = k-ITOdules
',W -
V m ' v G V = k-ITOdule; G = k-group-functor;
m = character
R
n
(p )
 = ring haranorphism with range R
X = k-scheme; p = prime number
,!
I, 1, 2.4
I, 1, 4.2
I, 1, 3.2, 4.2,
5.8 and 6.3
I, 1, 4.10
I, 1, 6.4
I, 1, 6.5
I, 1, 6.5
I, 1, 6.9 and 6.10
II, 1, 3.4
II, 1, 3.10
II, 2, 1.3
II, 2, 2.4
II, 7, Intrcxl.
II, 7, 1.1
IX Miscellaneous Signs
0 II, 5, 1.3
I, 2, 1.4
* I, 2, 1.1 and 1.3;
II, 1, Intrcxl. and 1.4; II, 4, 6.2
x II, 1, Intrcxl. and 2.8
+ II, 1, Intrcxl.
I I, 1, 4.1 and 6.8
J II, 4, 5.2
vi I, 1, 2.4
#- ,b I, 1, 3.1

TERMINOI.J::X;ICAL INDEX
We refer to chapter A,  b, No. c.d. by means of A,  b, c.d. Similarly, G.C.
refers to the general conventions. The symbol Gr intrcxluces the correspnding
term in the terminology of Grothendieck, in case his terminology is diffe-
rent fran ours.
additive
adjoint
affine
algebra
algebraic
associative
augmentation
bialgebra
biduality
Boolean
bracket
character
characteristic
Cartier
central
centre
additive group
adjoint representation
affine line
affine functor over another functor
affine rrorphism
affine scheme
affine S-scheme
X-algebra, S-algebra
0J<-algebra
algebraic curve
(Gr. courbe algebrique irreductible)
algebraic hull of a sub-Lie-algebra
algebraic scheme, locally algebraic
scheme (Gr. scherna de presentation finie,
schema localement de presentation finie)
algebraic subalgebra of a Lie-algebra
associative law of canr::osition
augmentation of a bialgebra
biduality haroJrorphism
Boolean space
Boolean group
bracket in the Lie-algebra of a group
character of a group-functor
characteristic subgroup
Cartier algebra, Cartier dual of a finite
caTlIT!Utative group
central subgroup
central extension
351
II, 1, 2.2
II, 4, 4.1
I, 1, 3.3 and
6.1
I, 2, 5.1
I, 1, 3.2
I, 1, 6.1
I, 2, 5.4
II, 1, 2.5
I, 5, 3.1
II, 6, 2.4
I, 3, 2.1
II, 6, 2.4
II, 1, 1.1
II, 1, 1.6
II, 1, 1.6
II, 1, 2.10
I, 1, 2.12
II, 1, 2.12
II, 4, 4.2
II, 1, 2.9
II, 1, 3.9
II, 1, 2.10
II, 1, 1.3
II, 3, 3.2
II, 1, 3.9

352
centralizer
clean
closed
closure
coefficient
ccmnutative
carodule
ccxrplete
canposition
connected
constant
constructible
convolution
coprcxluct
counit
covering
curve
derived
deviation
diagonal
diagonalizable
differential
dimension
TERMINOI.J::X;ICAL INDEX
clean rrorphism, clean scheme
closed eTIbedding
closed image of a rrorphism of schemes
closed rrorphism
locally closed subfunctor
closed subfunctor, subscheme
closed subscheme defined by an ideal
universally closed rrorphism
closure of a subscheme
(Gr. adherence schematique)
coefficient of a representation
cCXImUtative law of canr::osition
cancxlule over a bialgebra
ccmplete k-scheme (Gr. k-schema
propre et de presentation finie sur k)
law of canposi tion
geometrically connected scheme
scheme of connected canponents
group of connected canponents
constant rronoid
constant scheme
constant tenu of a differential operator
constructible subset of a scheme
convolution prcxluct of distributions
coprcxluct of a bialgebra
counit of a bialgebra
open covering of a :?{-functor
algebraic curve
(Gr. courbe algebrique irreductible)
derived group of an algebraic group
diagonal group
diagonal morphism of a scheme
diagonalizable group, rronoid
diagonalizable representation
differential operator
ITOdule of differentials
dimension of a tor::ological space, local
dimension at a r::oint
r
II, 1, 3.4
I, 4, 3.2
I, 2, 6.1
I, 2, 6.11
I, 1, 4.2
I, 2, 7.2
I, 2, 6.8
I, 2, 6.8
I, 5, 1.3
I, 2, 6.11
II, 2, 2.3
II, 1, 1.1
II, 2, 2.1
I, 5, 2.1
II, 1, 1.1
l, 4, 6.8
I, 4, 6.6
II, 5, 1.10
II, 1, 1.5c)
1, 1, 6.10
II, 4, 5.3
I, 3, 3.1
II, 4, 6.1
II, 1, 1.6
II, 1, 1.6
I, 1, 3.10
I, 5, 3.1
II, 5, 4.8
II, 4, 5.1 and
 4, 5.7
II, 1, 3.11
II, 2, 2.1
II, 1, 2.11
II, 2, 1.7
II, 4, 5.3
I, 4, 2.1
I, 3, 5.1

dimension
direct
distribution
daninant
dual
embedding
equivalent
etale
exr::o nential
extension
factor
faithful
faithfully
fibre
finite
finitely
flag
flat
free
Frobenius
functions
functor
TERMINOI.J::X;ICAL INDEX
dimension of a scheme, local dimension
at a r::oint
Krull dimension of a ring
direct image of an X-ITOdule
distribution on a scheme
algebra of distributions on a group-scheme
daninant rrorphism of schemes
Cartier dual of a finite caTlIT!Utative
group scheme
algebra of dual numbers
embedding
closed embedding
open embedding of geometric spaces
open embedding of :i-functors
equivalent H-extensions
etale algebra, rrorphism, scheme
exr::onential map
base extension, extension of scalars
H-extension
simple factor of a representation
faithful linear representation
faithfully flat rrorphism
faithfully flat k-scheme
fibre of a rrorphism of functors
finite rrorphism
finitely generated quasicoherent sheaf
finitely generated rrorphism of schemes
finitely presented rrorphism of schemes
locally finitely generated rrorphism
locally finitely presented rrorphism
scheme of flags
flat rrorphism, faithfully flat rrorphism
flat scheme, faithfully flat scheme
locally free rrorphism
locally free scheme
Frobenius rrorphism
ring of functions over a geometric space
ring of functions over a :i-functor
ring of functions over a k-functor
:i-functor
k-functor, S-functor
353
I, 3, 6.1
I, 3, 5.1
I, 2, 1.1
II, 4, 5.2
II, 4, 6.1
I, 1, 2.4 and
 3, 3.7
II, 1, 2.10
II, 4, 1.1
I, 2,
I, 2,
I, 1,
I, 1,
II, 3,
I, 4,
II, 6,
I, 1,
II, 3,
II, 2,
II,
I,
I,
I, 1,
 1,
7.1
6.1
1.4
3.6
2.1
3.2
3
6.5
2.1
1.5
5, 5.2
2, 3.1
2, 3.3
5.8 and
6.3
1.1
2, 2.6
3, 1.12
3, 1.6
3, 1.12
3, 1.6
I,
I,
I,
I,
I,
I,
I,
I,
I,
I, 5,
I, 2,
II, 7,
I,
I,
I,
I,
I,
-5,
2, 6.5
2, 3.1
2, 3.3
1.1
9.5
1.1
1, 2.5
1, 3.3
1, 6.1
1, 3.1
1, 6.1

354 TERMINOI.J::X;ICAL INDEX
functor underlying X-functor I, 1, 6.3
generated closed subgroup generated by a rational
r::oint II, 5, 4.5
generic generic r::oint of an irreducible subset I, 1, 2.10
geometric geometric space I, 1, 1.1
geometric realisation of a X-functor I, 1, 4.2
geometric realisation of a k-functor I, 1, 6.8
genu genu of a function over a geometric space I, 1, 1.1
Grassmannian I, 1, 3.4
group k-group-functor, k-group-scheme II, 1, 1.1
II-group II, 5, 1.7
height height of an inf ini tesimal group II, 7, 1.4
H-extension II, 3, 2.1
Hochschild Hochschild ccrnplex, Hochschild group II, 3, 1.1
hull algebraic hull of a sub-Lie-algebra II, 6, 2.4
image- image of a rrorphism of :i-functors I, 1, 4.2
image-functor of a rrorphism of :i-functors I, 1, 4.2
closed image of a rrorphism of schemes I, 2, 6.11
induced geometric space induced on a subset I, 1, 1.3
inessential inessential H-extension II, 3, 2.1
infinitesimal infinitesimal group-scheme II, 4, 7.1
injective injective rrorphism of X-functors I, 1, 4.2
inner inner autarorphism operation II, 1, 3.3d)
integral integral rrorphism I, 5, 1.1
invariant invariant derivation II, 4, 4.6
invariant differential operator II, 4, 6.5
invariant subgroup II, 1, 1.3
inverse inverse image of a sheaf I, 2, 1.3
irreducible irreducible linear representation II, 2, 1.5
irreducible tor::ological space I, 1, 2.10
isotypical isotypical linear representation II, 2, 1.5
Jacobi Jacobi identity II, 4, 4.3
Jacobson Jacobson fonnula II, 7, 3.2
kernel kernel of a double-arrow I, 1, 5.1
kernel of a hararorphism of group-functors II, 1, 1.3
Krull Krull dimension of a ring I, 3, 5.1
law law of canposition on a k-functor II, 1, 1.1

Lie
linear
local
ITOdel
rronoid
multiplicative
nationality
neighbourhood
neutral
nilr::o tent
nonnalizer
open
operator
operation
opr::osite
orbit
parti tion
r::o int
r::o wer
TERMINOI.J::X;ICAL INDEX
Lie-algebra of a group-scheme
Lie p-algebra
Lie p-algebra of a group-scheme
linear group, special linear group
linear representation
local dimension at a r::oint of a
local dimension at a r::oint of a
space
local embedding of k-schemes
local embedding of schemes
local functor
355
II, 4, 4.8
II, 7, 3.3
II, 7, 3.4
II, 1, 2.4
II, 2, 1.1
scheme I, 3, 6.1
tor::ological
I, 3, 5.1
I, 3, 4.4
I, 3, 4.3
I, 1, 3.11
ITOdel, k-ITOdel
X-ITOdule, S-ITOdule, ITOdule over a
ITOdule of an embedding
(Gr. faisceau cononnal)
k-G-ITOdule (oup-functor)
o -ITOdule
seaf of ITOdules over a geanetric
G.C.
:i-functor I,  2, 4.1
space
I, 4, 1.3
II, 2, 1.1
II, 1, 2.5
I, 2, 1.1
rronoid law, rronoid-functor, rronoid-scheme II, 1, 1.1
standard multiplicative group
(Gr. groupe multiplicatif) II, 1, 2.8
nationality of a flag
first neighbourhood
nth-neighbourhood
neutral canponent of an algebraic group
nilr::otent element of a Lie-algebra
open embedding of geometric spaces
open embedding of :i-functors
open rrorphism of :i-functors
open subfunctor
open subscheme
open covering of a :i-functor
differential operator
operation of a k-group on a k-scheme
oPr::osite rronoid
partition of unity in a ring
r::oint of a :i-functor
p th -j:XJWer in a Lie-algebra
I, 2, 6.5
I, 4, 1.1
II, 4, 5.5
II, 5, 1.3
II, 6, 3.7
II, 1, 3.4
I, 1, 1.4
I, 1, 3.6
I, 1, 4.2
I, 1, 3.6
I, 1, 3.11
I, 1, 3.10
II, 4, 5.3
II, 1, 3.1 and
3.2
II, 1, 1.1
II, 5, 3.3
I, 1, 3.10
I, 1, 4.2
II, 7, 2.1

356
j:XJWer
prime
projective
proper
pure
quasi-coherent
quasi-compact
quasi-separated
ramified
rank
rational
reduced
reducible
regular
representation
restriction
root
scheme
semi-sirrple
semi -direct
separated
set
simple
smooth
TERMINOI.J::X;ICAL INDEX
P th -j:XJWer in the Lie-algebra of
a group-functor
prime spectrum of a ring
prime spectrum
projective space, projective line
proper rrorphism of schemes
pure suhrodule
quasi -coherent ITOdule
quasi -coherent S-algebra
quasi -coherent sheaf of ITOdules
quasi -canpact rrorphism
quasi -canpact scheme
quasi-separated rrorphism
quasi-separated scheme
non-ramified rrorphism, scheme
rank of a locally free rrorphism
rational r::oint
reduced scheme, part of a scheme
reduced ring

II, 7, 3.3
I, 1, 2.3
I, 1, 2.9
I, 1, 3.4
I, 5, 2.1
II, 2, 1.3
Ii
,
'11
,w
)
I, 2, 4.1
I, 2, 5.4
I, 2, 1.7
I, 2, 2.3
I, 2, 1.8
I, 2, 2.2
I, 2, 1.8
.
I, 4, 3.2
I, 5, 1.1
I, 3, 6.7
I, 2, 6.11
I, 2, 6.13
n
!
Ip
:

I
1

f
;
canpletely reducible linear representation II, 2, 1.5
regular algebraic curve
regular representation
linear representation
regular representation
base restriction, restriction of scalars
Weil restriction
group of nth-roots of unity
scheme
(Gr. foncteur sur les anneaux representable
par un schema)
k-scheme
(Gr. foncteur sur les k-algebres represen-
table par un k-schema)
semi-simple representation
semi -direct prcxluct
separated rrorphism, separated :i-functor
II-set
I, 5, 3.1
II, 2, 1.2d)
II, 2, 1.1
II, 2, 1.2d)
I, 1, 6.4
I, 1, 6.6
II, 1, 2.8
I, 1, 3.11
I!
f
I, 1, 6.1
II, 2, 1.5
II, 1, 3.10
I, 2, 7.4
I, 4, 6.4
simple factor of a representation
simple representation II, 2, 1.5
smooth rrorphism, smooth scheme, k-smooth
scheme I, 4, 4.1

TERMINOI.J::X;ICAL INDEX 357
space space of FOints of a :i-functor I, 1, 4.2
special special open subset I, 1, 1.3
special linear group II, 1, 2.4
spectral spectral space (Gr. schema) I, 1, 2.9
spectrum prime spectrum of a ring I, 1, 2.3
prime spectrum I, 1, 2.9
spectrum of a quasi-coherent sheaf of
algebras I, 2, 5.5
stable stable suhrodule II, 2, 1.5
strict strict triagonal group II, 1, 3.11
structural structural projection of an S-functor I, 1, 6.3
subscheme I, 2, 5.1
subset subset of a :i-functor I, 1, 4.2
surj ecti ve surjective rrorphism of :i-functors I, 1, 4.2
tangent tangent space of Zariski I, 4, 4.15
translation translation operation II, 1, 3.3c)
transFOrter I, 2, 9.4 and
II, 1, 3.4
triagonal triagonal group, strict triagonal group II, 1, 3.11
unit unit section II, 1, 1.1
universally universally closed rrorphism I, 5, 1.3
value value of a function at a FOint I, 1, 1.1
Weil Weil restriction I, 1, 6.6
Zariski tangent space of Zariski I, 4, 4.15