University
Lecture
Series
Volume 37
Quadratic Algebras
Alexander Polishchuk
Leonid Positselski
American Mathematical Society
Providence, Rhode Island

Quadratic algebras, i.e., algebras defined by quadratic relations, often occur in various
areas of mathematics. One of the main problems in the study of these (and similarly
defined) algebras is how to control their size. A central notion in solving this problem is
the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then
appeared in many areas of mathematics, such as algebraic geometry, representation
theory, noncommutative geometry, i^-theory, number theory, and noncommutative linear
algebra.
The book offers a coherent exposition of the theory of quadratic and Koszul algebras,
including various definitions of Koszulness, duality theory, Poincare-Birkhoff-Witt-type
theorems for Koszul algebras, and the Koszul deformation principle. In the concluding
chapter of the book, they explain a surprising connection between Koszul algebras and
one-dependent discrete-time stochastic processes.

EDITORIAL COMMITTEE
Jerry L. Bona (Chair) Eric M. Friedlander
Adriano Garsia Nigel J. Higson
Peter Landweber
2000 Mathematics Subject Classification. Primary 16S37, 16S15, 16E05, 16E30, 16E45,
16W50, 13P10, 60G10.
For additional information and updates on this book, visit
www.ams.org/bookpages/ulect-37
Library of Congress Cataloging-in-Publication Data
Polishchuk, Alexander, 1971-
Quadratic algebras / Alexander Polishchuk, Leonid Positselski.
p. cm. — (University lecture series, ISSN 1047-3998 ; v. 37)
Includes bibliographical references.
ISBN 0-8218-3834-2 (acid-free paper)
1. Quadratic fields. 2. Associative rings. 3. Commutative rings. 4. Stochastic processes.
I. Positselski, Leonid, 1973- II. Title. III. University lecture series (Providence. R.I.) ; 37.
QA247.P596 2005
512'.4—dc22 2005048198
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Acquisitions Department, American Mathematical Society.
201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by
e-mail to reprint-permission@ams. org.
© 2005 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05

Contents
Introduction vii
Chapter 1. Preliminaries 1
0. Conventions and notation 1
1. Bar constructions 2
2. Quadratic algebras and modules б
3. Diagonal cohomology 7
4. Minimal resolutions 7
5. Low-dimensional cohomology 9
6. Lattices and distributivity 11
7. Lattices of vector spaces 15
Chapter 2. Koszul algebras and modules 19
1. Koszulness 19
2. Hilbert series 21
3. Koszul complexes 25
4. Distributivity and n-Koszulness 29
5. Homomorphisms of algebras and Koszulness. I 32
6. Homomorphisms of algebras and Koszulness. II 37
7. Koszul algebras in algebraic geometry 40
8. Infinitesimal Hopf algebra associated with a Koszul algebra 45
9. Koszul algebras and monoidal functors 49
10. Relative Koszulness of modules 53
Chapter 3. Operations on graded algebras and modules 55
1. Direct sums, free products and tensor products 55
2. Segre products and Veronese powers. I 59
3. Segre products and Veronese powers. II 63
4. Internal cohomomorphism 68
5. Koszulness cannot be checked using Hilbert series 77
Chapter 4. Poincare-Birkhoff-Witt Bases 81
1. PBW-bases 81
2. PBW-theorem 82
3. PBW-bases and Koszulness 84
4. PBW-bases and operations on quadratic algebras 85
5. PBW-bases and distributing bases 86
6. Hilbert series of PBW-algebras 87
7. Filtrations on quadratic algebras 88
8. Commutative PBW-bases 91

vi CONTENTS
9. Z-algebras 95
10. Z-PBW-bases 96
11. Three-dimensional Sklyanin algebras 98
Chapter 5. Nonhomogeneous Quadratic Algebras 101
1. Jacobi identity 101
2. Nonhomogeneous PBW-theorem 103
3. Nonhomogeneous quadratic modules 104
4. Nonhomogeneous quadratic duality 105
5. Examples 108
6. Nonhomogeneous duality and cohomology 111
7. Bar construction for CDG-algebras and modules 112
8. Homology of completed cobar-complexes 117
Chapter 6. Families of quadratic algebras and Hilbert series 119
1. Openness of distributivity 119
2. Deformations of Koszul algebras 120
3. Upper bound for the number of Koszul Hilbert series 122
4. Generic quadratic algebras 123
5. Examples with small dimAi and dim^2 125
6. Koszulness is not constructible 127
7. Families of quadratic algebras over schemes 128
Chapter 7. Hilbert series of Koszul algebras and one-dependent processes 133
1. Conjectures on Hilbert series of Koszul algebras 133
2. Koszul inequalities 135
3. Koszul duality and inequalities 138
4. One-dependent processes 139
5. PBW-algebras and two-block-factor processes 142
6. Operations on one-dependent processes 143
7. Hilbert space representations of one-dependent processes 146
8. Hilbert series of one-dependent processes 147
9. Hermitian construction of one-dependent processes 149
10. Modules over one-dependent processes 151
Appendix A. DG-algebras and Massey products 153
Bibliography
155

Introduction
The goal of this book is to introduce the reader to some recent developments
in the study of associative algebras defined by quadratic relations. More precisely,
we are interested in (not necessarily commutative) algebras over a field that can
be presented using a finite number of generators and (possibly nonhomogeneous)
quadratic relations. This book is devoted to some aspects of the theory of such
algebras, mostly evolving around the notions of Koszul algebra and Koszul
duality. Its content is a mixture of known results with a few original results that we
circulated since 1994 as a preprint of the same title.
One of the original motivations for the study of quadratic algebras came from
the theory of quantum groups (see [43, 77]). Namely, quadratic algebras provide a
convenient framework for "noncommutative spaces" on which quantum groups act
(see [78]). One of the basic problems that arose in this context is how to control
the growth of a quadratic algebra (e.g., measured by Hilbert series). A related
question is whether there are generalizations of the Poincare-Birkhoff-Witt theorem
(for universal enveloping algebras) to more general quadratic algebras. The core of
this book is our attempt to present some partial solutions. It turns out that one
can shed some light on questions of this kind using the remarkable notion of Koszul
algebra introduced by S. Priddy [104]. In fact, the study of this notion brought
some dramatic changes to the area. Loosely speaking, our experience shows that
general quadratic algebras behave as badly as possible, while for Koszul algebras
the situation is usually much nicer. As we hope to convince the reader, the study
of Hilbert series provides a good illustration of this principle.
Perhaps one of the important features of the theory of Koszul algebras is
duality: for each Koszul algebra there is a dual Koszul algebra (roughly speaking, it
is obtained by passing to the dual space of generators and the orthogonal space
of quadratic relations). This often leads to remarkable connections between
seemingly unrelated problems. For example, Koszul duality of the symmetric algebra
and the exterior algebra underlies the famous description of coherent sheaves on
projective spaces in terms of modules over the exterior algebra due to J. Bernstein,
I. Gelfand and S. Gelfand [27]. More generally, in a number of situations one can
prove an equivalence of derived categories of modules over Koszul dual algebras
(see [23, 11, 24, 51]). This topic is beyond the scope of our book although we will
discuss some more elementary aspects of Koszul duality.
The notion of Koszulness also proved to be a really impressive prediction tool.
In many examples a few observations may suggest that some quadratic algebra
is Koszul. Then this conjecture turns out to be related to some important and
nontrivial features of the setting. It is also quite amazing that many important
quadratic algebras naturally arising in various fields of mathematics are Koszul.
Examples known to us arise in the following areas:
vii

viii INTRODUCTION
(i) algebraic geometry—certain homogeneous coordinate algebras are Koszul (see
[29, 37, 39, 67, 50, 72, 73, 89, 96]);
(ii) representation theory—certain subcategories of the category О for a semisimple
complex Lie algebra are governed by Koszul algebras (see [19, 24]);
(iii) noncommutative geometry—the Koszulness condition arises naturally in the
theory of exceptional collections; the algebras describing certain noncommutative
deformations of projective spaces are Koszul (see [30, 31, 117]);
(iv) topology—Steenrod algebra, cohomology algebras of formal rational К [к, 1]-
spaces, holonomy algebras of supersolvable hyperplane arrangements, as well as
some algebras related to configuration spaces of surfaces are Koszul; the category
of perverse sheaves on a triangulated space is equivalent to modules over a Koszul
algebra (see [104, 88, 113, 28, 97, 127]);
(v) number theory—the Milnor X-theory ring of any field (tensored with Z//Z for
a prime I) is conjectured to be Koszul—this is a strengthening of the Bloch-Kato
conjecture relating Milnor X-theory with Galois cohomology (see [103, 102]);
(vi) noncommutative algebra—the universal algebra generated by pseudoroots of a
noncommutative polynomial is Koszul (see [111, 93]).
Checking the Koszul property usually requires some effort and the methods of
proof vary from one case to another. Although we do not try to give a systematic
exposition of these methods here, the reader will find a few sample techniques for
checking Koszulness (mostly in chapter 2).
As we have already mentioned, one of the central questions studied in our book
is how to generalize the Poincare-Birkhoff-Witt-theorem (PBW-theorem) to
quadratic algebras. Recall that the classical PBW-theorem for the universal enveloping
algebra U# of a Lie algebra g can be formulated in two different ways. In the first
formulation one starts with a basis of g and then the theorem states that certain
standard monomials in basis elements form a basis of U#. Another formulation
simply asserts that the associated graded algebra of U# with respect to the standard
filtration coincides with the symmetric algebra Sg. Thus, the first way to generalize
the PBW-theorem to other algebras is to modify the notion of standard
monomials. Assume that we have a graded quadratic algebra (i.e., quadratic relations are
homogeneous). Then using lexicographical order on the set of all monomials in
generators one can define a certain set of standard monomials (depending on quadratic
relations). The analogue of the PBW-theorem in this case states that if the
standard monomials form a basis in the grading component of degree 3 then the same
is also true for all grading components (so that we get a PBW-basis in our algebra).
This theorem is a particular case of the so-called diamond lemma in the theory
of Grobner bases developed in works on combinatorial algebra in the late 70s (see
[26, 35, 36]). Note that the universal enveloping algebra Uq can be homogenized
by adding an extra central generator, so that the classical PBW-theorem would fit
into this context.
Before stating the second generalization of the PBW-theorem let us say a few
words about the terminology adopted in the book. We use the term "quadratic
algebra" only in reference to algebras defined by homogeneous quadratic relations
(because with the exception of chapter 5 we consider only such algebras). Assigning
degree 1 to each generator one can view a quadratic algebra as a graded algebra A =
©n>o An such that Aq is the ground field and A is the quotient of the tensor algebra
of A\ by an ideal generated in degree 2. Note that sometimes (e.g., in applications

INTRODUCTION ix
to representation theory) it is necessary to consider more general quadratic algebras
such that Aq is not necessarily equal to the ground field but rather is a semisimple
algebra. We will briefly discuss algebras of this kind in section 9 of chapter 2.
Our second generalization of the PBW-theorem deals with a "nonhomogeneous
quadratic algebra", i.e., an algebra with a finite number of generators and non-
homogeneous quadratic defining relations. If A is such an algebra then one can
consider the natural filtration on A determined by the set of generators. Let us
denote by gvA the associated graded algebra. On the other hand, one can truncate
the relations in A leaving only their homogeneous quadratic parts. Let A^ be
the obtained quadratic algebra. The nonhomogeneous PBW-theorem states that
the natural map A^ —> grA is an isomorphism provided A^ is Koszul and a
certain self-consistence condition is satisfied (this result was proved independently by
A. Braverman and D. Gaitsgory [33]). This self-consistency condition is obtained
by looking at expressions of degree 3 in generators. In the case A = U# it coincides
with the Jacobi identity for the Lie bracket on g.
It is interesting that the notion of Koszulness appears also in the first
generalization of the PBW-theorem: quadratic algebras having a basis of standard
monomials, called PBW-algebras, are always Koszul (this observation goes back
to S. Priddy [104]). However, the converse is not true: Koszul algebras are not
necessarily PBW (see section 3 of chapter 4). In fact, the class of PBW-algebras is
substantially smaller than that of Koszul algebras and is much easier to study. For
example, the set of PBW-algebras with a given number of generators is constructible
in Zariski topology while the set of Koszul algebras is often not constructible (see
section 3 of chapter 4 and section 6 of chapter 6). On the other hand, there are
many parallel results for both classes of algebras. Firstly, both properties can be
formulated in terms of distributivity of certain lattices of vector spaces. Secondly,
various natural operations with quadratic algebras, such as quadratic duality, free
product, tensor product, Segre product and Veronese powers preserve both classes.
The comparison between the classes of Koszul and PBW-algebras is also an
important part of the present work. In our experience PBW-algebras often provide
a good testing ground for guessing the general pattern that might hold for all
Koszul algebras. Usually there is no problem with proving that a pattern holds for
PBW-algebras; however, the case of Koszul algebras is often much harder (if at all
accessible).
One of the most striking properties of Koszul algebras is the following.
Koszul Deformation Principle (V. Drinfeld [43]). If a formal family of graded
quadratic algebras A(t) is flat in the grading components of degree ^ 3 and the
algebra A(0) is Koszul then the family is flat in all degrees.
More precisely, a similar statement holds for local deformations (in Zariski
topology) if we consider only a finite number of grading components (see
Theorem 2.1 of chapter 6). The second version of the PBW-theorem considered above
can be easily deduced from this principle. Another unexpected consequence that
we derive from it is the finiteness of the number of Hilbert series of Koszul algebras
with a fixed number of generators (the analogous statement for quadratic algebras
is wrong). We conjecture that Hilbert series of Koszul algebras enjoy several
interesting properties that can be easily checked for PBW-algebras (although we prove
that these two sets of Hilbert series are different). For example, we conjecture that
the Hilbert series of a Koszul algebra is always rational.

x INTRODUCTION
The study of Hilbert series of Koszul algebras led to the discovery in [100] of
an interesting connection with the theory of discrete stochastic processes. Namely,
to every Koszul algebra A one can associate a one-dependent stationary stochastic
sequence of 0's and l's. It is convenient to encode probabilities of various events
in such a process by a linear functional ф : R{xo,xi} -^Eon the free algebra in
two variables, taking nonnegative values on all monomials and satisfying 0(1) = 1.
Then the condition of one-dependence is equivalent to the equation
Фи-(х0+х1)-д) = ф(/)ф(д),
where /, g £ Ш{хо,х\}. Abusing the terminology we call such a functional ф а
one-dependent process. It is easy to see that ф is uniquely determined by the values
(0(xiO). Now the one-dependent process associated with a Koszul algebra A is
defined by
фА(х^1) = ап/а^
where an = dim An. Nonnegativity of values of ф on all monomials is equivalent
to a certain system of polynomial inequalities for the numbers an. The fact that
these inequalities are indeed satisfied for a Koszul algebra seems to be a remarkable
coincidence. However, the analogy between the two theories does not end here. It
turns out that under this correspondence the subclass of PBW-algebras maps to the
set of so-called two-block-factor processes. The relation between all one-dependent
processes and the subclass of two-block-factors was intensively studied in the 90s
after it was proved in [2] that a one-dependent process does not have to be a two-
block factor (see [1, 118, 122]). This topic seems to be surprisingly similar to the
relation between Koszul and PBW-algebras. Motivated by this analogy we
conjecture that the Hilbert series associated with every one-dependent process admits a
meromorphic continuation to the entire complex plane. Rationality of Hilbert
series of Koszul algebras would follow from this (by a theorem of E. Borel [32]). We
also observe that the polynomial inequalities satisfied by the numbers (ф{х™)) form
a subset in the well-known system of inequalities defining the notion of a totally
positive sequence (also known as Poly a frequency sequence). It is known that the
generating series of a totally positive sequence admits a meromorphic continuation
(see [71]). This can be considered as another hint in favor of our conjecture.
Here is the more detailed outline of the content of the book.
Chapter 1 contains some basic definitions and results concerning cohomology of
graded algebras, quadratic algebras and distributivity of lattices. In particular, in
section 2 we define quadratic duality for quadratic algebras and quadratic modules
(we use the term "Koszul duality" when referring to this duality in the case of
Koszul algebras and Koszul modules).
In chapter 2 we describe various equivalent definitions of Koszulness, including
Backelin's criterion in terms of distributivity of lattices (see [15]). We give similar
equivalent definitions for a related notion of n-Koszulness that has an advantage
of being defined by a finite number of conditions. We also show that many results
about quadratic and Koszul algebras have natural analogues for quadratic and
Koszul modules. In section 5 we consider the problem of preservation of Koszulness
under homomorphisms of various types between graded algebras, generalizing some
results of Backelin and Froberg [20]. In section 7 we give examples of projective
varieties with Koszul homogeneous coordinate algebras. In section 8 we explain how
to associate to a Koszul algebra A a (graded) infinitesimal bialgebra (or e-bialgebra)

INTRODUCTION xi
Уд. This construction can be viewed as a categorification of the one-dependent
process Фа associated with A, because the values of Фа on monomials are given
by dimensions of certain multigrading components of Уд. In section 9 we consider
some generalizations of the notion of Koszulness including an important case of
graded algebras A = фп>0 Ап such that A0 is a semisimple algebra (in the rest of
the book we assume that Aq is the ground field). We also give an interpretation of
Koszul algebras in terms of monoidal functors from a certain universal (nonunital)
monoidal category.
In chapter 3 we consider several natural operations on quadratic algebras and
modules that preserve Koszulness and discuss the behavior of Hilbert series under
these operations. Following [20] we consider free sums, free products, along with
several types of tensor products, the Segre product AoB. the dual operation "black
circle product" A • В and Veronese powers A^n\ The operation A • В is also
closely related to the internal cohomomorphism operation introduced by Manin
(see [77, 79]). We prove that if one of the algebras is Koszul then the Hilbert
series of A • В can be computed in terms of those of A and В and show that this
is impossible if both algebras are not Koszul. An interesting application of these
operations is given in section 5, where we show, following D. Piontkovskii [92], that
Koszulness of a quadratic algebra A cannot be determined from the knowledge of
the Hilbert series of A and A\
Chapter 4 is devoted to PBW-algebras. We start by giving a proof of the
PBW-theorem for quadratic algebras that gives a criterion for the existence of
a PBW-basis (as we have mentioned before, this is really a particular case of the
diamond lemma). Then we prove that PBW-algebras are Koszul and give a criterion
of the PBW-property in terms of distributivity of lattices in the spirit of Backelin's
criterion of Koszulness. We also check that the class of PBW-algebras is stable
under quadratic duality and under all operations considered in chapter 3. Then
we discuss Hilbert series of PBW-algebras. We show that the Hilbert series of a
PBW-algebra is a generating function for the number of paths in a finite oriented
graph and hence is rational. In section 7 we prove a generalization of the PBW-
theorem involving flit rations with values in an ordered semigroup. In section 8 we
consider commutative PBW-algebras. We prove that they are Koszul and compute
their Hilbert series. We also present some examples showing that the sets of Hilbert
series of PBW-algebras and Koszul algebras are different. In section 9 we discuss
a generalization of the classes of Koszul and PBW-algebras from graded algebras
to Z-algebras. In section 11 we consider 3-dimensional elliptic Sklyanin algebras.
We prove that they are Koszul but do not admit a PBW-basis even viewed as
Z-algebras.
In chapter 5 we consider nonhomogeneous quadratic algebras. For these
algebras we prove in section 2 the PBW-theorem involving an analogue of the Jacobi
identity and Koszulness of the corresponding homogeneous quadratic algebra. We
also prove in section 3 a version of this theorem for nonhomogeneous quadratic
modules. In section 4 we consider an analogue of quadratic duality for the
nonhomogeneous case. It turns out that the dual object to a nonhomogeneous quadratic
algebra is a so-called CDG-algebra (curved DG-algebra). In section 5 we give some
examples of nonhomogeneous quadratic algebras and modules. In particular, we
list all solutions of the analogue of the Jacobi identity in the case of the
quadratic relations corresponding to a free commutatative superalgebra, and consider an

xii INTRODUCTION
example related to the PBW-theorem for quantum universal enveloping algebras
(Example 6). The remainder of this chapter is devoted to various cohomological
calculations related to nonhomogeneous quadratic duality.
Chapter 6 is devoted to the Koszul Deformation Principle for quadratic algebras
and some of its consequences, such as finiteness of the number of Hilbert series of
Koszul algebras with a fixed number of generators. Furthermore, in section 3 we
give an explicit bound on this number and in section 7 we prove that the number of
such Hilbert series is finite even if the ground field is allowed to vary. In section 4
we discuss some results on generic algebras among quadratic algebras with a given
number of generators and relations. In section 5 we consider examples of possible
Hilbert series for algebras with a small number of generators and relations. Section 6
contains counterexamples from [56] showing that the set of Koszul algebras is not
constructible and that the set of Hilbert series of quadratic algebras with a given
number of generators is infinite.
In chapter 7 we explain the connection between Koszul algebras and one-
dependent processes. We start by formulating several conjectures on Hilbert series
of Koszul algebras, such as the rationality conjecture. Then we derive a system of
polynomial inequalities satisfied by the numbers an = dim An for a Koszul algebra
A. The polynomials of an appearing in these inequalities express the dimensions
of multigrading components of the б-bialgebra Va- Then we show that these
inequalities allow one to associate a one-dependent process to the sequence (an). We
show that Koszul duality corresponds to the natural duality on one-dependent
processes and also introduce analogues of some other operations on Koszul algebras for
one-dependent processes. In section 5 we show that the one-dependent process
associated with a PBW-algebra is a two-block-factor and that every two-block-factor
can be approximated by those obtained from PBW-algebras. In section 7 we
review the notion of a Hilbert space representation of a one-dependent process due
to V. de Valk [121]. In section 8 we discuss the conjecture that the Hilbert series
of a one-dependent process can be extended meromorphically to the entire complex
plane. We show that this series always admits a meromorphic continuation to the
disk \z\ < 2 (it converges for \z\ < 1) and prove the conjecture for two-block-factor
processes. In section 9 we give a construction due to B. Tsirelson of a one-dependent
process associated with an arbitrary quadratic algebra and a Hermitian form on
the space of generators. In section 10 we consider an analogue for Koszul modules
of the construction of a one-dependent process from a Koszul algebra.
In the Appendix we recall some definitions concerning DG-algebras, DG-mod-
ules and Massey products.
Acknowledgments. First, we would like to thank A. Vaintrob whose question about
possible generalizations of the PBW-theorem to quadratic algebras started this
work in 1991. Also, we are grateful to J. Backelin, A. Braverman, J. Bernstein,
P. Etingof, V. Ginzburg, V. Ostrik, D. Piontkovskii, J.-E. Roos, A. Schwarz, B. Shel-
ton, B. Tsirelson, and S. Yuzvinsky for many interesting discussions and
suggestions. Special thanks are due to J. Backelin for pointing out several mistakes in the
manuscript. Finally, we are grateful to the referee for many useful suggestions.

CHAPTER 1
Preliminaries
In this chapter after setting up notation and reviewing some homological
algebra (including the bar construction) we give basic definitions concerning quadratic
algebras and quadratic duality. We review some general results on cohomology of
graded algebras and modules in sections 3-5. Sections 6 and 7 contain some basic
results on distributive lattices.
0. Conventions and notation
Throughout this work, by an algebra A we mean an associative algebra with unit
1a £ A over a fixed ground field k. The unit acts identically on all our A-modules.
A graded algebra is a graded vector space A = 0ieZ Ai with an algebra structure
such that Ai • Aj С Ai+j and 1^ G Ao- A graded {left) A-module is a graded vector
space M = 0iGZ Mi with an A-module structure such that Ai • Mj С Mi+j. For
a graded A-module M we denote by M(n) the same module with shifted grading:
M(n)i = Мг+п.
An augmented algebra A is an algebra equipped with a direct sum
decomposition A = к ф A+ such that Ik = к • I a is the line spanned by 1a and A+ is
a two-sided ideal in A. The ground field к is equipped with the left and right
A-module structures via the augmentation.
Vector spaces are usually assumed to be finite-dimensional. Graded vector
spaces V = 0nGZ Vn are assumed to be locally finite-dimensional, i.e., to have
finite-dimensional grading components. For such a graded vector space V we denote
by У* the graded dual vector space with components (V*)n = Vln.
We say that a graded vector space V = 0n Vn has polynomial growth if there
exists a constant С > 0 and a positive integer d such that dim Vn ^ С • nd for
n ^> 0. On the other hand, if there exists a constant с > 1 such that dim Vn ^ cn
for n ^> 0, then we say that V has exponential growth.
Starting from section 2 of chapter 1 we assume that our graded algebras A =
0^ Ai are locally finite-dimensional with Ai = 0 for i < 0 and Ao = к • 1a- We
always equip an algebra A of this kind with the natural augmentation such that
A+ = 0°^! Ai. We assume our graded A-modules M = 0^ Mi to be locally finite-
dimensional and bounded below (i.e., with Mi = 0 for г <ti 0). By a nonnegatively
graded A-module we mean a graded A-module M such that Mi = 0 for i < 0.
Many of our definitions (and results) for algebras have analogues for modules
over algebras. Sometimes, we will use the letter "(M)" to label such analogues.
For a vector space V, we denote by T(V) = 0г^о Тг(У), where T(V) = V®\
the free associative algebra (tensor algebra) generated by V. The free
commutative (symmetric) algebra generated by V is denoted S(V) = 0°^о^г(Ю> and the
free skew-commutative (exterior) algebra is denoted /\{V) = @°^0/\г{V). The
l

2
1. PRELIMINARIES
free associative (noncommutative polynomial) algebra generated by a set of
variables xi,...,xm over a field к is denoted k{x\,..., xm} and the free commutative
(polynomial) algebra generated by x\,..., хш is denoted k[x\,..., xm].
For vector spaces V and W over к (or algebras, or modules over them) the
tensor product over к is denoted simply by V 0 W.
For a set of vectors vi in a vector space V, we denote by (vi : i G /) С V
their linear span. For a vector space W, we denote by Q(W) = JJJ^ GU(W)
the Grassmannian variety of all vector subspaces U С W, where zz = dim [/. By
P^ = к U {oo} we denote the projective line over k.
With the exception of section 7 of chapter 6 we consider algebraic varieties over
a fixed field k. We say that x is a point of a variety X and we write x G X if x is a
k-valued point of X, where к is an algebraic closure of k. When using topological
notions for algebraic varieties we always mean these notions with respect to Zariski
topology.
For a real number x G M, the symbol \x~\ means the minimal integer which
is not smaller than x, while [x\ is the maximal integer not greater than x. With
the exception of sections 5 and 7 of chapter 7 we denote by [m, n] the segment of
integer numbers between m and n.
We use both chain and cochain complexes. In the former case we use lower
indexing (as in C.) and the differential is lowering the degree by one: Ci —> Ci-\.
In the latter case we use upper indexing (as in K*) and the differential is raising
the degree by one: Кг —> Кг+1. For a chain complex of graded vector spaces V.
(where the differential preserves the grading) we denote by HijV. the component
of grading j in the г'-th homology.
1. Bar constructions
In this section we consider bar and cobar constructions that provide explicit
realizations of the functors Tor and Ext. Although we are mostly interested in the
graded case we also consider these functors for nongraded algebras and modules
(to be used in chapter 5). The reader can skip this chapter and refer to it when
necessary.
For an algebra A and a pair of A-modules M, N we denote by HOMa(M, N)
the space of A-module homomorphisms and by EXT^(M, N) the corresponding
extension spaces (defined as derived functors of HOM^(M, TV)).
Assume that A is augmented: A = к ф А+. Then for every left A-module
M we have the following resolution by free A-modules, called the {normalized)
bar-resolution Bar.{A,M):
•. • ► A 0 A+ 0 A+ 0 M > A 0 A+ 0 M > A 0 M ► 0,
where Ваг{{А, M) = A 0 А®г 0 M and the differential is given by
<9(ao0- • -<g>ai<g)m) = ^(—l)sao0- • -0as_ias0- • -0a;0m+(- 1)г+1ао0- • -0агШ.
s=l
We view Ваг{(А, М) as an A-module by letting A act on the left. (There is a similar
construction for a nonaugmented algebra A, with A+ replaced by A everywhere,
called the nonnormalized bar-resolution.)

1. BAR CONSTRUCTIONS
3
Since Bar.(A, M) is a free resolution of M, we have
ЕХТгА(М, N) = H\EOMA{Ba~r.{A,M),N)).
In other words, EXT^(M, N) can be computed as the г-th cohomology of the cobar-
complex
COB9 {A, M, N) = ttOMA{Bo7r.{A, M),N).
Note that COB1 {A, M, N) ~ HOMk(Af 0 M, AT).
Now assume that Л is a graded algebra A and M, AT is a pair of graded A-
modules M. We will denote by Ext*A(M,A0 = 0i€ZExt^'(M, N) the derived
functor of the graded homomorphisms functor Ext^(M, N) = Hom^(M, N) =
0eZHom^(M, AT), where Hom^(M, AT) is the space of all homomorphisms
mapping Mk to Nk-j. The first grading i is called the homological grading and the
second j is called the internal one. As above we can compute these spaces using
the graded cobar-complex:
Ext*A(M, N) = Нг{СоЬ9{А,М, АО),
where
Cob9(A,M,N) = ttomA(Bo7r.{A,M),N) ~ Homk(Af9 0 M,N).
This identification is compatible with internal gradings.
For an algebra A, a right A-module R, and a left A-module L, we denote by
Torf (Д, L) the derived functors of the tensor product over A, so that Tor£ (ii,L) =
R®AL. If Л is a graded algebra and the modules R and L are graded then the spaces
Tor^ (Д, L) acquire the corresponding internal grading induced by the grading of
R®AL:
Tor?(R,L) = ®Tor?3(R,L).
For example, Tor0 • = (R <g>A L)j is spanned by elements x 0 y, where x £ Rk and
у G Z/j-ь Using the bar resolution above we can compute these spaces as homology
of the bar-complex:
Tovf {R,L) = Hl{Bar.{R, A, L)),
where
Bar.{R, A,L) = R <g>A Bar .{A, L).
Note the following duality between bar and cobar complexes:
COB9 {A, M, Rv) = Bar.{R, A, M)v,
where for a vector space V we set Vw = НОМцДУ, к) (in the left-hand side we use
the natural structure of a left A-module on Rw). This leads to the corresponding
duality between homology:
EXTA{M,RV) = Torf{R,M)v.
In the graded case we have a similar duality using graded duals:
Cob9 {A, M, iT) = Bar.(R, A, M)*
Ext2j'(M, Д*) = Тог£(Д, М)*,
where Я* is graded by (R*)k = (Д.*)*.

4
1. PRELIMINARIES
Let Л be a graded algebra, M be a graded left A-module. The spaces
Ext ^ (M, k) and Tor^- (к, М) are called respectively cohomology and homology spaces
of the module M. A particular case of the above duality is
Ext^(M,k) = Tor£(k,M)*.
We denote the relevant bar and cobar complexes by
Bar .(A, M) = Bar.{k, A M),
Cob9 (A, Ad) =Cob*(A,M,k) = Bar.(A,M)*.
In the case M = к we set
Bar.{A) = Bar.{A,k),
Cob*(A)=Cob*(A,k) = Bar.(Ay,
where Bari(A) = Af\
The Yoneda multiplication (composition) on the Ext-spaces can be described
as follows. Let M, TV, P be left A-modules and
c': Af' 0 N —>P, c": Af" 0 M —> N
be cocycles representing some classes f e EXT^(7V,P) and f" e EXT^'(M,iV)
respectively. Starting from c"', define the corresponding morphism of the resolutions
c": Bar.+in{A,M) > Bar.(A,N)
as
gj': A 0 Af 0 A®*" 0 M > A®Af®N
clq 0 x 0 у 0 m i—► ao 0 x 0 с" (у 0 m).
It is easy to verify that c" is a morphism of complexes of A-modules and the
composition Bar in (A, M) —> Bar${A, N) —> P coincides with the original cocycle
c". Therefore, the composition class £' о £" e ЕХТ^+г (M, TV) is represented by
the cocycle c' oc": Barv+i" (A, M) —► Bar^ (A, N) —► P, which is given explicitly
by
с' о с" : Af 0 Af" 0 M > P
x 0 у 0 m i—► c'(:r 0 c"(y 0m)).
The same computation works for graded Ext-spaces.
The dual comultiplication structure on the Tor-spaces can be explicitly
described as follows. Let P be a finite-dimensional left A-module. Then we have a
natural identity element idp G P 0k P*. Therefore, for every pair (i?, L), where
R is a right A-module and L is a left A-module, we have a natural map of vector
spaces, functorial in R and L
R®AL > (R 0A P) 0k (P* 0A L)
and hence the induced map
А: Тог^(ВД > Тогг^(Р,Р) 0кТо4(P*,L), г' + г" = г.
This map is dual to the Yoneda product
EXT^(P,Pv)0EXT2X(L,P) > EXT2A(L,#v).

1. BAR CONSTRUCTIONS
5
On the level of bar-complexes it is given by the formula
A: R®Af' ®Af" ®L > (R 0 Af' 0 P) 0 (P* 0 Af" 0 L)
r 0 x 0 у 0 I i—> r 0 x 0 idp 0y 0 /.
The Yoneda multiplication defines a graded algebra structure on EXT^(k, k)
= ©г^оЕХТА(к>к) and a structure of a graded left EXT^(k,k)-module on
EXT^(M,k) = 0г^оЕХТгл(М,к). Analogously, the above construction (for P =
R — к and L equal to к or M) makes Tor^(k.k) = ф°^0 Torf (к, к) into a graded
coalgebra and Tor^(k, M) — ®°^0 Torf (к, М) a graded left comodule over it. The
same structures appear for graded Ext and Tor. Moreover, in this case there is an
additional internal grading, so that we get a bigraded algebra and bigraded module
(resp., bigraded coalgebra and bigraded comodule).
Now assume that A and M are locally finite-dimensional with Ai — 0 for г < О
and Mi — 0 for г <С 0. Then we have natural isomorphisms
Cob\A) = {Afy = Af>\ Cob\A, M) = (Af 0 M)* = A*®1 0 AT,
where the first tensor component in Af1 is coupling with the first tensor component
in A*^\ etc. The cobar-differential on Cob9(A,M) takes the form
d(fi 0 • • • 0 fi 0 g) =
г
X](-1)S~1/i ® ''' ® di,A(/e) ® • • • ® /» ® 0 + (-l)Vi ® • • • 0 fi 0 dliM(g),
s=l
where /5 G A+, # G M*, and the map dijA: A+ —► A\®A\ (resp., di,M: M* —>
A+ 0 M*) is dual to the multiplication A+ 0 A+ —► A+ (resp., the action A+ 0
M —> M). A similar formula holds for the differential on Cob9{A). The algebra
structure on Cob9{A) and the left Cob9 (A)-module structure on Cob9 (A, M) are
given by the formulas
(Л ® • • • ® Л)(Л ® • • • ® fl> ® д) = Л ® • • • ® h ® Л ® • • • ® Л ® 9
and are easily verified to be compatible with the differential d by the usual Leibniz
rules, so that Cob9{A) is a DG-algebra and Cob9(A,M) is a DG-module over it
(see Appendix). Disregarding the differential, we can say that Cob (A) — T(A+)
and Cob(A,M) = Cob (A) 0 M* is the free associative graded algebra generated
by А*+ and the free graded left Cob (A)-module generated by M*. The induced
multiplicative structures on the cohomology spaces Ext^(k, k) and Ext^(M, k) are
immediately found to coincide with the Yoneda multiplication computed above.
Analogously, the coalgebra and comodule structures on the spaces Tor^(k, k)
and Tor^(k, M) are induced by the DG-coalgebra and DG-comodule structures on
the complexes Bar .{A) and Bar.(A,M) defined above.
In particular, we see that for the opposite graded algebra Aop the Ext-spaces
are the same as for A: Ext^op(k, k) = Ext^(k, k), and the Yoneda multiplication
just changes to the opposite one. (Another way to deduce the first property is to
note that Ext^(k, k) = Tor^-(k, к)" and the latter spaces for A and Aop coincide
by the definition.)

6
1. PRELIMINARIES
2. Quadratic algebras and modules
Recall that we assume our graded algebras A (resp., graded A-modules M)
to be locally finite-dimensional with Ai — 0 for i < 0 and Aq = к (resp., locally
finite-dimensional and bounded below).
Definition 1. A graded algebra A is called one-generated if the natural map p: T(Ai)
—> A from the tensor algebra generated by A\ is surjective. A one-generated
algebra is called quadratic if the kernel J a — kerp is generated as a two-sided ideal
in T(Ai) by its subspace Ia — Ja^ T2(Ai) С А\ 0 A\. Therefore, a quadratic
algebra A is determined by a vector space of generators V — A\ and an arbitrary
subspace of quadratic relations / С V 0 V. We denote this by A = {V, /}.
For every graded algebra A, there is a uniquely defined quadratic algebra qA
together with an algebra homomorphism qA —► A which is an isomorphism in
degree 1 and a monomorphism in degree 2. Namely, in the above notation one has
qA — {Ai, I a}- We call this algebra the quadratic part of A.
The quadratic dual algebra to a quadratic algebra A = {V, /} is defined by
A!- = {V*,/-1}, where V* is the dual vector space to V and I-1 C V* 0 V* is the
orthogonal complement to I with respect to the natural pairing (i>i 0г>2, v^v^) =
(vi,vl)(v2,vZ) between V 0 V and V*0 V*.
Definition 2(M). Let A — {V, /} be a quadratic algebra. A left A-module M is called
quadratic if Mi — 0 for i < 0, the natural map A 0 Mo —> M is surjective, and
its kernel Jm is generated as an A-submodule in A 0 Mq by the subspace Км —
Jm П A\ 0 Mq. In other words, a quadratic A-module is determined by a vector
space of generators И = Mq and an arbitrary subspace of relations К с V 0 H.
We denote this by M = (Я, if) = (Я, K)A.
Note that we do not consider quadratic modules over nonquadratic algebras.
For every graded algebra A and a nonnegatively graded A-module M there is
a natural quadratic module q^M over the quadratic algebra qA together with a
morphism q^M —> M of modules over qA which is an isomorphism in degree 0
and a monomorphism in degree 1. In other words, one has qAM = {Mq,Km)<±a-
We call this module the quadratic part of M.
For a quadratic module M = (Я, К) a over a quadratic algebra A — {V, /} we
define the quadratic dual module M' — M'A over the quadratic algebra A] dual to
A by M\ = {Н*,К^)Ач where К1- С V* 0 Я* is the orthogonal complement to
К CV®H.
Examples. 1. If A = §(V) is the symmetric algebra then A- = /\(V*) is the
exterior algebra with the dual space of generators.
2. If A = T(V) is the tensor algebra then A1 = Ik 0 У* is an algebra with A}{ = 0
for г ^ 2.
3. For any quadratic algebra A the duality for modules sends free A-modules to
trivial A!-modules and vice versa: k^ = A1 and AlA — k.
4. If A is a commutative quadratic algebra then quadratic relations in A are linear
combinations of odd supercommutators of generators. Hence, A1 is isomorphic to
the universal enveloping algebra U(g) of a graded Lie superalgebra g (where g is
quadratic, i.e., generated by gi with defining relation of degree 2).
5. Similarly to the previous example, the dual of a skew-commutative quadratic
algebra is the universal enveloping algebra of a usual Lie algebra (also quadratic).
For example, let У be a finite-dimensional vector space with a skew-symmetric form

4. MINIMAL RESOLUTIONS
7
uj ф 0. Then we can consider the quadratic algebra A — /\(V*)/(cu). The dual
quadratic algebra A- is isomorphic to the universal enveloping algebra £/(()), where
f) = \)ш is the graded Lie algebra generated by f)i = V and by one more element
с of degree 2 with defining relations [v,v'\ — oj(v, v')c. It is easy to see that if
the rank of и is ^ 4 then these relations imply that с is central, so in this case
f) is the Heisenberg central extension of У by Ik associated with u. For example,
if uj is nondegenerate and dim У = 2n ^ 4, then f) is the standard Heisenberg
Lie algebra and A can be presented as follows: it is generated by 2n elements
Xi,... , xn, yi,... , yn with the relations
[xi,Xj] = [yuyj] = 0 for 1 ^ г, j ^ n,
[xi,2/j] = 0 for г^ j,
[zi,2/i] = [22,2/2] = ... = [zn,2/n].
6. If M is a graded module over a graded algebra A then we can introduce an
algebra structure on Am = Лф M(—1) by the rule (a, ra)(a'\m!) = (aa',am'). It
is easy to see that if A is a quadratic algebra and M is a quadratic A-module then
Am is again a quadratic algebra. This observation can often be used to deduce
module analogues of results on quadratic algebras.
3. Diagonal cohomology
The following result which is essentially due to S. Priddy and C. Lofwall [104,
75], gives a cohomological interpretation of quadratic duality.
Proposition 3.1. For any graded algebra A and a nonnegatively graded A-
module M all nonzero cohomology spaces Ext^ (M, k) are concentrated in the region
i < j. The diagonal subalgebra 0Ext^(k,k) of the algebra 0Ext^(k, к) is always
a quadratic algebra and the diagonal submodule 0 Ext^(M, k) is a quadratic module
over it. More precisely,
0ExtJi(k,k) ~ (q^)!, 0Ext*i(M,k) ~ (qAM)\qAy
Proof: As we have seen in section 2, these Ext-spaces can be computed as the
cohomology of the cobar-complexes
Cob^{A)= Y, Ak®---®Ak
Cobij (A,M)= Yl Al ® • • • ® Al ® Mi ■
fcH \-ki+l=j, fcs^l, l>0
All the assertions follow easily from this. □
Note that if A is one-generated then the diagonal part of Ext^(k, k) coincides
with the subalgebra generated by Ext^(k,k). To generate non-diagonal pieces one
needs higher Massey products (see [82] and Appendix).
4. Minimal resolutions
In this section we prove that every graded module (bounded below) over a
(nonnegatively) graded algebra admits a minimal free resolution, unique up to an
isomorphism.
We start with the following version of Nakayama's lemma for noncommutative
graded algebras.

8
1. PRELIMINARIES
Lemma 4.1. Let A be a graded algebra and M a graded A-module. Then a
graded vector subspace X С M generates M as an A-module iff the composition
X —> M —► Ik 0 a М is surjective.
Proof: X generates M iff the natural morphism f : A® X —► M is surjective. It
is clear that this implies surjectivity of the map / : X —► к 0 a M. Conversely,
assume that / is surjective and let us show that / is surjective. We can argue by
induction in n that the degree n component fn is surjective. This is trivially true
for n <C 0. Assume that fi is surjective for all i < n. Given an element m G Mn
there exists x G Xn such that m — x G A+M. Hence, by the assumption m — x
belongs to the image of /, so m is also in the image of /. □
It follows immediately from the above lemma that inside any generating sub-
space X С М one can find a smaller generating subspace X' С X such that the
map X' —► Ik 0 a M is an isomorphism.
Definition. A bounded above complex of free graded A-modules
• - • > P2 > Pi > P0 > 0
is called minimal if all the induced maps Ik 0 a Д+i —► к 0а Pi vanish.
For every graded A-module M, one can construct a minimal free graded A-
module resolution of M in the following way. Using Lemma 4.1 choose a generating
subspace Xq С M such that Xo — Ik 0a M. Consider the corresponding morphism
A 0 Xq —> M and let Mi С А 0 Xq be its kernel. Choose a generating subspace
Xi С Mi such that X\ ~ Ik 0a Mi . It is easy to see that continuing in this manner
we will find a minimal resolution of M consisting of the modules Pi = A 0 X{.
Since any two free resolutions of the same module M are connected by a chain map
inducing the identity on M, it follows that a minimal resolution is unique up to a
nonunique isomorphism.
In fact, the following more general statement is true.
Proposition 4.2. Every bounded above complex F# of free graded A-modules
admits a decomposition Fm = P#0T# into the direct sum of two subcomplexes of free
graded A-modules, where the complex P. is minimal and the complex T. is acyclic.
Proof: Assume that our complex
• • • > **г+1 > ?i > Pi-\ > • • •
satisfies di — Ik 0 di — 0 for all i < n for some n G Z. This is certainly true for
sufficiently small n since F. is bounded above. We are going to show that in this
case there exists a decomposition F. ~ F'm ФГ, into a direct sum of subcomplexes of
free graded modules, where T# is an acyclic complex concentrated in degrees n + 1
and n, and the differentials df{ in F'm satisfy d'i — 0 for i < n + 1. Iteration of this
process would give the required decomposition. To construct F'm and Tm as above
let us consider the A-module M = Fn/dn+i(Fn+i). Let Fi = A 0 Xi for i e Z.
Since Xn generates M there exists a graded subspace Y С Хп such that the map
Y —► Ik 0a M is an isomorphism. Set F'n — Y 0 A. We claim that the composed
map / : Fn+i —> Fn —> Xn/Y 0 A is surjective. Indeed, the map F'n —> M
is surjective by Lemma 4.1, hence Fn — F'n + dn+i(Fn+i) and therefore the map
/ : Fn+i —> Xn/Y 0 A is surjective. Let F^+1 С Pn+i be the kernel of /. Note
that i^+1 = 3П+1(У) 0 A. Let us set F[ = Fi for i ф n,n + 1. Since the map

5. LOW-DIMENSIONAL COHOMOLOGY
9
Fn+2 —► F<n+i factors through i^+1 we obtain a subcomplex F'm С Fm. Moreover,
we have a split exact triple of complexes
>K+2 >^+i—^ >F^ >F^ >...
> Fn+2 > Fn+i — > Fn > Fn_i > • • •
> 0 > Xn/Y 0 A —^-> Xn/Y 0 A > 0 > • • •
We claim that d'n+i = 0. Indeed, by the choice of Y we have a direct sum
decomposition Xn = dn+i(Xn+i) 0 Y. Therefore, dn^_1(Y) coincides with the kernel of
di+i and the map
к®a K+i = dull(Y)^Y = k®A F'n
is zero. It is also clear that d'n — 0 and d'i — 0 for i < n. It remains to choose a
free submodule Ti+i С Fi+i that projects isomorphically to X/Y 0 A and to set
Г* = di+i(Ti+i). □
Remark. The decomposition considered in the above proposition is unique up
to an automorphism of F,. One can also check that a morphism between minimal
complexes inducing an isomorphism on homology is an isomorphism itself.
The following class of minimal resolutions will play an important role for the
notion of Koszulness.
Definition. A resolution
• • • > P2 > Pi > Pq > M > 0
of a graded A-module M by free graded A-modules is called a linear free resolution
if each Pi is generated in degree i.
A linear free resolution can be written in the form
••• > V2®A(-2) > Vi®A(-l) ► V0®A > M > 0,
where Vi are vector spaces (of degree zero).
It is clear that linear free resolutions are minimal. Note also that M admits a
linear free resolution iff Tor^(Ik, M) = 0 for i ф j (equivalently, Ext2j(M,Ik) = 0
for г ф j). Finally, we observe that if a module M admits a linear free resolution
then it is unique up to unique isomorphism. Indeed, since Pi is generated in degree
г, it follows that an endomorphism of P. is determined by its action on к <g>A P»,
i.e., on the spaces To^ (к, М). But this action is trivial for any endomorphism of
Рщ inducing the identity on M.
In fact, as we will see in section 3 for every nonnegatively graded A-module M
one can construct a certain complex of free A-modules that will coincide with the
linear free resolution when M admits one.
5. Low-dimensional cohomology
In this section we prove the well-known result that the spaces Torf (к, к) and
Tor^(k, k) describe respectively minimal homogeneous generators and relations of

10
1. PRELIMINARIES
a graded algebra A, while the spaces Tor^(k, M) and Torf (к, М) correspond to
minimal generators and relations of a graded A-module M.
Proposition 5.1. Let A be a graded algebra. Assume that a graded A-module
M is represented as the cokernel of a morphism of free graded A-modules,
A®Y > A®X > M > 0,
where X = ®?ez^j and Y — (&je%Yj are graded vector spaces. Then the spaces
Tor0 -(к, М) — (M/A+M)j are naturally identified with certain quotients of the
spaces Xj, while the spaces Torlj-(k, M) are subquotients of the spaces Yj.
Furthermore, for any module M there exists a presentation of the above type such that we
have natural isomorphisms Xj ~ Tor0j(k, M) and Yj ~ Tor^(k, M).
Proof: Let us extend the resolution one step further to the left, that is, choose
a morphism A ® Z —> A 0 Y such that the resulting complex is exact at the
term A 0 Y. Tensoring with the right A-module k, we get a complex of graded
vector spaces Z —► Y -^-> X. By the definition, we have Tor^(k, M) ~ coker(/?
and Tor: (к, М) ~ ker^/im^. As was explained in section 4 one can choose a
representing morphism A 0 Y —> A 0 X for M such that (p = ip — 0. This proves
the last statement. □
Proposition 5.2. Assume that a graded algebra A is represented as the
quotient of the free algebra T(X) generated by a graded vector space X by the ideal
generated by a graded subspace Y С T(X),
T(X) 0 У 0 T(X) > T(X) > A > 0.
Then the spaces Tor: -(к,к) = (A+/A^)j are naturally identified with certain
quotients of the spaces Xj, while the spaces Tor2> -(к, к) are subquotients of the spaces
Yj. For every graded algebra A there exists a presentation of this type such that
Xj ~ Tor^-(k,k) and Yj ~ Tor^-(k,k).
Proof: More generally, assume that an augmented algebra A is the quotient algebra
of an augmented algebra В by a two-sided ideal J. Then we claim that there is a
natural five-term exact sequence
Torf(k,k) > Tor£(k,k) > J/(B+J + JB+)
> Torf (k,k) > Torf(k,k) > 0.
Indeed, it is obtained from the following exact triple of complexes, two of which are
just initial fragments of the bar-complexes for A and B:
0 ► J ► B+ > A+ > 0
I 1 1
0 ► B+0J0J(8)B+ ► B+ <g> B+ > A+ <g> A+ > 0
1 1 I
0 ► kernel ► B+ <g> £+ <g> B+ > A+ <g> A+ <g> A+ > 0
In the case В = T(X) and J = (У), we have Torf (k, k) = X and Torf (к, к) =
0. On the other hand, the space Y maps surjectively onto J'/{B+J + J#+), so the
first assertion follows.

6. LATTICES AND DISTRIBUTIVITY
11
It remains to construct a presentation of A with the required minimality
property. First, arguing as in Lemma 4.1, it is easy to check that a homogeneous vector
subspace X с А+ generates the algebra A iff it maps surjectively on the quotient
space А+/А\. Therefore, one can choose a generating subspace X such that it
maps isomorphically to this quotient. Then it follows from the above five-term
exact sequence that Tor2 (k, k) ~ J/(B+J + JB+). It remains to use Lemma 4.1
for the B-B-himodvle J to find an appropriate minimal relations space Y. □
Using duality between Tor^-(k, M) and Ext^(M, k) we deduce the following
Corollary 5.3. Let A be a graded algebra, M be a nonnegatively graded A-
module.
(i) A is one-generated iff Ext^(k, k) = 0 for j > I;
(ii) A is quadratic iff Ext^7 (k, k) = 0 for j > i and i — 1,2.
(ИМ) Assume that A is quadratic. Then M is quadratic iff Ext^7 (M, к) = О for
j > i and i — 0,1. □
Alternatively, the relation between minimal generators and relations and Tor-
spaces can be described as follows. For a graded algebra A and integer к > 1
we denote by A^ the graded algebra with the same generators and relations of
degree < к as in A, and with no generators or defining relations of degree > k.
Thus, we have Tor^- (k, k) = Tor^- (k, k) = 0 for j ^ к and there is a morphism
of graded algebras A^ —> A which is an isomorphism in degree < k. Analogously,
for a graded A-module M and integer к denote by M^ the graded A-module with
Tor^- (M w, k) = Tor^- (M <fc>, к) = О equipped with a morphism of graded modules
]\/[{k) —> yi which is an isomorphism in degree < k. Then there are natural exact
sequences
0 > Tor£fc(k,k) > A{kk) > Ak > Tor^fc(k,k) > 0
0 > Tor^fc(M,k) ► M{kk) ► Mk > Tor£fc(M,k) > 0.
6. Lattices and distributivity
Here we present some results from the general lattice theory. Most of them are
due to B. Jonsson [69] and Musti-Buttafuoco [86].
A lattice is a discrete set Ct endowed with two idempotent (i.e., a* a = a),
commutative, and associative binary operations Л, V: ft x Ct —> Ct satisfying the
following absorption identities:
aA{aVb) = a, {aAb)Vb = b.
We write a ^ b, if the equivalent conditions aAb = aova\/b = b hold. The dual
lattice fl° is obtained by switching the operations Л and V.
A lattice is called distributive if it satisfies the following distributivity identity:
a A (bVc) = {aAb) V (a Ac).
We will see that dual lattices ft and fi° are distributive simultaneously. However,
the distributivity condition above for fixed elements a, b, с is not self-dual in general,
except in the case of a modular lattice.
A lattice is called modular, if the distributivity identity holds for any triple of
its elements a, 6, с such that a ^ c. Equivalently, one should have
a^ с => aA(bVc) = (aAb)Vc.

12
1. PRELIMINARIES
This condition is clearly self-dual.
Lemma 6.1. [69] For any triple of elements x, y, z of a modular lattice ft, the
following two dual distributivity conditions are equivalent and do not change after
permuting the elements x, y, z:
x A (y V z) = (x A y) V (x A z), x V (y A z) = (x V y) A (x V z).
Proof [69]: If the first equation holds, then by modularity and absorption we have
(z V x) A (z V y) = z V (x A (z V y)) = z V (x A z) V (x A y) = z V (x A y).
The other implications can be obtained by duality and permutation of x, y, z. □
From now on we assume our lattices to be modular. A triple of elements x, y,
z of a lattice ft is called distributive if it satisfies the equivalent conditions of the
above lemma.
A sublattice of a lattice ft is a subset closed under both operations V and Л.
The sublattice generated by a subset X С ft consists of all elements of ft that can
be obtained from the elements of X using these operations. Note that a finitely
generated distributive lattice is finite. The analogous statement for modular lattices
is not true [64].
Our main goal is to find a system of equations on the elements of X that would
guarantee distributivity of the sublattice generated by X. Temporarily, let us make
the following definition.
Definition. A subset X с ft is called distributive if for any pair of finite subsets У,
Z С X and any element x G X the triple
Vyeyy> *' /\*ezZ
is distributive. It follows easily from modularity that it is enough to consider non-
intersecting x, У, and Z С X.
Note that if X is distributive then for any pair of finite subsets У, Z С X one
has
<V„,»)M/U')-V,«>aA,„«)-
We are going to prove that this condition is actually sufficient for distributivity of
the sublattice generated by X (see Theorem 6.3 below). The proof will be based
on the following technical result.
Proposition 6.2. Let X g ft be an N -element subset in a modular lattice ft
such that any of its proper subsets is distributive. Then for every 2 < к ^ TV — 1
the following two conditions are equivalent:
(a) For any decomposition X = У' U У" U Z of the set X into a disjoint union
of nonempty subsets У, У", Z such that #У + ФУ" = к, the triple
v y'^y v у"£у" I \Z^Z
is distributive.
(b) For any decomposition X = У U Zf U Z" of the set X into a disjoint union
of nonempty subsets У, Z', Z" such that #У = к — 1, the triple
V у, Л *;, Л *"
is distributive.

6. LATTICES AND DISTRIBUTIVITY
13
Moreover, if one of these distributivity conditions holds for some choice of a
decomposition then it holds for all decompositions, i.e., (a) and (b) are satisfied (for fixed
k). Finally, X is distributive iff these conditions are satisfied for all 2 ^ к ^ TV — 1.
Proof: First, we note that for У = У, Z — Z" and У" = Z' = {x} corresponding
conditions (a) and (b) coincide. Using the assumption of distributivity for proper
subsets of X let us show that condition (a) does not change after rearranging
elements between the subsets У and У"', having the subset Z fixed. Indeed, the
left-hand side of the distributivity equation
(V i/'vW у")лЛ z = V у л Л *
V V y,ey, у V упеу„ у ) I \zez V yeyuy» y I \zez
does not change under such a rearranging. For the right-hand side we have
(V у'лЛ z) v (W у"лЛ z)
\Vy<ey<y /\zez > ^Чу"еУ"у '\z€Z }
= V (У' Л Л z) V W (у" Л Л z) = W (у Л A z)
V у,€У, \у I \zeZ I V уп€Уп У* I \zez ) V уеуиу" уу I \zez }
by the distributivity of У U Z and У" UZ. Analogously, one can show that
condition (b) depends only on the subset У. Using these equivalences one can rearrange
any decomposition (a) or (b) of the set X into any other. For the proof of the last
assertion note that if X is distributive then condition (a) holds for any
decomposition with У" consisting of one element. □
Theorem 6.3. [86] A subset x±,... ,xpj e П of a modular lattice ft generates a
distributive sublattice if and only if for any sequence of indices 1 ^ i\ < • • • < ц ^. N
and any number 2 ^ к ^ I — 1 the triple
X{1 V * * * V Xik_1, Xik , Xik+1 /\ • • • /\ X{t
is distributive. In particular, if any proper subset of x\,... ,xn generates a
distributive sublattice, then the whole set x\,... ,хдг has the same property iff for any
2^k^N -1 the triple
xi V • • • V Xfc-i, Xk, xk+i Л • • • Л xN
is distributive.
Proof: The "only if" part is trivial, so we only have to prove the "if" part. Arguing
by induction in N we can assume that any proper subset of xi,..., xn generates a
distributive sublattice. Applying Proposition 6.2 we immediately see that the set
Xi,... ,хдг is distributive (in the sense of the definition given above). It is enough
to prove that any subset Д,..., fs of the sublattice generated by xi,..., хдг is
distributive (in fact, it is enough to prove this for all three-element subsets).
Let us define the complexity of an element / G £1 of the sublattice generated
by Xi,..., хдг, as the minimal number of the operation signs Л and V used in an
expression of / in terms of Xi. The complexity of a subset Д,..., fs is by definition
the sum of complexities of all ft. We use induction in the complexity of a subset
/i,..., fs and for subsets of the same complexity the induction in s.
So suppose we are given a subset Д,..., fs and Д = g V h is the first step of a
minimal expression for f\. By the induction assumption, the set g, h, /2,..., fs is
distributive and any proper subset of Д,..., fs is also distributive. It remains to
observe that condition (a) of Proposition 6.2 holds for any decomposition {Д,..., fs}
= У U У" U Z with /i G У because it reduces to a similar condition for the
distributive set g, /1, /2,..., fs. □

14
1. PRELIMINARIES
The above theorem shows that a subset of a modular lattice is distributive iff
it generates a distributive sublattice. Here are some other corollaries.
Corollary 6.4. [69] Let Q, be a modular lattice and X С SI be a subset. Then
the following two conditions are equivalent:
(a) the set X is distributive;
(b) any subset X' с X containing no pairs of elements x±, x2 such that
X\ ^ X2, is distributive.
Proof: Assume that (b) holds. Clearly, we can also assume that X is finite and (by
induction) that all its proper subsets are distributive. If X contains no elements
X\ ^ £2, there is nothing to prove. Otherwise, let X\ ^ £2 be such a pair. Then
distributivity of X follows from the fact that condition (a) of Proposition 6.2 holds
for all decompositions with xi,X2 £ У (where к > 2) and for all decompositions
with У = {xi} and У = {x2} (k = 2). □
Corollary 6.5. Let u,x\,... ,хдг be elements of a modular lattice £1. Then
the following conditions are equivalent:
(a) the set u, Xi,..., xn is distributive;
(b) both sets £1,..., xn and uAxi,..., uAxn are distributive and the equation
u A (\AG/X0 = VzG/^AXi)
holds for any subset I C [1, N];
(b*) both sets xi,..., xn and u\/xi,..., uVxn are distributive and the equation
holds for any subset Id [1, N};
(c) both sets и Л Х\,..., и Л xn and и V x\,..., и V xn are distributive and all
triples distributive for 1 ^ i < j ^ N.
Proof: (b) ^=> (a): By induction, we can assume that any proper subset of the
set 7i, #i,... ,хдг is distributive. Now the condition of Theorem 6.3 for xn+i = и
and к = N essentially coincides with the equation in (b), while the conditions for
к < N can be expressed in terms of и Л #i,..., и Л хдг.
(с) =^ (a): By induction, we can assume that the union of any proper subset
of £1,..., xn with и is distributive. Then the last condition of (c) is only needed
in the trivial case N = 2. As we have noticed above the conditions of Theorem 6.3
applied to x\,..., xn, u, for к < N can be expressed in terms of гх Л xi,..., гх Л хдг •
Similarly, the conditions of this theorem applied to щ X\,..., хдг, for к > 2 can be
expressed in terms of u\/xi,..., u\Jxn- Therefore, condition (a) of Proposition 6.2
holds for all k. □
Corollary 6.6. Let xi,..., xn and щ ^ • • • ^ щ be elements of a modular
lattice Q, such that щ ^ Xi ^ щ for all 1 < i ^ N. Then the following conditions
are equivalent:
(a) the set щ,..., щ, х\,..., xn is distributive;
(b) all the sets us Л Xi,..., us Л хдг are distributive and the equation
US Л {\/ге1Хг) = \/ге1(и8/\Хг)
holds for any subset I С [1, N] and 1 < s ^ I;

7. LATTICES OF VECTOR SPACES
15
(b*) all the sets us V Xi,..., us V хдг are distributive and the equation
Us v (f\ieIxt) = AZG/^Vx*)
holds for any subset I С [l,N] and 0 ^ s ^ I - 1;
(с) а// йе sets 7is Л X\ V 7is_i, ... , us А хдг V 7is_i are distributive and for
every 1 ^ г < j ^ N one of the following two sets of equations holds:
us A (x{ V Xj) V us-i = (us A Xi) V (us Axj)\/ us-\, 1 < s ^ Z;
7is Л (Xf Л Xj) V 7is_i = 7is Л (Xi V 7is_i) Л (Xj V 7is_i), 1^5^/
ftte expressions without brackets make sense due to the modularity
identity).
Proof: The equivalence of (a), (b), and (b*) follows immediately from
Corollaries 6.4 and 6.5. Let us prove the only nontrivial part (c) =>> (a). Suppose the first
set of equations holds for some г, j. Then we can prove by induction in s that the
triple distributive for every s. Indeed, in the induction step one has to
apply A(xi\/Xj) to the 5-th equation and use modularity and the induction
assumption. A dual argument shows distributivity of us,Xi,Xj under the assumption that
the second set of equations holds. Next, we prove by induction in s that the subsets
us A Xi,..., us A xn are distributive. The induction step follows easily by applying
Corollary 6.5 (c) => (a) to the elements и = us-i,usAxi,..., us Лхдт. Dually, we
can check that all the subsets us Vxi,..., us V xn are distributive. Applying
Corollary 6.5 (c) => (a) to the set us, Xi,... , хдг we derive its distributivity. Finally, by
Corollary 6.4, this implies distributivity of the entire set щ,..., щ, Xi,..., хдг. □
7. Lattices of vector spaces
Let W be a vector space. The set Qw of all its linear subspaces is a lattice with
respect to the operations of sum and intersection: X\/Y = X+Y and XAY = XC\Y.
This lattice is modular but not distributive: the equation (X+Y)nZ = XnZ+YnZ
does not hold in general. In fact, it is not difficult to classify all triples of vector
subspaces X, У, Z с W up to isomorphism. All indecomposable triples but one
are distributive and have dimly = 1. The only nondistributive indecomposable
triple is that of three lines in a plane.
We will say that a collection of subspaces Xi, ... , Хдг С W is distributive if
it generates a distributive lattice of subspaces of W.
The dual lattice f^ is naturally identified with the lattice Qw* of all vector
subspaces of the dual vector space W*: a subspace X с W corresponds to its
orthogonal complement Xх С W*. It follows that the orthogonal complement
collection Xj1, ... , Хдг С W* is distributive iff the collection Xb ..., Хдг с W is.
It is easy to see that the direct sum collection X{ 0 X", ... , X^ 0 X^ с W 0
W" is distributive iff both collections X{,..., X^ С W and X{',..., X^ с W"
are distributive. If both spaces W and W" are nonzero then a similar statement
holds with the direct sum collection replaced by the collection of Nf + N" subspaces
X^ 0 W", W 0 X'l с W 0 W". Indeed, the "only if part is trivial and the "if"
part is a consequence of the following result.
Proposition 7.1. Let W be a vector space and Xb ..., Хдг cW be a collection
of its subspaces. Then the following conditions are equivalent:
(a) the collection Xb ..., Хдг is distributive;

16
1. PRELIMINARIES
(b) there exists a direct sum decomposition W = 0„G7^ Wv of the vector space
W such that each of the subspaces Xi is the sum of a set of subspaces Wv.
(c) there exists a basis {wa : a e A} of the vector space W such that each of
the subspaces Xi is the linear span of a set of vectors wa.
Proof: (b) =>> (a) and (b) <^=> (c) are clear. Let us prove (a) =>> (b). For every
subset 77 С [1, N] let us choose a subspace Wn С Пгет? ^ such that
Пге^ = ^((Пге/0п(£^)).
Then we claim that
More generally, we can prove by descending induction in a subset £ С [1, N] that
Indeed, suppose this is true for all strictly larger sets £' D £. Then using the
definition of W^ and the distributivity identity one can easily derive the above
equation for £. Note that in the case £ = 0 this equation states that the subspaces
Wrj generate W.
It remains to prove that the subspaces (W^), rj С [1,N], are linearly
independent. Assume that J2sw,ns = 0 for a set of nonzero vectors wris G И^в, where
all subsets rjs are distinct. Choose so such that the subset ?7So с [1,ЛГ] does not
contain any other subsets rjs. Then we have
This is a contradiction since W^ does not intersect Г-^ Xj by the definition.
□
We say that a basis of W satisfying property (c) above is distributing the
collection Xi,..., Xn С W.
Using the above proposition we immediately see that for any pair of distributive
collections X{,..., X'N С W and X",..., X'^ С W" the tensor product collection
X[ 0 Xf С W7 0 W" is also distributive. Note that the converse statement is not
true: take X^.X^X'^ to be any nondistributive collection and X% = 0.
Proposition 7.2. Let W be a vector space and Xb ..., XN cW be a collection
of subspaces such that any proper subset Xi,..., Xk, • • •, Xn is distributive. Then
the following conditions are equivalent:
(a) the collection X\,..., Хдт is distributive;
(b) the following complex of vector spaces K9(W\ X\,..., Хдг) is exact
0 —> Xi n • • • n xN —>x2n---nxN —> x3 n • • • n xN/Xi —>
-*V(*1 + ■ • ■ +X;v_2) — W/(Xi + • • • +Xjv-l) — W/(Xi + • • • + Хдг) —► 0
where we denote Y/Z = Y/Y П Z;

7. LATTICES OF VECTOR SPACES
17
(c) the following complex of vector spaces B9(W; Xi,..., Хдг)
w - ®w/xt ^ • • • - 0ti<...<t_ ИУЕЛ? *. -
zs e:mc£ everywhere except for the leftmost term;
(с*) £fte following complex of vector spaces B*(W; Xi,..., Хдг)
о-п.*.— etl<...<tN_4n^^.-—et^-^
is exact everywhere except for the rightmost term.
Proof: It is immediate to verify that the conditions of exactness of the complex
K9(W; Xi,..., Xn) coincide with the equations of Theorem 6.3 characterizing
distributive sets in modular lattices, so (a) <^=> (b) follows. Let us prove (а) <<=> (с).
There is a natural exact sequence of complexes
0 > B.(W/Xi; (X2+Xi)/Xb ... , (Xn+XJ/XJ
> B.(W;XU...,XN) > B.(W;X2,...,XN)[-1] > 0,
where [—1] denotes the shift of homological degree. Since we assume that any proper
subcollection is distributive, the third complex is exact in homological degree ф N.
Note that HNB.(W\ Хъ ..., Хдг) = X\ П- • -DXN. It follows easily that the second
complex is exact at the desired terms iff the first complex is exact in degree Ф N — 1
and the connecting map
X2 П • • • П XN > (X2 + Xi) П • • • П (XN + X1)/X1
is surjective. Using induction in N we conclude that exactness of the first complex
is equivalent to distributivity of the collection X\ + X2, ... , Xi + Xn • It remains
to apply Corollary 6.5, (a) «Ф=> (b*), with и corresponding to X\. П
In conclusion, let us reformulate the statement of Corollary 6S, (a) <=> (c)
for the case of lattices of vector spaces. Assume we are given a vector space W
equipped with a filtration
0 = F0W С FxW С • • • С Ft-xW С FtW = W
and a collection of subspaces Xb ..., XN С W. Let gvFW = 0S=1 FsW/Fs-iW
be the associated graded space. We can consider the associated collection of sub-
spaces grFXb ..., grFXN С gvFW, where grFX = ©s FSWnl/Fs_iWnX. The
following result can be considered as a version of the PB W-theorem (discussed later
in the book) for collections of subspaces.
Corollary 7.3. Let 0 = F0W с F{W С • • • с F,_iW С F{W = W be a
filtered vector space and let X\,..., Xn CW be a collection of subspaces. Then the
following two conditions are equivalent:
(a) the set of subspaces FqW, ... , FiW, Xi, ... , Хдг с W is distributive;
(b) the associated graded collection grFXi, ... , grFXjv in the associated
graded vector space grFW is distributive and for any 1 ^ i < j ^ N
one of the two equivalent conditions holds:
grF(Xz + ^) = grFXz+grFX, or gr^X.nX^gr^ngr^,
Proof: The main statement follows from Corollary 6.6, (a) <<=> (c). The
equivalence of two equations in (b) follows from the proof of that corollary. □

CHAPTER 2
Koszul algebras and modules
In this chapter we discuss the notion of Koszulness, central for this book. In
section 1 we give definitions of a Koszul algebra and of a Koszul module. We should
stress that our setting is not the most general since we assume that Aq coincides
with the ground field (except for section 9). An excellent introduction to
Koszulness without this assumption can be found in sections 1 and 2 of [24]. In section 2
we consider a numerical identity between Hilbert series that gives an important
necessary condition of Koszulness. In section 3 we define Koszul complexes and
in section 4 we use them to prove Backelin's criterion of Koszulness in terms of
distributivity of certain lattices. Then in sections 5 and 6 we explain how one can
control Koszulness under certain types of homomorphisms between algebras. In
section 7 we consider examples of projective varieties whose homogeneous
coordinate algebras are Koszul. In section 8 we describe a construction of a graded
infinitesimal Hopf algebra associated with a Koszul algebra that will be used later
in chapter 7. In section 9 we consider the definition of a Koszul algebra in a more
general context of monoidal abelian categories and prove that it amounts to having
a monoidal functor from a certain fixed monoidal category (without unit). Finally,
in section 10 we consider the notion of a relatively Koszul pair of modules over an
algebra.
1. Koszulness
Recall that we always assume our graded algebras A (resp., A-modules) to be
nonnegatively graded and locally finite-dimensional with Aq = Ik (resp. locally
finite-dimensional and bounded below).
Definition 1. A graded algebra A is called Koszul if the following equivalent
conditions hold:
(a) Ext^'(k,k) = 0for гф j;
(b) A is one-generated and the algebra Ext^(k, Ik) is generated by Ext^(k, k);
(c) A is quadratic and Ext^ (к, к) ~ A1;
(d) the algebra Ext^(k, k) equipped with internal grading is one-generated.
Definition 2(M). Let A be a Koszul algebra. Then a graded A-module M is called
Koszul if the following equivalent conditions hold:
(a) Extzj (M, Ik) = 0 for i ф j;
(b) M is generated by M0 and the Ext^(k, k)-module Ext^(M, Ik) is generated
by Ext^(M,k);
(c) М is quadratic and Ext^(M,k) ~ M'A\
(d) the Ext^(k, k)-module Ext^(M, k) equipped with internal grading is
generated in degree zero.
19

20
2. KOSZUL ALGEBRAS AND MODULES
The equivalence follows easily from the results of sections 3 and 5 of chapter 1.
Additional equivalent conditions will be given in sections 3 and 4. We will see in
section 3 that quadratic dual algebras or modules are Koszul simultaneously. Note
also that algebras A and Aop with opposite multiplication are Koszul
simultaneously, since their Ext-spaces are naturally isomorphic (see section 1 of chapter 1).
We leave for the reader to define the notions of being quadratic (resp. Koszul)
for a right A-module R and to check that this is equivalent to imposing the same
conditions on the corresponding left Aop-module Rop.
The condition (a) in Definition 2 is equivalent to the condition that the module
M has a linear free resolution, i.e., a resolution by free A-modules
... —> P2 —> Pi —> P0 —> M —> 0
such that Pi is generated in degree i. The algebra A is Koszul iff Ik admits a linear
free resolution.
Example. The symmetric algebra and the exterior algebra of a vector space V
are Koszul. Indeed, consider the standard Koszul complex
... —> Д V*) ® S(V) —> Д V*) ® S(V) —> V* <8> S(V) —> S(V)
with the differential a* ® a \—> Z^Li(a* л ei) ® (eia), where (е*) and (e*) are
dual bases of V and V*. It is easy to see that this complex is isomorphic to the
tensor product of n complexes of the form k[x] —> k[x] : / i—> xf. Hence, it
is a linear free resolution of the trivial §(V)-module Ik. The space /\(V*) ® §(V)
with the above differential can also be viewed as a linear free resolution of Ik as a
A(V*)-module.
It is clear that for a given Koszul algebra A the category of Koszul modules
over A is stable under extensions. We can also consider the truncation operations
on a nonnegatively graded A-module M. Namely, for every r )0we define the
truncated module AfM = M^r(r), so that AtJjT1 = Mr+n for n ^ 0 and Mjf1 = 0 for
n<0.
Proposition 1.1. Let A be a one-generated algebra. If an A-module M is
generated in degree zero then the same is true for M^ with r ^ 0. Assume in addition
that A and M are quadratic or Koszul Then the same is true for M^. More
precisely, one has Tor£(MM,k) = 0 z/Tor^+r(M,Ik) = 0 and Tor^ljfc(k,k) = 0
for allj + l^k^j + r.
Proof: Since the operation M —> M^ is the r-th iteration of the operation
M —> M^, it suffices to consider the case r = 1. In this case the assertion about
homology follows from the long exact sequence corresponding to the exact triple
0 > M[1](-l) > M > M0 (8) Ik > 0
D
Avramov and Eisenbud showed in [14] that for every finitely generated module
M over a commutative Koszul algebra the truncated module M^ is Koszul for
r ^> 0 (we will reprove this in Corollary 7.2). This result was generalized to some
noncommutative Koszul algebras in [70].

2. HILBERT SERIES
21
2. Hilbert series
Let V — ®kex Vk be a graded vector space. Then its Hilbert series is a formal
power series defined by
hv(z) = y52{dimVk)zk.
kez
For a graded module M over a graded algebra A we consider also the double
Poincare series
PaMu*z) = Yl (dimExt^(M,k))<uV.
Note that the spaces Extu (M, k) are finite-dimensional as cohomology of the locally
finite-dimensional complex Cob9(A, M) (see section 1 in Chapter 1). For the trivial
A-module Ik we denote PA(u, z) — PA^{u, z).
Proposition 2.1. For any graded algebra A and graded A-module M, the
following numerical relation holds:
hA(z)PAM(-^,z) = hM{z).
In particular, we have
(2.1) hA{z)PA{-l,z) = l.
Proof: Let F9 be a minimal free graded A-module resolution of M. By the
definition, we have Ext^(M, Ik) = (k 0 Fi)* and therefore hp^z) — hExti (M,k)(z)hA(z).
The desired relation follows immediately by passing to the Euler characteristic of
the complex F9. □
Remark 1. The above proposition implies that the dimensions (dimAn) of
grading components of A are determined by the dimensions of the homology spaces
Tor^(k,k). Let us show what happens on the level of vector spaces for n — 2 and
n — 3. In the case n — 2 consider the subspace A\A\ С A2. Then we have an
isomorphism
A2/A1A1 ~Torf?2(k,k).
and an exact sequence
0 > Tor£2(k,k) -^-> Tor^^k^Tor^^k) > АгАг > 0,
where Д' is the comultiplication map (see section 3). In the case n — 3 we
consider the natural filtration A\A\A\ С {A\A2 + A2A\) С А$. Then we have an
isomorphism
Tor^k,k) - A3/(AiA2 + A2Ai),
and exact sequences
0 -> ker(A") -> Tor£3 ^ Tor^(k, k) <g> Torf?2(k, k) 0 Tor^?2(k, k) <g> Torf?1(k, k)
-> (AiA2 4- A2A1)/A1A1A1 -> 0,
0 -> ker(A") -A coker(^) -> A1A1A1 -> 0,
0 -> Tor£3(k, k) -> Torf?1(k, k) 0 Tor£2(k, k) 0 Tor242(k, k) 0 Torf?1(k, k)
^ Torf i(k,k)03 -> coker(#) -> 0,

22
2. KOSZUL ALGEBRAS AND MODULES
where g — A! <S> id + id 0Д' and / is the map dual to the triple Massey product
(see Appendix). Considering dimensions in these exact sequences one gets the same
expression for dim A3 in terms of dim Tor^- (Ik, Ik) as from Proposition 2.1.
Corollary 2.2. For quadratic dual Koszul algebras A and A one has
(2.2) hA{z)hAi{-z) = l.
For a Koszul module M over a Koszul algebra A one has
hM<x(z) = hA*{z)hM{-z),
where M\ is the quadratic dual module over A1.
Proof: One just has to observe that a graded algebra A is Koszul iff PA(u,z) —
hA< (uz), and an A-module M is Koszul iff Ра,м(щ z) — hM* (uz). □
Example 1. The above corollary provides a way to check that certain algebras
are not Koszul. For example, let A be the universal enveloping algebra of the
Heisenberg Lie algebra associated with a symplectic vector (V,u) of dimension
2n ^ 4 (this algebra is quadratic, see Example 5 of section 2 of chapter 1). Then A
has a central degree 2 element t such that В = A/(t) is isomorphic to the symmetric
algebra of V, and t is not a zero divisor by the PBW theorem. This implies the
following equality for Hilbert series:
, / ч _ hB{z) 1
W)- i_z2 - (l-zyn{i_z2y
Hence, Ha(—z)~l — (1 + z)2n(l — z2) has the negative top degree coefficient. By
Corollary 2.2 this implies that A cannot be Koszul. A similar argument works for
the universal enveloping algebra of the Heisenberg Lie algebra associated with a
possibly degenerate skew-symmetric form uj provided that the rank of и is ^ 4
(otherwise this algebra is not quadratic).
Example 2. More generally, let A — U(g) be the universal enveloping algebra
of a graded Lie algebra g = фг>1 &• Then we claim that Koszulness of A implies
that either Q = Qi (and hence, A = S(fli)) or the algebra g has exponential growth.
Indeed, by the PBW theorem the Hilbert series of A is equal to
м.)-Пгг^7.
1 (1 _ Zi\G
where a* = dim9^ If A is Koszul then by Corollary 2.2 one has
(2.3) hA.(-z) = hA{z)-1=l[{l-zir.
But A' is the quotient of the exterior algebra, so h^ is a polynomial. Writing
Ид* {—z) — (1 — ol\z) ... (1 — amz), where ctj G C, and taking the logarithm of both
sides of (2.3) we get the equality
771
5^1og(l - a5z) = ^ailog(l - zl).
Comparing the coefficients with zn for n ^ 1 we get the identity
771
£ «? = $>-•
j=l d\n

2. HILBERT SERIES
23
This leads to inequalities
nan ^ У^ off ^ п У^ aj
j=l 2=1
If \otj\ > 1 for some j then the second inequality shows that g has exponential
growth. On the other hand, if \otj\ ^ 1 for all j then the first inequality shows that
an = 0 for n > m. We claim that in this case a2 — 0. Indeed, equation (2.3) shows
that for a2 > 0 one has hA\(l) — 0, which is a contradiction.
Example 3. Now let A — U(L) be the universal enveloping algebra of a graded
Lie superalgebra L — ®i>YLi (where the parity is induced by the Z-grading).
Then we claim that Koszulness of A implies that either L has exponential growth,
or Z/3 = 0 and dimZ/2 ^ dimLi. Indeed, in this case we have
hA(z) = ]J(l-(-zy)^
t-1^
2>1
where аг- = dimL;. Since the algebra A' is commutative and has a\ generators, we
obtain an equality of the form
hAz) = (1~а*')-"£~ат') = № - **)(-1)1<ч.
where aj G C, or equivalently,
(2.4) (1 - aiz)... (1 - amz) = Ц{1- z^-1^.
г>2
As in the previous example, taking the logarithm and comparing the coefficients
with zn we derive the identity
771
i=l d\n,d>l
This leads to inequalities
771 771 П
j=l j=l г=1
As before, these inequalities imply that either L has exponential growth or it is
finite-dimensional. Let us show that in the latter case one has аз = 0. Assume that
аз t^ 0 and let n be the maximal odd number such that an ф 0. Then the right-hand
side of (2.4) has a pole at a primitive n-th root of unity, which is a contradiction.
Finally, the condition that (1 — z2)a2/(1 — z)ai has positive coefficients implies that
a>2
^Gilt was unknown for some time whether any quadratic algebra A satisfying
numerical relation (2.2) is Koszul. Counterexamples were found independently
in [106] and [101]. In section 5 of chapter 3 we will present an even more remarkable
counterexample due to Piontkovskii [92] showing that it is impossible to decide
whether a quadratic algebra A is Koszul knowing only Hilbert series of A and A[.
We conclude this section with two results that show that in certain special
situations relation (2.2) does imply Koszulness. The first of these results is related
to the Golod-Shafarevich inequality (see [63]). For a power series f(z) G M[[z]] we
denote by \f(z)\ the series obtained by deleting all terms starting from the first

24
2. KOSZUL ALGEBRAS AND MODULES
negative term. Also, for two power series we write f(z) ^ g(z) if this inequality
holds coefficient-wise.
Proposition 2.3. For any quadratic algebra A with m generators and r
relations one has
(2.5) hA{z) > Kl-mz + rz2)"1!.
Also, if
hA(z) = (1 -mz + rz2)'1
then A is Koszul
Proof: For any quadratic algebra A — {V, 1} we have an exact sequence
/<g>A(-2) > V®A{-1) > A > к > 0.
Therefore, we get the inequality (1 — mz + rz2)hA{z) — 1^0. If the equality holds,
then the above sequence is also exact at the term /0A(-2); hence, we get a linear
free resolution for Ik. We claim that in any case we get the required inequality.
Indeed, it suffices to check that for a pair of power series f(z),g(z) G Щ[ж]], where
g(0) > 0, the inequalities f(z) ^ 0 and f(z)g(z) ^ 1 imply that f(z) ^ \g(z)~l\.
Indeed, assume that the first negative coefficient of g(z)~l appears with zN. Then
we just have to observe that |g(z)_1| = g(z)~l mod (zN) and that f(z) ^ g(z)~l
mod (zN). D
Corollary 2.4. Let A be a quadratic algebra such that either As — 0 or
A^ — 0. Assume that relation (2.2) holds. Then A is Koszul
The following theorem is a slight generalization of Theorem 2.2 of [112]
(however, the proof is the same). Recall that a graded algebra A is called Gorenstein if
Ext^(k, A) is one-dimensional.
Theorem 2.5. Let A be a graded Gorenstein algebra of finite cohomological
dimension d. Then Ha(z) is a rational function. Furthermore, the following
conditions are equivalent:
(i) A is Koszul;
(ii) Ext^(k, A) has internal degree d;
(Hi) Ha has degree —d.
Proof: Let
0 —> Pd —> ... —> P2 —> Pi —> Po —> к —> 0
be a minimal free resolution of Ik as a left A-module. The Gorenstein condition
implies that the dual complex of right A-modules
(2.6) P0V — P? — ... — PU — ^V
(where P2V = Hom^Pi, A)) has cohomology Ik. Note that the complex (2.6) is also
minimal. In particular, the map P^_i —> Рд cannot be surjective. Hence, the
complex (2.6) is a minimal resolution of k(n) as a right A-module for some sGZ.
This implies that P^ ~ A{n) and generators of P^-i live in degrees ^ — n + i. In
other words, Pd ~ A(—ri) and Pd-i has generators in degrees < n — i. It follows
that hpd(z) = znhA(z) while hp^^z) = pd-i{z)hA{z) for some polynomialspd-i(z)
such that degpd-i ^ n — i. Since P. is a resolution of Ik we obtain the identity
d
1 = hA(z) ■ ((-l)dzn + Y^i-tf-'Pd-ii*))-
2=1

3. KOSZUL COMPLEXES
25
Therefore, Ha{z) is rational and has degree — n. From this argument one can
immediately deduce the equivalence of (ii) and (iii). If A is Koszul then Pd is
generated in degree d, hence n = d. This proves the implication (i) => (ii).
Conversely, if n — d then Pd-i has generators in degrees d — г, so Ik has a linear free
resolution, i.e., A is Koszul. □
Corollary 2.6. Let A be a quadratic Gorenstein algebra of finite cohomolog-
ical dimension satisfying relation (2.2). Then A is Koszul.
Remark 2. One can show that a graded algebra A is Gorenstein of finite
homological dimension iff its cohomology algebra Ext^(k, Ik) is Frobenius, i.e.,
dim Ext^ (Ik, Ik) = 0 for г > d, dimExt^(Ik, Ik) = 1, and the multiplication maps
Ext2A(Ik, Ik) <g> Ext ^~* (Ik, Ik) ► Ext^(k, Ik)
are nondegenerate pairings. Hence, a Koszul algebra is Gorenstein of finite homo-
logical dimension iff its quadratic dual algebra is Frobenius.
Remark 3. Another situation when Koszulness can be derived from other homo-
logical properties together with information about the Hilbert series is considered in
Theorem 2 of [53] that implies that every commutative quadratic extremal
Gorenstein algebra is Koszul (here extremality means that deg/i^ = —dimЛ + 2). For
example, this gives Koszulness of the following algebras: (i) Ik[x^-, 1 ^ г, j ^ 3]//,
where / is generated by all 2 x 2-minors; (ii) Ik[x^-, 1 < г, j ^ 5]//, where xij — —Xji
and / is generated by the Pfaffians of all principal 4 x 4-submatrices (see [109]).
3. Koszul complexes
The following construction generalizing the standard Koszul complex (see
Example in section 1) is due to S. Priddy [104].
Let A and A1 be dual quadratic algebras. Let us equip the tensor product A® A[
with the standard algebra structure (so that subalgebras A and A[ commute) and
consider the identity element ел £ A\ ® A!\. One can check easily that e\ — 0
in the algebra A 0 A!. Indeed, let A — {V,I}. Then the multiplication map
(Ai 0 A[)®2 —> Л2 0 A?2 can be identified with the natural map
Hom(V02, Vm) -> Hom(/, V®2/I).
Under this identification the element ел 0 ел corresponds to the identity element
in Hom(V®2, V®2)] hence, the above map sends it to zero.
More generally, for any graded algebra A consider the algebra В — (q^4)!, dual
to the quadratic part of A. Then the image of the element eqA under the natural
homomorphism qA —> A defines an element ел G A\ 0 B\ with the same property
e\ — 0 in A <S> B. Let R (resp., L) be a graded right (resp., left) A-module (resp.,
Б-module). Note that the graded dual space L* has a natural structure of a right
Б-module. Hence, the tensor product R <g> L* is a right A 0 Б-module. Now the
action of ел £ A<S> В equips R 0 L* with a differential д. This complex of vector
spaces is called the Koszul complex of L and R and is denoted KA(R, L).
It will be convenient for us to use two different notations for the natural bi-
grading on KA(R, L). Namely, we set
K£{R,L) = pqKA(R,L) = Rq®L;, for i = p, j=p + q.
The differential д maps pqK to p_i?(?+iif and Кц to Ki-ij. Note that for the
subscript (p, q) the roles of R and L are symmetric. On the other hand, using (i,j)

26
2. KOSZUL ALGEBRAS AND MODULES
we can consider KA(R, L) as a complex of graded vector spaces, with г being the
homological grading and j the internal one.
For a left nonnegatively graded A-module M denote by N — (Qa^OLa) tne
quadratic dual left I?-module and set
'K.(A,M) = K?(A,N),
where A is viewed as a right A-module. Then 'Km (A, M) is a complex of free graded
left A-modules
• • • > A <g> Щ у A <g> 7V2* ► A <g> TV* > A <g> 7V0* > 0,
where N* is equipped with internal degree i. Note that Nq = M0 is the space of
degree 0 generators of M and N£ — Км is the space of quadratic relations in M.
It is easy to see that the natural A-module morphism A 0 Mq —> M factors
through H$'Km(A,M). Hence, we can extend the above complex by adding M
on the right. Let fK9(A,M) denote the obtained complex, so that fKi(A,M) —
'Ki{A,M) for i ф -1 and '#_i(A,M) = M.
Proposition 3.1. Let M be a nonnegatively graded module over a graded
algebra A. Then M admits a linear free resolution iff the complex fK9(A,M) is a
resolution of M, i.e., the complex fK.(A,M) is exact. More precisely, for any
a, b ^ 0 the following two conditions are equivalent:
(i) Tor£(k, M) = 0 for all i^j, i < a, j - г ^ b;
(ii) Hi-ij 'K.(A, M) = 0 for all i ^ a, j -i O-
Furthermore, there are natural morphisms^j : Нг-ij 'K9(A, M) —> Tor^-(k, M)
for all i < j. If the above equivalent conditions hold then фа,а+ь+1 is infective and
V>a+i,a+6+i is surjective.
Proof: There is a natural morphism of complexes of free A-modules
(3.1) 'K.(A.M) > Bar.(A,M), a 0 x i—>a®p(x),
where p: N* —> Afг 0 M0 is the embedding dual to the action map Bfг 0 N0 —>
Ni. Let Cm be the cone of this morphism. Then C9 is also a complex of free
A-modules. Also, since Bar9(A,M) is a resolution of M, we have Н^{Ст) —
Hi-ijfK.(A,M). On the other hand, the complex Ik 0a fK(A,M) has a zero
differential and the morphism (3.1) tensored with Ik induces an isomorphism of
Hi(k®A 'K(A, M) with Нц(к®А Bar(A, M)) = Тогг^(к, М) (see section 3 of
chapter 1). Hence, we have Ну(к®А С.) = Tor4(k, M) for i ф j and Нц(к®А С.) = 0.
Now we define ^ j as the maps гф1*- : Hij(C.) —> Я^(к0д С.) induced by the
natural morphism of complexes of graded vector spaces C9 —> к 0 a C9.
It remains to prove that for an arbitrary bounded above complex C. of free
graded A-modules and a pair of integers (a, 6), the vanishing of Hij(C9) for i ^ a
and j — i ^ b is equivalent to the vanishing of H{j(k 0a Cm) in the same region,
and that in this case фа,а+ь+1 (resp., ^a+i>a+b+i) is injective (resp., surjective).
According to Proposition 4.2 we can assume the complex C. to be minimal (since
an acyclic complex of free A-modules remains acyclic after tensoring with Ik). It
is easy to deduce from minimality that the morphism ip{ v is injective (resp.,
surjective) if Hij(k 0a C.) = 0 for i — io and j < j0 (resp., for i — i0 — 1 and
j < jo)- Using this observation about injectivity one can immediately check that
the vanishing of Яу(к(8)дС.) for г ^ a, j — i ^b implies the vanishing of Н^{Ст) in

3. KOSZUL COMPLEXES
27
the same region. To prove the converse we can use induction in a. Note that since
Cm is bounded above, the assertion is trivial for a <^ 0. Assume that Hij(Cm) — 0 for
i ^ a and j — i^b. By the induction assumption this implies that Нгз(к®лС9) — 0
for г ^ a — 1 and j — i ^ b. Using the above observation we derive that ф1 • is
surjective for i ^ a and j — г ^ 6, and the assertion follows. D
The Koszul complex of a graded algebra Л is denned as Km(A) — fK9(A,k).
Since N — k!qA — (qA)\ it has the following form:
• • • ► A <g> A3* ► A <g> Л2* ► A (8) Ai* > A > 0.
Here A-{ — A\ is the space of degree 1 generators of A and A2* = I a is the space
of quadratic relations of A.
Corollary 3.2. Let A be a graded algebra, M be a graded A-module.
(i) A is quadratic iff HijK9(A) = 0 for j > 0 and i — 0,1.
(zMj Assume that A is quadratic. Then M is quadratic iff H{ 'Кщ (А, М) — 0 /or
г = — 1 and 0.
(ii) A is Koszul iff the Koszul complex K9(A) is acyclic in all positive internal
degrees.
(UM) Assume that A is Koszul. Then M is Koszul iff the Koszul complex 'Km (A, M)
is a resolution of M.
(Hi) One has Ext^(k, к) = О for i ф j and j ^ n iff K.(A) is acyclic in all positive
internal degrees ^ n.
(iiiM) One has Ext^ (M, A:) = 0 for i ф j and j ^ n iff the complex 'К. (А, М) is
acyclic in all internal degrees ^ n. D
Let Л be a quadratic algebra A. Then for a left Л'-module L and a right
A-module R we have a natural isomorphism
pqK (R,L) = qpK (L,R)*
and the corresponding differentials are dual to each other. Therefore, for any p and
q one has pqHKA(R, L) = 0 iff qpHKAl°P(L, R)* = 0.
Corollary 3.3. Quadratic dual algebras A and-A- are Koszul simultaneously.
More precisely, for fixed a and b one has Ext^ (Ik, k) = 0 for all i — 1 ^ a, 1 <
j — i + 1 < b iff Ext^, (k, k) = 0 for all i — 1 < b, 1 < j — i + 1 ^ a. In addition,
there is a natural pairing
(•,.) :Ext^+1,c+d(k,k)(g)Ext^t1,c+d(k,k) ► к
for any c, d ^ 2. It is nondegenerate on the left (i.e., (x, y) — 0 for all у implies x =
0) if the above condition holds for a — c—\, b — d. The pairing is nondegenerate
on the right if the above condition holds for a — c, b — d—\.
Proof: As we have observed above, the complexes К (A) — KA(A, A[) and K(A]op)
— KA (A\ A) are dual to each other. The pairing (•, •) corresponds to the map
ф£1;с+А^Ас+чГ-- ExtcA+1'c+d(k,k)^TorcA+liC+<i(k,k)* > HCtC+dK.(Ay
~ Hd<c+dK.(A^) ► Tor£;riC+<i(k,k) ~ Tor^liC+d(k,k) ~ Ext't^M)*,
where the morphisms ф^ were defined in Proposition 3.1. All the statements follow
immediately from this proposition. □

28
2. KOSZUL ALGEBRAS AND MODULES
Assume for a moment that an algebra A is Koszul. Then the complex K9 (A) —
KA(A, A1) is a free graded resolution of the left A-module k. Similarly, the complex
K.(Aop) = KA°P(A, A1) is a resolution of the right A-module k. Therefore, in this
case for every A-module M the homology of the complex
"K.{A, M) := KfP(M, A!) = KfP{A, A[) ®A M :
• • • > A3* <8> M > Л2* <g> M > A1!* <g> M > M > 0
is isomorphic to Tor^-(k, M). Recall that M is a Koszul module iff this homology
is concentrated on the diagonal. Moreover, the diagonal homology is isomorphic
to Мд (see Proposition 3.1 of chapter 1). Let "Km(A,M) denote the cone of the
natural embedding Мд —> "Km(A, M), where Мд is considered as a complex with
zero differential. It follows that a module M is Koszul iff the complex "Km(A,M)
is acyclic.
Proposition 3.4. For any graded algebra A and graded A-module M, there
are natural morphisms ipi !• : Нц "K9(A, M) —> Tor^ -(к, M). If for some a and b
one has Tor^-(Ik, к) = О for all i — \ ^ a, 1 < j — г + l ^ b, then these morphisms are
infective for all i ^ a, j — i ^b and surjective for all i — 1 < a, 1 < j — i + 1 ^ b.
Proof: The morphisms tpij are obtained by applying the hyper-Tor functors
Tor^(-,M) to the morphism of complexes of right A-modules K.(Aop) —> k.
Note that the cocone of this morphism coincides with the complex 'K9(Aop,k) —
K. (Aop). Hence, it suffices to show the vanishing of Tor£- (K. (Aop), M) for all i ^ a
and j — г ^ b under our assumptions on Tor^- (к, к). But this follows immediately
from the hyperhomology spectral sequence
E2pq = Tor^(НдК.(А°П,М) =» Toi£+q(K.(A°p), M)
and Proposition 3.1 applied to M = k. □
Corollary 3.5.(M). Quadratic dual modules M and MA over quadratic dual
Koszul algebras A and A are Koszul simultaneously. More precisely, if A is Koszul,
then one has Ext'j(M, k) = 0 for all i - 1 ^ a, 0 < j - i ^ b iff Ext'j, (M!,к) = О
for all г — 1 ^ b, 0 < j — i ^ a. More generally, for a quadratic algebra A and a
quadratic A-module M there is a natural pairing
Ext^+1'c+d+1(M,k)0Ext^t1'c+d+1(M!,k) ► к
for any c, d ^ 1. It is nondegenerate on the left (resp., right) if Ext^(k, к) = О for
i — 1 ^ c, 1 < j — i + 1 ^ d+1 (resp., г — 1 ^ с + 1, 1 < j — i + 1 ^ d) and the
above conditions hold for a — c—\, b — d. (resp., a = c, b = d — 1).
Proof: Note that the complexes fK{A,M) = KA(A,Ml) and "if(A!,M!) =
KA (M!, A) are dual to each other. The desired pairing corresponds to the map
^cV^X+J*: ExtcA+1'c+d+1(M,k) ► Hc,c+d+1'K.(A,M)*
~Hd+1,c+d+1"K.(A\M]) > Ext^+^M'.k)*,
where the morphisms ф and (p were defined in Propositions 3.1 and 3.4. All the
assertions follow from these propositions. □
Remark 1. In the simplest nontrivial cases the pairing from Corollary 3.3
provides an isomorphism ExtA' (к, к) ~ ExtA\ (к, к)* and a surjective morphism

4. DISTRIBUTIVITY AND n-KOSZULNESS
29
ExtA' (к, к) —> Ext^', (к, к)* for any quadratic algebra A. One can check that the
kernel of the latter morphism coincides with the image of the Yoneda multiplication
map
Ext^(k, k) <g> Ext^'4(k, k) 0 Ext^'4(k, k) <g> Ext^(k, k) > Ext^5(k, k).
The map from each of the direct summands can have a kernel and their images
can intersect each other. For this reason it is impossible in general to recover the
dimension of Extл5, (к, к) from the dimensions of Ext^(k, k) [101]. Similarly, from
Corollary 3.5 for a quadratic M over a quadratic algebra A we get an
isomorphism Ext^3(M,k) ~ Ext^Mj^k)* and a surjective morphism Ext^'4(M,k) —>
ExtA', (Мд,к)* whose kernel coincides with the image of the multiplication
Ext^(k, k) <g> Ext^'3(M, k) 0 Ext^'4(k, k) <g> Ext^'°(M, k) > Ext^'4(M, k).
For a Koszul algebra A the second summand vanishes. More generally, one can
show that the images of the Yoneda multiplication are always contained in the
kernels of the pairings from Corollaries 3.3 and 3.5.
Remark 2. There exist also natural complexes computing Hochschild and cyclic
homology of a Koszul algebra (see [49], [123]). Furthermore, it is shown in [124]
that for a Koszul Gorenstein algebra of finite global dimension there is a duality
between Hochschild cohomology and Hochschild homology (with some twist).
4. Distributivity and n-Koszulness
The following criterion is due to J. Backelin [15]. The generalization for
modules was proved in [24].
Theorem 4.1. A quadratic algebra A = {V, 1} is Koszul iff for all n ^ 0 the
collection of sub spaces
Xi = V®1'1 <g> / <g> V®"--1-1 с У0П, i = 1,..., n - 1,
is distributive. More precisely, the following conditions are equivalent:
(a) Ext2j(k, к) = О for all i<j^ n;
(b) the Koszul complex K9 (A) is acyclic in positive internal degrees < n;
(c) the collection (Xi,... ,Xn_i) in V®n is distributive.
Proof:
(b) 4=Ф (с). One can immediately check that the complex K9(V®n', Xi,..., Xn-i)
from Proposition 7.2 of chapter 1 coincides with the (internal) degree-n component
of the Koszul complex K9(A). On the other hand, it is easy to see that any
proper subcollection of the collection of subspaces У®2-1 <g> I <g> y®™-*-1 in y®n [s
distributive iff similar collections of subspaces in У®-7 are distributive for all j < n.
Now the assertion follows by induction in n.
(а) Ф=> (с). This immediately follows from Proposition 7.2 of chapter 1 by
observing that the complex B9(V®n;Xi,... ,Xn-i) can be identified with the degree-n
component of the bar-complex Bar. (A). □
Note that the equivalence of (a) and (b) holds for not necessarily quadratic
algebras A (by Corollary 3.2).
Definition 1. A graded algebra A is called n-Koszul if it satisfies the equivalent
conditions (a) and (b) of Theorem 4.1. This is also equivalent to the condition that

30
2. KOSZUL ALGEBRAS AND MODULES
the algebra
ExtA(k,k)/( 0 Ext2j(k,Ik))
equipped with internal grading is one-generated (the equivalence follows from
Proposition 3.1).
For example, any graded algebra is 1-Koszul, any one-generated algebra is 2-
Koszul, and any quadratic algebra is 3-Koszul. An algebra is Koszul iff it is n-Koszul
for every n ^ 1.
Since the subspaces X[,..., X]n_1 С A®n = y*®n corresponding to the
quadratic dual algebra A are the orthogonal complements to the subspaces Xi,...,
Xn-i С V®n, it follows that A and A1 are n-Koszul simultaneously (this can also be
deduced from Corollary 3.3). Note also that the complex B9(V®n,Xi,... , Xn_i)
from Proposition 7.2 of chapter 1 can be identified with the degree-n part of the
cobar-complex Cob (A).
Theorem 4.2.(M). A quadratic module M = (H,K)a over a Koszul algebra
A = {V, 1} is Koszul iff for any n ^ 0 the collection of subspaces
_ (v^-^I^V®71-1-1®!!, z = l,...,n-l
1 ~ [V^-^K, i = n
in V®n ® H is distributive. More generally, assume only that A is n-Koszul. Then
the following conditions are equivalent:
(a) Ext'j (M, k) = 0 for all i <j ^ n;
(b') the Koszul complex fK9(A,M) is acyclic in internal degrees ^ n;
(b") the Koszul complex "Km(A,M) is acyclic in internal degrees ^ n;
(c) the collection (Yi,..., Yn) in V®n ® H is distributive.
Proof: Set W = V®n (g) H. We are going to use Proposition 7.2 of chapter 1
for the collection (Yi,...,Yn) of subspaces in W. One can easily see that the
complex B9(W;Yi,... ,Yn) coincides with the degree-n part of the bar-complex
Bar.(A,M), and that the complex K.{W\ Уь ..., Yn) (resp., K.{W; Yn,... ,Yi))
coincides with the degree-n part of the complex fK9(A,M) (resp., "K9(A,M)).
Using n-Koszulness of A and Theorem 4.1 one can easily show that any proper
subcollection of X\,...,Xn is distributive iff similar collections of subspaces in
У®-7 (g) H are distributive for all j < n. It remains to apply Proposition 7.2 of
chapter 1 together with induction in n. □
An important class of quadratic algebras is formed by monomial quadratic
algebras. By definition these are algebras of the form
A = k{xi,... ,xn}/{xiXj, {i,j)eS),
where S С [l,n] x [l,n] is any subset of pairs of indices. Quadratic monomial
modules over such an algebra A are modules of the form
M = (Aei 0 ... 0 Aem)/ ^ Ax{e^
for some T С [l,n] x [l,m].
Corollary 4.3. A monomial quadratic algebra is Koszul. A monomial
quadratic module over a monomial quadratic algebra is Koszul.

4. DISTRIBUTIVITY AND n-KOSZULNESS
31
Proof: In this case the natural monomial bases of the relevant spaces are
distributive. □
Note that equivalence of (a), (b') and (b") in Theorem 4.2 holds for a not
necessarily quadratic module M (by Corollary 3.2 and Proposition 3.4).
Definition 2(M). A graded module M over an n-Koszul algebra A is called n-Koszul
if the equivalent conditions (а), (b'), or (b") of Theorem 4.2 are satisfied (see
Corollary 3.2). This is also equivalent to the condition that the ExtA(k,k)-module
ExtA(M,k)/( 0 Ext^'(M,k))
i^Q,j>n
is generated in degree zero.
For example, any module generated in degree 0 is 1-Koszul and any quadratic
module is 2-Koszul. As before, using lattices one can immediately see that quadratic
dual modules (over dual n-Koszul algebras) are n-Koszul simultaneously.
Clearly, the distributivity condition (c) of Theorem 4.2 implies n-Koszulness
of the algebra A itself, provided H ф 0. The following result shows how one can
deduce n-Koszulness of A imposing homological conditions on both quadratic dual
modules M and M\
Proposition 4.4. Let A = {V,/} be a quadratic algebra, M = (H,K)A a
nonzero quadratic A-module, and M- the dual quadratic A-module. Then for any
n ^ 0 the following conditions are equivalent:
(a) Ext2j(M, k) = 0 and Ext2j, (M!, к) = О for all i<j^ n;
(b) both Koszul complexes 'K9(A,M) and "Km(A,M) are acyclic in internal
degrees ^ n;
(c) the algebra A is n-Koszul and the A-module M is n-Koszul
Proof: According to Proposition 3.1, our vanishing condition for Ext^(M, k) is
equivalent to exactness of the complex of 'K9(A, M). Since the complex "K9(A, M)
is dual to fK9(A,M-), its exactness is equivalent to our vanishing condition on
ExtA-(M!,k). This proves (а) <<=> (b). The implication (c) => (a) follows from
the observation that the quadratic dual module to an n-Koszul module is n-Koszul.
It remains to prove that (b) => (c). We are going to use notation from the proof of
Theorem 4.2. Recall that the complex K.{W\Y\,..., Yn) (resp., K.{W\ Yn,..., Yi))
can be identified with the degree-n part of ,K9(A,M) (resp., "Km(A,M)). Using
induction in n we can assume that any proper subcollection of Yi,..., Yn-i is
distributive together with Yn. As was noticed in the proof of Proposition 7.2, the
conditions of exactness of K9(W; Yi,..., Yn) (resp., K9(W\ Yn,..., Y{)) coincide
with the lattice equations on Y1?..., Yn (resp., Yn,..., Yi) from Theorem 6.3. Now
we can apply Corollary 6.5, (c) => (a) for N = n — 1 and и corresponding to
Yn. □
Let us conclude with one more result relating (n— l)-Koszulness, n-Koszulness
and quadratic duality.
Proposition 4.5. Let A be an (n — \)-Koszul quadratic algebra. Then for
every 2 < i < n, there is a natural perfect pairing
Ext2/(k, k) 0 Ext^7'+2'n(k, k) —> k.

32
2. KOSZUL ALGEBRAS AND MODULES
If A is an n-Koszul quadratic algebra and M is an (n — 1)-Koszul quadratic module
over A then for every 1 < i < n there is a natural perfect pairing
Ext^n(M,k) ® Ext^72+1'n(M!,k) —> k.
Proof: This follows immediately from Corollaries 3.3 and 3.5. D
5. Homomorphisms of algebras and Koszulness. I
In this section we start to discuss the relation between Koszulness of graded
algebras connected by certain special kinds of homomorphisms A —> B. The
motivation for this discussion comes mainly from two situations studied in [20]
when Б is a quotient of A by an ideal generated by a normal element of degree 1 or
of degree 2. Below we present some generalizations of the results obtained in [20].
It is convenient to consider the following relative version of Koszulness.
Definition. A homomorphism A —> В of graded algebras is called left Koszul (resp.,
right Koszul) if В has a linear free resolution as a left (resp., right) module over A.
Note that a Koszul homomorphism is necessarily surjective (since В is
generated by Bo = к as an A-module). Also, a graded algebra A is Koszul iff the
augmentation homomorphism A —> Ik is Koszul.
Proposition 5.1. Composition of left (resp., right) Koszul homomorphisms is
again a left (resp., right) Koszul homomorphism.
Proof: Let A —> В —> С be a pair of left Koszul homomorphisms. Consider the
standard spectral sequence
(5.1) E2pq = Tor* (Tbr?(k, B), C) => Tor£+g(k,C).
Since A —> В is Koszul, the space Tor^(k, B) is concentrated in internal degree
q. In particular, the right action of В on it is trivial. Hence,
£p29~Tor,A(k,£)®To.f(k,C).
Since В —> С is Koszul, this space is concentrated in internal degree p + q; hence,
A —> С is Koszul. D
Our next result involves a slightly more general class of homomorphisms (cf.
Thm. 5 of [102]).
Theorem 5.2. Let A —> В be a homomorphism of graded algebras such that
Тот{ -(к, В) = 0 for j > г + 1, and let В —> С be an arbitrary homomorphism
of graded algebras such that the composed homomorphism A —> С is left Koszul.
Then В —> С is left Koszul.
Proof: Below by degree we always mean the internal degree. Consider again the
spectral sequence (5.1). Let us prove by induction in n that Torn (к, С) has internal
degree n. This is true for n = 0 since surjectivity of A —> С implies surjectivity
of В —> С. Assume now that the assertion is true for all n' < n. Note that
the graded right Б-module Tor^(k, B) is an extension of its grading components
of degrees q and q + 1 (endowed with the trivial ^-module structures). It follows
that for every p < n the space E% is concentrated in degrees p + q and p + q + 1.
Next, we claim that the term E^0 = Torf (Tor^(k, В), С) is concentrated in
degree ^ n. Indeed, the differential d^ in our spectral sequence has the form
n _ zpN . z^./V v ipN

5. HOMOMORPHISMS OF ALGEBRAS AND KOSZULNESS. I
33
Since E^_N N_x is a subquotient of E^_N N_±, it is concentrated in degrees n — 1
and n. On the other hand, the limit term is concentrated in degree n (by Koszulness
of A —> C), so it follows that E\ 0 is concentrated in degrees n — 1 and n, which
proves our claim.
By our assumption the right Б-module Tor^(k, B) is concentrated in degrees
0 and 1, so we have an exact triple of right Б-modules
0 —>Tor£1(k,B)<g>k(-l) —>Тог£(1к,Б) —>k—> 0.
Let us consider the fragment of the associated long exact sequence:
...—> Tor*(Tor£(k, B), C) —> Tor*(k, C) —*
Тог^д(к,В) ® Torf.^-lJ.C) —> ...
By the induction assumption the space Tor^_1(k(—1), C) has internal degree n.
Hence, from this sequence we deduce that Tor^(k, C) lives in degrees < n. But it
also lives in degrees ^ n for trivial reasons. Therefore, it has to be concentrated in
degree n. □
Corollary 5.3. Let A —> В be a left Koszul homomorphism of graded
algebras and let В —> С be an arbitrary homomorphism. Then В —> С is left Koszul
iff the composed homomorphism A —> С is left Koszul.
Proof: The "only if follows from Proposition 5.1 and the "if follows from Theorem
5.2. □
Taking С = Ik in the above corollary we obtain the following well-known result.
Corollary 5.4. Let A —> В be a left Koszul homomorphism of graded
algebras. Then A is Koszul iff В is Koszul.
Corollary 5.5. Let M be a Koszul left module over a Koszul algebra A. Let
us equip the graded space Am = ^0 M(—1) with the algebra structure by setting
(a,m) • (af,m') = (aaf,am'). Then the algebra Am is Koszul.
Proof: One can check easily that the conditions of Theorem 5.2 are satisfied for
the embedding A —> Am (we take В = Am and С = к). □
Example 1. Recall that an element t e A is called normal if At = tA. Let us
say that t is left (resp., right) normal if tA С At (resp., At С tA), i.e., At (resp.,
tA) is a two-sided ideal in A. Assume that t is a left normal element in A\ that is
not a right zero divisor (i.e., at ф 0 for а ф 0). Then the algebra В = A/At has a
very simple linear free resolution as a left module over A:
0 —> At —> A —> В —► 0.
Hence, in this case the homomorphism A —> В is Koszul. From the above corollary
we get that in this situation A is Koszul iff В is Koszul (cf. [20] Thm. 4 (e); [112]
Thm. 1.5). This statement is often used in the commutative situation (see Lemma
7.3).
Example 2. Let У be a vector space and let q б §2(У) be a nonzero element.
Then the algebra §(V)/(q) is Koszul (see Example 1 of the next section for a
more general statement). Indeed, this can be proved by induction in dimF. If
dim У = 1, then this is clear. The induction step follows easily from the previous
example: pick v £ V such that q is not divisible by v in S(V). Then v is not a
zero divisor in S(V), so the homomorphism S(V)/(q) —> S(V)/(v, q) is Koszul. But

34
2. KOSZUL ALGEBRAS AND MODULES
S{V)/{v,q) ~ §(V')/(q'), where V = V/kv and q' e S2{V) is the image of q, so
this algebra is Koszul by the induction assumption.
Example 3. Assume that an algebra A contains a pair of elements t\ and £2 of
degree 1 such that At\ + At2 is a two-sided ideal and that a relation s±ti = ^2
holds for some Si, S2 £ ^i- Assume in addition that x = stfi — $2^2 is not a right
zero divisor and that Ax = At\ П At2. Then the algebra В — A/(Ati + At2) has a
linear free resolution as a left A-module of the form:
0 —> Ax —> At!®At2 —> A —> В —> 0.
Hence, in this case the homomorphism A —> В is Koszul. Some homomorphisms of
this type can be obtained as compositions of two homomorphisms as in Example
1. However, not all of them can be represented in this form. For example, consider
the quadratic algebra A generated by degree one elements si, 52, £1, £2 with defining
relations stfi = s2t2 and tiSj = Utj for all i = 1,2 and j = 1,2. Then the above
conditions are satisfied. In this case В is the free algebra in si and 52.
The following result gives a useful criterion of Koszulness of a homomorphism
(see sections 7 and 11 for applications).
Theorem 5.6. Let f : A —> В be a surjective homomorphism of graded
algebras. Assume that В is Koszul and there exists a complex Km of left (resp.,
right) free A-modules of the form
... —> V2 <8> A{-2) —> Vi <8> A(-l) -^ A
(so that Ki = Vi ® A(—i)), where Vi are finite-dimensional vector spaces, such that
im(<5) = ker(/) and Hi(Km)j = 0 for i > 1 and j > г + 1. Then f is left (resp.,
right) Koszul.
Proof: Assume that we are dealing with left modules. First, let us consider the
spectral sequence
E$tq = Tor£(k,Kq) => Tor£+q(k,K.).
Note that E^q = 0 for p > 0 and the differential d1 on Eq^ ls zero due to the
form of Km. Hence, the sequence degenerates at E1 and Tor^(k, K9) ~ Ik ®a Kn
is concentrated in internal degree n. Next, let us prove by induction in n that
Torf (k, B)0 = 0 for i < n and j > i. This is true for n = 0 since / is surjective.
Assume that the assertion is true for n — 1. Then the spectral sequence (5.1) with
С = k shows that Torf (к, к) is concentrated in degree i for all i < n. Now let us
consider another spectral sequence
££,=Тог£(к,Я,(ВД =► Tor^+9(k,i^.)-
Since for q ^ 1 the A-module Hq(K9) is an extension of k(—q) and k(—q — 1), we
obtain that for p < n and q ^ 1 the term E^q is concentrated in internal degrees
p + q and p + q + 1. This implies that for r ^ 2 the differential dr : E^L_rr_1 —> E£ 0
has zero components of internal degree ^ n + 1. Hence, all components of internal
degree ^ n + 1 in E^ 0 = Tor^(k, B) "survive" in the spectral sequence and give a
contribution to E£° = Tor^(k, Km). But the latter space is concentrated in internal
degree n, so we obtain that Tor^(k, B)j = 0 for j > n. □

5. HOMOMORPHISMS OF ALGEBRAS AND KOSZULNESS. I
35
Corollary 5.7 (cf. [96], Thm. 4). Let A be a graded algebra such that there
exists a complex Km of right free A-modules of the form
> V2 ® A{-2) —> Vi <8> A(-l) —> A
where Vi are finite-dimensional vector spaces, such that Hi(Km)j = 0 for j ^ г > 2
and also for (i,j) G {(0,1), (1,1), (1, 2)}. Then the quadratic part of A is Koszul.
Proof: The condition Hi(K9)j = 0 for i ^ 1 and j = i or j = i + 1 implies that we
have isomorphisms Vi — A1*, so that the complex K9 is identified with the Koszul
complex of A. It is easy to see that the dual complex A <g)A* considered as a complex
of ^'-modules satisfies the assumptions of Theorem 5.6, where as homomorphism
/ we take the augmentation homomorphsim A1 —> k. This implies that A is
Koszul. □
The following simple result is very useful in relating homological properties of
a homomorphism of Koszul algebras with those of the induced homomorphism of
dual Koszul algebras.
Theorem 5.8 ([102], Thm. 6). Let f : A —> В be a homomorphism of Koszul
algebras and let f : B] —> A1 be the dual homomorphism. Then there are natural
isomorphisms
То4(к,В)*~Тог^.(Л!,к),
where В is viewed as a left A-module and A as a right B--module. Furthermore,
the direct sum of these isomorphisms over all (i,j) is compatible with the natural
left A-action (resp. right В-action).
Proof: This follows easily from the fact that both spaces are appropriate bigrading
components of the cohomology of the complex A ® В (see section 3). □
As an application of the above theorem, we can interpret Koszulness of a
homomorphism between Koszul algebras in terms of the dual Koszul algebras.
Corollary 5.9. Let f : A —> В be a homomorphism of Koszul algebras.
Then f is left Koszul iff A1 is a free right Bl-module. If this is the case then the
dual morphism B- —> A is infective, and В and A/AB+ are dual Koszul modules
over A and A.
Proof: Both assertions follow easily from the theorem. For the first assertion one
has also to use the fact that A is a free ^'-module iff Tor^-(A!,k) = 0 for i ф 0.
For the second assertion we apply the isomorphisms of Theorem 5.8 for i — j. □
Corollary 5.10. Let f : A —> В be a homomorphism of commutative Koszul
algebras. Then f is Koszul iff the dual homomorphism /■ : B- —> A' is infective.
Proof: The "only if part is immediate from the above theorem. For the converse,
we observe that the quadrataic dual algebras to A and В are universal enveloping
algebras of graded Lie superalgebras: A' = U(g) and Bl = £/(fj) (see Example 4 in
section 2 of chapter 1). Furthermore, the homomorphism f : B] —> A is induced
by a Lie algebra homomorphism fj —> g. By the PBW theorem if /• is injective
then A is a free right Б'-module. It remains to apply the above theorem. □
Here is another useful criterion of Koszulness of the quotient of a Koszul algebra
by a number of linear relations. In the commutative case it does not give anything
new, since in this case the homomorphism A —> В is Koszul. However, in the
noncommutative case it deals with a new type of homomorphisms.

36
2. KOSZUL ALGEBRAS AND MODULES
Proposition 5.11. Let В = A/H where H is two-sided ideal generated in
degree 1, free as a right A-module. Assume also that H/HA+(1) has a free linear
resolution as a left B-module. Then A is Koszul iff В is Koszul. In this case the
homomorphism B[ —> A! is infective.
Proof: Consider the spectral sequence
£& = Tor£(k,Tor*(B,k)) =» Toi£+,(k,k).
From the exact sequence 0 —► H —> A —> В —► 0 we get an isomorphism
Тог^ВД-Тог^ВД
for q ^ 1. But H is a free right A-module, so the latter space is zero unless
q = 1. Hence the term E2 of our spectral sequence is concentrated on two lines:
for q = 0 we get E20 = Tor* (k,k) and for q = 1 we get Е2г = Tor* (к, Н/НА+).
Our assumption on H/HA+ is equivalent to the condition that the space Е2г is
concentrated in the internal degree p + 1. It follows that all the differentials
d2:TorpB+2(k,k)=E2p+2fi—> Е2рЛ
vanish. Therefore, the spectral sequence degenerates at the term E2 and gives rise
to exact sequences
0 —> E2_hl —> Toi£(k,k) —> TorB(k,k) —> 0.
Since Е2_г1 has internal degree p this shows that A is Koszul iff В is Koszul. We
also see that the map Bp —> A]p is injective being dual to the map Torp (к, к) —>
Torf(k,k). D
Following [7] we call a homogeneous element t G Ad strongly free if A has the
same Hilbert series as the free product of A/(t) and k[t] (where degt = d), i.e.,
hA/(t)(z)~1=hA(z)-1 + zd
(see equation (1.2) of chapter 3). Let ТУ С A be a graded complement to (t) С А
(so W is just a graded subspace of A). It is easy to see that the sequence
W ®k A(-l) —> A —> A/(t) —> 0
is exact, where the left arrow is w (8) a i—> wta. Furthermore, the comparison of
Hilbert series shows that t is strongly free iff the map W ®k A(—1) —>• (t) above
is an isomorphism. Now we can deduce the following result due to Backelin and
Froberg (see [20] Thm. 4(e) (i)).
Corollary 5.12. Let В = A/{t), where t e A± is strongly free. Then A is
Koszul iff В is Koszul.
Proof: We apply the above proposition to H = (£). As we have seen above, the
assumption that t is strongly free implies that H is free as a right A-module and
that one has an isomorphism H/HA+ ~ A/(t)(-l). D
For example, if В is any Koszul algebra then the free product of В with k[t]
(where degt = 1) is also Koszul. In section 1 of chapter 3 we will show that the
free product of any two Koszul algebras is Koszul.

6. HOMOMORPHISMS OF ALGEBRAS AND KOSZULNESS. II
37
6. Homomorphisms of algebras and Koszulness. II
In this section we prove several results about Koszulness in the case when В is
a quotient of A by a number of quadratic relations.
Proposition 6.1 (cf. [20], Thm. 7(a)). Let A be a Koszul algebra and I С А
be a two-sided ideal such that 1(2) is a Koszul left A-module. Then the algebra В
is also Koszul.
Proof: This follows immediately from Theorem 5.2. We only have to observe that
since 1(2) has a linear free resolution as an A-module, it follows that В has a
resolution of the form
... —> V2 <g> A(-3) —> Vi <g> A(-2) —> A —> В —> 0
where V{ are vector spaces (of degree 0). Therefore, Tor^ -(к, В) = 0 for i ^ 1 and
3 Ф i +1. ' □
Example 1. Let us consider the case A = k[#i,... ,xn]. Then according to the
work of P. Schenzel [109] the assumption of the above proposition holds when A/1
is a quadratic extremal Cohen-Macaulay (CM) ring. Note that A/1 is a CM-ring iff
its projective dimension over A is equal to n — dim A/1. Extremality here means that
deg ft^//(z) = —dimA/1 + 1. For example, A/1 is a quadratic extremal CM-ring
in the following examples: (i) / is generated by all maximal minors in the 2 x n-
matrix of indeterminates; (ii) / is generated by all 2 x 2-minors in the symmetric
3 x 3-matrix of indeterminates.
From the above proposition we immediately recover the following results of [20]
(Thm. 4(e)(iii),(iv)).
Corollary 6.2. Let A be a Koszul algebra and let В = A/tA where t e A2
satisfies tA+ = A+t — 0. Then В is also Koszul.
Corollary 6.3. Let A be a Koszul algebra and let В = A/At where t G A2 is
a right-normal element which is not a right zero-divisor. Then В is also Koszul.
Example 2. The above corollary shows that if A is a Koszul algebra then
В = A/(t) is also Koszul, provided that t G A2 is a central element that is not a
zero-divisor. Starting with the symmetric algebra and iterating this assertion we
derive Koszulness of k[x\,... , xn]/(qi,... ,qr) for any regular sequence of quadrics
(gi,... ,qr). Note that the converse to the above statement is not true: Koszulness
of В does not necessarily imply Koszulness of A. Indeed, let A be the universal
enveloping algebra of the Heisenberg Lie algebra associated with the symplectic
vector (V,lj) of dimension 2n ^ 4. This algebra is quadratic (see Example 5 of
section 2 of chapter 1) but not Koszul (see Example 1 of section 2). By definition,
A has a central degree 2 element t such that В = A/(t) is isomorphic to the
symmetric algebra of V, and t is not a zero divisor by the usual PBW theorem.
On the other hand, as was proved in [20] (Thm. 4(e) (iv)), if A is commutative
and В = A/(t), where t G A2 is not a zero divisor, then Koszulness of В does imply
Koszulness of A. We will prove a slightly stronger assertion in Proposition 6.5
below.
Example 3. In the skew-commutative case quadrics are usually not Koszul.
Namely, if V is a vector space and uj G Д (V) is a nonzero element then the algebra
A — /\(V)/(u) is Koszul iff a; is decomposable, i.e., uj = v\ Av2 for some V\,v2 G V.

38
2. KOSZUL ALGEBRAS AND MODULES
Indeed, if и is not decomposable then A] is isomorphic to the enveloping algebra
of the (possibly degenerate) Heisenberg Lie algebra associated with u;, that has a
non-Koszul Hilbert series as we have seen in Example 1 of section 2. Koszulness of
A in the case when и is decomposable can be checked in many ways. For example,
it follows from Theorem 8.1 of chapter 4.
Lemma 6.4. Let A be a quadratic algebra, t G A2 be a nonzero element, В =
A/(i). Then there is a natural line kt G B2 such that A- = Bm/(t'). Assume in
addition that t is central in A. Then the element t is central in B- iff the equality
xt = 0 in A for x E A\ implies that x = 0.
Proof: Let A = {V,IA}, В = {VJB}. Then IA С IB and the element t is a
generator of the 1-dimensional space Ib/Ia С A2. We define tl G B2 to be the dual
of the functional
ф:1в—> Ib/Ia ^ k.
Then A!2 = B2/ktl. The element t! is central in Bl iff for every reIB®VnV®IB
one has
ф (g> id(r) = id <8>ф(г)
in V. Let t G Iв be any element projecting to t G Ib/Ia- Then Ib = Ia + kt and
for every v G V one has
I® v - v <S> te I a <S> V + V <S> I a
since t is central in A. Now let r G Is ® V П V (8) /б be an arbitrary element. Then
we can write
Г =t®Vi+Xi =V2®t + X2
where Vi,v2 G V, X\ £ Ia®V , x2 £V®Ia- Furthermore, one has
ф <S> id(r) = t>i, id(g)0(r) = г>2.
Thus, t! is central in Б! iff the condition
T®vi-V2®-telA®V + V®IA
for г>1,г>2 G V implies г>1 = V2- But
f (8> vi - v2 (8> Г= F(8> (vi - г>2) mod JA (8> V + У (8) /л-
Hence, t! is central iff the condition
T®v e lA®V + V ®IA
for г> G V implies г> = 0. D
Example 4. Let V be a vector space, q G S2(F) a nonzero element. Applying
the above lemma to A = S(V) and t = q we see that the dual quadratic algebra to
В = §(V)/(g) has a central element t! of degree 2 such that the quotient Bl/(tl) is
the exterior algebra of V*. In fact, Б! is the graded version of the Clifford algebra
associated with q. Namely, it is generated by the degree-1 space V* along with a
central degree-2 element t! and has defining relations
for all £1,^2 £ V'*, where £1 • £2 is the symmetric pairing on V* associated with q.
Proposition 6.5. Let A be a commutative quadratic algebra, t G A2 be an
element such that the equality xt = 0 in A for x £ A\ implies x = 0. Assume that
the algebra В = A/(t) is KoszuL Then A is KoszuL

6. HOMOMORPHISMS OF ALGEBRAS AND KOSZULNESS. II
39
Proof: According to the above lemma we have A[ = Bl/(t[) where t! G B\ is a
central element. Note that В is the universal enveloping algebra U(g) of a graded
Lie superalgebra g and tl is a nonzero element of 92 С U(q). Therefore, by the
PBW-theorem t is not a zero-divisor. Applying Corollary 6.3 we deduce that A is
Koszul and hence A is Koszul. D
Next, we have the following analogue of Corollary 5.9.
Theorem 6.6. Let f : A —> В be a homomorphism of Koszul algebras such
that f induces an isomorphism Ay^^Bx. Let f : B] —> A be the dual
homomorphism. Then ker(/)(2) is a Koszul left A-module iff H := ker{/!)(2) is a free
right Bl -module. In this case ker(/)(2) and H/HB+ are dual Koszul modules over
A and A'.
Proof: "only if". Assume that ker(/)(2) is a Koszul left A-module. Let us
consider the spectral sequence
^ = Ext^op(TorfP(E,k),k) => Ext£P9(k,k).
We have E%0 = Ext^op (к, к) ~ Б!ор and
E2pq ~ Ext^op(TorfP1(ker(/),k),k) ~ Torf P1(ker(/),k) ® Bl*>
for q ^ 1. It follows that all the terms E* with q ^ 1 have internal degree p + q + 1
and hence get killed in the limit. Furthermore, the only nonzero differentials in the
spectral sequence have E*0 as a target. On the other hand, the natural morphism
©р-^ро — ^!°P —> ^!°P coincides with f\ Hence, we get a filtration on ker(/!)
with the associated graded object
ф E2pqc?TorfP(ker(f),k)®B'-°v.
"if". Let us rename A, B[ and f to Б, A and /, respectively. We know that
H = ker(/)(2) is a free right A-module and we want to deduce that ker(/!)(2) is a
Koszul left Б'-module. Let us consider the spectral sequence
E2pq = Ext? (Tor^ (B, k), k) =» Ext*+g (к, к).
Since H(—2) —> A is a minimal free resolution of Б as a right A-module, we have
Тог^(Б,к) = 0 for q > 1 and Torf(£,k) ~ H/HA+{-2). Hence, the spectral
sequence degenerates in the term E3 and we get an exact sequence
0 —> Extb(tf/tfA+,k)(-2) —>B] —> A! —> 0.
It follows that for every p ^ 0 the space Ext^(H/HA+,k) is concentrated in the
internal degree p. Hence H/HA+ is a Koszul Б-module and ker(/!)(2) is the Koszul
dual Б'-module. D
Here is an analogue of Proposition 5.11 for degree-2 quotients.
Proposition 6.7. Let f : A —> В be a homomorphism of graded algebras
such that A\ = B\ and let H = ker(/)(2). Assume that
(i) A is Koszul;
(ii) H is free as a right A-module;
(Hi) H/HA+ has a linear free resolution as a left B-module.
Then В is Koszul.

40
2. KOSZUL ALGEBRAS AND MODULES
Proof: Consider again the spectral sequence
4, = Tor£(k,T<<(B,k)) =» Tor^+9(k,k).
Since H(—2) —> A is a minimal free resolution for В as a right A-module, we have
Тог^(Б,к) = 0 for q > 1 and Torf (B,k) ~ H/HA+(-2). Therefore, the sequence
degenerates in the term E$ and the only nonzero differentials cfe have the form
d2 : E2pl = TorB(k,H/HA+)(-2) —> E2p+2fi = Тог*+2(к,к).
Since H/HA+ has a linear free resolution over В we see that E^ has internal degree
p + 2. Since A is Koszul this implies that #2+2,0 nas internal degree p + 2, so В is
also Koszul. □
Finally, we have the following degree-2 analogue of Corollary 5.12.
Corollary 6.8. Let A be a quadratic algebra and let В = A/{t), where t £ A2
is strongly free. Then A is Koszul iff В is Koszul
Proof: The conditions of the above proposition are easily checked, so if A is Koszul
then В is also Koszul. To prove the converse let us consider the dual quadratic
algebras. We have A[ = Bl/(t[). We claim that t]B[ = B[tl = 0 in B\ Indeed, in
terms of the original algebras this means that /д ® V П V ® /л =/в0^ПУ^/в,
where V is the space of generators of A (and B), I a and Iв are the spaces of
quadratic relations for A and B. But this equality follows immediately from the
equality
dim A3 = dim B3 + 2 dim V
which holds since t is strongly free. Now applying Corollary 6.2 we deduce that if
B[ is Koszul then A[ is also Koszul. □
7. Koszul algebras in algebraic geometry
Many examples of commutative quadratic algebras appear as homogeneous
coordinate rings of projective varieties. Recall that for a closed subscheme X с Pn
the homogeneous coordinate ring Rx is defined as the quotient of k[xo,. •. ,xn] by
the homogeneous ideal Ix generated by all homogeneous forms vanishing on X. In
this section we are going to discuss several examples of Koszul algebras of this type.
We start with a couple of general observations providing some links between
Koszul algebras and projective geometry. The first of them is a geometric criterion
for a module of the form M = 0^o H°(X, F(i)) to be Koszul.
Proposition 7.1 (cf. [98], Lemma 2.1). Let R be a commutative Koszul
algebra. Consider the corresponding projective scheme X = Proj(i2). Then for a
coherent sheaf F on X such that Hl(X,F(—i)) = 0 for all i ^ 1 the R-module
M = ®.^0 tf°(X,F(i)) is Koszul
Proof: Localizing the Koszul complex of R we get an exact complex of sheaves on
X of the form
... —> 4* 0 Ox{-2) —> M* 0 Ox(-l) —> Ox —> 0.
Let us tensor it with F(j) and consider the spectral sequence computing the hy-
percohomology of the obtained complex:
El>q = P}lp®H*(F(j+p)) => 0.

7. KOSZUL ALGEBRAS IN ALGEBRAIC GEOMETRY
41
The assumption on F implies that Ep = 0 for q > 0 and j + p + q > 0. Since
the differential dr has bidegree (r, — r + 1), it follows that all the terms Ep0 with
p > — j "survive" in our spectral sequence. Hence, Ep0 = 0 for p > —j. It remains
to observe that the complex (Ep0, d\) coincides with the component of the internal
degree j of the complex
... —► i# ® M(-2) —► #•* ® Af (-1) —> M
obtained by tensoring the Koszul complex of R with M. Therefore, we get an
isomorphism Ep0 ~ Tor^p (k, M)j and our assertion follows from the above
vanishing. □
Corollary 7.2 ([14]). Let M be a finitely generated module over a
commutative Koszul algebra R. Then for all sufficiently large n the truncated module
M^ = M^n(n) is Koszul
Proof: Let F be a coherent sheaf on X = Proj(i?) obtained from M by localization.
Then for sufficiently large n we have AfW = 0^o H°(X, F(n + i)). Also for n > 0
we will have Hl(X,F(n — i)) =0 for all i ^ 1. It remains to apply the above
proposition. D
The next lemma relates the Koszulness property for the coordinate ring of X
with that of its hyperplane section. Recall that a closed subscheme X С Pn is
called protectively normal if for every i ^ 0 the natural map i^°(Pn, Of>n(i)) —>
H°(X,Ox{i)) is surjective.
Lemma 7.3. Let X cFn be a closed subscheme and let H G Pn be a hyperplane.
Assume that X is protectively normal and that no irreducible component of X (with
reduced scheme structure) is contained in H. Then Rx is Koszul iff RxnH is
Koszul.
Proof: Let / = 0 be the equation of H. Then for every i^Owe have an exact
sequence
0 - Ox(i - 1) Л Ox(i) - ОхпнЦ) - О.
Projective normality of X implies that RxnH coincides with the image of the
homomorphism of algebras
Rx = 0Я°(Х,^й) -> 0Я°(1ПЯ,0хпяй).
Hence, RxnH — Rx/(f) as graded algebras. Since / has degree one, Koszulness of
Rx and Rx/(f) are equivalent (see Example 1 of section 5). □
Remark 1. The assumption of projective normality cannot be omitted in the
above lemma. For example, let X be a smooth rational curve of degree 4 in P3.
Then for a generic plane ЯеР2 the algebra RxnH is Koszul (e.g., by Lemma 7.4
below, since X П H is 4 points in general linear position in P2). However, Rx is
not even quadratic because there is a unique quadric passing through X.
Now we are going to discuss some concrete examples of projective schemes
with Koszul coordinate algebras. The simplest class of such examples is provided
by complete intersection of quadrics (see Example 2 of section 6). Considering
Hilbert series one can easily see that the Koszul dual to these algebras are universal
enveloping algebras of quadratic Lie superalgebras L such that L3 = 0 (see Example
3 of section 2).

42
2. KOSZUL ALGEBRAS AND MODULES
Next, let us consider the case when X С Pn is a finite set of points. The theorem
of Kempf [73] asserts that Rx is Koszul provided that \X\ ^ 2n and points of X
are in general linear position. Theorem 7.5 below is a slight generalization of this
result. We start with the crucial case when \X\ — 2n.
Lemma 7.4. Let X be a set of 2n distinct points in Pn. Assume that X =
Xi UX2, where \Xi\ = \X2\ = n and points in X\ (resp., X2) span a hyperplane H\
(resp., H2) in Pn such that H\ П X = Xi (resp., H2 П X = X2,). Г/ien the algebra
Rx is Koszul and hnx(z) = (1 + z)(l + (n — l)z)/(l — z).
Proof: Let Л = Rx- Note that trivializing Орп (1) near X we get natural evaluation
maps Ai —> /cx. We claim that these maps are isomorphisms for i ^ 2. Indeed,
it suffices to show that for every point x G X there exists a quadric g such that
g(x) 7^ 0 and q(X \ {x}) = 0. Say, xEli. Then we can take q = I • h, where / is
a linear form such that l(x) Ф 0 and /(Xi \ {x}) = 0.
Let Д=0 (resp., f2 = 0) be the equation of Hi (resp., #2)- Let us consider
the natural surjective homomorphism A —► В := Rx1- We claim that its kernel
coincides with f\A. Indeed, since our homomorphism factors through A/f\A, it
suffices to prove that dimAi/f\Ai-\ = n for i ^ 1. This is clear for i = 1 since
dim Ax =n + l. Also we have {/ G Ai : Д/ = 0 G A2} = k/2, since Д does not
vanish on X2. Therefore, dim/iAi = n and
dimA2//iAi = 2n — dim/iAi = n.
Finally, it is easy to see that for i > 2 one has /iA-i = к*2 С kx = ^ which
implies our claim. Similar arguments show that the complex
... A{-3) -^ A{-2) -^ A(-l) -^ A
is a resolution of В as an A-module. Hence, the homomorphism A —> В is Koszul.
It is easy to see that В is isomorphic to the monomial quadratic algebra with n
generators £1,... , tn and relations Utj = 0 for all i < j (since the points in Xi are
linearly independent). Hence, В is Koszul by Corollary 4.3. It remains to apply
Corollary 5.4 to conclude that A is also Koszul. □
Theorem 7.5 ([98], Cor. 0.2). Let X с Pn be a finite set of points. Assume
that X = Xi U X2, where X\ and X2 are linearly independent and
Li П X2 = L2 П Xi = 0,
where Li с Pn is the linear subspace spanned by Si (i = 1, 2). Then Rx is Koszul.
Proof: We can assume that both X\ and X2 are nonempty. Hence, |Xi| ^ n and
IX21 ^ n. Let us enlarge Xi and X2 to n-tuples by adding generic points. Then we
obtain a set of 2n points X containing X, such that X satisfies the assumptions of
Lemma 7.4. It follows from Lemma 7.4 that the natural map
H°(Fn,Opn(i))-+H°(X,Ox(i))
is surjective for i ^ 2. Now let J С Rj^ be the kernel of the natural homomorphism
R^ —> Rx and let J С O^ be the ideal sheaf corresponding to X С X. The
above surjectivity implies that J^2 = ф^>2 Н°(Х, J{i))- Since by Lemma 7.4
the algebra R^ is Koszul, we can apply Proposition 7.1 to the sheaf J(2) on X

7. KOSZUL ALGEBRAS IN ALGEBRAIC GEOMETRY
43
to deduce that «/^2(2) is a Koszul module. Since Rj? is Koszul from the exact
sequence
0 —> J^2 —> J —> Л ® k(-l) —> 0
we derive that Torf*(J,lk)^ = 0 for j > i + 2. Therefore, Tovf^(Rx,k)j = 0 for
j > i + 1. It remains to apply Theorem 5.2 to the homomorphism R^r —> Rx. □
Remark 2. The above theorem implies the Koszulness property for 2n points
in general linear position in Pn (first established by Kempf in [73]). Another proof
of Kempf's theorem using the notion of Koszul nitration is given in [40]. In [39],
sec.4, it is shown that for points in general linear position this result is optimal:
there exist 9 points in P4 in general linear position, whose coordinate algebra is
quadratic but not Koszul. On the other hand, there is a generalization of Theorem
7.5 proved in [98] that gives a criterion of Koszulness for a set of more than 2n
points in Pn under certain conditions on linear spans of various subsets of this set.
Remark 3. Let us say that a finite set of points X С Pn is Koszul if the algebra
Rx is Koszul. Theorem 4.2 of [40] states that the set of s points in Pn with generic
coordinates is Koszul iff s ^ 1 + n + n2/4. The "only if part can be strengthened
as follows: if a Koszul set of s points X С Pn imposes independent conditions on
quadrics then s ^ 1 + n + n2/4. Indeed, let A be the quotient of Rx by a generic
linear form. Then Ha(z) = 1 + nz + (s — n — l)z2 (here we use the assumption that
X imposes independent conditions on quadrics). But A is still a Koszul algebra, so
/i^(z)-1 should have nonnegative coefficients. The latter condition is equivalent to
the required inequality s — n — 1 < n2/4.
Corollary 7.6. Assume that the characteristic of the ground field is zero. Let
С С Pn be a projectively normal irreducible curve of degree ^2n — 2, not contained
in any hyperplane. Then Re is Koszul.
Proof: It is well known that points of a generic hyperplane section of С will be in
general linear position (see [10], III.l), so we can apply Lemma 7.3 and Theorem
7.5. □
Another remarkable example of a Koszul homogeneous coordinate algebra is
the canonical ring of a curve.
Theorem 7.7 ([50]). Let С С P5_1 be a smooth curve of genus g embedded by
the canonical linear series \K\ (so С is not hyperelliptic). Assume that С is not
trigonal and not a plane quintic. Then Re is Koszul.
Proof: It is known that under our assumptions on С there exists a divisor D =
Pi + ... + Pg-i of degree g — 1 such that h°(D) = 2 and both linear series \D\
and \K — D\ have no base points (see [65]). Note that the linear span L of points
pi,... ,pg-i has codimension h°(K — D) — 2. We claim that these points are in
general linear position in L. Indeed, it is enough to show that any g — 2 of them
span L, i.e. for every i — 1,... ,g — 1 one has h°(K — D + pi) — 2. But this is
equivalent to the equality h°(D —pi) — 1 which holds since pi is not a base point of
\D\. Since g ^ 5 we have g — 1 < 2(# — 3), hence, by Theorem 7.5 the algebra Rd is
Koszul (see Remark 2 above). Now we are going to check that the assumptions of
Theorem 5.6 are satisfied for the homomorphism Rc —> Rd- This would imply that
this homomorphism is Koszul and hence the algebra Rc is Koszul by Corollary 5.4.
Set V = H°(D), U = H°(K - D) (here and below the global sections are taken on

44
2. KOSZUL ALGEBRAS AND MODULES
C). Since the linear systems \D\ and \K — D\ are base point free, we have exact
sequences
0 —> Oc(-D) -^V®Oc-^ Oc(D) -+ 0,
0 —> Oc(D - K) —-> /7 0 Oc —-> Oc(^ - D) —-> 0.
Tensoring these sequences with powers of the canonical bundle and passing to global
sections we obtain exact sequences
0—>H°(nK-D) -^V®H°(nK) ^H°(nK + D),
0 —-> Я°((п -l)K + D)-^V® H°(nK) -^ H°((n + 1)JT - D).
Moreover, we have Hl(nK - D) = Я:((п - 1)ЛГ + D) = 0 for n ^ 2, hence, an
and /?n are surjective for n ^ 2. Considering dimensions of the spaces in the above
sequences for n — 1 we see that a\ and j3\ are also surjective. Now let us consider
homomorphisms of ii^-modules
f :V®RC—^U® Дс(1), g:U®Rc^V® Rc(l)
with grading components given by 6n+i о an and an+i о /3n. Let also
h : U ® Rc(-1)—+Re
be the homomorphism with grading components given by 6n_i followed by the
embedding H°(nK — D) —> H°(nK). The above surjectivity properties of an and
/3n easily imply that the complex
K. : ... -Д U 0 Дс(-З) ^70 Дс(-2) -^ t/ ® Дс(-1) "^ ^c
satisfies H0(K.) = Я^, Hi(K.)j = 0 for г ^ 1 and j > г + 1. □
The above proof is taken from [96]. It is different from the original proof in [50].
Yet another proof can be found in [90]. The fact that under the assumptions of the
theorem the algebra Re is quadratic is a classical theorem due to Petri (see [10]).
One can ask whether the homogeneous coordinate algebra of a projectively normal
smooth connected complex curve is Koszul provided it is quadratic. However, the
answer turns out to be negative (see [116], Thm.3.1 and [39], sec.4).
Other examples of projective varieties with Koszul coordinate rings are:
(1) abelian varieties embedded into projective spaces using Ln where L is an ample
line bundle and n ^ 4 (see [74]);
(2) Schubert varieties (see [29] and [67]);
(3) smooth projective varieties of dimension n embedded by К ® Ad <S> B, where
К is the canonical class, A is very ample and В is numerically effective, d ^ n + 1
(with one simple exception, see [89]);
(4) some toric varieties (see [34], [91]). Note that there is a conjecture that the
coordinate algebra of every projectively normal smooth toric variety is quadratic
(see [115], Conj. 13.19). Perhaps one should also expect that these algebras are
Koszul.
Also, we will show in chapter 3 that the operations of taking the image of
a projective variety by a Veronese embedding and taking the Segre product of
projective varieties preserve Koszulness. Furthermore, as Backelin proved in [16],
for any projective variety the Veronese embedding of a sufficiently high degree has
Koszul homogeneous coordinate ring (see Corollary 3.4 of chapter 3, for another
proof see [46]). A similar result with "Koszul" replaced by "quadratic" appears in
[85] and [20].

8. INFINITESIMAL HOPF ALGEBRA ASSOCIATED WITH A KOSZUL ALGEBRA 45
Last but not least, let us mention the remarkable class of elliptic Sklyanin
algebras defined by Feigin and Odesskii [48]. These are (noncommutative) quadratic
algebras associated with elliptic curves (plus some additional data). Their most
remarkable feature is that they have the same Hilbert series as the algebras of
polynomials. In addition they are Koszul and satisfy other nice homological
properties (see [117]). We will consider three-dimensional Sklyanin algebras in section
11 of chapter 4.
8. Infinitesimal Hopf algebra associated with a Koszul algebra
In this section we will show that to every Koszul algebra A one can associate
a graded infinitesimal Hopf algebra containing A as a subalgebra and A[* as a sub-
coalgebra. Let us first recall the notion of infinitesimal bialgebra introduced by
S. Joni and G. Rota in [68], sec. XII and studied by M. Aguiar [3, 4].
In the following definition and the subsequent discussion by an algebra (resp.,
coalgebra) we mean an associative algebra (resp., coassociative coalgebra) over k,
not necessarily having a unit (resp., counit).
Definition. An infinitesimal bialgebra (abbreviated as e-bialgebra) is a triple
(£,/x, Д), where (B,/jl) is an algebra, (B,A) is a coalgebra, and Д is a
derivation of В with values in В ® Б, i.e.,
A(ab) = (a 0 1)Д(Ь) + Д(а)(1 <8> b)
for a, 6 G B.
There is a natural structure of an e-bialgebra on the bar-complex 0n>o A®n
of an associative algebra A (not necessarily unital) with the following maps \i and
Д:
/x([ai|... |am], [ai|... \a'n]) = [ax... |am_i|amai|a2| ... \a'n],
m — \
A([ai|...|am]) = ^[ai|...|a,](g) [ai+i|... \am].
i=l
The axioms are easily checked. Note that we have A(A) — A(k) = 0, so there is
no counit for Д (there is an obvious unit for /л).
Now let A = 0n>o An be a graded algebra with Aq — k. Then we have the
above structure of e-bialgebra on the bar-complex Bar9(A) — 0n>o A+n. Using
the grading on A we can define a natural sub-e-bialgebra in Bar9(A). Namely, for
every (non-empty) collection of positive integers (ni,... ,nr) we define a subspace
V(ni,... ,nr) С Bars(A) by setting
V(ni,... ,nr) = (Ani (g)...(g)Anr)nker(<9),
where д : Barr(A) —> Barr-i(A) is the bar-differential (see section 1). We define
the Z-grading on
V = VA:= 0 V{nu...,nr)
by setting deg V(ni,... , nr) — n\ +... + nr. Note that every grading component of
V is finite-dimensional. Let us denote V(ln) — V(l,... ,1) (1 is repeated n times).
Proposition 8.1. The subspace V с Bar .(A) is a graded sub e-bialgebra.
Furthermore, 0n>1 V(n) С V is a subalgebra isomorphic to A+ and 0n V^(ln) С
V is a sub-coalgebra isomorphic to A+.

46
2. KOSZUL ALGEBRAS AND MODULES
The proof is a straightforward consequence of the definitions.
Recall (see [3]) that an e-bialgebra (B,/jl, A) is called an €-Hopf algebra if it
has an antipode, i.e., a к-linear map S : В —> В satisfying the equations
ii о (S (8 id) о A = fi о (id (8)5) о А = -S - id.
In the case of the e-bialgebra Уд we have /л о A = 0, hence, we can set S — — id.
Thus, Va is an €-Hopf algebra.
Next, we are going to study Уд in the case when A is Koszul.
LEMMA 8.2. (1) There is a bijective correspondence between the set of all
sequences n\, ... , nr ^ 1, r ^ 1 with fixed n\ + • • • + nr — n and the set of all
subsets J С [l,n—l] defined by the rule
Лц,...,пг = [l,n]\{nb rii +n2, ... , rii H hnr} С [l,n- 1].
(2) One has Jnb...,nr С JTOlr..)TOs if and only if there are some 1 ^ t\ < • • • <
ts-i < r such that т\—п\Л + ntl, ... , ms — nts_1-\-i H \-nr.
(3) For any quadratic algebra A — {У, R), positive integers n\ + • • • + nr — n,
and the subset J — Jni,...,nr С [l,n — 1] there is a natural isomorphism
v®n/J2R(jn) ^л,!®---®^,
jeJ
where R^n) = У^"1 (g) Я® У071"-7"1.
Proof: The statements (1) and (2) are elementary; (3) follows easily from the fact
that
An = y^/E^n)-
3 = 1
П
Proposition 8.3. If the algebra A is m-Koszul then the following sequence is
exact for every (щ,... , nr) such that n\ + ... + nr ^ m:
0 —> y(ni,... ,nr) —> C1(ni,... ,nr) —> ... —> Cr(ni,... ,nr) —)• 0
where
Сг(щ,... ,nr) =
0 Ani+...+nei 0 Ansi+1+...+ns2 (8 ... ® AnSr_.+1+...+nr,
Йе differential on C#(ni,... ,nr) г5 induced by the bar-differential and
V(m,... ,nr) —> Ani (8...® AUr = C^ni,... ,nr)
is the natural embedding.
Proof: Let qA be the quadratic part of the algebra A. Then the homomorphism
qA —► A is an isomorphism in degree ^ m. Hence, we can assume that A is a
quadratic algebra. Let A — {У, R}, where R С У (8 У is the space of quadratic
relations. Using the notation of Lemma 8.2 we set J — Jni,... ,Пг С [1, n — 1], where
n = Tii + ... + nr. Recall that according to this lemma we have
W = y0n/^^n) ~ Ani 0 ... 0 Anr.
J6J

8. INFINITESIMAL HOPF ALGEBRA ASSOCIATED WITH A KOSZUL ALGEBRA 47
Now the images Xi of 1ц for i 0 J form a distributive collection of subspaces
in W (here we use the assumption that A is m-Koszul). One can easily check
that the complex C#(ni,... , nr) coincides with the complex considered in
Proposition 7.2(c) of chapter 1 for this collection of subspaces in W. This implies that
C#(ni,... , nr) is exact everywhere except for the first term. It remains to
observe that V(ni,... , nr) coincides with the kernel of the map Cl(n\,... , nr) —►
C2(ni,... , nr) by the definition. □
Corollary 8.4. If A is a Koszul algebra then for every ni,... ,nr and every
1 ^ s < r we have the following exact sequence:
0 —> У(пь...,пг) —► F(nb...,ns) (g)F(ns+b...,nr)
—► y(ni,...,ns_i,ns+ns+1,ns+2...,nr) —► 0,
Proof: From the definition of complexes C#(ni,... , nr) one can immediately
observe that for 1 ^ s < r there is a natural exact triple of complexes
0 —► C#(nb...,ns + ns+i,... ,nr)[-l] —> C#(nb...,nr)
—► C#(nb...,ns)(8)C#(ns+b... ,nr) —>0.
Now the required exact sequences are obtained by considering the corresponding
long exact sequence of cohomology. □
Remark 1. The exact complex of Proposition 8.3 allows us to express the
dimensions of V(ni,...,rifc) as some universal polynomials in dimensions of the grading
components of the algebra A. In section 2 of chapter 7 we will consider
corresponding polynomial inequalities on dim An. There is a nice interpretation of these
formulas in terms of noncommutative generating series (see chapter 7, section 8,
Prop. 8.2).
Our interest to bialgebras Уд is due to their nice behavior under Koszul duality
proved in the next proposition. Below we use again the notation of Lemma 8.2.
We denote by V^ the restricted dual to VA< using the grading on it.
Proposition 8.5. There is an isomorphism of graded e-bialgebras Va — V£
sending Уд(п1,... ,nr) to V^ (mi,... ,ms)*7 where «/mi,...,ms is the complement to
Jnu... ,nr С [1, n — 1] (where n — n\ + ... + nr — m\ + ... + ms).
Proof: Let us fix n and abbreviate iC- to Rj. For every J С [l,n — 1] we set
Rj = J2jej Rj and RJ — C)jej Rj- ^есаи tnat tne quadratic relations for A- are
defined by R[ = R± С (У*)02. It follows that for every J one has
(Rj)1- = Ru, (i?J)x - R'j.
Let us set V(J) — Va(tii, ... ,nr), where J = Jni,... ,nr- Then Lemma 8.2(3) implies
that
(8.1) V(J) = f]RJU{i}/Rj.
Using the distributivity of the lattice generated by (Rj) we compute
П Rju{i} = f](Rj + Rt) = Rj + RJ°,
where Jc С [l,n — 1] denotes the complement to J. Therefore,
V{J) = {Rj + RJC)/Rj ^ RJC/(Rj П RJC).

48
2. KOSZUL ALGEBRAS AND MODULES
Passing to orthogonal complements we get
V{J)" ^ (Rj П i?JC)x/GRjy = (Ru + Rljc)/R}JC
which gives the identification of V(J)* with the component of VA> corresponding
to Jc.
It remains check that this isomorphism is compatible with multiplication and
comultiplication. For J — Jnb...,nr let us set nj — щ + • • • + nr. Consider the
following component of the multiplication on Va'
V(I)®V(J)^V(K),
where К — /U (nj + J) U {nj}. It is easy to check that it is induced by the natural
isomorphism
RIC®RJC^RK'
that follows from the equality Kc — Iе U (nj + Jc). Similarly, for К — IU (nj + J)
the component of the comultiplication
V(K) —> V(I) <g> V{J)
is induced by the embedding
Using this interpretation one can easily check that the above isomorphism V£ ~VA*
is compatible with e-bialgebra structures. □
Remark 2. From the above proof one can easily see that as a vector space Va
can be identified with the associated graded space of 0n>1 V®n with respect to
the decreasing filtration F9 defined as follows:
FmV®n= ^ RJ'
\J\=m
More precisely, there is a canonical isomorphism
Fr-^/FrV9"* 0 VA(ni,...,nr).
ni-\-...-\-nr=n
Furthermore, (Fm) (resp., F._i) is an algebra (resp., coalgebra) filtration on
Фп>1 V®n an(^ the above isomorphism is compatible with multiplication (resp.,
comultiplication).
Example. Let A = k{xi,... ,xn}/(xiXj, (i,j) G 5), where S С [l,n]2 is a
subset. Then we can identify Va with the augmentation ideal in the free algebra
k{xi,... ,xn}, equipped with the following multiplication and comultiplication:
м*п...*«,®xh...xim) = h■■■x^h••■*'- [*fc,il| ?*'
[0, (»fe,Ji)e5;
ZA^Xfi . . . Xffc J — / ^ ^ii • • • %ir ^ *^ir-)_i • • • %ik •
r:(ir,ir+i)£5
It is an interesting problem to find what are possible minimal subsets J such
that the vanishing V(J) =0 can occur for some Koszul algebra. There is a related
numerical problem for one-dependent stochastic sequences (see section 7 of
chapter 7). It suggests that with some simple exceptions the only possibilities should
be [l,n] and 0.

9. KOSZUL ALGEBRAS AND MONOIDAL FUNCTORS
49
Finally, we observe that there is a slight generalization of the above construction
involving a Koszul module. Namely, if M is a Koszul module over a Koszul algebra
A then we can introduce a subspace
У am = 0 У am (ni,... , nr) С Ват. (А, М)
by setting
Ул,м(пь... >nr) = {Ani <g> ... <g> Anr_1 (g)Mnr)nker(<9),
where ni,... ,nr_i ^ 1 and nr ^ 0. One can check that У am IS an infinitesimal
Hopf module over Va (see section 2 of [4]). Furthermore, we have exact sequences
0 —► Уам(п1т- >nr) —> Уа(п1,... ,ns) ® Уа,м(п5+1,... ,nr)
► ^4,м(^Ь- • • ,П5_1,П5 +ns+i,ns+2,- •• ,ПГ) ► 0
similar to those of Corollary 8.4.
9. Koszul algebras and monoidal functors
In this section we sketch a more general approach to the notion of a Koszul
algebra that includes the generalization to the case of graded algebras A for which
Aq is arbitrary and An is not necessarily of finite type over Aq. We also show that a
Koszul algebra can be interpreted as a (nonunital) monoidal functor from a certain
universal (nonunital) monoidal abelian category to the category of vector spaces.
The infinitesimal Hopf algebra considered in the previous section records the values
of this monoidal functor on simple objects. Note that the Koszulness property is
preserved under exact monoidal functors. One application of this observation is the
recent proof in [76] of Koszulness of preprojective algebras of quivers (introduced
in [60]) for quivers not of Dynkin type.
Below we work with nonunital monoidal categories, i.e., categories С with the
functor о : С х С —► С equipped with the associativity isomorphisms satisfying the
pentagon constraint, but not necessarily possessing a unit object. When considering
monoidal abelian categories we require the functor о to be exact in each argument.
First, let us observe that the definition of a Koszul algebra in terms of
distributive lattices makes sense in an arbitrary monoidal abelian category C. Indeed, since
the notion of distributivity can be considered for subobjects of an object in any
abelian category, we can make the following
Definition. Let С be a monoidal abelian category. A Koszul algebra in С is a pair
(V, R), where V is an object of C, R is a subobject of V о V, such that for every
n ^ 3 the collection of subobjects
(9.1) RoVo...oV,VoRoV ...V,... ,У 0...0V0R
in Т/°(п+1) is distributive, i.e., generates a distributive lattice.
Koszul duality in this general context may be viewed as a duality between
Koszul algebras in С and Koszul algebras in Copp. Namely, the dual of a Koszul
algebra (V, R) in С is the Koszul algebra (V, Q) in Copp, where Q is the quotient of
V о V by R in С (hence, a subobject of У о V in Copp). If the category С is equipped
with an exact monoidal duality functor Copp —► С then we can apply this functor
to (V, Q) to get a Koszul algebra in C. For example, the usual Koszul duality
corresponds to applying the standard duality on the category of finite-dimensional
vector spaces.

50
2. KOSZUL ALGEBRAS AND MODULES
Remark 1. In the case when С has a unit object \q one can define a graded
algebra A — 0n>o An in С associated with a pair (V, R) by setting Aq = 1^, A\ —
V, A2 = Vo2/R, A3 = Vo3/(R oV + V oR), etc. It is easy to see that Koszulness
in this case is still equivalent to the condition that the object 1q considered as an
A-module admits a linear free resolution.
Example 1. Let Aq be a semisimple ring and let Aq — Mod— Aq denote the
category of Ao-bimodules. It has a natural monoidal structure given by the tensor
product over Aq. Then a Koszul algebra in Aq — Mod—Aq can be viewed as a
graded algebra A — 0п>оДг, such that Aq viewed as a left A-module has a
linear free resolution. Most of the theory of Koszul algebras that we presented
in the case when Aq is a field can be generalized to this setup (see [24]). Let
A0 - Modz/ -Aq СЛ0- Mod -Aq (resp., A0 - Modr/ -Aq С A0 - Mod -A0) be
the full subcategory of Ao-bimodules that are finitely generated as left (resp., right)
Ao-modules. Then we have a natural duality
(A0 - Modz/ -Ao)®=^A0 - Modr/ -Aq : V h—> V\
where V* — Hom^o-ModC^ ^o) with the bimodule structure given by af(v) —
f(va), fa(v) — f(v)a for / G V*, a G Aq, v G V. Moreover, it is easy to check that
this duality respects tensor products, hence, it induces the duality between Koszul
algebras in Aq — Modz* — Aq and Koszul algebras in Aq — Mod7^ — Aq (called left
finite and right finite Koszul algebras, respectively). The dual algebra to a left finite
Koszul algebra A can be identified with Ext^(A0, A0)opp (see [24], Thm. 2.10.1).
Example 2. An important particular case of the previous example is when
Aq — kn, the direct sum of n copies of a ground field. Then An is a bimodule
over kn, i.e., a vector space equipped with a decomposition An — @iAAn)ij. For
such an algebra let Ak denote the graded algebra with Ak — Ik and A^ — Ai for
i > 0. An explicit lattice consideration shows that the algebras A and Ak are
Koszul simultaneously. Note that in the case when An are finite-dimensional one
can define a matrix-valued Hilbert series of such an algebra as
hA{t) = ^(dim(An)^)r.
The matrix-valued analogue of the relation (2.2) still holds in this case. Algebras
of this kind appear as endomorphisms of strongly exceptional collections in
triangulated categories. Recall that such collections (£i,... ,En) are characterized
by the condition that Ext*(£'i, Ej) = 0 for г > j, Extn(£;, Ej) = 0 for г ^ j
and n > 0, and Нот(Ег,Ег) = Ik for all i. The corresponding endomorphism
algebra A = End(®i£';) = 0f- Hom(^,^) has a natural grading given by
^n = 0,ч=пНот(^,^) (note that A is an example of a mixed algebra in the
terminology of [23]). Let Si be the simple A-module corresponding to the г-th
summand in the decomposition Aq = 0^=1 Hom(£'i, Ei). Then Koszulness of A is
equivalent to the condition Ext^(Si, Sj) — 0 for n ф i — j. This leads to a natural
interpretation of Koszulness of A in terms of mutations (see [30], Cor. 7.2).
Next, we are going to explain a way of thinking about Koszul algebras in Ik-
linear monoidal categories in terms of monoidal functors from a certain universal
category. Let Cuben denote the category of n-dimensional commutative cubes of
finite-dimensional vector spaces over Ik. An object of Cuben is a collection of finite-
dimensional vector spaces (Vj,I С [1,^]) and morphisms ajtj : Vj —> Vj for

9. KOSZUL ALGEBRAS AND MONOIDAL FUNCTORS
51
I С J, such that for / с J С К one has olj^k — &j,k ° &i,j- For n = 0 we define
Cubeo to be the category of finite-dimensional к-vector spaces.
There is a natural tensor product operation
(g) : Cubem x Cuben —> Cubem+n
given by
(£/.) ® (V.) = (W.),
where Wju(j+m) = Uj®Vj and Ic[l,m],Jc[l,n]. Now let P(l) be the object of
Cubei corresponding to the diagram Ik -^—> Ik. Then we can define a new monoidal
structure on Cube. := 0n>o Cuben by setting
Х*У = Х0Р(1)0У,
so that Cubem * Cuben С Cubem+n+i. Note that this new monoidal structure has
no unit object.
For every subset / С [1, n] we denote by S1 = 57(n) the simple object of Cuben
with Sj = к and Sj — 0 for J ф I. Let P1'(n) = P1 be the projective cover of
57, so that Pj = к tor I С J and Pj = 0 for / (jL J. For г = 1,... , n we denote
pi _ р{г} j^ is ciear that P1,... , Pn is a distributive collection of subobjects in
P0. We claim that this collection is universal in the following sense.
Lemma 9.1. A collection of subobjects Ri,... ,Rn С X of an object in a
k-linear abelian category С is distributive iff there exists an exact functor F :
Cuben —> С such that F(P0) = X and F(Pl —> P0) = Rz ^ X.
Proof: The "if part follows immediately from the fact that (P1,... , Pn) is
distributive in P0. Conversely, for an arbitrary collection of subobjects (Pi,... , Rn)
in X we define the exact functor from Cuben to the category of complexes over С
by sending an object (Vj, J С [1, n]) to the following complex:
C{V.)1 = ®jux=[i,n],|jnK\=iVj ®X/J2 Rj,
where the differential components are of the form
Vj ®X/ J2 Rj -^ Vj'®*/ Y. Ri>
jeK jeK'
for J С J7, К С К'', and | J' \ J\ + \K' \ K\ — 1, where the above map differs from
the natural one by the sign e — (—1)IJHK \K\. Proposition 7.2 (c) (or rather, its
analogue for abelian categories) implies that for every simple object S G Cuben
one has HlC(S) — 0 for i > 0. Hence, this is true for every object of Cuben and
the functor Vm i—> H°C(V.) is exact. One can also easily check that Н°С(Рг) ~
Ri. □
Remark 2. In the case when С = Vect^ the category of finite-dimensional
k-vector spaces, the exact functor F : Cuben —> Vect^ corresponding to a
distributive collection (Pi,... ,Pn) of subspaces in X is represent able. Namely, we
have F(V9) = Homcuben(-Pj^«)> where the projective object P G Cuben is defined
as follows. For every / с [l,n] we set Pj = (X/^^jRi)* with natural maps
Pj —> Pj for I C J. Note that we can define this object Pj e Cuben for an
arbitrary collection (Pi,... , Pn). The above lemma implies that Pj is projective
iff the collection is distributive.

52
2. KOSZUL ALGEBRAS AND MODULES
Theorem 9.2. Let С be a k-linear monoidal abelian category. Then the category
of Koszul algebras in С is equivalent to the category of exact monoidal functors
(Cube.,*) ->C.
Proof: Let 5(0) = Ik be the ground field viewed as an object of Cubeo- Then
5(0) * 5(0) = P0(l) G Cubex and we have an exact sequence
0 —> Px(l) —> P0(l) —> 50(1) —> 0.
Now for an exact monoidal functor F : (©n>0 Cn, *) —> С we set
У = F(5(0)), Я = F(Pl(l)) С F(P0(1)) ^УоУ,
We claim that the quadratic algebra (V, R) is Koszul. Indeed, note that 5(0)*(n+1)
= P0(n) G Cuben and
5(0)*(i"1} * Р\1) * 5(0)*(n~i} ~ Р\п) С Р\п).
This immediately implies that the collection of subspaces (9.1) is distributive.
Conversely, if (V,R) is a Koszul algebra in С then for every n by Lemma 9.1
we have an exact functor Fn : Cuben —> С sending P0(n) to y°(n+1) and Рг(п) to
R(n+i) .= yo(i-i) 0jRoyo(n-i) We claim that the induced functor (Fn) : Cube. -►
С has a monoidal structure. Indeed, if U. G Cubem and W. G Cuben then we have
a natural morphism of complexes
C(U.) о C(W.) ^ C(U. *W.)
with components
[Uj ® t/°^+i7 J2 4m+1)] о [Wj, ® y°^+i7 j] 4?+1)] —
c/j 0 Wj. ® y°(-+-+2)/ ^ я(-+п+2)>
iGKU(X'+m+l)
One can easily check that this morphism is a quasi-isomorphism, hence it induces
an isomorphism
Fm(U.) о Fn(W.)=^Fm+n+l{U. * W.).
П
Remark 4. The infinitesimal bialgebra Уд associated with a Koszul algebra
A — (У, P) (see section 8) can be defined also in the more general context considered
above. Moreover, it has a natural interpretation in terms of the corresponding
monoidal functor F : Cube. —> C. Namely, we claim that it is the direct sum of the
values of F on all simple objects of Cube.. Indeed, the construction of Lemma 9.1
shows that for / С [1,тг] one has
F(SJ) = кег(1Л<п+1)/ J2 fljn+1) — 0 Уо(п+1)/[^П+1)+Е^П+1)]
(where we use the notation from the proof of Theorem 9.2). These are exactly the
components of Уд (see (8.1)).
Remark 4. According to the theorem any exact monoidal functor G : Cube. —>
Cube, generates an operation on Koszul algebras which acts on the corresponding
monoidal functors by F i—> FoG. Also, G itself can be viewed as a Koszul algebra
in Cube.. Similarly, a Koszul algebra in Cube. 0k Cube, gives rise to a binary
operation on Koszul algebras. Here is an example of a Koszul algebra in Cube.

10. RELATIVE KOSZULNESS OF MODULES
53
associated with a positive integer d: V = P®(d), R = ker(P®(d) —► S0(d)). The
corresponding operation is the (d+ l)-th Veronese power (see chapter 3).
Example 3. Malkin, Ostrik and Vybornov in [76] use a certain simple Koszul
algebra in the category of representations of the quantum group SLq(2) (where q is
not a root of unity) to derive Koszulness of the preprojective algebra of a quiver not
of Dynkin type (without loops). Here is a sketch of their proof (note that a different
proof of a particular case is given in [81]). Fix q G k* and let (^Oi^-y^n be a
collection of finite-dimensional vector spaces equipped with nondegenerate bilinear
forms Eij : V^ 0 Vji —> к such that for every г one has
n
YJ^{ElJ{EJl)-1) = -q-q-\
i=i
Since E^ identifies Vji with V*j we have a canonical tensor e^ G Vij®Vji. Consider
the quadratic algebra П with П0 = kn, Hi = ф. • Vi3 and with n defining relations
n
У c-ijCji U, о — l....,7T/.
i=i
The main assertion is that the algebra П is Koszul provided q is not a root of unity
(all preprojective algebras of quivers that are not of Dynkin type appear in this
family of algebras). The main idea of the proof is that by the work of Etingof and
Ostrik [47] the above data gives a tensor functor from the tensor category Cq of
representations of SLq(2) to the category of kn-bimodules. One easily checks that
П is the image under this functor of a certain quadratic algebra П in Cq. When q
is not a root of unity the algebra П turns out to be the analogue of the algebra of
polynomials in two variables: П = 0n>o Ln where Ln is the irreducible (n + 1)-
dimensional representation of SLq(2). Its Koszulness is checked in the same way
as for the usual algebra of polynomials: one uses exactness of the Koszul complex
0 —> П(-2) —> Li ® П(-1) —> П —> lCq —> 0.
Since exact monoidal functors preserve Koszulness, this implies that the algebra П
is also Koszul.
10. Relative Koszulness of modules
In this section we explain the notion of relative Koszulness introduced by
R. Bezrukavnikov (see [29]).
Proposition 10.1. Let A be a graded algebra. R (resp., L) a nonnegatively
graded right (resp., left) A-module. Then nonzero spaces ToqJR. L) are
concentrated in the region i ^ j and for the diagonal part one has
Tor^№L)^(^!0A-L!):
where L! = (qAL)\qA) and д"' = (<U°p#)(qA°p)'
Proof: The spaces Тог^-(Я, L) are computed as homology of the bar-complex
Barm(R,A,L), where
Barij{R,A,L)= ^2 RP®Akl®---®Aki®Lq.
p+fciH \-ki-\-q=j,ks^l,p,q^0

54
2. KOSZUL ALGEBRAS AND MODULES
So, the complex itself is concentrated in the region required and the diagonal Tor-
spaces are just the kernels of the morphisms Ro 0 Afг 0 L$ —> R 0 A®1'1 0 L. □
Theorem 10.2. Let A = {V, 1} be a Koszul algebra, R = (Я', К')а°* a Koszul
right A-module, and L = (Hn', K")a o» Koszul left A-module. Then for every n ^ 0
the following conditions are equivalent:
(a) Тог£п(Я, L) = 0 for all 0 < г < n;
(b) the comultiplication maps Torf (R,L) —> Тог^(Я, к) 0Tor^,(Ik, L) are
injective in the internal degree n for all i' + i" = i, where i1', г" ^ 0;
(c) t/ie degree-n component of the Koszul complex K^(R,LlA)
(10.1)
0 > Ro 0 L-* ► - • • > Яп_! 0 L'* > Rn 0 Lq* > 0
is exact at all terms except for the first and the last;
(d) the collection of subspaces
(K'tSV®"-1®!!", г = 0
Xi = < н' 0 v®*-10/0 v®n-1-10 я", i = i,..., n -1
[я/0У^п-10К//, г = п
m Я' 0 V®n 0 Я" zs distributive.
Proof: (a) ==> (b). This follows from Proposition 10.1.
(b) ==> (a). Since Я and L are Koszul modules, the spaces Тог^-,(Д,к) and
Tor£ -,,(k,L) are concentrated on the diagonals.
(a) Ф=> (c). The complex K^(A,Ll) is a free graded resolution of a Koszul A-
module L. Therefore, the homology spaces of the complex K^(R,L[) = R<8>a
K^{A,V) are isomorphic to Тог^(Я, L).
(c) Ф=> (d). Set W = Hf 0 V®n 0 Я" Since the modules R and L are Koszul, any
proper subcollection of the collection Xo,..., Xn in W is distributive. Now we
observe that the Koszul complex (10.1) coincides with the complex K. (W; Xo, ..., Xn)
with the first and the last terms deleted (see Proposition 7.2 of chapter 1).
(а) Ф=> (d). The degree-n component of the bar-complex Barm(R,A,L) coincides
with the complex B.(W; Xo,..., Xn) with the rightmost term deleted (see
Proposition 7.2 of chapter 1). □
Definition. If the equivalent conditions of the above theorem are satisfied for all
n > 0 then we say that the pair (Я, L) is relatively Koszul
A nice example of a relatively Koszul pair of modules is given in [29], Rem. 2.5.
In this example A, R and L are homogeneous coordinate rings of X, У and Z,
where X is the projectivization of the orbit of the highest weight vector, У С X is
a Schubert variety and Z С X is the opposite Schubert variety.

CHAPTER 3
Operations on graded algebras and modules
The class of Koszul algebras is closed under a large set of operations on graded
algebras [20, 77, 78]. In section 1 we will consider the simplest of these operations
such as the direct sum, the free product and a family of tensor products. Then
in section 2 we will discuss Segre products and Veronese powers. Almost all the
results of sections 1 and 2 are due to J. Backelin and R. Froberg [20]. In section 3
we study homological properties of Segre products and Veronese powers following
the ideas of J. Backelin [16]. In section 4 we discuss the internal cohomomorphism
operation (introduced by Manin) closely related to the Segre product. Generalizing
computations of [101] we show that this operation can be used to describe coho-
mology of the Segre product of two algebras one of which is Koszul. Finally, in
section 5, following Piontkovskii [92], we apply our study of operations to produce
a pair of quadratic algebras A and A! such that A is Koszul, A! is not Koszul, but
Ha{z) = hA'{z) and hA\{z) = hA,<(z).
1. Direct sums, free products and tensor products
Let A and В be graded algebras (as before, we assume that A{ — B^ — 0 for
i < 0 and Aq = Bo = k). Then there are essentially three different constructions of
graded algebras A • В with (A * B)i = A\ ® B1.
Definition 1. The direct sum (categorical product) A\lB is the graded algebra with
(А П B)q = Ik and (А П B)i = Ai@ B{ for i > 0, where the products A+ • Б+ and
Б+ • A+ are set to be zero. The free product (categorical coproduct) A U В is the
associative algebra generated freely by A and B. Explicitly, we have
(1.1) AUB = 0 А%£1®{В+®А+)т®В®£\
i^0;£i,e2e{0,l}
Finally, there is a family of q-tensor products A®q В with q G P£. By definition,
A (g)9 В is the quotient algebra of A U В by the relations ba — qbaab — 0 for a G Aa
and b G B^ (for q — oo the relation becomes ab — 0). We will use a special notation
for the one-sided products А Л В := i0° В and Ач В := A (g)°° В.
For q ф oo the multiplication map A <g> В —> A <g>q В is an isomorphism of
graded vector spaces. In terms of this identification, the multiplication in A <g>q В
is given by the formula (oi (g)9 b1)(a2 ®q b2) = qblE^(a1a2 <S>q bib2). Note that
the complete family of algebras A ®q B, gePj[ cannot be viewed as a family of
multiplications on the same graded vector space: the grading components {A®qB)n
form a nontrivial vector bundle 0J+fc=n Aj 0 5fc0 0(jk) over P£.
All of the above operations have a computable effect on Hilbert series. Namely,
one can immediately check that
hAnB{z) = hA{z) + hB(z) - 1,
55

56 3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
hA®*B{z) = hA(z)hB{z).
Also, from (1.1) one can easily derive that
(1.2) hAuB(z) = {hA{z)~l + hB(z)-1 - I)"1.
Definition 2(M). Let M and N be nonnegatively graded modules over graded
algebras A and В equipped with isomorphisms Mq ~ N0 ~ H, where H is a fixed
vector space. For each of the operations • = П, U, Л, V, we are going to construct
a nonnegatively graded A*B-mod\i\e M*#iV with (M*#7V)0 = H. Let us denote
M+ = ®Zi Mi (г«Ф, N+ = ®Zi Ni).
(i) For • = П we set
M Пн N = H © M+ 0 7V+ = M 0 N+ = M+ 0 N.
The А П Б-module structure is defined by letting A act by zero on 7V+ and В act by
zero on M+. Note that there is an embedding of АПБ-modules M\1HN —► M@N,
where M (resp., N) is an А П Б-module with the trivial action of В (resp.. A).
(ii) For i = Uwe set
MUHN ={BU M+) 0 Я 0 (A U ЛГ+),
where for an A-module P (resp., Б-module Q) we denote В \J P = (A\J В) (8 a Р
(resp., AUQ = {AUB)®B <?)• It is easy to see that BUP = P®(AU (Б+ ® Р))
as an A-module. Hence, we get an identification
M U# N = M 0 (A U (Б+ О M+ 0 JV+))
that allows us to view M U# iV as an A-module. Similarly, we get a Б-module
structure from the identification
M UH N = (B U (M+ 9i+0 JV+)) 0 N.
In fact, these actions come from an A U Я-module structure. There is a natural
surjection of АиБ-modules AuN@BuM —> MUHN. Its kernel is the free AUB-
submodule generated by the image of the embedding H —► Mo0iVo : h \—> (h, —h).
(iii) For * = Л we set
M Ля N = M ф А (8 7V+ = M+ 0 A+ (8 7V+ 0 AT,
where A acts on A 0 ЛГ+ as on the free module generated by ЛГ+ and В acts on
M+ фА+0 ЛГ+ by zero. As an A-module (resp., Б-module) M Д# N can be
identified with a quotient (resp., submodule) of the А Л Б-module M 0 A <g> N.
Similarly, for • = V we set
M V# N = M 0 Б+ (g) M+ 0 N+ = В (g) M+ 0 iV,
so that M Уя iV - iV Ля М as a module over АуБ-БаА
Definition 3(M). For a graded A-module M, a graded Б-module N and any g G Ik*
we define an A <g>q Б-module structure M <8>q N on the graded vector space M <g> N
by the rule (a <S>q b)(m <g>q n) = qbrn(am <g>q bn). Note that from our point of view
the analogous construction for q = 0 is badly behaved. For example, it is easy to
see that the A (8° Б-module M <g>° N is never quadratic, unless the A-module M is
free or the Б-module N is trivial.

1. DIRECT SUMS, FREE PRODUCTS AND TENSOR PRODUCTS 57
Proposition 1.1. For any graded algebras A and В and any q ePj, we have
natural isomorphisms of bigraded algebras
Ext^nB (Ik, Ik) ~ Ext^ (Ik, Ik) U Ext^ (Ik, Ik),
Ext^uB (Ik, Ik) ~ Ext^ (к, к) П Ext^ (Ik, Ik),
Ext^,B(l,k) ~ Ext^(k,Ik) 09 ' Ext^(k,k).
Now let M (resp., N) be a nonnegatively graded A-module (resp., В-module)
equipped with an identification M0 ~ H (resp., Nq ~ H) for a fixed vector space
H. Then for any q G Ik* there are natural isomorphisms of bigraded modules
Ех^пв(МПя TV,к) ~ Ext*A(M,k) U#* Ext^(iV,Ik),
Ext^uB(M ия N,k) ~ Ext^(M,k) Пя* Ext^(iV,Ik),
Ext^AjB(M л# N,k) ^ Ext*A(M,fc) Уя* Extb(iV,Ik),
Ext*A(gfqB{M 09 TV, к) ~ Ext^(M,k) 09'1 Ext*B{N,k),
where Ext^(M, Ik) and Ext*B(N,k) are considered as modules over Ext^(Ik, Ik) and
Ext^(k, Ik), respectively. The symbol 0Z denotes the multiplication rule
(6 ®z T7i)(6 ®z m) = (-ir^%i6 Г vim),
where £ G Z^o is the homological grading of £ and £ G Z^o is the internal grading.
Sketch of proof: First of all, it is not difficult to extend our operations to differential
algebras and modules and check that they commute with passing to the cohomology.
Then one should construct natural morphisms of differential algebras
Cob9(A*B) —> Cob9 (A)* Cob9 (B)
and morphisms of differential modules
Cob9 (A*B,M *я N) —> Cob9(A, M) *я* Cob9(В, N),
Cob9(A ®q Б, M®q N) —> Cob9(A, M) 09'1 Cob9(B, N),
where Cob9{A,M) and Cob9(B,N) are considered as differential modules over
Cob9(A) and Cob9(B) and the correspondence * i—► * is denned by the statements
of the proposition. The last step is to check that the morphisms of complexes
constructed induce isomorphisms of the cohomology.
Let us consider in detail the most difficult case of the algebra A <S>q В and the
module M 09 N. The so-called "shuffle product" morphism
sh : Barv (A, M) 0 Ban» (Б, N) —> Вагг,+г„ (A ®q Б, М 09 N)
is given by the rule
sh : (ai 0 • • • 0 a*/ 0 ra) 0 (6i 0 • • • 0 bi>> 0 n)
i—> g-(6i+-+^,,)^ai 0 • • • 0 a? 0 bi 0 • • • 0 hi» 0 ra 0 n
_ д-(Ь1+-+ь,„)т-^ь1а1 ф ... ® a.,_1 0 6: 0 a^/ 0 62 0 • • • 0 h" 0 m 0 n + • • • ,
where the summation is over all the (г J ) permutations of the tensor components
ai, ... , ay and bi, ... , bi" preserving the order within each group (i.e., over
shuffles). The components m and n always stay at the very end. The rule for the
coefficients is the following: any permutation of components as and bt generates

58
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
the factor —q~asbt in the coefficient. The permutation of m through b\ ® • • • ® by
generates the factor that appears with the first summand in the above formula.
The desired morphism for the cobar-complexes is the dual map to sh. It is
straightforward to check that it commutes with the differential and the multiplication.
The following trick helps to prove that it is a quasi-isomorphism. Using the same
formula as above one can define a morphism
(1.3) Bar .{A, M) ® Bar.{B, N) —> Bar .{A ®q В, М ®9 N)
of the bar-resolutions. The tensor product of complexes on the left has a
natural A <S>q Б-module structure. Moreover, both sides are free resolutions of the
A <S>q Б-module M ®q N. Hence, (1.3) is a homotopy equivalence. It remains to
observe that our morphism of cobar-complexes is obtained by applying the functor
Нот.А®яв(—?к) to the morphism (1.3).
For other operations *, the construction of the morphism on the level of cobar-
complexes is clear. For An В and Mn#iV we even get an isomorphism of complexes.
In the remaining cases an explicit homotopy can be constructed. For • = U one
can also use the same argument as above, since the operation M U# N takes free
graded modules to free graded modules. □
Corollary 1.2. We keep the notation of Proposition (1.1). Each of the
algebras A\1B, AuB, A®q B, q eFl is quadratic (resp., Koszul) iff both algebras A
and В are quadratic (resp., Koszul). The quadratic dual algebras are given by the
formulas {АЙВ)1 = AlUB[ and {A®q B)' = A1 (gr*-1 B[. If A and В are quadratic
(resp., Koszul) then each of the modules M П# N, M U# N, M Д# N is
quadratic (resp., Koszul) iff both modules M and N are quadratic or (resp., Koszul). If
M ф 0 and N ф 0 then the analogous statement is true for the A (g)9 В-module
M& N, where q G Ik*. □
For example, for each of the operations * = П, U, Л we have
(tf, K')A *н (Я, K")B ~ (Я, K' © K")AieB,
where K' 0 К" С (А\ 0 B{) (g> H. Note that for all the above operations except
for (g)9, where q G Ik*, it is easy to verify the statements of the above corollary
explicitly: first, one needs to check that these operations preserve the classes of
quadratic algebras and modules, and then use Backelin's criterion of Koszulness
in terms of distributive lattices (Theorems 4.1 and 4.2 of chapter 2). Indeed, the
space (A*B)fn = (A\ 0 Bi)®n is the direct sum of several tensor product spaces,
including Afn and Bfn. It is easy to see that each of the subspaces X{ С (A*Bi)®n
from Theorem 4.1 of chapter 2 is a direct sum of the corresponding subspaces in the
direct summands of (A 0 B)fn. The subspaces we get in Afn and Bfn coincide
with the subspaces Xi for the algebras A and B. The collections of subspaces in
other summands are distributive provided that the collections of subspaces in A®*
and Bf^ corresponding to the algebras A and В are distributive for j < n (see the
remark on distributivity of direct sum collections in the beginning of section 7 of
chapter 1).
Koszulness of the tensor product of Koszul algebras A <S>q В can be derived
from Koszulness of A 0° В using the standard spectral sequence associated with
a filtration (see Example 3 in section 7 of chapter 4). We do not know any
simple lattice-theoretic explanation for Koszulness of the tensor product of modules
M®q N.

2. SEGRE PRODUCTS AND VERONESE POWERS. I
59
2. Segre products and Veronese powers. I
Definition 1. The Segre product of two graded algebras A and В is the graded algebra
А о В = 0^LO An ® Bn. For a pair of graded modules M and N over A and B,
respectively, their Segre product is the graded АоБ-module MoN = ®nGZ Mn0iVn.
By definition, А о В (resp., MoN) is a subalgebra of the tensor product А® В
(resp., submodule of M 0 N). However, these embeddings are not morphisms of
graded algebras or modules, since they double the grading.
One can define similarly g-Segre products AoqВ с A®qВ for q G Ik*; however,
for simplicity we restrict ourselves to the case q — 1.
Proposition 2.1. (i) If both algebras A and В are one-generated, quadratic,
or Koszul, then the algebra AoB is of the same type. More generally, if the algebra
A is one-generated then the generator spaces of the algebra AoB are given by the
formula Тог1?°Б(1к, к) = Ап 0 Torf n (Ik, Ik). If both algebras are one-generated then
there are natural exact triples for the relation spaces
0 > Tor£jk,k)0Tor*n(k,k) > Tor£°B(k,k)
> Tor£Jk, k) 0 Бп ® An 0 Tor^Jk, Ik) > 0.
(ii)(M) If both A-module M and В-module N are generated in degree zero (resp.,
quadratic, resp., Koszul) then the same is true for the А о В-module MoN. More
precisely, if the algebra A is one-generated and the A-module M is generated in
degree zero then the generator spaces of the module MoN are given by the formula
Tor£°nB(M о TV, Ik) = Mn 0 Torjfn(iV,k). If both algebras are one-generated and
both modules are generated in degree zero then there are natural exact triples
0 > Tor^n(M,k)0Torfn(iV,k) > Tor*°nB(MoN, к)
> Tor^n(M, к) 0 Nn 0 Mn 0 Torf n(N, k) > 0.
Proof: (i) For any graded algebra С one has Torfn(k,k) = (C+/C+C+)n. Hence,
in the case when A is one-generated we have
/ i к > 1 \
Torf °nB(k, k) = An ® Bn/ V ; " AjAk ® BjBk
= An 0 Sn/ ( X)i+fc=n An ® ^'Sfc ) = An ® Torfn(^^)-
For the rest of the proof we assume that both A and В are one-generated.
Assume that A and Б have no denning relations of degree k. Let us check that the
same is true for the algebra AoB. For a graded algebra С let us denote
Jf :=kev(Cfl -^Сг).
Note that С has no defining relations of degree к iff the natural map
(2.1) d ® Jfcc_! © Jf_! <8> Ci —> j£
is surjective. Now for every i > 1 we have an exact sequence
о —> jf 0 jf —> JzAoB —> J? 0 Bi e ^ 0 jf —► o.
Using these sequences it is easy to show that surjectivity of (2.1) for both algebras
С = A and С = В implies its surjectivity for С = А о В.
For a graded algebra С let C^ denote the graded algebra with the same
generators and relations of degree < n as in С and with no generators or denning

60
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
relations of degree > n. The above argument applied to A^ and B^n\ where
n > 2, shows that A^ о В^ has no denning relations of degree > n, and hence
(АоБ)(п) ~ A(n) оБ(п>
for every n > 2. Note that for a one-generated algebra С there is an exact sequence
0 —> Tor£n(k,k) —> C<"> —> Cn —► 0
(see the end of section 5 of chapter 1). Applying this to С = А о В we see that
Tor^n(Ik, Ik) is isomorphic to the kernel of the natural map An
Using the above exact sequence for С = A and С = В we obtain the required exact
sequence for the relation spaces.
Finally, to prove Koszulness, we use Backelin's criterion (see ch. 2, Theorems
4.1 and 4.2). Distributivity of the relevant collections can be easily checked using
Proposition 7.1 of chapter 1.
The proof of (ii) is completely analogous. □
Note that in contrast with operations considered in section 1, Koszulness of the
Segre product А о В does not imply Koszulness of A and B. Corollary 4.9 below
describes completely the situation assuming that one of the factors is Koszul.
Definition 2. For any integer d > 1, the Veronese subalgebra of degree d of a
graded algebra A is the graded algebra A^ with the components An ' = Adn. The
Veronese submodule of a graded A-module M is the graded A^ -module M^ =
0nGZ Mdn. By definition, A^ is a subalgebra of A and M^ is an A^-submodule
of M but these embeddings are not compatible with the grading.
Proposition 2.2. (i) Veronese subalgebras of a one-generated (resp.,
quadratic, resp., Koszul) algebra A are still one-generated (resp., quadratic, resp., Koszul).
Moreover, if A is one-generated and has no defining relations of degree > (k — l)d+l
then A^ has no defining relations of degree > k. In particular, if A is one-generated
and has no defining relations of degree > d + 1 then A^ is quadratic.
(ii)(M) Assume that A is one-generated. If an A-module M is generated in degree
zero (resp., A and M are quadratic, resp., A and M are Koszul) then all Veronese
submodules M^ are also generated in degree zero (resp., quadratic, resp., Koszul).
If M is generated in degrees ^ kd then M^ is generated in degrees ^ k. Finally,
assume that M has no defining relations of degree > Id, and either M is generated
in degree zero, or M is generated in degrees ^ kd and A has no defining relations
of degree > (I — k)d + 1. Then M^ has no defining relations of degree > I.
Proof: Assume that Л is a one-generated algebra. Then all the maps Ai <S> Aj —►
Ai+j are surjective; hence the algebra A^ is also one-generated. Assume in
addition that A has no defining relations of degree > (k — \)d + 1. Let us show
A^ has no defining relations of degree > k. Using the notation from the proof of
Proposition 2.1 we have to show surjectivity of the morphisms
for all n>k. The natural projections Afdn —> Afn = A[d)®n map j£d surjectively
onto j£ • Therefore, it suffices to check that the morphisms
(2.2) Af ® J?n.1)d 0 Jfn_1)d ® Afd > J*d

2. SEGRE PRODUCTS AND VERONESE POWERS. I
61
are surjective for n > k. Our assumption about A implies that the morphisms
A1®j£l_1®j£l_1®A1 — Ji
are surjective form> (k — l)d+l. This easily implies surjectivity of the morphisms
® t=o Л®' ® Jm"s ® Л®°~Ь * Jm
for m — s ^ (fc — l)d+ 1. It remains to observe that for s = 2d — 1 and m = nd the
latter morphism factors through (2.2).
Note that if Л is one-generated then an A-module M is generated in degree ^ n
iff the maps A\ ® Mm_! —► Mm are surjective for m > n. In this case the maps
A$ (g) Mm_s —► Mm are surjective for m — s ^ n. To show that the module M^
is generated in degrees ^ fc, take 5 = d and m ^ (k + l)d and use the assumption
that M is generated in degrees < kd.
Furthermore, for an A-module M generated in degree zero consider the A-
module M' = ker(A (g> M0 —> M). Then the relation spaces of the module M can
be identified with the generator spaces of the module M'. Since M'^ = M^', we
get the desired estimate for the relations of the Veronese submodule of the module
generated in degree zero.
Now assume that we have a surjective morphism A<g>X —► M with the kernel
M'', where the graded vector space X is concentrated in the degrees ^ kd and the
A-module M' is generated in degrees ^ Id. Consider the composition of surjective
morphisms
Q AW®A.3modd®X3 ^ (A®X)W _£-> M<d>,
where 0 ^ —j mod d ^ d— 1. We have to prove that its kernel M^' is generated in
degrees ^ /. Clearly, it suffices to prove that both A^-modules ker(/) and ker(g)
are generated in degrees ^ /.
We have ker(g) ~ M'^\ hence it is generated in degrees ^ / (since M' is
generated in degrees ^ Id). For the module L = ker(/) we have
L = ф L(-3 mod d) ® Xj,
J
where L(r) = ker(A^ <g> Ar —> A). It remains to show that the A^-modules L(r)
are generated in degrees ^ I — к provided A has no defining relations of degree >
(I — k)d + 1. Let us check that the maps Ad ® Z/(r)n_i —► L(r)n are surjective for
n > l — k. The restriction of the multiplication map Af)nd+r —> And <S> Ar provides
a surjective map J^d+r —> L(r)n- For n > I — к the multiplication map
\j) A\s ® J(n-i)d+i ® A > J-nd+r
s+t=d+r-l
is also surjective. Using the embedding L(r)n —► And <S> Ar it is easy to see that
the composition Afs ® ^_iw+i ® ^f* —* ^{r)n is zero for t ^ r, and factors
through Ad ® b(d)n_x for s ^ d. At least one of these conditions holds for any
s + t = d + r — l,so the assertion follows.
To prove Koszulness we apply Backelin's criterion. For a quadratic algebra A
and a quadratic A-module M, the collection of subspaces X{ , ... , Xn of the
vector space W^ = A^n®M0 corresponding to the algebra A^ and module M(d)

62
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
is connected with the collection of subspaces X\, ... , Xnd С W = Afn 0 M0
corresponding to the algebra A and the module M in the following way:
W{d) = W/Y" Xk- X(d) = Xdi mod V Xk.
Clearly, such a procedure preserves distributivity. □
Remark 1. The complexity of the operations А о В and A^ in comparison
with the operations considered in section 1 is manifested by the fact that it is
in general impossible to recover the spaces Ех^оБ(к, к) or Extj?(d) (Ik, Ik) from the
Ext-spaces of the algebras A and B. For example, the space Ext^;?2) (Ik, Ik) for a one-
generated algebra A can be described as follows. It has a natural 3-step nitration
0 С Fi с F2 С F3 = Ext^2) (Ik, Ik) such that Fx ~ Ext^2i(k, Ik), F2/F1 is isomorphic
to the kernel of the multiplication map
Ext^(k,k) ®Ех^-1(1к,1к) ® Ext^2i_1(k,k) ® Ext^(k,k) > Ext^2i(k,k),
and F3/F2 is isomorphic to the kernel of the Massey product morphism from the
kernel of the multiplication map
Ext^1 (к, к) (8 Ext^'2-7'-2 (к, к) (8 Ext^1 (к, к)
> Ех^д(к,к) ® Ext^"1^,^) ©Ext^'^k) (8Ext^(k,k)
to Ext^2-7(Ik,Ik) (see Appendix).
Now let us describe the space
Тог^°Б(к, к) = (A3 ® B3)/(A1A2 ® BXB2 + A2A1 + B2BX)
(cf. Remark 1 in section 2 of chapter 2). It has a natural subspace coming from
{AiA2 + A2Ai) (8 (BiB2 + B2Bi) С А3 (8 B3. The quotient by this subspace can
be identified with the kernel of the map
A3 (8 B3 > Tor^3 (к, к) (8 Torf 3 (к, к),
so its dimension is determined by the dimensions of the homology spaces of A
and B. However, the subspace itself is isomorphic to
{A1A2/A2A1) <g> {B2B1/B1B2) © {A2A1/AlA2) ® {B1B2/B2B1),
where we denote U/V = (U + V)/V\ It is easy to see that the space AiA2/A2Ai
is isomorphic to the cokernel of the comultiplication map
Tor£3(k,k) > Tor^(k,k)(8Tor^2(k,k).
On the other hand, as we will show in the next section, the estimates for
the generators and relations of Veronese subalgebras and submodules obtained in
Proposition 2.2 can be generalized to higher homology spaces Tor$ (k,k) and
Tor^ (M^,k). The situation here is similar (although much more complicated)
to the case of truncated modules M^ considered in section 1 of chapter 2. Roughly
speaking, we will show that the algebras A^ and modules M^ for a one-generated
algebra A and an A-module M generated in degree zero become closer and closer
to being Koszul as d grows to infinity (one can consider Proposition 1.1 in section
1 of chapter 2 as a toy version of this result). Note that the condition that A is
one-generated cannot be omitted here: for the free associative algebra A with one
generator of degree 1 and one generator of degree 2 all the algebras A^d\ d ^ 2
are infinitely generated. Similarly, we will estabish in the next section estimates for

3. SEGRE PRODUCTS AND VERONESE POWERS. II
63
the vanishing region of higher Tor-spaces of the Segre product algebra А о В and
module M о N.
Remark 2. For a commutative or skew-commutative one-generated algebra A
one can get slightly stronger estimates for the degrees of relations of the Veronese
subalgebras and submodules. Namely, the Veronese subalgebra A^ has no defining
relations of degree > /с, к ^ 2, provided the algebra A has no defining relations of
degree > kd. The module M^ has no defining relations of degree > / provided
M has no generators of degree > (I — \)d and no denning relations of degree > Id
and a(M) + b(A) ^ Id, where a(M) is the maximal degree of a generator of M not
divisible by d and b(A) is the maximal degree of a defining relation of A. The proof
is analogous to that of Proposition 2.2, with the following changes. To prove the first
assertion one should consider the algebra Л as a quotient algebra of the symmetric
algebra S(Ai) (or the corresponding exterior algebra). Since the symmetric algebra
is quadratic, it suffices to estimate the degrees of generators of the corresponding
ideal in S(Ai). For the second statement one has to show that the A^-module
0n And+r has no relations of degree > t, t ^ 1 (that is, the module L(r) has no
generators of degree > £), if the algebra A has no relations of degree > td + r. To
do this consider the algebra A^ as a quotient algebra of S(Ai)^^ and the module
©n And+r as a quotient module of 0n §nd+r(Ai) over §(Ai)^). Then observe that
the latter module is quadratic and estimate the degrees of generators of the kernel.
3. Segre products and Veronese powers. II
In this section we obtain general estimates for the vanishing region of the
homology spaces of Veronese subalgebras and submodules and Segre products of algebras
and modules. We use the following numerical measure introduced by J. Backelin.
Definition 1. The rate of growth of the homology spaces of a graded algebra A (or
simply the rate of A) is defined as the finite or infinite supremum
rate(A) = sup { ^j | Тог^.(к,к) ф 0 }.
If A is not one-generated then by definition rate(A) = +oo.
Theorem 3.1. For any graded algebra A and positive integer d one has
rate(^>) < \Щ&
I d
Proof: Obviously, we can assume that A is one-generated. Consider the spectral
sequence
E2pq = Tor£(Tor£(d)(k, A),k) =► Tor^(k,k); <Tpq: ££, — E^r^r_,
associated with the homomorphism of augmented algebras /: A^ —> A. Note
that the homomorphism / multiplies the grading by d. Therefore, the internal
grading of terms E^q should correspond to the internal grading of the Tor over
A(d) multiplied by d. In particular, it follows that the internal grading of E^q
is concentrated in degrees divisible by d. On the other hand, the A^-module
A splits into a direct sum of modules numbered by residues modulo d, and the
submodule corresponding to the residue 0 is free. Therefore, the internal grading
of Tor^ (Ik, A) lives entirely outside of values divisible by d for all q > 0.

64
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
Denote by b(q) the maximal internal grading of Tor^ (Ik, A) and by c(q) the
(d) ,
maximal internal grading of E% q = Torg (Ik, A) ®A Ik. Since A is one-generated,
it follows easily that \c(q)/d] = \b(q)/d) for any q. Finally, denote by hgA(p) the
maximal internal grading of Tor^(Ik, Ik), and analogously for A^d\ Notice that the
maximal internal grading of E*q does not exceed hgA(p) + b(q).
It follows from the divisibility observations above that the terms E^q are killed
in the spectral sequence for all q > 0. So we have
c(q) < max hgA(s + 1) + b(q - s).
Dividing by d and applying the upper integral parts, we obtain
d I
c(q)
^ max
< max (
hgA(s + l) + b(q-s)
d
max I
[Kq-
Ha(*
-s) + ll\
d \)
»+1) — 1
+
b(q - s)
since b(q — s) is never divisible by d. Analogously,
hgA{p) + b{n - p)
I *
hgA(d) (n) ^ max -
^ max
a
hgA(p)-l| , [b{n-p) + l
+
d
■1)
= max
=.(i!
НаЬ)-1 I + [b(n-p)
since [x + y\ ^ [x\ + |~y] for all real x, y.
Now let us prove by induction in q that \b(q)/d] ^ g|~rate(A)/d] + 1. By
definition, hgA(s + 1) — 1 < s • rate(A). Using the induction assumption for q — s
we get
d
^ max (
I l^s^g V
5 • rate(A)
+ (Q ~ s)
rate(A)
^ max I 5
rate (A)
+ Ы -s)
d
rate(A)
■rate(A)
d
-hi.
Finally, we can obtain the desired estimate for hgA(d)(n):
hgA(^) (n) < max ( "
< max
-л
{p - 1) rate(A)
bgA(p)-l | \b(n-p)
d J I d
rate(A)-
+ (n - p)
d
-hi
) <(n-l)
rate(A)
d
+ 1.
П
Definintion 2(M). Let A be a graded algebra and M a nonnegatively graded left
A-module. We denote by hgAM(n) the maximal internal grading of Tor^(k, M)
and define rate(A, M) as supp(hgAM(p)/p). If M is not generated in degree zero
then by definition rate (A, M) = +oo.

3. SEGRE PRODUCTS AND VERONESE POWERS. II
65
Proposition 3.2.(M). Assume that rate(A) ^ c\ andhgAM(p) ^ c\p + c2
for some constants c±, c^ and all p. Then b.gA(d)^M{d)(n) < \ci/d\n+ \c2/d\. In
particular, when c^ — 0 we obtain the estimate
-max(rate(A),rate(>l,M))-
rate(A(d),M(d)) <
d
Tor^(k,M).
Proof: Consider the spectral sequence
^ = Tor^(Tor^(d)(k,A),M) -^ AWp+g,
Notice that Tor^(d) (Ik, M(d)) is a direct summand of Tor^(d> (к, М); hence
dhgA(d);M(d)(n) < maxo<P^n(hgA5M(p) + b(n-p)),
where we keep the notation 6(g) for the maximal internal grading of Tor (
from the previous proof. Calculating as above, we obtain
Ьблл/Ср)-1 | , [b(n-p)'
,A)
O^p^n \
ClP + C2 -1
+
< max
O^p^r
(L
-h(n-p)
№1
+
, 1)
\ (\CiP + C2
1 = max }
J O^p^n V
< max
O^p^n
+
)-rti
+ {n-p)
+
П
The condition rate(A) < d is sufficient for A^ to be Koszul but by no means
is it necessary. Using the above proposition we get another sufficient condition.
Corollary 3.3. Let В —> A be a homomorphism of graded algebras. Assume
that В is Koszul and h.gBA(p) < (p + \)d for all p. Then A^ is Koszul
Proof: Indeed, from Proposition 3.2 we obtain that hgB(d) а(*)(р) ^ P + 1 for
all p. Since B^ is Koszul, this implies Koszulness of A^ by Theorem 5.2 of
chapter 2. □
For example, if A is commutative and one-generated then we can take В =
S(Ai). Since ra,te(B,A) is finite, we derive the following result.
Corollary 3.4. [16] Let A be a commutative one-generated graded algebra.
Then A^ is Koszul /or d » 0.
Further discussion of homological properties of Veronese subrings can be found
in [16, 17, 20, 46].
Now we turn to homology estimates for Segre products.
Theorem 3.5. For any graded algebras A and B, one has
rate(A о В) ^ max{rate A, rate B}.
Proof: We can assume that both A and В are one-generated. Consider the spectral
sequence
(3.1) E2pq = Tor^B(Torf B(k, A 0 B),k) =► Tor£f (к,к)
corresponding to the homomorphism of algebras f:AoB —> A<S> B. The tensor
product A <S> В is equipped with a natural internal bigrading (a, /?), and the Segre
product А о В is exactly the part of А® В where a — (3. So / is a homomorphism of
bigraded algebras, where on the first algebra both gradings coincide. Therefore, Ervq

66
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
is a bigraded spectral sequence and on the limit term E™ two gradings coincide,
E°° ap = 0 for а Ф /?, since the limit term corresponds to the homology of А о В.
On the other hand, we have the decomposition
A®B = @( 0 Aa®Bp)
ieZ a-p=i
of А® В viewed as an AoB-modu\e, where the summand corresponding to a—(3 — 0
is a free АоБ-module. It follows that Тог^оБ(к, А®В)аф vanishes for a — (3 except
for the case q = 0 = a = (3. Hence, for any q > 0 the A®B-modu\e Тог^оБ(1к, А&В)
splits into the direct sum of Тог^оБ(1к, А 0 B)a<p and Тог^оБ(1к, А 0 B)a>p.
We will need two more spectral sequences:
'E2vq = Tor*(Torf B(k, A®B),k) =► Tbr#£(k, A) = Tor£f (к, к) ®к Л and
"E2pq = ToT£(Torf°B(k,A® B),k) =► Tor^f(k,B)=Tor^f(k,k)®kB.
The limit term '.E££ is concentrated in the internal bidegrees a ^ /3. Hence, the
direct summand Tor^(Тог^оБ(к, А®В)а<р,к) of 'E%q is killed somewhere in 'Erpq
for all p ^ 0, q ^ 1.
Now let us define
a(g) = max{a | 3(3 > a : Тог^оБ(1к, А 0 B)Q/3 ф 0}
= max{a | 3(3 > a : Тог^(Тог^оБ(1к, A 0 B)Q/3,k) ^ 0}
and analogously
b{q) = max{/? | 3a > /? : Тог^оБ(1к, Л 0 B)Q/3 ф 0}
- max{/3 | 3a > /3 : Тог£(Тог^оБ(1к, А 0 B)Q/3, к) ^ 0}.
Assume that for given a < b the component Tor^ (Тог^оБ(к, А 0 £)аь,к) is
nonzero. Since it gets killed in the spectral sequence 'Er ^ there should exist
s = r - 1 such that 'E2s+lq_sab = Torf+1(Tbr£f (к,Л ® Б),к)аЬ / 0. The
latter component splits into a direct sum of two parts according to the splitting of
Tor£!f(k,i4®5) into pieces with internal bigrading a < /3 and a > (3. Hence,
one of these parts has to be nonzero. If Torf+1(Tor^!f (к, А 0 B)a<p,k)ab Ф 0
then we conclude that a < a(q — s), since applying Tor over В does not change the
а-grading. If Torf+1(Tor£!f (k, A 0 B)a>/3, k)ab ^ 0 then 6 < hgB{q - s) + % - 5)
(where we keep the notation hgA(g) from the proof of Theorem 1). In any case we
obtain
a(q) ^ max max{ a(q — s), b(q — s) + hgB(s + 1) — 1}.
As h.gA(s + 1) — 1 ^ srate(A) and the same for i?, it follows immediately by
induction simultaneous for a(q) and b(q) that
a(q), b(q) ^ gmax{rate(A),rate(i?)}.
Now let us prove the desired estimate for hgAoB(n). Here we will use the
spectral sequence Ervq (see (3.1)). Let с be a number such that E^cc Ф 0. Then
the component E^cc = Тог^Б(Тог^оБ(к, А 0 S),k)cc should also be nonzero.
As above, for q ф 0 this component splits into two direct summands corresponding
to the two direct summands of Тог^оБ(к, A 0 B). In the case q = 0 one needs an
estimate for the internal bigrading of Тог^®^(к, к), which can be easily obtained.

3. SEGRE PRODUCTS AND VERONESE POWERS. II
67
In the case q ф 0 assume that Тог^®Б(Тог^оБ(1к, А ® B)a<p,k)cc ф 0. It follows
that the tensor product of ТоГр®Б(&, к) and Тог^оБ(1к, А ® B)a<p will also have
nonzero component in bigrading (c,c). Hence с ^ maxi^t<p(hg^(£) + a(q)) and
generally
hgAosW ^ ™а* max{hgA(£) + a(n-p),hgB(£) + b(n-p)}.
Recall that hgA(s) — 1 ^ («5 — 1) rate(A) and we finally obtain the desired estimate
hgAoB(n) — 1 ^ (n — 1) max{rate(A),rate(i?)}.
□
Proposition 3.6.(M). Let A and В be graded algebras, M (resp., N) be a
nonnegatively graded A-module (resp., В-module). Assume that
max(rate(A),rate(i?)) ^ c\ and
max(hgAM(p), hgBN(p)) ^ сгр + c2
for some constants c\ and c2. Then hgAoB моы{п) ^ cin + c2- I71 particular, in
the case c2 — 0 we have
rate(A о Б, М о N) ^ max{rate(A), rate(A, M), rate(B), rate(B, N)}.
Proof: Let us keep the notation a(q) and b(q) from the previous proof. Clearly, we
can assume that A and В are one-generated. Consider the spectral sequence
E*q = Tor£®B{TorfoB{k, A ®B),M®N) => Tor££(M <8> W).
Clearly, this spectral sequence carries the bigrading (a,/?) used in the proof of
Theorem 3.5. As before we use the splitting of the А о Б-module M ® N into
the direct sum of submodules obtained by fixing the value of a — (3. Note that
the summand corresponding to a = /3 is M о N. Therefore, hgAoBMoiV(n) is the
supremum over p-\- q = n of 7 such that E^ ф 0. Thus, we have to show that for
such p, g, 7 one has 7 ^ cin + c2- We are going to deduce this from the nonvanishing
of Ep . As before, we decompose the term E^q into the direct sum of two parts
corresponding to the direct decomposition of the A ® Б-module Тог^оБ(к, А® В)
into two summands, one with a < (3, the other with (3 < a. At least one of these
summands of E% should have a nonzero component of internal bidegree (7,7), say
Тог£®в(Тог£°Б(к, A <g> B)Q</3, M <g> W)7,7 ф 0.
As in the proof of Theorem 3.5 this implies that the tensor product of Тог^оБ (к, А®
B)a<p with Тог^<8>Б(1к, M®N) has a nonzero component of bidegree (7,7). Using
the formula Тог^Б(к, M ® N) = Tor^(k, M) ® Torf (Ik, N) we conclude that
7 ^ max hg(A, M)(t) + a{q) ^ at + c2 + Ci<? ^ cin + c2.
□

68
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
4. Internal cohomomorphism
After defining the operation cohom(A, B) (due to Yu. I. Manin [77, 79]) and
establishing its relation to the Segre product, we proceed to study the cohomology
of cohom(A B). We show that in the case when В is Koszul, passing to cohomology
switches the operations cohom(A, B) and А о В (see Theorem 4.6).
Definition 1. For any integer n ) 0 we define the category 0n as follows. An
object of 0n is a sequence of integers 0 = (0i,..., 0/(0)) with /(0) ^ 0, 9i > 1, and
^2i=1 6i — n. Morphisms have the form
0 = (01, ...» 0/(0)) ► 0; = (01 H l-0ii, • ••» 0iz(e/)_i + l ^ Ъ0/(0)).
There is at most one morphism between any two objects. For any n, m ^ 0 there
is a canonical functor pn?m: 6n x 0m —> 6n+m defined by
(0i» •••^0/(0/)) x (0iV- • J0/V')) '—* (0i»--40zV)'^i'•'•'^V'))'
Also, for any n G Z we define the category On with objects 0 = (0i, • • •, 0/(0)),
where /(0) ^ 1, 0* ^ 1 for i < 1(6), 0/(0) G Z, and J^li 0г = п. Morphisms are
defined in the same way as above. Note that for any n ^ 0 and m G Z we have a
natural functor pn>m: 0n x 0m —> 6n+m.
For any small category1 E let us consider the category of covariant functors
from E to the category Vect of finite-dimensional vector spaces over Ik. For a pair
of such functors Ф, Ф: E —► Vect the set НотЕ(Ф, Ф) of morphisms of functors
Ф —> Ф has a natural structure of a vector space over Ik (infinite-dimensional, in
general). We have a natural isomorphism
НотЕ(Ф, Ф) ~ СопотЕ(Ф, Ф)\
where
СопотЕ(Ф, Ф) = coker(ff) Ф(аг) 0 Ф(а2)* -> £ft Ф(а) 0 Ф(а)*).
A covariant functor 7г: Е' —> Е" induces natural linear maps
7г*: СопотЕ,(7гоФ, 7г о Ф) > СопотЕ"(Ф, Ф)
for any pair of functors Ф, Ф: E" —> Vect.
For a pair of functors as above Ф': E' —> Vect and Ф": E" —> Vect, we
can define the functor Ф' 0 Ф"': E' 0 E" ► Vect sending a' x a" to Ф'(<7') 0
Ф"(а"). It is easy to check that for a quadruple of functors Ф', Ф': E' —> Vect and
Ф", Ф": E" —> Vect there is a natural isomorphism
СоЬотЕ/(Ф/,Ф/)0СоЬотЕ,,(Ф//,Ф//) ~ Сопот^^Ф' 0 Ф", Ф' 0 Ф").
Definition 2. For a graded algebra A we define a sequence of functors
ф£°: ©n > Vect
by setting Ф^(01,..., 0/(0)) = ^ 0 • • • 0 ^0i(0) where the action of ф(п) on
morphisms is defined using the multiplication in A. Similarly, for a graded A-module
M we have a sequence of functors
*1&I ■ ®n > VeCt
1i.e., a category in which objects form a set.

4. INTERNAL COHOMOMORPHISM 69
such that ФЦ,(0i,..., 6m) = A01 <8> • • • <8> A*^^ 0 М*1(в).
Clearly, there are natural functor isomorphisms
Now we are ready to define the cohomomorphism operations on graded algebras
and modules.
Definition 3. For a pair of graded algebras A and В the internal cohomomorphism
(or universal coacting) algebra cohom(A,i?) is defined by
cohom(A,B)n = СоЬоте-(Ф^п),Ф^)), п ^ 0.
The multiplication is given by the composition
СоЬоте-(Ф^),Ф^))0СоЬоте-(Ф^т),Ф^т))
~ Сопотв"х0~(Ф^п) <g> Ф^т), Ф^п) <g> Ф^т))
~ СоЬотипХит(Ф^ ; о рп?т, Ф^ ; о рщш)
{Р^^ Сопоте"+™(Ф^п+т\ф^+т)).
Similarly, for a graded A-module M and a Б-module JV, one can define the internal
cohomomorphism module сопотЛ'Б(М, N) over cohom(A B) such that
cohomA,B(M, N)n = СоЬот^фЦ,, Ф £°M), n e Z.
Note that cohom(A, Б)0 = Ik and cohom(A, S)i = Ax ® 5J. If Mi = Nt = 0
for г < 0 then cohom^,B(M, W)o = Mo 0 Nq . Since the category 0n has a finite
number of objects, the vector spaces cohom(A, B)n are always finite-dimensional.
The same is true for the vector spaces cohom(M, N)n, because the functors Ф^м
have nonzero values on a finite number of objects of 0n.
Proposition 4.1. (i) For any triple of graded algebras А, В, С one has a
natural bisection between the sets of graded algebra homomorphisms
(4.1) Нот(Д В о С) ~ Нот(сопот(Д Б), С).
In particular, for a pair of graded algebras A and В there is a canonical morphism
(4.2) A > Bocohom(A,B)
corresponding to the identity endomorphism of cohom(A, B).
(ii)(M) For any triple of graded modules M, N, and Q over the algebras A, B,
and cohom(A, B), respectively, one has a natural isomorphism
Hom^M, NoQ) ~Rom°cohom{^B){cohomA.B{M,N), Q),
where N о Q is equipped with an A-module structure via the homomorphism (4.2).
In particular, for any graded modules M and N over A and В there is a natural
morphism of graded A-modules
M ► N о cohomAiB(M,N)
corresponding to the identity endomorphism of cohomа,в {М, N).

70
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
Proof: A morphism of graded algebras /: A —> В о С is given by a sequence of
linear maps An —> Bn<g> Cn, or equivalently, linear maps
/n:C; > Rom(An,Bn).
On the other hand, a morphism g: cohom(A, B) —> С is presented by a collection
of maps СоЬотв«(Ф^г),Ф^1)) —» Cm, which is the same as
gm:Cm >Нотв~(Ф<Г),Ф<Г)).
Now given a sequence (gm) we define the corresponding maps fn as compositions
of gn with the natural maps
Ноте-(Ф^п),Ф^)) > Rom(An,Bn)
sending a morphism of functors to its value on the object во = (n) € Qn, l(во) = 1.
Conversely, for a sequence (fn) one can recover (gm) by looking at compositions
Cm ►Q1®"-®Ql(e) ^^^Hom^®.-^^,^^.-^^).
It is straightforward to check that the above correspondence leads to the
isomorphism (4.1). The proof for the case of modules is completely analogous. □
Remark. Dualizing the n-th component of the homomorphism (4.2) we get a
linear map
(4.3) AoB* ^cohom(A.B)
of graded vector spaces, where А о В* = 0n An ® Б*. Below we will show that
cohom(A, B) is generated as an algebra by the image of this morphism.
Corollary 4.2. (г) For any triple of graded algebras A, B, and С there are
natural (iso)morphisms of graded algebras
(4.4) cohom(cohom(A, Б), С) ~ cohom(A, В о С)
(4.5) cohom(A, С) ► cohom(£, С) о cohom(A,£)
(4.6) cohom(A, cohom(£,C)) > cohom(A, В) о С.
(ii)(M) For any triple of graded modules M, N, and P over A, B, and С there are
natural (iso)morphisms of graded modules
cohomcoho
т(А,в), с(сопотА,в(^5 -^0? P) — cohom^ Boc(M, N о P)
cohomA,c(M,P) ► coliomB,c(N,P) о cohoma,b(M,N)
cohomA>cohom(B5C)(M, cohom(VV,P)) > cohomAjB(M,N) о Р
compatible with the above homomorphisms of algebras.
Proof: Let X by either the left-hand side or the right-hand side of (4.4). Then
from the universal property given in Proposition 4.1 we get a natural isomorphism
Hom(X,D) = Rom(A, B0C0D). This immediately implies (4.4). Both morphisms
(4.5) and (4.6) correspond via the universal property to the composition
A ► Босопот(ДБ) ► С о сопот(Б, С) о cohom(A, В).
The case of modules is similar. □
In Proposition 4.3 below we will give an alternative description of the internal
cohomomorphism algebras and modules in terms of generators and relations (see

4. INTERNAL COHOMOMORPHISM 71
[77, 79]). In the case of quadratic algebras and modules this description becomes
especially simple and is related to the following construction.
Definition 4. The operation black circle product A • В := (А1 о В1)1 (see [77]),
quadratic dual to the Segre product of quadratic algebras, can also be defined in
terms of generators and quadratic relations as follows:
{V, I'} • {V\ I"} = {V ® V\ I' <8> /"}.
Note that this definition makes sense for quadratic algebras only. There is an
analogous operation for quadratic modules: MmN = (M^oJVB)^og,, or explicitly,
(tf', K')A • (H", K")B = (H' ® H", К' 0 K")A.B.
For a pair of graded vector spaces Z and T let us denote Z о Т — фп Zn®Tn.
Proposition 4.3. (i) For a quadratic algebra A and an arbitrary graded algebra
В there is a natural isomorphism
сопот(ДБ) ~ Am{qB)\
More generally, let A be a graded algebra generated by a graded vector subspace
X с A with the ideal of relations generated by a graded vector subspace Y С Т(Х).
Then for any graded algebra В one has
cohom(A, Б) ~ сокег(У о Б* -> Т(Х о Б*)),
where the map 7оБ*-> Т(Х о В*) has components
YnoB*n >0^ Xni®B*ni®---®Xn.®B*nt
given by the tensor products of the inclusion maps Yn —> Tn(X) and the comulti-
plication maps B^ —> Tn(B)\
(ii)(M) For a quadratic module M over a quadratic algebra A and a graded module
N over a graded algebra В there is a natural isomorphism o/cohom(A, B)-modules
сопотл,Б(М, N)~Mm (qiV)qB.
More generally, let M be a graded A-module generated by a graded subspace X с М
with the submodule of relations generated by a graded subspace Y с А® X. Then
for any graded В-module N one has
сопотл,Б(М, N) ~ сокег(У о N* -► cohom(A, B) 0 {X о N*))y
where the map Y о N* —> cohom(A, B) 0 (X о N*) is given by the composition
у о AT* —L_> (А о Б*) ^ (X о AT*) —^ cohom(A, B) 0 (X о N*),
where f is induced by the embedding Y —► A®X and the coaction B* —> B*<g>N*,
while g comes from (4.3).
Proof: It suffices to check that the quotient algebra (resp., module) satisfies the
universal property from Proposition 4.1. There is a bijective correspondence between
homomorphisms of graded algebras T(X о Б*) —► С and maps of graded vector
spaces X —> В о С. It is easy to see that a homomorphism T(X о Б*) —> С kills
the image of УоБ* iff the corresponding map of vector spaces can be extended to an
algebra homomorphism A —> В о С. The argument for modules is analogous. □

72
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
Corollary 4.4. (i) For any graded algebras A and В there are natural maps
of the Ext-spaces
(4-7) Ext^hom(A,B) (k' k) > ExtA (k' fe) ® Bi
compatible with products. The map (4.7) is an isomorphism ifi = 1 and a monomor-
phism if i = 2. If an algebra В is one-generated then (4.7) is also an isomorphism
for i = 2.
(ii)(M) For any pair consisting of a graded A-module M and a graded В-module TV
there are natural maps
(4.8) Ext^hom(AfB)(cohomAfB(M,iV), k) > Extjj (M, k) ® TV,-
compatible with products by elements of the spaces appearing in (4.7). These maps
are isomorphisms for i = 0 and monomorphisms for i = 1.
Proof: For any graded algebras В and С and graded modules TV and P over В and
C, respectively, there are natural morphisms of bar-complexes
Bar.(BoC) > В о Bar.(C)
Bar.(BoC,NoP) > NoBar.(C,P),
where we use the internal grading on the bar-complexes when forming Segre
products, so that В о Bar.(C) = 0. Bj ®Barij(C). The above morphisms are defined
using the natural multiplication maps Bfl —► В and В®г ® TV —► TV. The desired
maps between Ext-spaces are induced by the morphisms of cobar-complexes dual
to the compositions
Bar.(A) > Bar.(В о cohom(A, B)) > В о Bar.(сопот(А,Б)),
Bar.(A,M) > Bar.(B о cohom(A,Б), N о cohomA,B(M,N))
> N о Bar.(cohom(A,£), cohomA,B(M,N)).
It is easy to check compatibility of the obtained maps with products.
To check the statements on the behavior of our maps for i = 1 and г = 2 we
will use Proposition 4.3. Let us choose a minimal generator space X С A and let
Y С T(X) be a minimal relation space, so that X c^ Torf (Ik, k) and Y c^ Tor^(k, k)
(see Proposition 5.2 of chapter 1). Then the composition
УоБ* > Т(ХоБ*)+/Т(ХоБ*)2_ = ХоБ*
is zero, hence ХоБ*^ Tor^° ° ' (к,к). It is also easy to see that the subspace
imY о В* сТ(Х о В*) is a minimal generating subspace for the two-sided ideal it
generates, hence im7o5* ~ Tor^0 ° ' (k,k). Note that if В is one-generated
then the map Y о Б* —> T(X о Б*), so in this case У о Б* ~ Tor^ohom(^) ^ ^
One can check that these identifications are compatible with the maps of Ext spaces
defined above. The proof of the assertion for modules is similar. □
Next, we are going to compute the cohomology of the algebras А о В and
сопот(А,Б) (resp., modules M о N and cohom^,s(T^, TV)) assuming that the
algebra В (resp., the module N) is Koszul.
We start with an auxiliary lemma. Let Vect* denote the category of bounded
complexes of finite-dimensional vector spaces. For any functor Ф* : E —► Vect* let
Н(Ф*) denote the corresponding cohomology functor (taking values in the category
of graded vector spaces). The construction of the spaces СопотЕ(Ф, Ф) can be

4. INTERNAL COHOMOMORPHISM
73
extended to the case of functors to Vect9 in the obvious way. Below for a set S we
denote by k(S) the vector space over Ik with the basis S.
Lemma 4.5. Let E be a finite partially ordered set considered as a category with
morphisms o\ —► (72 corresponding to ordered pairs g\ ^ (72 • Then for a functor
Ф: E —> Vect the following conditions are equivalent:
(a) For any functor Ф* : E —> Vect9 there is a natural isomorphism of graded
vector spaces
Я(СопотЕ(Ф#, Ф)) ~ СопотЕ(Я(Ф#), Ф).
(b) The following three conditions hold:
(i) all the maps Ф(о"1 —> аг): Ф(о"1) —> Ф(о"2) &ге surjective;
(ii) /or (Jo ^ <Ji, (Jq > <J2 one /ms
кегФ(сг0 —> (Ji) + кегФ(а0 —> сг2) = П кегФ(а0 —> a3);
1 'СГ1,СГ2^СГЗ
(Hi) /or any object do G E t/ie collection of subspaces
(кегФ(а0 —> (Ji) С Ф(а0), a0 ^ (7X G E)
in Ф(сго) г5 distributive.
(c) ТЛе functor Ф zs isomorphic to a direct sum of functors of the form
Фа°(а) = k(Mors(a, (7o))* fwe ca// these functors "corepresentable").
Proof: It is more convenient to consider the dual situation using the functor Ф* :
Eop —> Vect : a i—> Ф(а)*. We have to check the equivalence of the following
conditons for Ф*:
(a) For any functor Ф*: Eop —> Vect9 there is a natural isomorphism of
graded vector spaces
Я(НотЕ°Р(Ф*,Ф*))-НотЕ°Р(Ф*,Я(Ф#)).
(b) The following three conditions hold:
(i) all the maps Ф*(<71 —► a2): Ф*(о"1) —► Ф*(сг2) are injective;
(ii) for (7o ^ (7i, (7o ^ cr2 one has
ппФ*((71)Пт1Ф*((72) = У^ шФ*((7з) с Ф*((70);
(iii) for any object (7q the collection of subspaces
(1тФ*((7! > (70) С Ф*((70), (70 ^ (7i )
in Ф*((7о) is distributive .
(c) The functor Ф* is isomorphic to a direct sum of represent able functors,
i.e., functors of the form Ф*о(а) = k(Mor£oP((jo, a)).
(с) ==> (a). This follows immediately from the natural isomorphism
НотБ°Р(Ф;о,Ф')^Ф'(а0).
(a) => (b). The condition (a) means that the functor Ф* is a projective object
in the abelian category Funct(Eop, Vect) of functors Eop —> Vect. It is easy to
see that Funct(Eop, Vect) is equivalent to the category of modules over a certain
ring and that the free module with one generator over this ring corresponds to the
functor 0a GE Ф*0. Since a projective module is a direct summand of a free one,
the validity of condition (b) for Ф* follows immediately from its validity for the
functors Ф* .

74
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
(b) => (c). This is a generalization of Proposition 7.1 of chapter 1, (a) => (b), and
the proof is analogous. For simplicity of notation we identify \£*(si) with its image
under the embedding \£*(ai) —> Ф*(сг2) for G\ ^ g2. For every gq G E let us choose
a direct complement Уао С Ф*(сг0) to the subspace ^cri<cr0 ^*(ai) c ^*(ао)- We
have a natural morphism of functors
To prove that it is an isomorphism we have to check that the map 0a<ao Va —>
Ф*(ао) is an isomorphism for any gq G E. It is easy to prove by increasing induction
in Go that this map is surjective. It remains to prove its injectivity. Assume that we
have J2i=i vi — 0 m ^r*(cro) for a collection (ai,... , afc), where Gi < Go for all г, and
some nonzero vectors Vi G Vai. Without loss of generality we can assume that there
is no г ^ 2 with 04 ^ аг. Then the inclusion г;х G **(<Ji) П ^2г^2 ^*(аг) С #*(cr0)
together with conditions (ii) and (iii) imply that v\ G ^2a<a ^*(сг) С Ф*(ао),
which is a contradiction. □
Theorem 4.6. (i) For a graded algebra A and a Koszul algebra В one has
natural isomorphisms of bigraded algebras
ЕхЬлов(КIk) — cohom (Ext^ (Ik, Ik), Б),
Extcohom(A5jB)(Ik,Ik) ~ ExtA(k,lk)oS,
u>/iere t/ie operations in the right-hand side use internal grading on Ext л (к, к).
(ii)(M) Let A and В be as in (i). For a graded A-module M and a Koszul B-
module TV one has natural isomorphisms of bigraded modules
E*tAoB(MoN, к) ~ cohomExtA(kjk)5jB(ExtA(M,Ik), TV),
Extcohom(A)jB)(cohomA>jB(M,A^),Ik) ~ ExtA(M,Ik) о TV.
Proof: First, let us prove the formulas for ExtAos(k,Ik) and ЕхЬаов(М ° N, Ik).
We claim that for any graded algebras A and В and modules M and TV there are
natural isomorphisms of differential algebras and modules
Cob0 (A oB)- cohom (Cofr* (A), B),
Cob*(А о Б, Mo TV) ~ cohomCob'(A), в(СоЪ*(А, М), TV),
where we extend the operation cohom to DG algebras and DG-modules in the
obvious way. Indeed, let us ignore the differentials for a moment and consider
Cob(A) and Cob (A, M) as a graded algebra and a graded module over it using the
internal grading. Then we can write Cob(A) = T(A*+) and Cob(A,M) = T{A*+) <g>
M*. Hence, from Proposition 4.3 we get
cohom(T(A;), В) ~ T(A; о Б*) ~ ЩА о В)\) and
cohomT(A;),s(T(A;)(g)M*, TV) ~ cohom(T(A;), B) <8> (M* о TV*)
~T((Ao£);)®(M®TV)*.
It is not difficult to check that these isomorphisms commute with the differentials.
Now if the algebra В and the Б-module TV are Koszul then the operations
cohom (C*,B) and cohom с • ,в{Р*, N) commute with passing to cohomology of
a DG-algebra Cm and a DG-module PV Indeed, this follows from Lemma 4.5,
(b) => (a), applied to the functors Ф^ : 9n —> Vect and Ф^ N: 0n —► Vect.

4. INTERNAL COHOMOMORPHISM
75
One just has to observe that condition (b.i) means that the algebra В (resp., B-
module TV) is one-generated (resp., generated in degree zero), (b.ii) means that В
and TV are quadratic, and (b.iii) is equivalent to Backelin's Koszulness condition
for В and TV.
It remains to prove that morphisms (4.7) and (4.8) constructed in Corollary 4.4
are isomorphisms provided В and TV are Koszul. Since cohom д# (Ik, B) = Ik, it
suffices to do this for (4.8). Consider the bar-resolution Bar.{A,M) of the A-
module M. Similarly to the proof of Corollary 4.4 we have a morphism of complexes
of A-modules
Bar.{A,M) > Bar.(B о cohom(A,B), TV о cohom A}B{M,N))
> TV о Bar.(cohom(A,P), cohom A:B{M,N)).
By Proposition 4.1 it induces a morphism of complexes of cohom (A, P)-modules
(4-9) _ _
сойоту,в{Bar.{A,M), TV) ► Bar.(cohom(А,P), cohom^,в(М, TV)).
As we have seen above for Koszul В and TV the operation P* i—> cohom A^B {P*, N)
commutes with passing to cohomology. Hence, the complex
cohomА:в {Bar. (A,M), TV)
is a cohom (A, P)-module resolution of cohom^s^, TV). Furthermore, since
Bar{A,M) ~ A ® Bar{A,M) as a bigraded A-module, by Proposition 4.3 we get
an isomorphism of bigraded cohom (A, P)-modules
сопотдв {Bar (A,M), TV) ~ cohom (A, B) <g> {Bar (A, M) о TV*).
Therefore, (4.9) is a morphism of free resolutions of the module cohom^^M, TV),
so it should be a homotopy equivalence. Applying the functor Homcoh0m(A,B) (—> 4
to (4.9) we get the morphism of cobar-complexes from the proof of Corollary 4.4.
Hence, the induced map on cohomology is an isomorphism. □
Corollary 4.7. (i) For any graded algebra A and a Koszul algebra В the
Hilbert series of the algebra cohom (A,B) is given by the formula
hCohom(A,B){z) = {h^1 О hB){z)~l,
where we denote (/ о g){z) = £\ f^z* for f{z) = £V flzi and g{z) = ^дггг. If
in addition A is quadratic then
(4.10) hA.B{z) = {h-A' о he'Xz)-1 = {h~Al о h-Bl){-z)-\
(ii)(M) Let A and В be as in (i). For any graded A-module M and a Koszul
B-module TV one has
KohomA.B(M,N){z) = {hA о hB){z)~l ■ {h^lhM о hN)(z).
If in addition A and M are quadratic then
hM.N{z) = (h^1 о h]B1){-z)~1 • {h^fiM о h^lhN){-z).
Proof: Recall that we denote Pc{u> z) = Z^j u%z2;dimExt^(k, Ik). By Theorem 4.6
we have Pcohom(A,s)
(u,z) = PA{u,z) о hB{z). Hence, using Proposition 2.1 of
chapter 2 we get
hCQhom{A,B){z) =PCohom(A,B)(-M)~1 = CPa(-1, о) ° hB) {z)'1 = {h^1 О hB) {z)~l,

76
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
The formula for Нсо^ОША b{m.n) is derived similarly. To compute the Hilbert series
of A • В and M • TV we apply Proposition 4.3 and use the relation (/ о g)(—z) =
f(z)og(-z). □
Note that for a fixed Koszul algebra В and a Koszul Б-module TV not only
the dimensions of the grading components of сопот(А,Б) and cohom а,в(М, TV)
are determined by the dimensions of the components of A and M, but the vector
spaces cohom(A, B)n and cohom^s^, TV)n themselves can be recovered functo-
rially from the vector spaces Ai and Mi. Indeed, this follows from the fact that for
any functor 4!: E —> Vect satisfying the equivalent conditions of Lemma 4.5 and
for any functor Ф : E —> Vect the vector space СопотЕ(Ф, Ф) is determined by
the values of Ф on objects (since this is true for a corepresentable functor Ф).
Corollary 4.8. (i) Let us fix an integer n ^ 0 and a graded algebra В with
Bi = 0 for all i > n. Then the algebra сопот(А,Б) is determined by the quotient
algebra A^n = A/A>n = 0™=о^4г of A. Furthermore, if the algebra В is Koszul
then the cohomology of the algebras AoB and сопот(Д В) can be recovered from
the spaces Ext^(k, k) with j ^ n.
(ii)(M) Let us fix an integer n ^ 0. a graded algebra В and a graded В-module
TV with Ni — 0 for all i > n. Then the module cohom^^M, TV) is determined
by the quotient A-module M^n = M/M>n = @i<nMi. If В and TV are Koszul
then the cohomology of the А о В-module M о TV can be recovered from the
cohomology of the algebra A and the spaces Ext^(M, k) with j ^ n. The
cohomology of the cohom (A, B)-module cohom^^M, TV) can be recovered from the spaces
Ext^'(M,k) with j ^ n.
Proof: It is clear that the algebra А о В and the module M о TV are determined
by A^n and M^n under our assumptions. Proposition 4.1 immediately implies
the same assertion for сопот(Л, В) and cohom^^M, TV). The statements about
cohomology follow from Theorem 4.6. □
Corollary 4.9. (i) Let В be a Koszul algebra with Bn Ф 0 and Бп+1 = О
for some n > 0. and let A be a graded algebra. Then the algebra AoB (resp.,
cohom(A,B)) is Koszul iff A is n-Koszui
(ii)(M) Let В be a Koszul algebra, TV a Koszul В-module with Nn ф 0 and
TVn+1 = 0. Assume that a graded algebra A is n-Koszul and the algebras AoB and
cohom(A, B) are Koszul. Then a graded A-module TV is n-Koszul iff the А о В-
module M о TV (resp., cohom(A, B)-module coh.ou\ а,в {M,N)) is Koszul.
Proof: (i) By Theorem 4.6 we have Extcoh0m(A,B)(^Ik) — Ext^(k, к) о В. Since
Bi ф 0 for 0 ^ i ^ n and Bi = 0 for i > n, this implies that cohom(A, B) is Koszul
iff A is n-Koszul. To prove the similar result for AoB we use the fact that the algebra
Л о Б is Koszul iff the algebra Ext^osfr k) is one-generated with respect to the
internal grading. By Theorem 4.6 we have Ext^o£(k, к) ~ cohom(Ext^(k,к), В).
To find generators of the latter algebra we use Theorem 4.6 again:
Extcohom(ExtA(M0,5) " ExtExtA(k>k)(k'k) ® ВГ
It follows that cohom(ExtA(k, k),B) is one-generated iff the algebra
ExtA(k,k)/( 0 Extzj(k,k))
i^lj>n

5. KOSZULNESS CANNOT BE CHECKED USING HILBERT SERIES 77
is one-generated with respect to the internal grading, which is equivalent to n-
Koszulness of A. The proof of (ii) is analogous. □
For arbitrary quadratic algebras A and В the Hilbert series of A • В in general
cannot be recovered from the Hilbert series of A and В (and therefore, the same
is true about cohom(A, B)). More precisely, we have the following result that was
stated in [101].
Proposition 4.10. // quadratic algebras A and В are not 4-Koszul then
dim(A • B)^ is bigger than the dimension given by formula (4.10).
Proof: First, let us explain another point of view on the formula expressing Ha.b
in terms of Ha and Kb in the case when one of the algebras is Koszul. For an n-
tuple of subspaces (Vi,..., Vn) in a vector space V consider the function on subsets
I C [l,n] given by
AT(V1,...,Vn)(I)=d)mV/Y,Vi.
iei
If (Wi,..., Wn) is another n-tuple of subspaces in a vector space W then one can
consider the n-tuple of tensor products (Vi ® W\,..., Vn ® Wn) in V ® W. It is easy
to see that if (Vi,-..., Vn) is distributive then there exist universal formulas
expressing values of M{V\ ® Wi,..., Vn ® Wn) as polynomials in values of J\f(Vi,..., Vn)
and J\f(Wi,..., Wn). Indeed, this follows from the fact that every distributive n-
tuple (Vi,..., Vn) is a direct sum of one-dimensional n-tuples and the multiplicities
of each isomorphism type of one-dimensional n-tuples are universal linear
functions of values of AA(Vi,.... Vn). Specializing to the case of n-tuples of the form
(Я^п),..., R%\) in V®71, where R С V02 we obtain the formula for dim(A • B)n
in the Koszul case. Thus, our statement is a consequence of the following claim:
for every pair of non-distributive triples of subspaces (Vi, V2, V3) and (Wi, W2, VV3)
in V and W the dimension of
V 0 W/(V! 0 W± + V2 (8) W2 + V3 0 W3)
is bigger than the value of the universal expression for this dimension in the
distributive case, evaluated at the values of N(V\, V2, V3) and Af(Wi, W2, W3). Since
every indecomposable triple of subspaces is either one-dimensional, or the triple of
distinct lines in the plane (the only non-distributive indecomposable triple), it
suffices to prove the above claim for one pair of non-distributive triples of subspaces.
To this end let us take (Vi, V2, V3) to be a triple of distinct planes passing through
a line in a three-dimensional space V. Then it is easy to check that
dim V02/(V102 + V®2 + V302) = 1
(since the same is true for a triple of distinct lines in a plane). On the other hand,
consider a generic triple of planes (V]0, V2°, V3°) in V. Then it is distributive and
F®2/((^10)®2 + (V20)02 + (V3T3) = 0.
Since Af(Vu V2, V3) = Af(V?, V2°, V3°) this proves our claim. □
5. Koszulness cannot be checked using Hilbert series
As we have seen in section 2 of chapter 2 for every Koszul algebra A one
has hA{z)hA'(—z) = 1. Examples of quadratic non-Koszul algebras satisfying this
relation were constructed in [101] and [106]. Below we will present a theorem due

78 3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
to Piontkovskii [92] showing that in fact it is impossible to tell looking only at
Hilbert series of A and A- whether A is Koszul or not.
Using identity (2.1) from chapter 2 one can easily see that for an (n — l)-Koszul
quadratic algebra A one has
hA(z)hA*(-z) = l + 0(zn).
The following lemma gives a way to construct (n — l)-Koszul but not n-Koszul
algebras for which 0(zn) can be replaced with o(zn) in the above identity.
Lemma 5.1. Let n > 4 be an odd integer, and let D be a quadratic algebra
which is (n — 1)-Koszul but not n-Koszul. Then the quadratic algebra В = D ® D[
is (n — 1) -Koszul but not n-Koszul and satisfies
(5.1) hB(z)hB*(-z) = l + o(zn).
Proof: Proposition 1.1 easily implies that В is (n — l)-Koszul but not n-Koszul.
Consider the formal series f(z) = hB(z)hB*(—z). Since B[ = D] (g)-1 D, we have
f(z) = hD(z)hD*(z)hD'(-z)hD(-z),
hence f(—z) = f(z). On the other hand, В is (n — l)-Koszul, so f(z) = 1 + 0(zn).
Since n is odd, this is possible only when f(z) = 1 + o(zn). □
The following lemma is the technical heart of the construction.
Lemma 5.2. Let n ^ 0 be an integer, and let В be a quadratic algebra such that
(5.1) holds. Then there exist quadratic monomial aglebras M and TV such that for
TV' = M U В one has
hN*{z)-hN,i{z) = o{zn),
hN,(z)hN,<(-z) = l + o(zn),
hN(z) -hN>(z) = o(zn).
Proof: If n ^ 2 then the assertion is clear (take M = Ik and TV any quadratic
monomial algebra that has the same number of generators and relations as Б), so
we can assume that n ^ 3. Note also that since TV is Koszul, we have 1in(z) =
/ijv'(—z)_1. Hence, the last equality follows from the first two.
The main building blocks in the construction of M and TV are the following
two series of quadratic algebras:
Es = k{xb ... ,xs}/(xix2,... ,xs-2Xs-i,xs-ixs),
Gs = k{xb... ,xs}/(xix2,... ,xs_2xs_i,xs_2xs),
where s ^ 3. One can easily compute the Hilbert series of the dual algebras
(cf. section 6 of chapter 4):
hEa* (z) = l+sz+(s- \)z2 + ... + 22s-1 + zs,
hGs< (z) = 1 + sz + (s - l)z2 + ... + 2zs_1.
Therefore,
(5.2) hEa*{z)-hGa*{z) = zs.
Let us pick a monomial quadratic algebra Q that has the same number of
generators and relations as B. Then
hB.(z) = hQ.(z) + z3P(z)+o(zn)

5. KOSZULNESS CANNOT BE CHECKED USING HILBERT SERIES 79
where P is a polynomial of degree n — 3 with integer coefficients. Let us write
z3P(z) = aiz11 + ... + apz** - hizJ1 - ... - brzjr,
where a* and bj are positive integers. Let us define our monomial quadratic algebras
M and TV as the following free products:
M = (Gh)Uai U ... U (Gl?)Ua? U (EJ1)ubl U ... U (£^)u*\
TV = Q U (Eh)Uai u ... U (£>)Uap u (GJ,1)ubl U ... U (Gjr)ubr.
We claim that with this choice of M and TV the required equalities with Hilbert
series are satisfied. Indeed, since TV'- = M! ПБ!, we have
(5.3) hNn (z) = hMi (z) + h& (z) - 1.
Similarly, we can express Hilbert series hM* and hN\ in terms of hEs>, hGs< and /ig>.
This leads to
p r
hN,\ (z) - hN' {z) = hB> (z) + ^ ашНСг^ (z) + ^ Ьг/i^v (г)
m=i г=1
- hQ[ (z) ~ 5Z arnhEi™< (z) - ^ blhGn[ {z)
771=1 Z=l
P Г
= Z3P(Z) - ^ OmZ*™ + ^ blzJl + °(^) = ^П)>
77i=i г=1
where we used (5.2). This proves the first equality. To prove the second equality
we note that by (1.2) one has
hN>(z) = (hM(z)-1 + hB(z)-1-l)-1.
Combining this with (5.3) we get
hN>(z)hN,'(-z) = (hMiz)'1 ^Нв^)-1 -l)-\hM(zyl + hB<(-z)-l) = l + o(zn),
since hB\(—z) = Hb{z)~1 + o(zn) by our assumption. □
Theorem 5.3. There exist quadratic algebras A and A! such that Ha{z) =
fiA'(z), hA\(z) = hA>\(z), and A is Koszul but A' is not.
Proof: The construction has as an input an odd integer n > 4 and a quadratic
algebra D such that D is (n — l)-Koszul but not n-Koszul. For example, one
can take n = 5 and the algebra D with three generators x,y,z and two relations:
xy = 0, yx + z2 = 0. It is not difficult to check that this algebra is 4-Koszul but
not 5-Koszul (see [15]).
Starting from D as above we set В = D ® D\ By Lemma 5.1 the algebra В
satisfies the assumptions of Lemma 5.2. Therefore, we can find monomial quadratic
algebras M and TV satisfying the conclusion of this lemma. Now we set A = TV о С
and A' = TV' о С, where TV; = M U В and С is any Koszul algebra with Cn+i = 0
and Cn Ф 0 (say, the exterior algebra with n generators). Note that A is Koszul,
being the Segre product of Koszul algebras (see Corollary 1.2). On the other hand,
TV; = M U В is not n-Koszul since В is not n-Koszul. By Corollary 4.9 this implies
that A! is not Koszul.

80
3. OPERATIONS ON GRADED ALGEBRAS AND MODULES
Since Cn+i = 0, the relation Hn{z) — hjsf{z) = o(zn) implies that Iia(z) =
h,Af(z). On the other hand, since C! is Koszul we can apply Corollary 4.7 to
compute the Hilbert series of A[ = Nl • C! :
hA> {z) = hN'mCi (z) = (h^} о ftc)"1 W-
Similarly, /i^/'(z) = (ft^. о /ic)_1(2;). Using the relation hN<(z) — hN,<(z) = o(zn)
we deduce that h^iz) = hA>[(z). □

CHAPTER 4
Poincare—Birkhoff— Witt Bases
In this chapter we study quadratic algebras for which a (homogeneous)
analogue of the PBW-theorem holds (called PBW-algebras). In another terminology
these are algebras admitting generators for which the noncommutative Grobner
basis of relations consists of elements of degree 2 (see [26, 36]). These algebras
form an important class of Koszul algebras preserved under various operations on
quadratic algebras. We consider some generalizations of this class in section 7.
These include the case of commutative PBW-algebras (also known as G-quadratic,
see [38]) considered in more detail in section 8. We also define a notion of Z-PBW
basis by replacing graded algebras with a wider class of Z-algebras (see section 9
and 10). Finally, in section 11 we show that a three-dimensional Sklyanin algebra
is Koszul but does not admit a Z-PBW-basis.
1. PBW-bases
Let A = {V, /} = T(V)/J be a quadratic algebra with a fixed basis {#i,..., xm}
of the space of generators V. For a multiindex a = (ii,..., in), where ik G [1, m],
we denote by xa the monomial x^x^ ... Xin G Т(У). For a = 0 we set x0 = 1.
Let us consider the lexicographical order on the set of multiindices of length n:
(ii,..., in) < (ji,..., jn) iff there exists к such that i\ = ji, ... , ik-i = jk-i
and ik < jk- The following lemma is the starting point for the construction of a
PBW-basis.
Lemma 1.1. Let W be a vector space with a basis wa numbered by a totally
ordered set of indices Л, and let К с W be a subspace. Consider the subset Sk С A
consisting of all a G A such that W& cannot be presented as a linear combination of
wp with f3 < a modulo K. Then the images of the elements wa with a G Sk form
a basis of W/K. The subset S = Sk is also uniquely characterized by the property
that there is a basis of К of the form
up = wp - 2_^ cPo№ol, P G S = A\S.
oc<(3
There exists a unique basis of К of this form with the additional property that
cpa = 0 for a ^ S. □
Note that the correspondence К <—► (5, сра) is nothing else but the Schubert
stratification of the Grassmannian G(W).
Now let us equip the space V ®V with the basis consisting of the monomials
XiYXi2. Applying the above lemma to the subspace of quadratic relations I CV&V
we obtain the set of pairs of indices S = Si С [l,m]2. Hence, the relations in A
81

82 4. POINCARE-BIRKHOFF-WITT BASES
can be written in the following form:
#ii#i2 = /-^ °ild2XhxJ2i (н,г2) G S.
0'и'2)<(н,г2)
(hJ2)es
Definition. Consider the following sets of multiindices:
5(n) = {(гь...,гп) | (гьг2) e S, (г2,г3) G 5, ..., (гп_ьгп) eS}, n > 2.
We also set S^ = [l,m] and S^ = {0}. Elements #i, ... , xm G V are called
PBW-generators of Л if the monomials (xa) with a G Un>o ^n^ f°rm а basis of
Л (called a PBW-basis of A). A PBW-algebra is a quadratic algebra admitting a
PBW-basis.
Note that in any case the monomials (ха,а G Un>o^n^) linearly span the
entire A. Indeed, every monomial x^ .. . x*n with (ikiik+i) £ S can be expressed
as a linear combination of smaller monomials modulo Jn. Let us denote by 7r: V <S>
V —> V ® V the projection on the first summand in the direct decomposition
V®V=(xa\aeS)@I. Explicitly,
Xixxi2, {h,h) € S
E(il,i2)<(il,i2) 411A2XhXJ21 {ЧЛ2) ^ S.
UiJ2)es
An element x G Tn(V) is congruent modulo J to the element
• • • 7Г 7Г • • • 7Г ' 7Г 7Г • • • 7Г ' X,
where 7r*'fc+1 = id0/c_1 ®тг ® id®n-fc-1 : ТП(У) —> ТП(У). This infinite product
is well defined since 7Г decreases the order. It maps Tn(V) to (xa, a G S^}.
Similarly, one can consider the infinite composition
. . . ^3,13 + 1^2,12 + 1^11,11 + 1
associated with any sequence ii, г2, гз, ... containing every index 1, ... , n — 1
infinitely many times. The PBW condition is satisfied iff all such products of
projections are equal to each other.
Remark. The PBW property depends on the choice of generators and on their
ordering. For example, (x, y) are PBW-generators of the algebra k{x, y}/(x2 — xy),
while (y,x) are not (see [119]). However, it is not difficult to see that the subset S
and the PBW property are determined by the filtration (xi) С • • • С (xi,..., xm_i)
С V spanned by xi,... ,xm (see section 7 below). Also, the existence of a PBW-
basis can depend on the ground field. For example, consider the quadratic algebra
Т(^)/(Я with one relation / G V®2, where dim У > 1 and / £ U®2 for any proper
subspace U С V. This algebra admits a set of PBW-generators iff the corresponding
quadratic form £ 1—> (£ ® £, /) on V* represents zero.
2. PBW-theorem
The following result is a particular case of the diamond lemma for noncommu-
tative Grobner bases [26, 36]. We keep the notation of the previous section.
Theorem 2.1. If the cubic monomials (x^x^x^, (г\, 22, гз) е S^) are linearly
independent in A3 then the same is true in any degree n. Therefore, in this case
the elements (xi,... ,xm) are PBW-generators of A.
TTyX^X^ )

2. PBW-THEOREM
83
Equivalently, elements (xi,..., xm) are PBW-generators of a quadratic algebra
A iff the following equation holds:
(O "П ^12 23^12 23„Д2 23
(Z.1J • • • 7Г 7Г 7Г = • • • 7Г 7Г 7Г .
Sketch of proof: Let us denote
j^a = ^ Tn(y)^a n v®k~l ® / ® y^-^-1 =
(х^Дх7 | 77 G 5, Д = x11 - ^2 cr){XCi Pvi < «)
and similarly define J<a. We can prove by induction in a that the monomials
(x& I /3 G S(n\ /3 ^ a) are linearly independent modulo J<a. Indeed, it suffices
to show that x^y^x1 — x@ y^'X1 G J<a for any /З777 = <* = P'tfl'• This is clear if
the sub words rj and 7/ do not intersect each other. Otherwise, it follows from the
condition in degree 3. □
Another proof: Consider a sequence ii, 22, гз, ... as above and let
n _ ... ^13,13+1^12,12+1^11,11+1 _ ^/^iijii+i
be the corresponding product. It is enough to prove that nn^^1 = п for any j.
We use induction in a: assuming that mr^^1 agrees with 7Г on Tn(V)<a for any j
and any 7Г as above we have to show that 7T7r7''J+1xa = 7rxa. If 7r7''J+1xa — ха then
there is nothing to prove, so we can assume that 7r7'J+1xa G Tn(V)<a. Then by
the induction hypothesis
(2.2) 7Г7Г^'+1Ха = TrV^ + V'^x" = TtV'^x".
We claim that it suffices to consider the case when 7ггьг1+1ха G Тп(У)<а. Indeed,
otherwise 7rxa = 7т'ха and using (2.2) we can replace n by nf. Decomposing further
7г' as nf = 7Г//7Гг2,г2+1 we can g0 on until after several steps we obtain it = jf. Now
applying 7Г to the equation
. . .^+1^1^1+1^ + 1^ = . . .^ii.ii+l^+l^ii.ii + l^a
(that follows from (2.1)) and using the induction assumption we obtain 7Г7Г7'-7+1ха
on the left and 7Г7гг1'г1+1ха = 7rxa on the right. □
Example 1. Let SP be a skew-polynomial algebra, i.e., a quadratic algebra with
generators Xi,..., xn and relations
XjXi — QijXj,Xj, 2 <. J,
where (g^, 1 ^ г < j ^ n) is a collection of nonzero constants. Then xi,..., xn
is the set of PBW-generators. The corresponding PBW-basis of SP consists of all
monomials of the form x™1 ... x™n, where mi,..., mn > 0. For example, every
basis of a vector space V gives PBW-generators of the symmetric algebra S(V).
Example 2. The classical Poincare-Birkhoff-Witt theorem can be deduced from
Theorem 2.1 as follows. Let g be a finite-dimensional Lie algebra. Consider the
graded version [7(g) of the universal enveloping algebra generated in degree one by g
together with an additional central generator t and with relations xy — yx = t[x, y]
for x, у G g. If xi,..., xm is a basis of g then the elements (t, xi,..., xm) are

84
4. POINCARE-BIRKHOFF-WITT BASES
PBW-generators of [7(g) and the corresponding PBW-basis consists of the
monomials (^x^1 •••х^тг), where /с, кг ^ 0. Indeed, the PBW condition in degree 3
is easily seen to be equivalent to the Jacobi identity. Using the isomorphism
U(q) = U(g)/(t — 1) we obtain that the monomials (x11 • • • x1^1) form a basis in
U(q). (See also Theorem 2.1 of chapter 5 and Remark 2 in section 2 of chapter 6.)
3. PBW-bases and Koszulness
The following result is due to S. Priddy [104].
THEOREM 3.1. A quadratic PBW-algebra is Koszul
Proof: Define a multiindex-valued filtration on the algebra A by the rule FaAn =
(x& | /3 < a), where length a = n. Since the elements (xi,...,xm) generate a
PBW-basis of A, the associated graded algebra grFA = 0a Fa/Fa' (where a'
precedes a in the multiindex order) is quadratic. Thus, gvFA is a quadratic monomial
algebra, so it is Koszul by Corollary 4.3 of chapter 2. Now the assertion follows
Fa л
easily from the spectral sequence E\ = Torgr (к, к) =>> Tor (к, k) associated
with the corresponding multiindex-valued filtration of the bar-complex computing
the Tor (cf. proof of Theorem 7.1). Another proof will be given in section 5. □
Example. Here is a (minimal in the numbers of generators and relations for
algebras over an algebraically closed field) example due to J. Backelin [18] of a
Koszul algebra that has no PBW-basis:
x2 + yz = 0
x2 + azy = 0, а ф 0, 1.
The proof of Koszulness is as follows. One can compute the Hilbert series Jia(z) =
^2n(dim An)zn using the noncommutative Grobner bases technique [26, 36]. This
computation shows that /i^(z)~ is a polynomial of degree 2. Hence, A is Koszul
by Proposition 2.3 of chapter 2. We will see in section 10 that the algebra of this
example has a so-called Z-PBW-basis which also implies Koszulness.
Remark. The set VBW(V) of quadratic PBW-algebras with a fixed space of
generators V is a Zariski constructible subset in the Grassmannian variety of
quadratic algebras G(V®2). Indeed, let us fix a basis x = (xi,..., хш) of V and consider
the Schubert stratification G(V®2) = UsC[i,m]><2 <&s{V®2]x®2) defined in Lemma
1.1. Let VBW(V;x;S) С VBW{V) denote the set of quadratic algebras having
PBW-generators with the corresponding set 5. Then TBW(V\ x\ S)
is a closed algebraic subvariety of Gs{V®2',x®2), preserved by the action of the
upper triangular subgroup B(m) С GL(m). Consider the finite disjoint union
TBW(V-,x) = \\STBW(V]X',S). Then the set TBW(V) can be presented as the
image of the algebraic morphism
GL(m) xB(rn)VBW(V;x) > G(0,
hence it is constructible. On the other hand, it turns out that the set of all Koszul
algebras with m generators is not constructible for m ^ 3 (see section 6 of
chapter 6).

4. PBW-BASES AND OPERATIONS ON QUADRATIC ALGEBRAS 85
4. PBW-bases and operations on quadratic algebras
As we have shown in chapter 3, a number of natural operations on quadratic
algebras preserves Koszulness. In this section we will check that these operations
also preserve the class of PBW-algebras.
Theorem 4.1. If (xi,... , xm) are PBW-generators of a quadratic algebra A
then the elements (x^,... ,x\) of the dual basis are PBW-generators of the dual
quadratic algebra A\
Proof: Let A0 denote the monomial algebra with the generators (xi,..., xm) and
relations XiXj = 0 for (i,j) G S. Note that (xi,... , xm) are PBW-generators of A
iff dim As = dim A®. The analogous monomial algebra A10 for the quadratic dual
algebra A! with the set of generators (x^,..., x\) corresponds to the complementary
subset S[ = S. Hence, A!0 coincides with the quadratic dual to A0. For any
quadratic algebra A = {VJ} we have A3 = V®3/{I <g> V + V <g> I) and A3* =
I (g)V dV ® I. Therefore, the dimension of A3 is given by the formula
dim A3 = (dim A1)3 - 2(dim Ai)(dim A2) + (dim A3).
Applying this to A and A0 we derive that dim A3 = dim A3 iff dim A3 = dim A3°.
Another proof can be derived from the proof of Proposition 5.1. □
Next, we are going to show that the class of PBW-algebras is closed under all
binary operations on quadratic algebras considered in chapter 3.
Proposition 4.2. Let (xi,..., xm) and (2/1,..., yi) be PBW-generators of
quadratic algebras A and B, respectively. Then the basis (xi,..., xm, 7/1,..., эд) of the
vector space A\ ©Б1 is a set of PBW-generators in each of the algebras А П B,
AUB, and A & B, where q e P£. Also, the basis
(xi ®2/i,...,xi ®2/i,x2 ®Уъ...,ят®2/1,...,жт®2/г)
of the vector space Ai®B\ is a set of PBW-generators in each of the algebras AoB
and A • B.
Proof: This follows from the fact that the set of PBW-monomials in each of these
algebras (except for A • B) coincides with the explicit basis constructed from the
PBW-bases of A and B. For example, Sa®b — SA^SB^{(i,j) | x* £ Ai, y3■, e Bi},
hence, S^B = [Jk+l=n S^ x 5^}. For A • В we use Theorem 4.1. □
The PBW-property is also preserved by the Veronese power construction.
Proposition 4.3. Let (xi,... ,xm) be PBW-generators of a quadratic algebra
A. Then the elements of the PBW-basis {xa)aeS(d) of Ad taken in lexicographical
order are PBW-generators of the Veronese subalgebra A^dK
Proof: This is an immediate consequence of the fact that the set of multiindices
can be described as follows:
SW = {(ab...,an) e(Sd)n : (a,, ai+i) G S(2d) for all г = 1,... ,n - 1}.
□
Remark. A more interesting interaction between the Veronese construction and
the PBW-property is connected with noncommutative Grobner bases (see [36]).
Namely, let A be a one-generated algebra. Assume that the Grobner basis of the
ideal of relations with respect to generators (xi,... ,xm) in A\ is finite. Then for

86
4. POINCARE-BIRKHOFF-WITT BASES
all sufficiently large d the set of normal monomials in Ad ordered lexicographically
is a set of PBW-generators of A^d\ More precisely, it suffices to take d bigger than
the degree of all elements in the Grobner basis.
5. PBW-bases and distributing bases
The following result explains the relation between PBW-bases and the criterion
of Koszulness in terms of distributive lattices.
Proposition 5.1. A basis x\, ... , хш of a vector space V generates a PBW-
basis of a quadratic algebra A = {V, /} iff for any n ^ 0 there exists a basis
(Уп..лп) of the vector space V®n distributing the collection of sub spaces V®k~l®I<&
y®n-k-\ £- y®n^ sucjl ^a£ ^y^ ^j differs from the monomial basis (x^®- • *®#in)
by the action of an upper triangular matrix:
2/i!...in =xil®--®xin- Yl 4l:%xJi ® ''' ® xjn.
(ii--in)<(ii--in)
Proof: "If". Apply Lemma 1.1 to the subspace V®k~l <g> I <g> V^-*"1 с У®п,
where У ®n is equipped with the lexicographically ordered basis (x^ <S> • • • <S> #;n) •
We have two bases of this subspace: one consisting of some subset of the basis
{yii...in) and the other consisting of the elements
xix <g> • • • <g> (xik ® x^-+1 - n(xik ® a:ifc+1)) ® • • • ® xin
with (г&, fcfc+i) G 5. Using the uniqueness part of Lemma 1.1 we derive that у%х..лп
belongs to V®k~l <g> 7 <g> V®n-k~l iff (ik, ik+l) G S. Therefore, the elements
(Vii...in I 3fe: (zfc,zfc+i) G 5)
form a basis of Y%Zl V®k~l <g> 7 <g> y®n-/c-i It remains to observe that An =
у®п/ £ y®k-i 0 j 0 ^®n-fc-i and t0 apply Lemma 1.1 again.
"Only if". Consider the dual quadratic algebra A1. It has a PBW-basis generated
by (x^,...,xj) with the corresponding set 5! = 5. Let тт[: V*®h —> У*0/г be
the projection onto the subspace (xa, a G S ) along the relations subspace J^.
Then the dual operator рн = 7^* : У0/г —► У®'1 is a projection onto f]t V®b~l ®
7 0 ^ы_1. For every multiindex a = (ii,... , гп) consider the partition of the
segment [1, n] into the union [1, fci] U • • • U [ks + 1, n] of the maximal subsegments
[fc, I] with the property that for all i G [fc, Z — 1] one has (г, г + 1) G 5. It is easy to
see that the elements
Уoc = (pk! ^Рк2-кг ®'-®Pn-ks){xa)
form the required distributing basis of V®n. □
Remark. The above criterion looks similar to the Backelin's criterion of
Koszulness in terms of distributivity of the collection (V®k~l <g) 7 ® V^n'k~l) in У®п
for all n. The crucial difference is that for the PBW-condition we are requiring
the existence of an upper triangular distributing basis. By Theorem 2.1 it suffices
to check that this condition is satisfied for n = 3. On the other hand, in Backe-
lin's criterion the distributivity condition is empty for n — 3. The first nontrivial
Koszulness condition occurs in degree 4 and no finite degree condition is sufficient:
for any n ^ 3 there exists a quadratic algebra with 3 generators which is n-Koszul,
but not (n + l)-Koszul (see section 4 of chapter 2 and section 6 of chapter 6).

6. HILBERT SERIES OF PBW-ALGEBRAS
87
6. Hilbert series of PBW-algebras
Clearly, the Hilbert series of a PBW-algebra A is equal to that of the
corresponding monomial algebra A0 and depends only on the subset S = Si С [l,m]2.
It is convenient to view S as the set of edges of an oriented graph Gs with the
set of vertices [l,m]. Then dimAn = \S^\ is equal to the number of paths of
length n — 1 in Gs (where n ^ 1). Here by a path of length n in Gs we mean a
sequence of edges of the form vq —» v\ —> V2 —» ... —» vn-i —> vn. Thus, the study
of Hilbert series of PBW-algebras reduces to the study of generating series of the
numbers of paths in oriented graphs. Below we present some basic results about
these generating series (see also [119]).
Let Г be a finite oriented graph with the set of vertices [l,ra] (we allow Г to
have two or more edges sharing the same source and target). With Г we associate
an m x m adjacency matrix Мг = {гпц), where mij is the number of edges going
from г to j. Note that in the case of the graphs Gs above the adjacency matrix
consists of O's and l's. Let an(T) be the number of paths of length n — 1 in Г,
where n ^ 1. For example, a\(T) = га, а2(Г) is the number of edges in Г. We also
set ао(Г) = 1 and form a generating series
The following result gives a way to compute this series.
Proposition 6.1. Let E denote the m x m matrix with all entries equal to 1.
Then
det(l + z(£-Mr))
r[Z)~ det(l-zMr) '
Proof: Let us set M = Mr. It is easy to see that
an(T)=tr(Mn-1E)
for n ^ 1. Therefore,
hr(z) = 1 + tr(zE + z2ME + z3M2E + ...) = 1 + tr((l - zM)~lzE).
Since det(l + T) = 1 + tr(T) for any rank-one matrix T we can rewrite the above
expression as
hr(z) = det(l + (1 - zM)~lzE) = det((l - zM)~l(l + z(E - M))).
П
For any S С [l,m]2 let Ms denote the m x m matrix with l's at all entries
belonging to S and O's at all other entries.
Corollary 6.2. The Hilbert series of a PBW-algebra A is rational. More
precisely,
det(l + zMs)
hA(z) =
det(l-zMs)"
Now we are going to show that the rate of growth of the sequence ап(Г) is
determined by the configuration of (oriented) cycles in Г. A cycle of length n is a
path of the form v\ —> V2 —> ... —> Vn —> vi m Г such that all vertices v\,... ,vn are
distinct (thus, a loop v —> v is a cycle of length 1). We say that two cycles intersect
if they pass through the same vertex. Let 71 and 72 be a pair of non-intersecting

88 4. POINCARE-BIRKHOFF-WITT BASES
cycles. We say that 72 is adjacent to 71 and write 71 —> 72 if there exists an edge
v\ —> V2 such that 71 passes through i>i and 72 passes through v2. A cftam of cycles
of length d is a sequence 70 —» 71 —» ... —» 7<j.
Proposition 6.3. (%) One /шз ап(Г) = 0 for n > 0 г/f Г /ms no cycles.
(ii) If T has a pair of distinct intersecting cycles then ап(Г) grows exponentially,
i.e., there exists a constant с > 1 such that ап(Г) ^ cn /or n ^> 0.
(mj i/T /ms no intersecting cycles then an(T) grows polynomially. More precisely,
if d is the maximal length of a chain of cycles in Г then there exist constants
c\ > C2 > 0 such that
cind ^ ап(Г) > c2nd
for n ^> 0.
Proof: (i) The proof is straightforward.
(ii) Let 71 and 72 be cycles of lengths П\ ^ П2 passing through a common vertex v.
Let us consider only paths that go along 71 and 72 sometimes switching from one
cycle to another when passing through v. Counting the number of such paths we
immediately get an estimate
(iii) First, we observe that in this case for every pair of adjacent distinct cycles
7i -> 72 there exists a unique edge v\ —> v2 such that 71 passes through v\ and
72 passes through v2. Indeed, if there were two such edges then we would be able
to construct a new cycle intersecting 71 and 72. Now let us fix a chain of cycles
7o -> 7i —> • • • —> Id and a vertex i>o on 70. Let us denote by pn the number of
paths of length n that start at vq, go around 70 several times then switch to 71,
etc. and finally end on 7^. It suffices to prove that there exists c\ > c2 > 0 such
that
c\nd ^ pn > c2nd
for n ^> 0. It is clear that if we decrease the length of one of the cycles by 1 the
number of paths pn can only increase. Therefore, it is enough to consider the case
when all cycles 71,..., 7d have the same length 5. Let t be the minimal length of a
path satisfying the above restrictions. Then
Pn = #{(no,. • • ,nd) e Z%1 \no + ...nd= l^J} = (^J + d^j,
so our assertion follows. □
7. Filtrations on quadratic algebras
In this section we generalize the PBW-theorem to the case of filtrations with
values in an arbitrary graded ordered semigroup.
Definition. A graded ordered semigroup Г is a (noncommutative) semigroup with a
unit e equipped with a homomorphism g: Г —> N and a collection of total orders
on the fibers Гп = g~l(n) satisfying the following condition:
a < /3 => cry < /З7, 7a < 7/9 for а,^Гп, 7 e Гк
(the strict inequality here is essential for the definition of the associated graded
object). In addition we assume that #-1(0) = {e}.
A Y-valued filtration on a graded algebra Л is a collection of subspaces (Fa =
FocAn С An, a € Гп) for every n ^ 0 such that

7. FILTRATIONS ON QUADRATIC ALGEBRAS
89
(i) a ^ /3 implies Fa С Fp;
(ii) Flri = An for the maximal element 7n G Гп;
(iii) Fa ■ Fp С Fap for arbitrary a, /3 G Г.
The associated Г-graded algebra for а Г-filtered algebra Л is defined in the usual
way: grF^. = 0aer Fa/Fa>, where a' precedes a in the total order (if a is minimal
in Гп then we take iv = 0). Note that the algebra grF^. is one-generated iff the
filtration F is one-generated, i.e., FaAn = ^ _f <a F^V • • • FinV, where is G IV
A one-generated filtration is determined by its restriction to A\.
If Л is a quadratic algebra equipped with a one-generated Г-valued filtration
F then Koszulness of grF^. implies Koszulness of A. In fact, the following stronger
version of this assertion holds.
Theorem 7.1. Let A be a quadratic algebra equipped with a one-generated Г-
valued filtration. Assume that the associated Г-graded algebra grFA satisfies the
following conditions:
(1) grF^. has no defining relations of degree 3;
(2) the quadratic part q(grF^.) of grFA is Koszul.
Then grFA is quadratic (and hence Koszul by (2)), and A is Koszul.
Proof: First, let us check that grF^. is quadratic. We are going to prove by
induction in n that grF^. has no defining relations in degrees from 3 to n. Assume
that this is true for n — 1 and consider the following fragment C. (A) of the Koszul
complex of A:
Лп_3 <g> Лз* > An-2 ® / > An-i <g> V > An > 0.
Note that it is equipped with the induced filtration with values in Гп.
Condition (1) implies that grFA!g = (grF^)3*, hence the associated Tn-graded complex
coincides with C.(grF^). We have HoCm(grFA) — 0 since grF^ is one-generated,
and HiCm(A) — 0 since A is quadratic. We also claim that H.2C9(grFA) = 0.
Indeed, by the induction assumption the homomorphism q(grF^.) —> grF^. is an
isomorphism in degrees < n. Therefore, our claim follows from the assumption that
q(grF^) is Koszul. Now the spectral sequence associated with the filtered complex
C9(A) immediately shows that HiCm(grFA) = 0. Hence, grF^ has no relations
in degree n. This proves that grF^. is quadratic. To verify Koszulness of A we
consider the induced Г-valued filtration on the bar-complex of A:
The associated Г-graded complex is isomorphic to the bar-complex of grF^, so we
get a spectral sequence E1 = TorgrFA(k,]k) => TorA(k,]k). Since Тог^л(к,к) =
0 for i ^ j, the same is true for TorA(к, к). □
Remark 1. Note that to prove only that grF^ is quadratic one can replace
(2) with the weaker assumption that the Koszul complex of q(grF^) is exact in
homological degree 3 (cf. Remark 1 in section 7 of chapter 5 and Remark 1 in
section 2 of chapter 6).
Example 1. To derive the PBW-Theorem 2.1 from Theorem 7.1 we take Г to
be the free noncommutative semigroup generated by the symbols 1, ... , m e Г\
and consider the Г-filtration generated by the filtration (xi) С (xi,x2) С ••• С

90
4. POINCARE-BIRKHOFF-WITT BASES
(xi,..., xm) = A\ associated with the basis xi,..., xm. It is clear that the PBW-
monomials form a basis in An iff the associated Г-graded algebra grF^ has no
defining relations in degree /c, where 3 ^ к ^ n. In this case the quadratic part
of the associated Г-graded algebra is always Koszul since it is a monomial algebra
(see Corollary 4.3 of chapter 2).
Example 2. Let /: A —> С be a homomorphism of quadratic algebras such
that Cif(Ai) С f(A\)C\. Assume that A and C/f{A\)C are Koszul algebras and
the left action of Л on С is free in degrees ^ 3, i.e., Ext^(C, k) = 0 for i > 0
and j ^ 3. Then С is a Koszul algebra and a free left Л-module. Indeed, this
follows from Theorem 7.1 applied to the filtration of С with values in the free
semigroup with two generators a < 7 (and the lexicographical order 0:7 < 7a)
defined by Fa = f(A\) and F1 — C\. Indeed, the only nonzero components of grFC
are gr^j kC = f(Ai)Ck/f{Ai+i)Ck-i, hence grFC is isomorphic to the one-sided
product A<S>° (C/f(Ai)C) in degrees ^ 3. Therefore, the conditions of Theorem 7.1
are satisfied and we conclude that grFC ~ A 0° (C/f(Ai)C). For example, if С is a
quadratic algebra and t G C\ is such an element that Ct С tC, C/tC is Koszul, and
the left multiplication by t is injective on C\ and C2, then t is not a left zero divisor
in C. For a central element t, this statement is equivalent to the nonhomogeneous
PBW-Theorem 2.1 of chapter 5.
Example 3. The twisted tensor product А 0 я В of two quadratic algebras
A and В with respect to an operator R: Bi 0 A\ —> A\ 0 B\ is defined as
the quadratic algebra with the generators space A\ 0 B\ and the relations space
I A + Ib + (b 0 a — R(b 0 a)). Applying Example 2 to the natural morphism
A —> A®RB we find that the multiplication map of vector spaces A<g>B —> A®RB
is an isomorphism iff the condition in degree 3
R23R12{B1^IA) ClA®Bl
Я12Я23(/Б0Л!) Cii0/B
is satisfied. It is easy to see that the Koszulness condition is not needed here.
Example 4. Consider the case when Г = Nm, 7(ai,..., am) = a\ + • • • + am,
and the Г-filtration on A is generated by the complete flag in A\ associated with a
basis #i,..., хш. Then the quadratic part В of the associated Г-graded algebra is
defined by relations of the following form: for any г there may be a relation x2 — 0
and for any i ^ j one has the following four possibilities:
(1) XiXj = qijXjXi with some qij G Ik*;
(2) either XiXj = 0, or XjXi = 0;
(4) no relation.
Let yi = x\ be the generators of B\ It is shown by Froberg in [52] that such an
algebra В is Koszul iff for any two monomials xa G В and y& G B[ there exists an
index i and a multiindex 7 such that either xa = cx1Xi and yiyP ^ 0 or y@ = cyiy1
and xaXi 7^ 0, where с G Ik*. We give a proof of this fact in the following three
cases when this condition is satisfied automatically:
(i) there are no relations of type (1) (B is a monomial algebra);
(ii) there are no relations of types (2) and (4) (B is a quotient of a skew-polynomial
algebra by monomial relations);
(iii) there are no relations of types (2) and (3).

8. COMMUTATIVE PBW-BASES 91
In case (i) Koszulness of В follows from Corollary 4.3 of chapter 2. In case (ii)
Koszulness of В will be proved in Theorem 8.1 below. Finally, case (iii) is dual to
case (ii).
The simplest example of an algebra of the above type for which Froberg's
condition is not satisfied (and hence the algebra is not Koszul) is the algebra with
generators Xi,X2>#3 and relations X1X2 = Х2#ъ х\хъ = ^3^1 = 0.
The following statement generalizes the results of sections 3 and 5 to arbitrary
(one-generated) Г-valued filtrations.
Proposition 7.2. Let A = {V, /} be a quadratic algebra equipped with a one-
generated Г-valued filtration F. Then the following two conditions are equivalent:
(a) the algebra grFA is Koszul;
(b) for every n ^ 0 the collection of all the subspaces V®k~l ®/®у®п-А:-1 <-
V®n together with the subspaces
Fa{V®n)= Y, FhV ® ■ ■ ■ ® FinV, а€Г„,
in V®n is distributive.
Proof: This follows from the Backelin's criterion of Koszulness. For the part (b) =>
(a) one has to check first that the algebra grF^ is quadratic. For the implication
(a) => (b) one can apply Corollary 7.3 of chapter 1, (b) => (a) (the condition
grF(Xi + Xj) — grFXi + g*FXj needed in this result follows from the absence of
cubic defining relations in grF^). □
Remark 2. Actually, the above argument proving (a) => (b) uses only
Koszulness of q(grFA) and the absence of relations of degree 3. Hence, it provides another
proof of Theorem 7.1.
For a graded ordered semigroup Г, let Г° denote the same graded semigroup
with the opposite order. Let F be a one-generated Г-valued filtration on a quadratic
algebra A. Let us define the dual Г°-valued filtration F° on the quadratic dual
algebra A1 by the rule F°A\ = (Fi>Ai)±, where i' is the element preceding i G Ti
in the order of Г.
Corollary 7.3. Let F be a one-generated Г-valued filtration on a quadratic
algebra A. Then the associated Г-graded algebras grFA and grF A1 are Koszul
simultaneously.
Proof: One can either use Proposition 7.2, or observe that q(grF A[) ~ (qgrF^)!
and apply Theorem 7.1 and the argument from the proof of Theorem 4.1. □
8. Commutative PBW-bases
In this section we discuss the notion of a commutative PBW-basis for a
commutative algebra. This is not the same as a noncommutative PBW-basis for such
an algebra (although there is a relation between the two notions, see below). Note
that both notions fit into the context of section 7 but the corresponding semigroups
are different: in the latter case it is the free noncommutative semigroup with m
generators, and in the former case one has to replace it with the free commutative
semigroup Nm. On the other hand, commutative PBW-bases correspond to
quadratic (commutative) Grobner bases for the ideals of relations [35, 26]. Commutative
algebras admitting such a basis are called G-quadratic (see [38]).

92
4. POINCARE-BIRKHOFF-WITT BASES
The definition is completely parallel to the noncommutative case, with the only
difference that noncommutative monomials are replaced by the commutative ones
Let A = k[xi,... ,xm]/Jc be a commutative quadratic algebra
with a fixed set of generators. Let us order commutative multiindices by the inverse
lexicographical order: (ai,..., am) < (bi,..., bm) iff there is such к that a\ = fti,
... , afc_i = bk-i and a& > Ь&. Note that we can also identify the set of commutative
multiindices of degree n with Symn[l, ra], the set of unordered n-tuples of elements
of [l,ra]. Applying the construction of Lemma 1.1 to the subspace of quadratic
relations Ic С S2(F) we obtain the corresponding subset 5 С Sym2[l,m]. Let
g(n) q Symn[l,m] be the set of all multiindices which are not divisible by any
a G S. It is easy to see that the commutative PBW-monomials (x6, b G Un>o S^)
generate Л as a vector space. If they are also linearly independent then we say that
they form a commutative PBW-basis of A. If A admits such a basis then it is called
G-quadratic (or commutative PBW-algebra).
The commutative PBW-Theorem states that if the commutative
PBW-monomials form a basis in As then the same is true in any degree n. Also, any
commutative G-quadratic algebra is Koszul (see [72]). To derive these statements from
Theorem 7.1, let us consider the Nm-valued filtration Fa ~ (xb | b ^ a) on A (see
Example 4 of section 7). The quadratic part of the associated Nm-graded algebra is
isomorphic to the commutative monomial algebra k[xi,... ,xm]/(xa = 0 | a G 5).
It remains to use the following result due to Froberg [52] (our proof differs from
the proof given in [52]).
Theorem 8.1. Let SP be a skew-polynomial algebra, i.e., a quadratic algebra
with generators xb ..., хдг and relations
XjXi q^jXiXj , t ч J,
where {q%j,l ^ г < j ^ N) is a collection of nonzero constants. Then every
quotient-algebra of SP by monomial quadratic relations with respect to generators
#i,... ,хдг is Koszul.
Proof: Let A be such a quotient. We are going to prove a stronger statement: all
A-modules A/1, where / is a left ideal generated by some subset of Xf's, have free
linear resolutions as left Л-modules (in fact, / is a two-sided ideal but this is not
important for us). We use double induction. The first induction is in cohomological
degree: we want to prove that for every n and every ideal / as above one has
Tor£-(k, A/1) = 0 for г ^ n and j > i. For n = 0 this is clear. Now let us
assume that this is true for some n. To deduce the assertion for n + 1 we use the
induction in the number of generators of i\ For / = 0 the assertion is clearly true.
Assume that our assertion that Tor^- (k, A/I) = 0 for г ^ n + 1 and j > i holds for
every / generated by ^ m elements. Given an ideal / generated bym + 1 elements
Xix,..., £;m+1 we can consider the left ideal J С I generated by the m elements
Xi2,..., £;m+1. Then the exact sequence of Л-modules
0 —>I/J —> A/J —> A/I —> 0
leads to a long exact sequence
> Tor^+1(k, A/J) -^ Tor^+1(k, A/1) -^ Tor^(k, I/J) -^ ....
Since Tor^+1(k, A/J) is concentrated in the internal degree n + 1 by the induction
assumption, it suffices to prove that Tor^(k, I/ J) has internal degree ^ n + 1. But

8. COMMUTATIVE PBW-BASES
93
the Л-module I/J has one generator x^ of degree 1 (the image of x^) and the
relations XiX^ = 0, where either i G {i2, •. • ,im+i} °r XiX^ = 0 in A. Therefore,
I/ J ~ A/J'{—1) for some ideal J' of the same type as above. It remains to apply
the assumption of the external induction to the space
Remark 1. The collection of left ideals considered in the above proof is an
example of а Коszul filtration. For commutative algebras this notion was introduced
in [40]. Piontkovskii observed in [93] that one can also consider a similar notion
for noncommutative algebras.
Remark 2. Commutative Grobner bases of the ideals of relations between a
finite number of generators are always finite (see [35]). Thus, the commutative
version of Remark from section 4 states that for every commutative graded algebra
A generated by A\ the Veronese subalgebra A^ is G-quadratic for all sufficiently
large d. In particular, we obtain another proof of the fact that A^ is Koszul for
all d > 0 (see Corollary 3.4 of chapter 3, [16], [46]).
The natural question is whether there is any relation between commutative
and noncommutative PBW-bases of commutative algebras. It can be easily seen
that if a commutative algebra A has a PBW-basis in the noncommutative sense
(with respect to the lexicographical order) then it also has a commutative PBW-
basis with the same generators (with respect to the inverse lexicographical order
on commutative monomials). Now assume that A has a commutative PBW-basis.
Let 5 С Sym2[l,ra] be the corresponding subset of commutative monomials (such
that |5| = dim^2). Let us consider also the subset S С [l,ra]2 obtained by looking
at noncommutative quadratic monomials in A. Then 5 consists of all pairs (z,j)
such that (z,j) ^ (j, г) and p(i,j) G 5, where p is the natural projection [1, ra]2 —>
Sym2[l,m]. Our commutative PBW-generators define a noncommutative PBW-
basis iff the relation (i,j) G S is transitive:
(ij) e S and (j, k)eS => (i, к) g S.
In fact, commutative PBW-generators often do not generate a noncommutative
PBW-basis. Furthermore, below we will give examples of commutative
quadratic algebras that possess commutative PBW-bases but no noncommutative PBW-
bases.
One can view the subset 5 С Sym2[l,m] defining a commutative quadratic
monomial algebra A as the set of effective divisors of degree 2 on the set [l,m].
Here by divisors on [l,m] we mean formal linear combinations YlT=\ пг(г) where
щ Е Z. The degree of such a divisor is J2ni- Effective divisors are those with
щ ^ 0. For a pair of divisors D\ and D2 we write D\ ^ D2 if D2 — D\ is effective.
The support of a divisor YlT=\ п*(г) ls tne set °f ^ sucn tlmt щ ф 0. We have
dim^n = |S(n)|, where S^ is the set of effective divisors D on [l,m] of degree n
such that for every effective divisor E of degree 2 with E ^ D one has E G S.
The next result gives a formula for the Hilbert series of A. Let us say that a
subset H С [1, m] is S-complete if for every iJeH with г ф j one has (i) + (j) G S.
We denote by H(S) the set of all S-complete subsets of [l,m]. For every subset
H С [1, га] we denote by L{H) the number of i G H such that 2(г) G 5.

94
4. POINCARE-BIRKHOFF-WITT BASES
Proposition 8.2. For every commutative quadratic monomial algebra A one
has
z\h\
м*)= Е (i_2)L(*r
Proof: It is clear that the support of a divisor in 5^n^ is an 5-complete subset of
[l,ra]. Let Я с [l,m] be an 5-complete subset, if С Я be the subset of i such
that 2(i) e S (recall that \K\ = L(H)). Then divisors in 5(n) that have Я as a
support are exactly all divisors of the form
Ew + ZX'W'
where rtj ^ 0, YljeKno ~ n~\H\- This immediately gives the required formula. □
Example 1. Here is the promised example (found using a computer) of a
commutative monomial quadratic algebra with the Hilbert series that differs from that
of any noncommutative monomial algebra. In particular, this algebra has no non-
commutative PBW-bases. Namely, consider the algebra
A = k[xi,..., x7]/(xl, ...,Xj, xix4, x2x5, х3хб, х2х7, Х4Х7, x6x7).
Its Hilbert series is hA(z) — 1 + Iz + Ibz2 + llz3 + z4. Assume that there is a
noncommutative monomial algebra with the same Hilbert series. This would mean
that the corresponding oriented graph has 7 vertices, 15 edges, 11 oriented paths of
length 2, 1 path of length 3 and no paths of length > 3 (see section 6). This graph
cannot have loops or oriented cycles (because the algebra is finite-dimensional).
Therefore, the unique path of length 3 has to pass through 4 distinct vertices
^i?^2j^3?^4- It follows easily that each of the remaining 3 vertices wi,W2,W3 is
joined with the set {vi,..., v^} by ^ 2 edges. There may be also ^ 6 edges in the
subgraph with the vertices {vi,..., г^} and ^ 3 edges in the subgraph {wi,W2, W3}.
To obtain the total number of 15 edges all of these estimates have to be attained.
This implies that there is a unique oriented graph (without oriented cycles) that
has 7 vertices, 15 edges, and 1 path of length 3. However, this graph has 10 paths
of length 2 instead of 11.
Example 2. As we have seen above, Hilbert series of commutative and
noncommutative PBW-algebras are given by explicit rational functions determined by
the combinatorial structure of quadratic relations. It would be interesting to
determine how "big" is the subset of Hilbert series of PBW-algebras in the set of all
Hilbert series of Koszul algebras. Among 83 cases of various Hilbert and Poincare
series of commutative quadratic algebras with four generators listed in [107] we
found two examples of Koszul algebras whose Hilbert series differ from that of any
commutative or noncommutative PBW-algebra. These algebras are
A! — k[x, y, z, u]/(x2 + xy, x2 + zu, y2, z2, xz + yu, xu)
A" = k[x, г/, z, u]/(xy + yz, xy + z2 + yu, yu + zu, y2, xz)
with the Hilbert series
hA, {z) = l+4z + 4:z2+z3 +z* + z5 + z6 + z7 + -••
hA»{z) = 1 + 4г + bz2 + 4z3 + 5z4 + 6z5 + 7z6 + 8z7 + • • • .

9. Z-ALGEBRAS
95
Example 3. Even though not every Koszul commutative algebra is G-quadratic,
there are situations when these two notions coincide. For example, Backelin showed
in [15] that a commutative quadratic algebra A is Koszul provided dim^2 ^ 2.
Later Conca [38] proved that (if characteristic is ф 2 and the field is algebraic
closed) with one exception every such algebra is G-quadratic (the exception is
k[x, y, z]/(x2,xy, y2 — xz, yz)). Also, it is shown in [38] that a generic commutative
quadratic algebra with dim A2 < dimAi is G-quadratic.
Example 4. It is an interesting question for which projective varieties X С Pn
the homogeneous coordinate ring Rx is G-quadratic. For example, it is known to
be true in the following cases:
(1) X is a set of ^ 2n points in a general linear position in Pn (see Thm. 3.1 of
[39]);
(2) X is a canonical curve (nonhyperelliptic, nontrigonal, not a plane quintic) - see
[50] and [39], Thm. 5.1. In fact, in [50] even a noncommutative PBW-basis is
constructed.
On the other hand, it is known that if X is a generic complete n-dimensional
intersection of quadrics in Pe+n and n < 6 ~~ ^ then Rx is not G-quadratic
(see [46], Cor.20). For example, this is the case for a generic complete intersection
of 5 quadrics in P5.
9. Z-algebras
The notion of Z-algebra (introduced in [25, 23]) is a convenient generalization
of the notion of graded algebra. Perhaps, a better term would be (Z, ^)-algebra,
i.e., an algebra over the partially ordered set (Z, ^). Since we never explicitly
mention algebras over Z, hopefully, using the term Z-algebra should not lead to
confusion.
Definition 1. A (positively graded) Z-algebra A over the ground field Ik is defined by
a collection of finite-dimensional vector spaces (Аат,&,т £ ^) such that AGT — 0
for a < т and Aaa = Ik, equipped with associative product maps
Apa ® Aar > ApT, P,<7,T G Z.
An A-module ЛЛ is a collection of vector spaces (Л4т,т G Z) equipped with action
maps AaT ® Л4Т » -M-a that are compatible with the multiplication in A. The
abelian category of Д-modules is defined in the obvious way.
For a Z-algebra A the relation spaces lpT are defined from the exact sequences
0 —> lpT —> Ap,p-i (g) Ap-i,p-2 0 • • • 0 Дг+2,т+1 ®
where the maps mpT are induced by the product in Д.
Definition 2. A Z-algebra A is called quadratic if the product maps mpT are surjective
and the relation spaces XpT are generated by the quadratic relation spaces Xa+i)a_i:
ф APlP-i®' ' '(S)A7+2,a+l®^a+l,a-l<S)A7-l,a-2®- ' '<8)Лг+1,т > ?рт * 0.
р-1>а^т+1
The dual quadratic Z-algebra A[ has the generator spaces Л^+1 а = *Д*+1 а and
the quadratic relation spaces ^-+i,a-i = ^a-\-i,a-i c Aa+l a 0 Aaa_1.
For a Z-algebra A and for a G Z let ka denote the irreducible Д-module
defined by (ka)T — 0 for а ф т and (ka)a — k. We have the following analogue of
Proposition 3.1 of chapter 1.

96
4. POINCARE-BIRKHOFF-WITT BASES
Proposition 9.1. One has Ext^(ka,kT) = 0 for i > a — т and the diagonal
Z-algebra BaT — Ext^~r(ka,Ikr) is the quadratic Z-algebra dual to the quadratic
part of A. □
Generalizing Backelin's Koszulness criterion to Z-algebras we can make the
following
Definition-Proposition 9.2. A Z-algebra A is called Koszul if the following
equivalent conditions hold:
(1) Ext^(ka,К) = О for i^a-r;
(2) Л is quadratic and for any p ^ r G Z the collection of subspaces
Aowo-i® •••®Xa+i,a-i®---® Ar+i,T С Лр,р-1<8>-- •<8>Лг+1,т, р-1^ сг ^ т + 1
is distributive.
In particular, quadratic dual Z-algebras A and A are Koszul simultaneously, and
for a Koszul algebra A one has Ext^(ka,kr) ~ AaT. □
Example. There is a natural embedding of the category of graded algebras into
the category of Z-algebras. Namely, for a graded algebra A we define the associated
Z-algebra A1 by the rule A^T = Aa-T. Then the category of Az-modules coincides
with the category of graded A-modules. A graded algebra A is quadratic (resp.
Koszul) iff the corresponding Z-algebra Az is of the same type.
Remark. For n > 0 one can define an [n]-algebra A as a collection of vector
spaces (AaT,n ^ <r, r ^ 1) and of product maps of the same form as above.
Similarly, replacing the partially ordered set (Z, ^) with ([l,n],<) one can define all
the notions above for [n]-algebras.
Now for a graded algebra A we can define the associated [n]-algebra by the rule
A)?}- — Aa-T for n ^ a, r ^ 1. It is easy to see that the [n]-algebra A^ is Koszul
iff A is n-Koszul.
On the other hand, with an [n]-algebra Л one can associate the graded algebra
Л© by the rule Af = ®a_T=iAaT. Since A® — kn the notion of Koszulness
still makes sense for A® (see Example 2 of section 9 in chapter 2). Note that the
category of Л-modules is equivalent to the category of nongraded Д®-пк^и1е8. As
for the category of graded *4®-modules, it splits into the direct sum of Z copies of
the category of Л-modules [126]. It follows easily that ал [n]-algebra A is Koszul
iff the graded algebra Л® is Koszul.
Finally, by modifying the degree zero component of a graded algebra В with
Bo ~ ken we get the graded algebra Bk = k 0 (©i>0-Si)j and this operation
preserves Koszulness.
Considering the composition of the three operations above, A \—> A^ \—>
^[n],®,k^ we gej. a transformation on the class of graded algebras that kills the
homological information in internal degrees > n, while preserving it in internal
degrees ^ n. This kind of construction can be used instead of the operation A \—>
AoC, where С is the exterior algebra with n generators, to produce counterexamples
with Hilbert series of non-Koszul algebras (see section 5 of chapter 3).
10. Z-PBW-bases
While the Koszulness condition does not change when passing from a graded
algebra to the corresponding Z-algebra, the notion of PBW-basis becomes more
general.

10. Z-PBW-BASES
97
Definition. Let Л be a quadratic Z-algebra. Choose a family of bases
in the generator spaces *4a+i,a- Applying Lemma 1.1 to the space Ла+1,а 0*4a,a-i
with the basis x^+ 'a (8) a;J,a_ in the lexicographical order and the subspace of
quadratic relations Xa+i)a_i, we get the corresponding subset of indices Sa+1,a~1 С
[l,ma+i>a] x [l,ma>a_i]. We say that the family of bases A'a+1'a is a set of PBW-
generators for a quadratic Z-algebra A if the monomials of the form
xw"i * * * <£',г > where fr^1." ^"^ e Sff+1'ff_1 for all p-l^tr^r + 1,
form a basis of ApT for all p ^ r.
Proposition 10.1. For any quadratic Z-algebra A, the PBW condition above
for p — т — 3 implies the same property for all (p,r). Furthermore, a family of
bases Afa+1'a is a set of PBW-generators for A iff for any p ^ r t/геге exists a basis
(2/ip,p_i,...,iT+i,T) o/t/ie space *4p,p-i 0- • *0.At+i,t distributing the lattice generated
by the subspaces APlP-i 0 • • • 0 2a+i;a_i 0-- -0 Лт+1,т, sucft t/iat (2/ip>p_i,...,iT+i>T)
differs from the basis xP,p~_ 0 • • • 0 ж]" 'r 6y £/ie action of an upper triangular
matrix. A Z-algebra admitting a PBW-basis is Koszul. If A is a quadratic Z-algebra
admitting a PBW-basis then the same is true for the dual quadratic Z-algebra A.
Proof: The proof is similar to that of Theorem 2.1 and Proposition 5.1. □
Let A be a quadratic graded algebra and let Az be the corresponding Z-algebra.
By definition, a Z-PBW basis for A is a PBW-basis for Az. We say that A is a Z-
PBW algebra if it admits a Z-PBW basis. Thus, we obtain the following hierarchy
of quadratic graded algebras:
quadratic D Koszul D Z - PBW D PBW.
The example below shows that the third inclusion here is strict. We already know
that the first inclusion is strict (see Example 1 in section 2 of chapter 2). One
can show that the second inclusion is strict using generic algebras (see Remark 3
in section 4 of chapter 6). More refined counterexamples are provided by elliptic
Sklyanin algebras described in the next section.
Example. Consider the quadratic algebra mentioned in section 3:
[x2 + yz = 0
[x2 +azy = 0, a ^ 0, 1.
It is Koszul and has no PBW-basis. However, we claim that it has a Z-PBW-basis.
Indeed, consider the family of bases {xa+1'a,ya+1,a, za+1,a} defined by
(x = xa+1'a
) у = у°+1^+ЬаХа+1'а
yz = zGJr1^ + caxa+1'a
where ca+i = —l/ba, 6a+i = -l/(aca). One can check that it is a set of PBW-
generators for Az.
Remark. It is an open problem whether the set of quadratic algebras with a
given generator space admitting a Z-PBW-basis is Zariski constructible (as is the
case for usual PBW-algebras, see Remark in section 3).

98
4. POINCARE-BIRKHOFF-WITT BASES
11. Three-dimensional Sklyanin algebras
Let E be an elliptic curve, and let (Ca)aez be an arithmetic progression of
line bundles of degree 3 on E. This means that all the line bundles £a+i ® C~x
are isomorphic to a fixed degree 0 line bundle (and deg£a = 3). Following [31]
we associate with such a progression a Z-algebra A with Aa-\-i,a — H°(E, Ca) and
Лт+2,а — H°(E,Ca+i ® £a), where the multiplication Да+2,а+1 ® Лт+1,а —>
Ла+2,а is given by the product of sections. Note that over an algebraically closed
field one can always find a graded algebra A with Az ~ A. Indeed, one can choose
a cubic root of the line bundle Ca+\/Ca and consider the translation tx : E —► E
associated with the corresponding point x E E. Then one has an isomorphism
txCa ~ Ca+i for all <r E Z. This immediately leads to the description of A in the
form Az. Note that in the case Ca+\ ~ Ca the algebra A is isomorphic to k[x, y, z].
Theorem 11.1 ([31], Thm. 7.4). The Z-algebra A is Koszul and the
corresponding graded algebra A has the Hilbert series Ha(z) — (1 — z)~3.
Proof: The proof is somewhat similar to that of Theorem 7.7 from chapter 2. Let
(M.a) be a sequence of line bundles such that Л4а+1 ~ Л4а <S>Ca. We can associate
with such a sequence a Z-algebra В by setting BG^T = Hom(.Mr, Ma) and using the
natural composition as a product. Note that the quadratic part of В is naturally
isomorphic to A, so we have a homomorphism В —> A. We are going to check
that A is Koszul by applying the Z-algebra version of Corollary 5.7 of chapter 2.
Note that the natural maps H°(E, Ca) ® Oe —> £a are surjective, so we can
define rank-2 vector bundles Va from the exact sequences
(11.1) 0 —> Va —> H°(Ca) 0O£^4^O.
It is easy to see that the vector bundle £a+i ® Va is generated by global sections
and that there is an exact sequence
(11.2)
0 —- C+2 —- Я°(£а+1 ® Va) ®Oe-^ C*+i ®Va^0
(here we use an isomorphism £a+2 — ^a+i ^^a1)- Tensoring (11.1) with .Ma+n(S)
М.~+г, where n ^ 1, and taking global sections we get exact sequences
0 _> H°(Ma+n <8> ЛС+i ® Ъ) —> йа+п,а+1 (g) Я°(Дх) —> Sa+n,a —> 0.
Similarly, from (11.2) we get exact sequences
0 —> Sa+n,a+3 —> Sa+n,a+2 0 ff°(£<r+i 0 Kff) ^ Я°(Л4а+п <8> AC|i (8) УД
where the maps /n are surjective for n Ф 3. Note that we can combine the above
sequences into complexes
0 —> Sa+n,a+3 —- Sa+n,a+2 ® Я°(£а+1 0 Fff) ^Sa+n,a+1 ® Я°(£а) —>
Дт-+п,ст * 0,
exact for n ^ 3 and with the only cohomology at the term Sa+3)a+i ® H°(Ca) for
n = 3. It is easy to see that these complexes are grading components of complexes
of projective S-modules. Therefore, Koszulness of A follows from the Z-algebra
version of Corollary 5.7 of chapter 2. The above argument also shows that the
quadratic dual algebra to A has the Hilbert series 1 + 3z + 3z2 + z3 which implies
the formula for Ha by Corollary 2.2 of chapter 2. □

11 THREE-DIMENSIONAL SKLYANIN ALGEBRAS
99
Proposition 11.2. The above Z-algebra Л does not admit a PBW-basis (in
the lexicographical order) unless Ca ~ CG+\-
Proof: Assume the contrary, and let
Ja+ha = (хГ1,ст> С Ка+1,а = (x°+1",xZ+1'a) С Va+l,a = H°(E,Ca)
be the corresponding complete flags in the generator spaces V^+i>a = Да+1>а of A.
Let ga+2>a с {1,2,3}2 be the corresponding б-element subsets and 5 ' be their
3-element complements.
The PBW property means that for each a there is exactly one triple of indices
(a,b,c) e {1,2,3}3 such that (a,b) e £a+3'a+1 and (6, c) e 5a+2'a. Note that A
has no zero divisors, hence 5 ' С {2, 3} x {1,2,3} for all a and it follows that
(2,1), (3,1) e 5a+2,a. Also, if (3,3) e 5a+2,a then (3,3) <£ 5r+2,r for r = a - 1
and r = a + 1. Next, we claim that if (3,3) ф S ' then for both p — т and
p = т + 1 one has Kp+iiP = H°(Cp(—xp)) С H°(CP) for some points xp e E.
Indeed, this is just a reformulation of the fact that degree 1 zero divisors in Д!
correspond to points of E. Therefore, for all a we have Ka+i^a = H°(Ca(—xa))
with some xa E E. On the other hand, the fact that the dimension of the image of
the natural map
я°(£а+1(-ха+1))0Яо(/:а) —> я°(£а+1®Ах)
is equal to 5 implies that S* ,a П {1,2} x {1,2,3} = {(2,1)}. Hence, the third
element of 5 'is either (3,2) or (3,3). Applying the PBW condition again, we
obtain that £a+2'a = {(2,1), (3,1), (3, 2)} for all a.
In particular, we see that V^+2,a+i#a+i,a = ifa+2,a+iК-+1,а in Aa+2,<j'
Therefore, the subspaces H°(£a+i ® £a(—хт)) С H°(£a+i ® £a) coincide for r = a and
r = cj + 1. This implies that all the points xa coincide with each other: xa — x G E.
Let us denote by Za+i,a the divisor of zeros of a nonzero section from Ja+i>a.
conditions (2
spaces of Aa+2,a:
The conditions (2,1), (3,1) G S ' give the following inclusions between sub
Ka+2,<T+lJa+l,a С Ja+2,a+lV<T+l,a,
Va+2,a+lJa+l,a С Ха+2,а+1^а+1;а-
Hence, Za+2,a+i С Za+i)a + (x) and (x) С Za+i,a- It follows easily that Za+2,a+i =
Za_|_i)0., and therefore
□

CHAPTER 5
Nonhomogeneous Quadratic Algebras
In this chapter we consider algebras defined by nonhomogeneous quadratic
relations. Note that in principle nonhomogeneous quadratic relations can imply
nontrivial linear relations as well as other quadratic relations. It is easy to write
a necessary (cubic) self-consistence condition generalizing the Jacobi identity for
Lie brackets (see section 1). In section 2 we prove the analogue of the Poincare-
Birkhoff-Witt theorem stating that if the algebra defined by the quadratic parts
of the relations is Koszul then it coincides with the associated graded algebra of
the original nonhomogenous algebra provided the analogue of the Jacobi identity is
satisfied. In section 5 we consider some examples of this generalized Jacobi identity.
In section 4 we consider the analogue of quadratic duality in the nonhomogeneous
case. Following [99] we show that dual objects to nonhomogeneous quadratic
algebras are quadratic curved-DG-algebras (CDG-algebras). This notion appeared
also in the works of A. Schwarz [110] and A. Connes [41]. Interactions of
nonhomogeneous quadratic duality with various homology and cohomology functors are
studied in sections б and 7. In section 8 we consider homology of the completed
cobar-complex of a DG-algebra (or a DG-module). We show that for a Koszul
DG-algebra this homology can be viewed as the derived functor of the A+-adic
completion of the corresponding Koszul quadratic-linear algebra A.
1. Jacobi identity
For simplicity, we start with the case of an augmented algebra.
Definition 1. A quadratic-linear algebra, or an augmented nonhomogeneous quadratic
algebra is an augmented algebra A — Ik 0 A+ with a fixed space of generators V С
A+ such that the defining relations are quadratic-linear, i.e., the ideal of relations
J = ker(T(F) —> A) in T(V) is generated by the subspace J2 = Jn (V + V®2). A
morphism between quadratic-linear algebras (A, V) and (A*', V) is a homomorphism
of algebras A —> A' sending V to У.
Example 1. Let g be a finite-dimensional Lie algebra. Then the universal
enveloping algebra Ug can be viewed as a quadratic-linear algebra with the space
of generators g and the defining relations X ® Y — Y ® X — [X, Y], where I,7eg.
Definition 2. A nonhomogeneous quadratic algebra is an algebra A with a fixed
generating subspace V С A such that \a G V (where 1^ is the unit in A), and
the ideal of relations J С f(V) = T(F)/(1T - 1A) in f(V) is generated by the
subspace J2 = J П T2(V), where Ti(V) is the image of Tl(V) under the projection
T(V) —> T(V). A morphism between nonhomogeneous quadratic algebras {A,V)
and (A', V) is a homomorphism of algebras A —> A' sending У to У.
101

102
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
Example 2. Let 0—>1к-к—>g—>g—>0bea central extension of a finite-
dimensional Lie algebra. Then the quotient 1/$/(к — 1) can be viewed as a non-
homogeneous quadratic algebra with the space of generators g and the defining
relations X 0 Y - Y 0 X - [X, Y] - c{X, У), where X, Y e g, and c(X, Y) is a
2-cocycle describing the extension.
Example 3. A finite-dimensional algebra A can be viewed as a nonhomogeneous
quadratic algebra with the space of generators V = A and the defining relations
a 0 a! — ao! for a, a! G A.
For a vector space V with a fixed vector 1a £ V consider the modified tensor
algebra T(V) defined above. A subspace J2 С T2{V) appears as the space of
quadratic relations of a nonhomogeneous quadratic algebra A with generators V С
A and uni^l^E V iff J2 П f i(F) = 0 and the ideal J сТ(У) generated by J2
intersects T2(^) exactly at J2. For any subspace J2 С T2(V) we can consider
the quotient algebra A — T(V)/(J2). Note that the natural map V —> A is not
necessarily an embedding. However, its image generates A as an algebra, and we
can consider the corresponding filtration FnA — Vй (n-tuple product in A). On
the other hand, we have the quadratic algebra A^ = {V,I} with V = V/{1a)
and I — J2 mod T\{y). In this situation A is a nonhomogeneous quadratic algebra
with generators V and relations J2 iff the quadratic part of the associated graded
algebra grFA is isomorphic to A^°\ For example, if A is an algebra equipped with
a filtration Ik = FqA С F\A С • • • С A such that grFA is quadratic then A is a
nonhomogeneous quadratic algebra with V — F\A.
Let us choose a complementary subspace V С V to the line spanned by 1^.
Then we have a natural isomorphism f (У) ^ TJV), so that f 2(У) ~ V®2 0 У 0 к.
Every subspace J2 С Т2(У) such that J2 П Ti(y) = 0 is the graph of a linear map
ф: I —>V ®k, i.e.,
(1.1) J2 = {a + V(a) |ae/}.
Let us write ф — (</?, 0), where y>: / —» V, #: / —»Ik.
Proposition 1.1. //J2 С T2{V) is the space of quadratic relations of a non-
homogeneous quadratic algebra generated by V then the map ф = (<p, в) satisfies the
following condition:
(1.2) {ф12 - ф23){У 0/n/0F)c{a + ф{а) \ а e /},
w/iere У0/П/0Ус V®3, ф12 = <00id : /0V -> У0201/ (Vesp., V23 = id0^ :
V 0 / —>• У®2 0 V^. Л can fre rewritten as the system of equations
(1.3) (<p12 - ^23)(^ 0/П/0У)с/
(1.4) (tp о (<p12 - p23) - (в12 - 923)){V 0/П/0У) = О
(1.5) (0 о (ер12 - ip23))(V 0 / П / 0 V) = 0.
Proof: For xGF0/n/0l/we have x + V>12(z) G J and x + V>23(z) £ J- Hence,
ф12(х) -ф23{х) eJn f2(V) = J2. □
For a quadratic-linear algebra we have 0 — 0 and equations (1.4) and (1.5)
reduce to
(1.6) <p о {if12- if23) =0.

2. NONHOMOGENEOUS PBW-THEOREM
103
In the nonaugmented case the definition of ip and 9 depends on the choice of
a complementary subspace V С V. If V С V is another such subspace then there
is a map a: V —> к such that V = {v — a(v) \ v G V}. The corresponding maps
(<£>', 0') are related to (<£, 0) by
(1.7) y/ = <p + a 0 id + id 0a, в' = 9 + аор + а®а.
2. Nonhomogeneous PBW-theorem
The following theorem gives a sufficient criterion for the subspace J<i С V®2 0
У0 Ik to be the space of quadratic relations of a nonhomogeneous quadratic algebra
A. In addition, it describes the associated graded algebra of A with respect to the
natural filtration (see also [99, 33]). We keep the notation of Proposition 1.1.
Theorem 2.1. Let A^ = {V, J} be a Koszul algebra, and let ф: I —> V 0 k
be a map satisfying (1.2). Then the corresponding quotient algebra A — T(V)/(J2)
is a nonhomogeneous quadratic algebra with grFA ~ A^.
Proof: Let us define recursively Jn := VJn-i+Jn-{V С ТП(У), where n ^ 3. Then
the ideal generated by J^ in T(V) is J = IJn>2 ^i- It is easy to see that grFA ~ A^
iff Jr\Tn(V) — Jn for all n. Clearly, it suffices to check that Jn nTn_i(Vr) = Jn-i-
Note that condition (1.2) is equivalent to J3 П ^(F) = /2-
Set Д* = У®*"1 0/0 у®"-*"1 с V®n for г = 1,..., n - 1. For any element
x G Jn we have
n-l
я = ^(n + VM+1fi) mod Jn_i,
г=1
where r^ G i?i- If £ G Jn П Tn_i(y) then we have r\ H + rn_i = 0. We need to
show that Y^i=\ /0г,г+1гг £ Jn-i m this case.
We use the characterization of Koszul algebras in terms of distributive lattices
(see Theorem 4.1 of chapter 2). Distributivity of the collection (i?i,..., Rn-\)
implies that
Ri П (Дг+i + • • • + Rn-l) = Ri П i?i-(-i H h ДгП Rn-\.
Hence, any vector (ri,..., rn_i) G 0™Г]_ i?i such that ^2 и = 0 in F®n can be
presented as a sum of vectors of the same type with only two nonzero components
ri — —rj (cf. Proposition 7.2 of chapter 1, (a) => (c*))« It remains to show that
фг,г+1г — фз,з+1г £ jn_1 for any element r G RiilRj. If |г — j| = 1 then this follows
from (1.2). Otherwise, we have фг1г+1фзл+1 = фо~1^фг^1 ? and hence
^M+ir _ ^j,i+ir = (id+^'+^^^+V - (id+^i+1)^'+1r G Jn_i.
П
Another proof will be given in section 7 (see Proposition 7.2).
Example. Without the Koszulness assumption the statement of the above
theorem is wrong. For example, consider two nonhomogeneous quadratic algebras
generated by x, у and z with defining relations
jxy = l \xy = x + y
jx2+y2 = 0 °Г |x2+y2 = 0.

104
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
In both cases by an easy calculation in degree 4 one can deduce the relation xy — yx.
On the other hand, we have I ®V C\V ® I = 0, so the condition (1.2) is trivially
satisfied.
Theorem 2.1 justifies to some extent the following
Definition. A nonhomogeneous quadratic algebra A is said to be Koszul if the
corresponding quadratic algebra A^ (obtained by taking homogeneous parts of
quadratic relations) is Koszul.
3. Nonhomogeneous quadratic modules
In this section we prove analogues of Proposition 1.1 and Theorem 2.1 for
modules over nonhomogeneous quadratic algebras. г
Definition 1(M). A nonhomogeneous quadratic module over a nonhomogeneous
quadratic algebra iD^Dk-l^isa left A-module M together with a fixed generating
subspace Я С M such that the submodule of relations L — ker(A 0 Я —> M) is
generated as an A-module by the subspace L\ — L П (V 0 H) С A<S> H.
A subspace L\ С V 0 Я appears as the relation space of a nonhomogeneous
quadratic module generated by Я iff the submodule L с A 0 Я generated by L\
intersects V0 Я exactly at L\. For any subspace L\ С У 0 Я we can consider the
quotient A-module M = A<S>H/A-Li equipped with the filtration FnM — FnA-H,
On the other hand, we have the quadratic A^0^-module M^ = (H,K)A(0) where
К — L\ mod к^ЯсУ^Я. In this situation M is a nonhomogeneous quadratic
module with generators Я and relations L\ iff the quadratic part of the associated
graded module grFM is isomorphic to M^°\ In particular, if grF^ is a quadratic
algebra and F0M с F\M С • • • С M is a filtered A-module such that the grFA-
module grF M is quadratic, then M is a nonhomogeneous quadratic A-module with
H = F0M.
As before, we choose a complementary hyperplane V С V to the line к- 1а С V.
Then we have V ~ V 0 Ik and V 0 Я ~ Я 0 У 0 Я. The subspace Ьг с V 0 Я is
the graph of a linear map /л: К —> Я:
(3.1) Li ={c + ai(c) \ceK}.
Proposition 3.1.(M). Assume that J2 С T2(V) is the space of relations of a
nonhomogeneous quadratic algebra A generated by V and L\ с V 0 Я is the space
of relations of a nonhomogeneous quadratic A-module generated by H. Then the
maps ф and ii defined by (1.1) and (3.1) satisfy the condition
(3.2) (<ф12 - ^22>){V 0#П/0Я)с{с + д(с) | с е К},
where V®KnI®HcV®V®H7 ф12 = ф 0id : I 0 Я -> У 0 Я 0 Я (Vesp.,
/х23 = id®/z : 1/ 0 К —> 1/ 0 Я,). Setting ф = (<р,0), w/iere </? : / —> V and
# : / —> к, we can rewrite this condition as the system of equations
(3.3) (ip12 - v23)(V 0КП/0Я)сК
(3.4) (м о (ip12 - M23) - 912){V 0 К П / 0 Я) = 0.
2We are grateful to A. Braverman for pointing out to us the possibility of considering such
analogues.

4. NONHOMOGENEOUS QUADRATIC DUALITY
105
Proof: For yeV<g>KnI®H the image of у + ip12(y) is equal to zero in A 0 Я,
while the image of у + //23(у) belongs to L С A 0 Я. Hence, ф12(у) — M23(y) £
L(1V®H = Ll □
If we change the complementary hyperplane У С У to У = {г> — а(г>) | f G V},
where a : У —» к, then the map /j, gets replaced by
(3.5) // = /z + a0id.
Theorem 3.2.(M). Le£ A be a nonhomogeneous Koszul algebra with A^ —
{V,I}, and let M^ — (H,K)A(o) be a Koszul A^-module. Then for any map
/j,: К —> H satisfying (3.2) the corresponding quotient module M — A® H/A • L\
is a nonhomogeneous quadratic A-module with grFM = M^\
Proof: Consider the natural map f(V) 0 Я —> M and denote by P С f(V) 0 Я
its kernel. We have P — Un>i ^™> where P\ — L\ and Pn = VPn-\ + Jn® H for
n ^ 2 (with Jn С Tn(V) defined in the proof of Theorem 2.1). To prove the desired
isomorphism grFM ~ M(0) we have to check that P П (f П(У) 0 Я) = Рп for all
n. Note that condition (3.2) is equivalent to P^ П Ti(y) 0 Я = Р\. The rest of
the proof is parallel to the corresponding part of the proof of Theorem 2.1: one has
to use conditions (1.2) and (3.2) and distributivity of the collection of subspaces
(Si,..., Sn), where S{ = Щ 0 Я for i = 1,..., n - 1 and Sn = F®"-1 0 К с
F®n 0 Я. □
Similarly to the case of algebras we make the following definition.
Definition 2(M). A nonhomogeneous quadratic module M over a nonhomogeneous
Koszul algebra A is called Koszul if the quadratic module M^ over the Koszul
algebra A^ is Koszul.
4. Nonhomogeneous quadratic duality
Let A be a nonhomogeneous quadratic algebra with the corresponding
quadratic algebra A^ — {V,I}. Consider the dual quadratic algebra В — A^\ Recall
that we have B^ — I*. Define ip*: Bi —> B^ and в в £ B^ as dual to the maps
ip: I —> V and 9: I —»Ik. The relation dual to (1.3) has the form
(y?*12 - (^*23)(/±) СУ^/Ч/1^.
This means that ip* extends to an odd derivation of degree -hi on Я, i.e., a map
dB ■ В —> В of degree -hi such that
dB(xy) =dB(x)y-h(-l)xxdB(y), xGBx, У £ By.
Then (1.4) and (1.5) are equivalent to
d|(x) = [0B,x] for all хеБ, <2Б((9Б) = О
(it suffices to check the first relation for x G B\). In the augmented case we have
9B — 0 and d2B — 0. The transformation (1.7) corresponds to
(4.1) d'B(x) = dB{x) -h [a,x], ^ = flB + dB(<*) + a2
for aeBi, where [ , ] denotes the supercommutator.
Now let M be a nonhomogeneous quadratic A-module with the corresponding
quadratic A(0)-module M^ = (H,K)AW. Let TV = M^[ be the dual quadratic

106
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
Б-module. Recall that we have N0 = Я* and Ni~K*. Let /i* : N0 —> Nx be the
map dual to ц. The relation dual to (3.3) has the form
This means that /x* can be extended to an odd derivation of the Б-module N
compatible with the derivation ds of B, i.e., a map djy : TV —> N of degree -hi such
that
djsf{xu) — dB{x)u -h (—l)xxdjsr(u), x G B^, и £ Nu.
The equation (3.4) is equivalent to
d2N(u) = 9Bu for all и е N.
In particular, in the augmented case we have d2N — 0. The transformation (3.5)
corresponds to
dfN{u) — djsf(u) -h оси.
Definition l.(cf. [99], [110]) A curved DG-algebra (CDG-algebra) В is a triple
В = (В, ds, Ob), where В = ©°^0 Bi is a graded algebra, d^: Bi —> Д+i is an
odd derivation of degree -hi, and в в G #2 is an element such that d2B(x) — [9в,х]
anddB(<9B) =0.
A morphism of CDG-algebras g: С —» i? is given by a pair g — (g, a), where
g: С —» Б is a morphism of graded algebras and a G #i is an element such that
g(dcx) = dBg(x) + [a,#(x)] and #(0c) = #в -h^ce-hce2.
A CDG-algebra is called quadratic (resp., Koszul) if the underlying graded
algebra is quadratic (resp., Koszul).
Note that the transformation В = (В^в.Ов) '—► -В' = {B,d'B,6'B) given by
(4.1) leads to an isomorphism a = (id, a): £?' —> В in the category of CDG-
algebras.
A nonnegatively graded DG-algebra В — (B,ds) (see Appendix) can be viewed
as a CDG-algebra with 9в — 0. Morphisms of such DG-algebras are given by
morphisms of the corresponding CDG-algebras (g, a) with a — 0.
Proposition 4.1. (%) T/ie above construction defines an anti-equivalence of
the category of nonhomogeneous quadratic (resp., quadratic-linear) algebras with a
full subcategory of the category of quadratic CDG-algebras (resp., DG-algebras).
(ii) We have an anti-equivalence of the category of nonhomogeneous (resp. quadratic-
linear) Koszul algebras with the category of Koszul CDG-algebras (resp. DG-
algebras).
Proof: Given a nonhomogeneous quadratic algebra A D V, we choose a
complementary hyperplane FcFto the line Ik and define the corresponding CDG-algebra
(В^в,вв) as above. A morphism /: A! —> A!' of nonhomogeneous quadratic
algebras is determined by its restriction p: V —> V" to the generating subspace
Vf С A'. Let V С V' and V" С V" be complementary subspaces to the unit lines.
Then we can write p\y> — q — a, where q G Нот(У, Vй), and a G V*. It is easy to
check that the dual map g*: V"* —> V* extends to a homomorphism of the dual
quadratic algebras g: B" —> Bf and the pair g — (g,oc) defines a morphism of the
corresponding CDG-algebras. This functor is fully faithful since a map between
generating subspaces extends a morphism of nonhomogeneous quadratic algebras
iff it preserves the quadratic relations. Part (ii) follows from Theorem 2.1. (See
also [99].) □

4. NONHOMOGENEOUS QUADRATIC DUALITY
107
We will describe the class of quadratic CDG-algebras appearing in part (i) of
the above proposition in Proposition 7.2.
Definition 2(M). A (left) curved DG-module (CDG-module) TV = (N,dN) over a
CDG-algebra В — (В, dB,9e) is a graded left Б-module N equipped with an odd
derivation djy: TV; —> TV^+i compatible with dB and such that d2N(u) — бви. А
morphism of CDG-modules (TV, djv) —» (TV',djv) is a degree 0 morphism / : TV —>
TV7 of Б-modules such that djyf = fd^. A CDG-module TV over a quadratic (resp.,
Koszul) CDG-algebra В is called quadratic (resp., Koszul) if such is the underlying
graded Б-module TV.
Note that a DG-module over a nonnegatively graded DG-algebra В (see
Appendix) is the same as a CDG-module over the algebra В viewed as a CDG-algebra.
We say that a DG-module is quadratic (resp., Koszul) if the underlying graded
module over В is quadratic (resp., Koszul).
It is easy to see that the categories of CDG-modules over isomorphic CDG-
algebras are naturally equivalent. More generally, for any morphism of CDG-
algebras g = (g, a): С —> В and a CDG-module TV over В the pull-back of
TV along g is a CDG-module over С defined by #*TV = (#*TV, d%), where g*TV is
the same graded vector space TV equipped with a C-module structure via g and
d%{u) — disr(u) + an.
Proposition 4.2.(M). (i) For any nonhomogeneous quadratic algebra A the
category of nonhomogeneous quadratic A-modules is anti-equivalent to a full
subcategory in the category of quadratic CDG-modules over the dual quadratic CDG-
algebra B.
(ii) If in addition A is Koszul then the category of nonhomogeneous Koszul A-
modules is anti-equivalent to the category of Koszul CDG-modules over В. □
If A is a quadratic-linear algebra then one can replace in the above proposition
CDG-modules by DG-modules over the dual quadratic DG-algebra B.
Example. The trivial module Ik over a quadratic-linear algebra A is dual to the
free DG-module N — В with djy — dB- Note that there is no natural way to define
a CDG-module structure on the free Б-module TV = В for a CDG-algebra В with
9в y^ 0. On the other hand, for any CDG-algebra В one can consider the trivial
CDG-module TV = TV0 = Ik with dN = 0. It is dual to the free A-module M = A.
Note that the opposite algebra Aop to a nonhomogeneous quadratic algebra
A is also a nonhomogeneous quadratic algebra with the same generating space V.
A right A-module R with a fixed generating subspace H С R is called a right
nonhomogeneous quadratic module if the opposite module Rop over Aop is a left
nonhomogeneous quadratic Aop-module with respect to generators H. The CDG-
algebra dual to Aop is Bop — (Bop, ^в°р, #в°р), where (contrary to our conventions
in chapter 1) we equip Bop with the multiplication that incorporates the sign rule
xopyop = (-ify(*yx)°v, and set dBop{xop) = {dB{x))op and вВоР = -(<9Б)ор. We
define a right CDG-module S over Б as a right graded Б-module S equipped
with a differential ds of degree -hi such that ds{sx) — ds{s)x + (—l)ssdB{x)
and d's(s) — —s6b- There is a natural equivalence between the categories of right
CDG-modules over В and left CDG-modules over Bop given by the rule S \—> 5op,
where xopsop = (-l)55(sx)op and d5oP(<>op) = (ds{s))op. The right module version

108
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
of nonhomogeneous quadratic duality associates with a right nonhomogeneous A-
module R = H 0 A/{c + /x(c) | с е if} • A the right quadratic CDG-module
5 = (5, d5) over В such that S = Д<°)! and ds,o = M*-
5. Examples
Here we describe examples of solutions of our equations (1.2) and (3.2) and
the corresponding nonhomogeneous quadratic duality. In particular, Example 5
includes the case of Lie super algebras, while Example 6 is related to quantum
groups.
1. Any finite-dimensional algebra A can be viewed as a nonhomogeneous
quadratic algebra with V = A. In this case / = V 0 V, A(0) = к 0 V, and (1.2) is
just the associativity condition for the map ф: V 0 V —> V 0 к. Similarly, an
augmented algebra A can be viewed as a quadratic-linear algebra with V — A+.
The dual DG-algebra is the normalized cobar-complex COB*(A) = (&nA*_®n (see
section 1 of chapter 1). We can view a finite-dimensional A-module M as a
nonhomogeneous quadratic A-module with H ~ M. The dual DG-module over COB9(A)
can be identified with the cobar-complex COB*(A, M) = 0n A*_®n 0 M*.
2. More generally, we can view a finite-dimensional module M over any non-
homogeneous quadratic algebra A D V as a nonhomogeneous quadratic module
with H = M. The corresponding graded A^-module is trivial: M(0) = M0(0) =
H. In this case К = V 0 H and (3.4) is just the compatibility of the action
map fi: V 0 H —► H with quadratic relations in A. As the dual object we get a
CDG-module structure on the free Б-module N = В 0 M*. The differential djsr
can be described explicitly as follows. The vector space M* has a natural structure
of a right A-module, so we can view the tensor product В 0 M* as a right В 0 A-
module. Let ел € V* 0 V С В 0 A be the identity element. We can consider ел
as an operator on Б 0 M*. Then it is easy to verify that d^ = ds 0 id-he^ (for
d£ = 0 such complexes appear in section 3 of chapter 2).
Assume in addition that A is quadratic-linear, so that d2N = 0. The
complex Km(A,M) = Б* 0 M dual to N is sometimes called the nonhomogeneous
Koszul complex of a module M over a quadratic-linear algebra A (its definition
can be extended to the case when M is infinite-dimensional). If A is Koszul then
the homology of K9(A,M) is isomorphic to Tor^ (к, М) (as follows from
Proposition 6.1 below). In particular, for M = A we get a free resolution of the right
A-module к called the nonhomogeneous Koszul resolution. More general
nonhomogeneous Koszul complexes are considered in [104].
3. Let A<°) be the symmetric algebra S(V), so that I = /\2VcV®V. Then
(1.3) is satisfied automatically and (1.6) is equivalent to the Jacobi identity for
the skew-symmetric bracket given by <р: Д V —► V. Furthermore, any element
в £ B<i is central, so (1.4) is equivalent to (1.6), so <p is a Lie algebra structure
on V. Assume first that в = 0, i.e., A is quadratic-linear. In this case A is
isomorphic to the universal enveloping algebra U(V). The dual DG-algebra is
the standard cohomological complex C*(V). The standard complexes C*(V,M)
for У-modules M appear in the nonhomogeneous duality for modules (see the
previous example). In the general case в is a 2-cocycle on V and A is the enveloping
algebra of the corresponding central extension 0 —> кя —>• V —> V —► 0, i.e.,

5. EXAMPLES
109
A = U(V)/(*c— 1). Note that transformation (1.7) corresponds to adding to в the
coboundary 2-cocycle da.
4. Let A(0) = /\(V) be the exterior algebra, so that I = S2V С V0 V. Assume
first that chark ф 2. Equation (1.3) implies that <Ж02)®£-£®<Ж02) = 0 for f G
V. Therefore, <p(£2) = A(f )f for some linear function A: V —► k. If (p has this form
then equations (1.4) and (1.5) are trivially satisfied. Using transformation (1.7) with
a = —A/2 we can make <p = 0. Therefore, the algebra A is the Clifford algebra
associated with the quadratic form в on the vector space V. In characteristic 2 we
get a structure of a restricted Lie algebra on V, i.e., a Lie bracket together with
a quadratic map V —► V : x \—► x2 such that [x, y] = (x + y)2 — x2 — y2 and
[x, [x,y]] = [x2, y]. In this case the construction from 2 provides the well-known
Koszul resolution for 2-restricted Lie algebras (see [104]). In the nonaugmented
case we also get a cocycle с on У together with a quadratic map c: V —► к such
that c(x, y) = c(x + y) — c(x) — c(y) and c(x, [x, y\) = c(x2, y).
5. Let V = Vo © V\ be a super vector space and let / С V 0 У be the
space of quadratic relations of the free commutative superalgebra generated by V:
/ ~ Д2 V0 ® V0 0 Vi 0 §2 Vi, where the embedding of Vo ® Vl into V 0 У is given by
vo 0 Vi i—► vo 0 V\ — V\ 0 ^o • Let ((/?, 0): I —> V ф к be a map satisfying equations
(1.3), (1.4) and (1.5). It is easy to see that if this map preserves parity then our
equations just mean that <p defines a Lie superalgebra structure and в is an even
2-cocycle (in the characteristic 2 or 3 as well). Let us denote by (p^b and ваЬ the
components of <p and в with respect to the above decompositions of V and / (so
that ip°c° : /\2V0 -> Vc, ^cl : V0 0 Vi -> Vc, etc.). The linear condition (1.3) is
equivalent to the system
<Р?°(* Л у) 0 С - £ 0 <р?0(х Л у) G §2Vb
^оЧжО®£+ f Ф^оЧ^О £ (v0®vi -^10^о I vs G Vs),
^1e20e-C0^1e2GS2y1,
where х, т/ G Vo, f G Vi, and ж( = x 0 ( - £ 0 x. Now assume that chark ф 2
and dim Vi > 1. Then it follows that </?5° = 0, (pg1 = 0, and </?р(^2) = A(f)f as in
section 4. Using transformation (1.7) we can also make tp\l = 0. Then it follows
from (1.4) that в01 = 0, and therefore A is the enveloping algebra of an even central
extension of a Lie superalgebra.
Note that in the remaining cases dim Vi = 1 or char к = 2 the subspace / С V®2
is determined by the subspace V\ С V. Consider first the case dimVi = 1 and
chark ф 2. Then / = Д2 V ф Vf2. In this case the only nontrivial linear condition
is that (p(x 0 f — f 0 x) G V\ for any x eV and f G Vi. Let ([ ], m) and (c, /) be
the components of (/? and в with respect to the decomposition of /. Our conditions
mean that [ ] is a Lie algebra structure on V, V\ С V is an ideal, с is a 2-cocycle,
and the following equations are satisfied:
\x,m(£2)] =2m(£\x,£])-2c(x,Z)Z, c(x,m(£2)) = 2f(S[x,®,
where ж G V, f G Vi. The transformation (1.7) has the following form:
m\e) = m{e) + 2аШ, c'(x, y) = c(x, y) + a([x, y]),

110
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
Consider the central extension V of V defined by the cocycle c, and let U =
U(V)/(k — 1) be the corresponding enveloping algebra. Then our algebra A is
isomorphic to U/(z), where z = £2 + rn{^2) + /(£2) is a quadratic element in U.
The above equations are equivalent to the condition that z is normal, i.e., Uz = zll.
It is easy to see that any normal quadratic element of U with the symbol of the
form £2 G §2(V) for some (еУ arises in this way.
If char к = 2 then the linear conditions are trivial. Our data consist of the Lie
algebra structure on V together with a restriction map x i—> x2 (as in section 4)
defined for x G V\ and the appropriate 2-cocycle.
6. Let (Г, +) be an abelian monoid, Ф С Г be a finite subset. Consider the
algebra A with generators xa numbered by a G Ф and defining relations
XaXp = Яа,рХрХас + Ca,p%a+p,
where we set xa = 0 for a £ Ф and assume that qp^Qa.p = 1, #<*,<* = 1, cp,cx =
—Qp,aCa}p and ca?Q = 0. It is easy to see that the analogue of the Jacobi identity
in this case reduces to the following equations:
{Яа,-уЯр,-у — Яа+Р,-у)Са,Р = 0,
Я-у,аСа,Рса+Р,-у + Яа,рСр,-уСр+-у,а + Qp,yCl,ocCy-^ot,p = 0?
where a,/3,7 G Ф. Note that A^ is a PBW-algebra (see Example 1 of section 2 of
chapter 4), hence it is Koszul. Therefore, Theorem 2.1 implies that grFA ~ A^
provided the above equations are satisfied. For example, this is the case for the
quantum universal enveloping algebra Uqn С Uqsl(N), where as generators we take
the root vectors (see [108]).
7. Let A be a nonhomogeneous quadratic algebra with A^ = {V, /}, and let
E С V be a subspace. We want to study nonhomogeneous quadratic A-modules M
such that the associated quadratic module M^ is generated by a given vector space
H^0 with defining relations K = E®H cV®H, i.e., Af<°) ~ (А^/А^Е)®Н.
Let В be the dual quadratic CDG-algebra to A, and let N be the quadratic CDG-
module corresponding to such M. By definition, we have N = В/ВЕ±®Н*, where
E1- С B\ is the orthogonal complement to E. Let us assume that E1- is left-normal
in Б, i.e., E^B С BE^. Then the quotient module В = B/BE1- is a quadratic
algebra. Now for any x G E1- and и G H one has
0 = djsr(xu) = <Ib(x)u — xdjsr(u) = dB{x)u.
This implies that dB{E±) = 0. Thus, in order to have any CDG-module structures
on N one should have dsiE^) = 0. Assume that this is the case. Then we can
equip В with the CDG-algebra structure, such that the differential d^ is induced
by ds and the element 9g G B2 is the image of в в £ Въ- Let us call the obtained
CDG-algebra B. The CDG-module N over В is the pull-back of a CDG-module
over B. Dualizing the relation dB{E^) = 0 we get (p(E®2 П I) С Е. It follows
easily that there exists a nonhomogeneous quadratic algebra A dual to В together
with a homomorphism A —> A and an A-module structure on H. The original
A-module M is isomorphic to the induced module M — A 0^ H.

6. NONHOMOGENEOUS DUALITY AND COHOMOLOGY
111
6. Nonhomogeneous duality and cohomology
Now we are going to show that nonhomogeneous quadratic duality provides a
way to compute the cohomology of nonhomogeneous Koszul modules over Koszul
quadratic-linear algebras.
Proposition 6.1. Let A D A+ d V be a Koszul quadratic-linear algebra and
let M D H be a nonhomogeneous Koszul A-module. Let also В be the quadratic
dual DG-algebra to A and N the quadratic dual DG-module to M. Then there is a
natural quasi-isomorphism of DG-algebras
(6.1) COB9 (A) —>B
and a compatible quasi-isomorphism of DG-modules
(6.2) COB*(A,M) —>ЛГ.
Passing to cohomology we obtain an isomorphism of algebras
EXT^k.k) ~ tf*(B) := H*dB(B)
and a compatible isomorphism of modules
ЕХТда.к) ~ H*(N) := H*dN(N).
Proof: We have COB* (A) = Barm(A)y and (6.1) will be dual to a natural morphism
of DG-coalgebras
(6.3) B* ^Bar.(A)
constructed as follows. Since Barm{A) is the cofree coalgebra cogenerated by A+ =
Bari(A), the natural embedding B{ = V ^-> A+ extends uniquely to a morphism
of coalgebras Б* —> Barm{A). This morphism commutes with differentials due to
commutativity of the diagram
/ > A+ 0 A+
V >A+
where / С V®2 is the space of quadratic relations in A^\ m : A+ 0 A+ —> A+
is the multiplication map. Similarly, since Barm(A, M) is the cofree comodule over
Barm(A) cogenerated by M = BarQ(A, M), the identical embedding TVq = H —► M
extends uniquely to a comodule homomorphism N* —> Barm(A,M). Moreover,
we get a morphism of DG-comodules
(6.4) AT* —>Bar.(A,M)
as follows from commutativity of the diagram
К > A+ 0 M
H >A
+
where К С V 0 H is the space of quadratic relations in M^°\ a : A+ 0 M —► M
is the action map. We define (6.2) to be dual to (6.4). It remains to show that
(6.3) and (6.4) are quasi-isomorphisms. The increasing nitrations F on the algebra

112
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
A and A-module M induce an increasing filtration on the complex Bar9(A, M) by
the rule
Fj(Afi®M)= Y^ FjlA+®->-®FjxA+®FkM.
j\-\ \-ji + k=n
Set Fj-iNJ = 0 and F5N? = Nf. Then (6.4) is compatible with nitrations and the
induced morphism of associated graded complexes is N* —> Barm (grFA, grFM),
where N* is equipped with a zero differential. Since the algebra A^ = grFA and
the module M^ = grFM are Koszul, the latter map is a quasi-isomorphism (see
Proposition 3.1 of chapter 1). It follows that (6.4) is also a quasi-isomorphism. The
same argument works for (6.3). □
Remark 1. Let us define a Koszul quadratic-linear structure on an augmented
algebra Л as a choice of a generator subspace V С А+ с A with respect to
which A becomes nonhomogeneous Koszul. In this situation we say that У is a
Koszul generator subspace. The above result allows us to associate (functorially)
with an augmented algebra admitting a Koszul quadratic-linear structure a quasi-
isomorphism class of DG-algebras. Namely, we claim that the DG-algebra В above
does not depend on the choice of a Koszul generator subspace V С A. Indeed, if
V С V" С А+ are two Koszul generating subspaces for A then the identity map
(A, V) —► (A, V") is a morphism of quadratic-linear algebras, and therefore it
induces a morphism of the dual DG-algebras B" —> Br. By the above proposition,
it is a quasi-isomorphism. Also, it is easy to see that for any two Koszul generating
subspaces V', V" С А+ there exists a third one V such that V + V" cV (this
follows from Proposition 2.2 of chapter 3). Analogously, the quasi-isomorphism class
of the DG-module N is an invariant of an augmented algebra A and an A-module
M admitting nonhomogeneous Koszul structures.
The question of existence of a nonhomogeneous Koszul structure on a given
algebra A or module M seems to be very nontrivial. Let us only observe that it
should be viewed as a finiteness question: indeed, on any finite-dimensional algebra
A (resp., module M) there is an obvious nonhomogeneous Koszul structure given
by V = A (resp., H = M), see Examples 1 and 2 in section 5.
Remark 2. Conversely, one can ask whether the quasi-isomorphism class of
the DG-algebra В (resp., DG-module TV) determines the isomorphism class of
an augmented algebra A (resp., A-module M). Here is a simple counterexample.
Consider the quadratic-linear algebra A = k[x]/(x2 + x). Then we have A(°) =
k[x]/x2, В = k[£], and й#(£) = £2. Clearly, Щ (В) = к and the morphisms of
quadratic-linear algebras к —> A and A —> к induce quasi-isomorphisms of the
dual DG-algebras к —> В —> к. For the A-module M = ku, where x acts by
xu = u, we have H2N(N) = 0. (See also Example in section 8.)
As we will show in section 8, the quasi-isomorphism class of the dual DG-
algebra to A (resp., dual DG-module to M) determines the A+-adic completion
of A (resp., M) and its higher derived functors.
7. Bar construction for CDG-algebras and modules
For a CDG-algebra В = (B,dB,eB) let Bar{B) = ф^овТ denote the bar
construction of the graded algebra В (see section 1 of chapter 1). Recall that
it has a natural bigrading Barij(B) = (bi 0 • • • 0 hi \ ji + • • • + Ji = j), where
bt G Bjt. The CDG-algebra structure on В provides the coalgebra Bar(B) with

7. BAR CONSTRUCTION FOR CDG-ALGEBRAS AND MODULES 113
three differentials depending on the multiplication in B, the differential ds and the
element 6в, respectively:
2-1
^(-1)*+-+*+% (g)... (g) bsbw ® • • • ® Ъ,
г
^(_l)Ji + -+it-i+t-l6l 0 . . . 0 ^(6t) 0 ... 0 6,,
t=l
г+1
^(_l)ii+-+it-i+*-i6l 0 ... 0 6t_! 0 0B 0 6t 0 •.. 0 b{.
t=l
The first differential д coincides with the differential on Bar(B) denned in section 1
of chapter 1 up to the signs reflecting the grading of В (these signs do not affect
the homology spaces).
Now let Barm(B) = Ваг*(В^в,9в) denote the total complex with the
differential D = д + d + 6, where Barn(B) = (Bj_i=nBari3(B). It follows from the
CDG-algebra axioms that D2 = 0, so Bar*(B) is a DG-coalgebra. Let Cobm(B) be
the dual DG-algebra with Ш)Ш(В) = Вагш(ВУ, and let Cob.(B) С Cob .{В) be
the DG-subalgebra given by СоЬш(В) = Barm(B)* = ®^1=шВаг^{В)\ This
DG-subalgebra looks as follows:
CO
> Cob3{B) > Cob2(B) > 0B^®B2*0B;ei2 > 0£i*0i-
hJ2 j=o
Note that as a graded algebra Cob(B) can be identified (up to a linear change in
the grading) with the algebra Cob(B) defined in section 1 of chapter 1. However,
the completions COB(B) and Cob(B) are entirely different even as vector spaces.
The next result and Proposition 7 below provide a generalization of
Proposition 3.1 of chapter 1 to quadratic CDG-algebras.
Proposition 7.1. Let В = (B,dB,eB) be a quadratic CDG-algebra, {V,I} =
Bl the quadratic dual algebra to B. Consider the maps <p = d*B :: / —► V and
6 = 6b'- I —► к (cf. section A). Then there is a natural isomorphism H$Cobm(B) =
Т(У)/(J2), where J2 = {a + </?(a) + 9(a) \ a e J}.
The proof is straightforward and is left for the reader.
There is a natural increasing filtration FkCobrn(B) = @i^li=rn>13arij(B)* on
the complex Cob*(B). It induces the standard filtration FkA = Vk on A =
НоСоЬш(В). The associated graded complex grFCob.(B) can be identified with
the usual cobar-complex of the graded algebra Б, so its homology is isomorphic to
Ext^(k, k). Thus, we get a spectral sequence
(7.1) 'Elq=ExtB™(k,k) =► g^Hp+qCob.{B)
with the differentials drvq: Brv q —► E7p_r qJrr_l. Now we can characterize the class
of quadratic CDG-algebras dual to nonhomogeneous quadratic algebras.
Proposition 7.2. (i) A quadratic CDG-algebra В corresponds to a
nonhomogeneous quadratic algebra A iff the spectral sequence (7.1) degenerates at 'Е\1Л for
i = 0. 1, and 2 (i.e., 'El_iA = 'E™iyi for i = 0, 1, and 2). Furthermore, one has
grF A = AW iff this spectral sequence degenerates at 'El_iA for all г ^ 0.
d(h 0 •
d(bi 0 •
£(bi 0 •
• 0 h) =
♦ 0 bi) =
■ 0 Ь{) =

114
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
(ii) If В is Koszul then (7.1) degenerates at 'El and Нф$СоЬт(В) = О, so the
natural map Cobm(B) —> A is a quasi-isomorphism.
Proof: Note that if В corresponds to some nonhomogeneous quadratic algebra then
this algebra has to be A = HoCobm(B) (by Proposition 7.1). Now (i) follows from
the identifications 'Elifi = Ext2j(k,k) = A\0) and 'E™^ = grf H0Cob.{B) = grf A.
For (ii) we observe that if Б is a Koszul algebra then all terms in 'E1 with p + q ф О
vanish, so the sequence degenerates. □
Combining parts (i) and (ii) of the above proposition we obtain a new proof of
the nonhomogeneous PBW-Theorem 2.1.
Remark 1. The above argument shows that the assumption of Koszulness of
A(°) = B- in Theorem 2.1 can be replaced by the weaker condition Ext^ ,3 (к, к) =
0 for j ^ 0. On the other hand, our first proof of Theorem 2.1 uses the distributivity
condition which is easily seen to be equivalent to the vanishing of Ext '^0)(k,k)
for j > 3 (the same condition is also used in the proof of the PBW-theorem for
nitrations in section 7 of chapter 4 and in the proof of Theorem 2.1 in chapter 6).
These conditions on A^0>} and on В are actually dual to each other (see Corollary 3.3
of chapter 2).
Now let S (resp., TV) be a right (resp., left) CDG-module over a CDG-algebra
B. Consider the bar construction Bar(S, B, N) = 0°^o S<S>Bfl 0 N (see section 1
of chapter 1) with the standard bigrading Barij{S, Б, N) = {s 0 b\ 0 • • • 0 bi 0 n |
k + ji + • • • + ji + I = j), where s G Sk, bt G Bjt, and n G iVj. Generalizing the
above construction of Bar(B) we define the following differentials on Bar(S, B, N):
d{s 0 b 0 n) = sb 0 n + (~l)ks 0 d{b) 0 n + (-1)*+^+-+*+*5 0 Ьщ
d{s 0 b 0 n) = ds{s) 0 b 0 n + {-l)ks 0 d(b) 0 n + (-i)fc+^+-+i*+is 0 b 0 dN(n),
6(s 0 6 0 n) = 5 0 £(b) 0 n,
where 6 = b\ 0 • • • 0 b{ G Б^ 0 • • • 0 Bji and 56 = 561 0 • • • 0 b{, bn = b\ 0
• • • 0 bin. Let Bar'(S,B,N) denote the total complex with the differential D —
д + d+S, where Barn(S,B,N) = (Bj-i=nBarij(SiБ,N). Let Co~b.{S,B,N)
denote the dual complex with Cobm(S, B, N) = Barm(S, В, N)y, and let
Cob.(S,B,N) с Cob.{S,B,N) be the subcomplex given by Cobm{S,B,N) =
0^_^тБаг^(5,Б,А^)*. Note that we have Bar9(B) = Bar*(k,B,k). Let us also
set
Barm(B,N) = £ar#(k,B,7V),
Cob .{В, N) = Co~b.{k, B, TV),
Co&.(B,7V)=Co6.(k,B,7V),
where к denotes the trivial right CDG-module over В (see Example 2 in section 5).
Similarly, we set Barm(S,B) = Bar*(S,B,k), etc. For example, the complex
Cob.(B,N) has the form
>Cob2{B,N)
CO
—> ф B*®jl ® B| ® B*®J2 ® 7V0* ф 0 Bfj ® N{ —> 0 B*®J' О JV0*.
J1J2 i j=o

7. BAR CONSTRUCTION FOR CDG-ALGEBRAS AND MODULES 115
Proposition 7.3.(M). Let В be a quadratic CDG-algebra. Set {V,/} = Bl,
A = HoCobm(B) (see Proposition 1.1). Let N = (N,djsr) be a quadratic CDG-
module over В and let (H, K)[y,i} = N[b be the quadratic dual module to N.
Consider the map /л = d*N 0 : К —> H. Then there is an isomorphism H$Cob9(B, N) =
A <g> H/(A • Li), where Lx = {c + /x(c) | с e К}. D
The increasing filtration FkCobm{B,N) = (BJj-i=m Ваг*Лв> NY on
the complex Cob9(B,N) induces the standard filtration F^M = VkH on M =
HoCob9(B,N). The homology of the associated graded complex is isomorphic to
Ext^iV, k), hence we get a spectral sequence
(7.2) '^;д=ЕхЦ9'р(ЛГ,к) =» gv^Hp+qCob.(B,N).
We have the following version of Proposition 7.2 for CDG-modules.
Proposition 7.4. (M). (i) Let A be a nonhomogeneous quadratic algebra and
В be the dual CDG-algebra. Then a quadratic CDG-module N over В corresponds
to a nonhomogeneous quadratic A-module M iff the spectral sequence (7.2)
degenerates at ,E\ii for i = 0 and 1. One has grFM = M^ iff this spectral sequence
degenerates at 'Е\{ i for alii ^ 0.
(ii) If В and N are Koszul then (7.2) degenerates at 'E1 and H^QCob9(B,N) = 0,
so the natural map Cob9(B,N) —> M is a quasi-isomorphism.
Note that the above proposition provides a new proof of Theorem 3.2. We leave
it for the reader to prove the analogue of Proposition 7.4 for right CDG-modules.
Remark 2(M). Using the above spectral sequence one can show that the
condition of Koszulness of M^ in Theorem 3.2 can be weakened to
(7.3) ExtJB1J{N, к) = О for all j.
In fact, for any quadratic CDG-module N over a quadratic CDG-algebra В such
that (7.3) holds, we get а Т(У)/( J2)-module M with the relations given by djsr and
an isomorphism of Л^°^-modules grFM ~ M^°\ On the other hand, the distrib-
utivity condition used in the proof of Theorem 3.2 is equivalent to the vanishing
of Ext^0)(k,k) for j > 3 and of Ext^0)(Af(°),k) for j > 2. It follows from
Corollary 3.5 of chapter 2 that this pair of conditions on A^ and M^ implies (7.3).
Conversely, (7.3) implies that Ext^0)(M^°\k) = 0 for j > 2 but not the vanishing
of ExtA'^0)(k,k). Thus, the approach via quadratic duality gives a slightly sharper
result.
Next, we will give a description of the Tor-spaces between nonhomogeneous
Koszul modules in terms of the cobar-complex associated with the dual CDG-
modules.
Proposition 7.5. Let В be a quadratic CDG-algebra and let S (resp. N)
be a right (resp., left) quadratic CDG-module over B. Set A = HoCob9(B) and
let R = H0Cob9(S,B) and M = HoCob9(B,N) be the corresponding right and
left modules over A, respectively. Then one has H0Cob9(S, B, N) ~ R<g>A M.
Moreover, if an algebra В and modules S and N are Koszul then there are natural
isomorphisms
HkCob9{S, B, N) ~ Тог£(Д, M)
for all k.

116
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
Proof: Both statements follow immediately from the next lemma applied to the
DG-algebra Cob.(B) and the DG-modules Cob.(S, B) and Cob.(B, N) over it. □
Lemma 7.6. Let E. = (••• —► E2 —► Ex —► E0) be a DG-algebra, P. =
(• • • —► P2 —> Pi —> Po) a right DG-module over E. and Q. = (• • • —►
Q2 —► Qi —> Qo) a left DG-module over E9. Let P. <8>e. Q» be the tensor
product complex (with the underlying graded vector space P <S>e Q)- Then one has
Ho(P9 ®e. Q%) — HqP9 ®h0e. HoQ9. Assume in addition that P (resp., Q) is a
free right (resp., left) module over E and HnP. = HnQ. = 0 for n^O. Then there
are natural isomorphisms
Hk(P. ®E. Q.) ~ Torfo£;-(Я0Р., H0Q.)
for all k.
Proof: The statement about Hq is straightforward. To compare higher homology
spaces with Tor-spaces let us consider the bar-complex Bar9(P9,E9, Q.) equipped
with the bigr&ding Bar{j(P., E9,Q9) = (p<8>ei®---<8>ei<8>g| fc+jH \-ji + l = j)
(where p G Pk, et G Ejt, q G Qi), the total differential D defined as above, and the
total (lower) grading i+ j. We have two spectral sequences
'Elpq = Tor fp(P,Q) => Hp+qBar.(P.,E.,Q.),
'Elq = Tor^E-(H,P.,HrQ.) => Hp+qBar.(P.,E.,Q.).
The first sequence shows that the natural morphism of complexes
Bar.(P.,E.,Q.) —> P. ®Em Q.
is a quasi-isomorphism. The second sequence implies that
HkBar.{P.,E.,Q.) ~Tor£°E-{H0P., H0Q.).
Combining these two facts we get the result. □
In the case when graded modules R^ and M^ are relatively Koszul over the
graded algebra A^ (see section 10 of chapter 2) one can simplify the answer for
Tor^(P,M) given in Proposition 7.1. Let us equip the tensor product S <S>b N
with the differential d®(s (g> n) = ds{s) (g> n + (—l)ss (g> djsr(n). Using the identities
^K5) = ~s®b and d2N(n) = ввп one can easily check that d® = 0. Let us denote
the obtained complex S ®в N.
Corollary 7.7. Let R (resp., M) be a right (resp., left) nonhomogeneous
Koszul module over a nonhomogeneous Koszul algebra A. Assume that the
corresponding graded modules R^ and M^ over
A(0)
are relatively Koszul. Let В be
the CDG-algebra dual to A, S (resp., N) the right (resp., left) CDG-module over
В dual to R (resp., M). Define the increasing filtration F on the vector space
Tor^(P, M) = R®AM by Fj{R (g)A M) = E/c+z=j [m(FkR ® FtM). Then there
are natural isomorphisms
grf (R ®A M) ~ (Д(°> ®A(o) M^)j for j > 0,
Torf{R, М)~1Г {S ®BN)* for i>0,
F0(R®aM)~H0(S®bN)*.
Proof: Consider the spectral sequence
'E$iq = Toi*qiP(S,Ny => gr£Hp+qCob.(S,B,N)

8. HOMOLOGY OF COMPLETED COBAR-COMPLEXES
117
associated with the natural filtration FkCobm(B, AT, S) = 0jf*Lm Bar{j (B, AT, S)*
of the complex Cob9(B,N, S). By the definition of relative Koszulness, we have
'E\,(? = 0 unless g = 0orp + g = 0. Now it is clear from the form of the spectral
sequence that it degenerates at E* for q ф 0 and at E'i for q = 0. Moreover,
the complex 'El 0 can be identified with (S <S>b AT)*. It follows that the natural
morphism of complexes (S®b N)* —> Cob9(S, B, N) induces an isomorphism on
the homology Щ with i > 0 and identifies H0(S®B N)* with F0H0Cob9(S, B, AT).
According to Proposition 7.5 the homology of the cobar-complex is isomorphic to
Tor^(i2, M). This immediately implies the isomorphism for Torf (R, M) for i > 0.
To describe gr?(R 0л М) we use in addition Proposition 10.1 of chapter 2. □
8. Homology of completed cobar-complexes
First, let us observe that if В = (В,(1в,0в) is a CDG-algebra with 6в Ф 0
then the complex Cob9(B) (resp., complex Cob.(B,N) for a CDG-module N) is
acyclic. Indeed, this follows easily from the fact that the unit in the DG-algebra
Cob.(B) is a coboundary.
Now let В = (Б, <1в) be a (nonnegatively graded) DG-algebra. We can view
its cobar-construction as a bicomplex, where CoblJ (B) = Barij(B)*. We
already used one spectral sequence (7.1) associated with this bicomplex. Now let
us consider the second spectral sequence associated with it. In other words, we
want to look at the spectral sequence associated with the decreasing filtration
GlCob(B) = 02-" Barij(B)*. It is easy to see that this spectral sequence
converges to the homology of the completed cobar-complex:
"£2g = Ext-™B)(k,k) => gx-jHv+qCob.{B).
Indeed, we have HnCob9(B) = HnBar9(B)v and the dual spectral sequence for the
increasing filtration of Bar*(B) clearly converges. Analogously, for a DG-module
N over В we have a spectral sequence
"Elq = Exb-™B){H*{N)M => grpGHp+qm.(B,N)
associated with the decreasing filtration GlCob{B,N) = 0^z Barij (B, AT)*.
Using these spectral sequences we immediately derive the following result.
Proposition 8.1. A quasi-isomorphism of nonnegatively graded DG-algebras
f : B' —> B" induces a quasi-isomorphism Cob9(B") —> Cob9(B ). Let N
(resp., N") be a DG-module over B' (resp., B"). Then a quasi-isomorphism
N' —> N" compatible with f induces a quasi-isomorphism Cob9(B , N ) —>
Co~b.(B',N').
The decreasing nitrations induced by G on the algebra A = HoCob9(B) and
on the A-module M = H0Cob.(B,N) are the A+-adic nitrations GlA = A\ and
GlM = A\M. Furthermore, it is easy to see that taking completions of the cobar-
complexes leads to the A+-adic completions on HQ:
H0Co~b.(B) ~ A~= \imA/A\,
H0Cb~b.{B,N) ~ M^=limM/4M.
Remark. We see that the completions AT and M~ are determined by the quasi-
isomorphism classes of В and N (cf. Remark 2 in section 6). In fact, it is not

118
5. NONHOMOGENEOUS QUADRATIC ALGEBRAS
difficult to prove a more general statement. For any augmented algebra A, the A+-
adic completion A^ is determined by the class of the differential coalgebra Bar, (A)
up to morphisms inducing isomorphisms of Hi and H^- For any A-module M, the
completion M^ is determined by the class of the differential comodule Bar, (A, M)
up to morphisms inducing isomorphisms of Щ and Hi.
Now let A be a Koszul quadratic-linear algebra and let В be the dual DG-
algebra. Then Cob.(B) is a free DG-resolution of A (see Proposition 7.2(ii)).
Therefore, the homology spaces HiCob.(B) can be viewed as nonabelian derived functors
of the A+-adic completion functor A —> A". Combining Remark 1 of section б with
Proposition 8.1 we see that these homology spaces are invariants of the algebra A
(not depending on a Koszul quadratic-linear structure). The example below shows
that these invariants are quite nontrivial. Similarly, the homology H*Cobu(N) is an
invariant of an A-module M admitting a nonhomogeneous Koszul structure (where
N is Koszul dual to M).
Example. Let A = Ug be the enveloping algebra of a semisimple Lie algebra
g. Then the dual DG-algebra is the standard cohomological complex C*(g). The
inclusion of the subalgebra of invariant forms defines a quasi-isomorphism of C*(g)
with its cohomology algebra H*(g). It follows that the algebra H*Cob9(C*(g)) ~
H*Cob.(H*(g)) is isomorphic to the symmetric algebra with generators in even
degrees > 2 (it is isomorphic to the center of Ug). Thus, in this case higher derived
functors of the A+-completion carry interesting information, unlike the usual A+-
completion ЩСоЬ9(C*(g)) ~ Ug~ = k. On the other hand, if M is a nontrivial
irreducible g-module then the dual DG-module N = C*(g, M*) (see Example 2 in
section 5) is acyclic. Thus, in this case H*Cob.(C9(g),C9(g,M*)) = 0.

CHAPTER 6
Families of quadratic algebras and Hilbert series
In this chapter we consider families of quadratic algebras. The main result is
the Koszul Deformation Principle (Theorem 2.1) stating that in a neighborhood
of a Koszul algebra flatness of a deformation in grading components of degree 3
implies its flatness in (any finite number of) higher degrees. We deduce from this the
finiteness of the number of Hilbert series of Koszul algebras with a fixed number
of generators and give an explicit upper bound on this number (see section 3).
Furthermore, in section 7 we prove that there exists a uniform bound on this number
over all ground fields. In section 4 we determine which generic quadratic algebras
are Koszul and in section 5 consider some examples of possible Koszul Hilbert series
(when the number of generators and relations is small).
1. Openness of distributivity
Let W be a vector space over Ik and let i?i(x), ... , Rm(x) С W be algebraic
families of subspaces (of fixed dimensions) parametrized by an algebraic variety
X over k. In other words, we have a collection of morphisms from X to the
corresponding Grassmannians of subspaces in W.
Proposition 1.1. Assume that the dimensions of pairwise intersections Щ{х)Г\
Rj(x) are locally constant functions on X. Then the set
U = {x E X | (Ri(x),..., Rm(x)) is distributive}
is open in X. Moreover, the dimensions of all subspaces obtained from Ri(x) using
the operations of sum and intersection are locally constant functions on U.
Proof: First, assume that m — 3. Consider the inclusion
(1.1) Rx{x) П (Д2(ж) + R3(x)) D Ri(x) П R2{x) + Ri(x) n R3(x).
Assume that it becomes an equality for some xo E X. Let l(x) > r(x) denote
the dimensions of the left-hand side and the right-hand side of (1.1), respectively.
Then there exists an open neighborhood UXo of x$ such that for x E UXo one has
l(x) ^ l(xo) and r(x) > r(xo). Since l(xo) — r(x0), we obtain that for x E UXo
the above inclusion is actually an equality and l(x) = r(x) is constant on UXo. The
statements for other intersections and sums follow easily from this.
Now let us consider the case m > 3. Assume that (Ri(xo),..., Дт(#о)) is
distributive for some xo E X. Let T be a word in the free lattice generated by
Ri. We claim that there exists an open neighborhood Ut of x0 such that the
dimension of T(x) is constant on Ut- We can argue by induction in the length of
Г. If the length of Г is at most 2 then this is true by our assumption. Now let
T — T\ * (T2 * T3), where * denotes intersection or sum. The induction assumption
implies that the dimensions of ТДх) and T{(x) П Tj(x) are constant on some open
119

120 6. FAMILIES OF QUADRATIC ALGEBRAS AND HILBERT SERIES
neighborhood of Xq. Applying the case m = 3 considered before we derive the
existence of a neighborhood Ut of Xq such that the dimension of T(x) is constant
on Ut- This finishes the proof of our claim. Recall that by Theorem 6.3 to check
distributivity of a finitely generated lattice one has to check only a finite number of
distributivity conditions. Hence, there exists an open neighborhood of xq on which
the collection (i?i(x),..., Rm(x)) is distributive. Since such a lattice is finite, the
dimensions of all elements of this lattice are constant near x$. П
2. Deformations of Koszul algebras
The following important result is essentially due to V. Drinfeld [43]. Recall that
a quadratic algebra A is said to be n-Koszul if Ext^ (k, k) = 0 for i < j ^ n, or,
equivalently, the relation lattice in Afn is distributive (see section 4 of chapter 2).
The advantage of the notion of n-Koszulness is that it is given by a finite number
of conditions (unlike Koszulness).
Theorem 2.1. Let A(x) = {V,I(x)} be a family of quadratic algebras with the
fixed generating space V and the space of quadratic relations I(x) parametrized by
points of an algebraic variety X (so that x —> I(x) is a morphism from X to the
Grassmannian Gr(V®2)). Assume that A(xo) is an n-Koszul algebra for some x$ G
X and that dim A3 (x) is constant on X. Then there exists an open neighborhood U
of xo such that for x G U the algebra A(x) is n-Koszul and dim^(x) = dim A{(xo)
for i ^ n.
Proof: We have An(x) = V^/iR^x) + • • • + i?n_i(x)), where R%(x) = V®1'1 <8>
I(x) (£> V®n~%~1. The assumption that dimA3(x) is constant on X implies that
the dimensions of all pairwise intersections Ri(x) П Rj(x) are also constant on X.
Hence, we can apply Proposition 1.1. □
Assume for simplicity that the ground field к is algebraically closed and
uncountable. We will get rid of this assumption in section 7. Let Qm?s = Gm2_s(km®2)
be the Grassmannian variety of quadratic algebras with A\ = km and dim A^ — s.
It has a stratification by the locally closed subvarieties Qm,s.u consisting of
quadratic algebras with dim A3 — u.
Corollary 2.2. The set of all n-Koszul algebras in Qm,s.u is open. Therefore,
the set of all Koszul algebras in Qm,s,u w a countable intersection of open subsets.
For i ^ n the restriction of the function dim Ai to the set of all n-Koszul algebras
in Qm,s,u is locally constant
Proof: Apply Theorem 2.1 to the natural family of quadratic algebras parametrized
by Qm,s,u- □
Example. The above corollary implies that if we have a one-parameter family
A(X) of quadratic algebras in Qm,s,u such that all algebras A(X) for Л ф Aq are
isomorphic to some fixed algebra A, then Koszulness of ^4(Aq) implies Koszulness
of A and in this case Ka = ^л(л0)- This observation is used in [113] to prove
Koszulness of the Orlik-Solomon algebra of a supersolvable hyperplane
arrangement (see [87]). This algebra is the quotient of the exterior algebra in generators
#i,... , xn by the relations of the form X{Xj —XiXk+XjXk = 0 for some set of triples
г < j < к (depending on the arrangement). Now one can define the family A(X)
by making the change of variables X{ —> Xxi so that the relations in A(X) become

2. DEFORMATIONS OF KOSZUL ALGEBRAS
121
XiXj — Xk JXiXk + Xk lXjXk — 0. It turns out that the Hilbert series is constant in
this family, so Koszulness of A(0) implies Koszulness of A(l).
Corollary 2.3. There is only a finite number of Hilbert series of Koszul
algebras with any fixed number of generators m = dim A\.
Proof: All these Hilbert series can be obtained from the algebras corresponding to
generic points of irreducible components of Qm;S)U (more precisely, those of them
that are Koszul). Therefore, the number of Hilbert series is bounded above by the
total number of irreducible components of the varieties Qm,s,u. □
Corollary 2.4. For every m > 0 and r > 0 there is only a finite number of
Hilbert series of Koszul modules M over Koszul algebras A, such that dimAi = m
and dim Mo = r.
Proof: Recall that to a pair (A, M) consisting of a Koszul algebra A and a Koszul
(left) module M over it we can associate a Koszul algebra Am = A 0 M(—1) (see
Corollary 5.5 of chapter 2). Since Iiam(z) — Ha{z) + zHm{z), our assertion follows
from the previous corollary. □
The above corollary has the following nice application.
Proposition 2.5. Let A be a Koszul algebra such that the sequence an =
dim An is bounded. Then this sequence has a periodic tail, i.e., there exists N > 0
and no ^ 0 such that ап+дг = an for all n ^ щ.
Proof: Consider the sequence of A-modules A^ = ®i>nAi, n ^ 0. By
Proposition 1.1 of chapter 2 each of these modules is Koszul. Since the number of generators
of A^ is equal to an, from the above corollary we deduce that there is only a finite
number of elements in the set of Hilbert series hA\n]. This implies the result. □
Remark 1. As in Theorem 2.1 of chapter 5 the Koszulness condition in
Theorem 2.1 can be weakened to the vanishing Ext^jL (k, k) = 0 for j > 3 (see Remark 1
in section 7 of chapter 5). It follows that one can use this condition in the statements
of Corollaries 2.2 and 2.3 instead of Koszulness.
Remark 2. Theorem 2.1 of chapter 5 can be derived from Theorem 2.1 as follows.
Let A be the algebra defined by a set of nonhomogeneous quadratic relations а +
ip(a) + h(a) = 0, a e I. Consider the family of quadratic algebras A(X) with the
generator space kt®V and the relations
(tx = xt, x G V; a + \<p(a)t + X2h(a)t2, a G /).
The analogue of the Jacobi identity implies that this deformation is flat in degree 3.
All the algebras A(X) with Л ф 0 are isomorphic to each other, hence they have the
same Hilbert series as A(0) provided that A(0) is Koszul. Taking the quotient by
t — 1 we conclude that grFA and A^ have equal Hilbert series.
Remark 3. It is also natural to consider the characterization of PBW-bases
using the third grading component (see sections 2 and 7 of chapter 4) as an analogue
of Theorem 2.1, since passing from a filtered object to the associated graded one
is somewhat analogous to passing from the generic point of a deformation to the
special one. For commutative PBW-bases, this analogy can be made more precise:
as was explained in Example 1 of section 8 of chapter 4, for any commutative
quadratic algebra with a fixed ordered set of generators the corresponding commutative

122
6. FAMILIES OF QUADRATIC ALGEBRAS AND HILBERT SERIES
monomial algebra can be obtained as a limit of a family of quadratic algebras
isomorphic to the original one (see also [72]). The same is true for Z-PBW-bases if
we consider deformations in the class of Z-algebras (cf. section 10 of chapter 4).
Furthermore, consider the "variety" JCh of Koszul algebras with the fixed space
of generators km and the Hilbert series h(z) — 1 + mz + • • •. Since the
algebraic group GL(m) is connected, its action preserves irreducible components of JCh-
Therefore, it makes sense to consider the component(s) containing a given Koszul
algebra A as invariants of the isomorphism class of A.
Now for the usual noncommutative PBW-bases, it is natural to ask whether the
associated monomial algebra will always belong to the same irreducible component
as the original PBW-algebra. The following example shows that the answer is
"no". Consider the quadratic algebra A with 2 generators #i, x<i and 2 relations
(#i +X2)2 = (#i — X2)2 = 0. Then #i, #2 are PBW-generators of A in the standard
order and the corresponding monomial algebra is x\ — 0, x<iX\ = 0. It is easy to
verify that these algebras belong to different irreducible components (see Example 2
in section 5).
On the other hand, we do not know any examples when JCh is not connected.
Remark 4. A generalization of Corollary 2.3 was recently obtained by
Piontkovskii [94]. Namely, he proved that for given integers n,a,b,c the set of
Hilbert series of graded algebras A with Aq = Ik, with ^ n generators of degree
^ a, relations concentrated in degrees ^ b, and Tor^(k, k) concentrated in degrees
^ c, is finite.
3. Upper bound for the number of Koszul Hilbert series
Using the results of section 2 it is not difficult to give an explicit bound for the
number of Hilbert series of Koszul algebras with fixed dim A\.
Lemma 3.1. Let X be an algebraic variety, U and W vector spaces, £ С U®Ox
a vector subbundle of rank e, and p: £ —> W ® Ox a morphism of vector bundles
over X. Then the closed subvariety Xk — {x G X : ткрх < к} can be presented as
an intersection of divisors from the linear system \ det~ £\, where det£ = Де £.
Proof: Consider the composition </? of the following maps:
Л О д О К д К д О /С д /С
£-^Д £®/\ £^/\ U®/\W®0.
It is clear that ткрх < к <^> рх = 0. Hence, Xk is the set of common zeros of
the space of sections /\e~h U* ®/\k W* —> #°(X, det"1 E). □
Proposition 3.2. The number of Hilbert series of Koszul algebras with dim A\
= m and dim A<i = s, where s ^ 1, is bounded above by
m3(2m)^2-*>(S(m2 - ,))! ^ ^ (a~J!_ ^ ^ (2ms)^2^.
Proof: Clearly, we can assume that 1 < 5 ^ ra2/2. Let us apply the above lemma
to X = Q^s = Gm2_5((Ikm)®2), W = V®3, U = V®3 ф V03 and the subbundle
£ С U® Ox with the fiber /0 V®V 0/ over/ G Gm2_s((Ikm)®2). We obtain that
the subvariety Qm,s,u is cut out by a linear subspace in the projective embedding
of X defined by the linear system |0x(2m)|, where Ox {I) is the ample line bundle
corresponding to the Pliicker embedding (with the fiber det-1/ over /). By the

4. GENERIC QUADRATIC ALGEBRAS
123
refined Bezout's theorem (see [58], Thm. 12.3) this implies that the projective
degree of Qm,s,u m this embedding is bounded above by
dego(2m) X = W^-Mm* ~ ^))'(w2^)!(S.^!_1)r
where we used the formula for the degree of the Grassmannian in the Plucker
embedding (see e.g. [58], Example 14.7.11(iii)). Now we observe that
!!•.•(*-1)!
(m2-s)!---(m2-l)!
1
^
Therefore, we have
(2m)<™2-°Hs(m2 - s))\
dego(2m) X ^ (T„2_e)!e(m2_s + 1)e-i-
By Corollary 2.2 the number of Hilbert series in question is bounded above by
m3 deg^m) -X" ? so to finish the proof it suffices to check the following inequality:
/o n m\s(m2-s))\ s(m2-s)
V ' ; (r^-s + l)*-1^2-*)!5 "
where 1 < s ^ m2/2. Assume first that 5 > 2. Then one can easily check that
ra3 ^ (ra2 — s + l)s_1, so (3.1) follows from the well-known inequality
(s(m2-s))\
< 5S(^2"S).
(m2 — s)!5
In the case 5 = 2 we can rewrite (3.1) in the form
m2 22(™2"2)(m2-2)!2
m2-l " (2m2-4)!m '
where m ^ 2. Note that for m = 2 this becomes an equality. It remains to note that
the right-hand side grows with m while the left-hand side decreases with m. □
Remark. The number of Hilbert series of PBW-algebras is obviously bounded
above by 2m , which is substantially less than the bound mm for Koszul algebras,
though still very large. One can see from the following table (calculated using a
computer) that the number of PBW-series does grow very fast:
m
PBW series
1
2
2
7
3
46
4
803
5
50650
4. Generic quadratic algebras
In this section we consider the variety of quadratic algebras with fixed dim A\ =
m and dimA2 = 5 (that can be identified with a certain Grassmannian). By a
generic quadratic algebra (with given m and 5) we mean a point of a countable
intersection of nonempty Zariski open subsets of this variety. Clearly, a generic
quadratic algebra has coefficient-wise minimal Hilbert series. It follows that the
open stratum Qm,5,u С Qm,5 corresponds to the minimal value of и = dim A3. The
next result is due to D. J. Anick [8].

124
6. FAMILIES OF QUADRATIC ALGEBRAS AND HILBERT SERIES
Proposition 4.1. The minimal possible value of dim A3 for quadratic algebras
with dim A\ = m and dim A<i = 5 is given by the formula:
_ J 0, if s^z£,
Umin-\2ms-m\ if s>^.
Proof: Because of the duality it is sufficient to consider the first case. We have to
show that there exists a quadratic algebra with 5 = [m2/2j and dim A3 = 0. Here
are explicit examples (taken from [8]):
/с{хь...,хп,у1,...,уп}/(хг%, XiXj -yiVj, ij = l,...,n)
for m = 2n and
k{xi,..., xn, 2/o, • • •, УгЛ/Оэд-ъ XiXj - Vi-iVj,
XiVn ~ УгУО, V0Xj ~ VnVj, 2/o, hj = 1, • • • ,n)
for ra = 2n + l. П
Remark 1. It is an open problem to find the minimal value of и = dim A3 for
Koszul algebras with fixed m ~ dim A\ and 5 = dim A2. Conjecturally, it coincides
with the minimum for PBW-algebras. For 5 ^ ra2/4 this is true (with и = 0)
by Thm. 3.1 of [40] stating that H(z) = 1 + mz + sz2 is the Hilbert series of a
Koszul algebra iff it is the Hilbert series of a PBW-algebra iff H(—z)~1 has positive
coefficients iff 5 ^ m2/4 (this also follows from the proof of Proposition 4.2 below).
Moreover, in this case there exists a commutative Koszul algebra with и = 0.
On the other hand, the maximal value of dim A3 for one-generated algebras
is obviously attained on PBW-algebras, since one has dim A3 ^ dim A3 for the
quadratic monomial algebra A0 associated with A. Thus, to find this maximal
value one has to find the maximum of the sum of all entries in M2 among all 0 — 1
matrices M of size m x m and with 5 entries equal to 1 (cf. section 6 of chapter 4).
This maximal value was found independently in the works [6, 5, 120]. For example,
one has и ^ s3//2 for s ^ m2/2 and и $ (s2 + 4s — l)/4 (unless 5 = 4 when и ^ 8)
for any ra. In addition, the problem of determining the minimal value of the sum
of all entries in M2 for a 0 — 1 matrix M of a given size and a given number of
l's was solved in [120]. Hence, this paper gives the minimal value of dim A3 for a
PBW-algebra with given dimAi = m and dimA2 = 5. In particular, it is shown
in [120] that as m —> 00 and s/m2 —> a G R, the minimal value of u/m3 (for a
PBW-algebra) tends to a piecewise algebraic function of a with break points at
a = (n ± l)/2n for n — 2, 3, ... (corresponding to the symmetric and exterior
algebras with n generators).
The main tool in studying generic relations is the Golod-Shafarevich inequality
(see Proposition 2.3 of chapter 2). The following well-known result is a typical
application of this inequality.
PROPOSITION 4.2. A generic quadratic algebra with dim Ai = m and dim A2 =
5 is Koszul iff one of the inequalities holds:
m2 3m2
5 ^ —- or 5 ^ ——.
4 4
Proof: It suffices to consider the case 5 ^ m2/2. Then for a generic quadratic
algebra we have Jia(z) — 1 + mz + sz2. It is easy to check that the power series
(\+mz + sz2)~l has nonnegative coefficients iff s ^ m2/4. Hence, a generic algebra

5. EXAMPLES WITH SMALL dimAi AND dim A2
125
with ra2/4 < s ^ ra2/2 is not Koszul. For s ^ ra2/4 we claim that hA\(z) =
(1 — mz + sz2)-1 for a generic quadratic algebra. Indeed, this value of the Hilbert
series is attained at the monomial algebra corresponding to a bichromatic graph
S С [1, |_m/2j] x [|_m/2j +1, m]. It remains to use Proposition 2.3 of chapter 2. □
Remark 2. We have seen that the Hilbert series of a generic quadratic algebra is
equal to 1+mz+sz2 for s ^ m2/2 and to (l+mz+(m2—s)z2)~1 for 5 ^ 3m2/4. It is
a natural conjecture1 that the Golod-Shafarevich estimate is attained for all 5, i.e.,
the Hilbert series of a generic quadratic algebra is equal to |(1 — mz + (m2 — s)z2)~l\.
This would imply that a generic algebra with 5 < 3m2/4 is finite-dimensional
(more precisely, that An = 0 iff s/m2 ^ 1 — \ cos-2 ^pj). However, the analogous
statement for non-quadratic one-generated graded algebras in not true [8].
Remark 3. The simplest way to see the difference between the classes of PBW
and Koszul algebras is to consider the analogue of Proposition 4.2 for PBW-
algebras. We claim that a generic algebra A with dimAi = m and dim A2 = s
admits a PBW-basis iff 5 ^ m — 1 or 5 ^ m2 — ra + 1. One may assume that
s ^ m2/2, so that dim A3 = 0. Then the PBW condition for a set of
generators #i, ... , хш means that the corresponding set S does not contain any (i,j)
and (j, k) simultaneously. In particular, we must have (1,1) ^ S, that is x\ G /.
Clearly, a generic subspace of relations / С V®2 intersects the m-dimensional cone
{x2 : x G V} at a nonzero point iff 5 = codim/ ^ m — 1. The reader will easily
check that for 5 ^ m — 1 a generic algebra does admit a set of PBW-generators
with the corresponding set S = {(1, 2),..., (1, 5 + 1)}.
A similar argument shows that a generic algebra with 2m —2 < 5 < m2 — 2m+ 2
has no Z-PBW-basis in any order (see section 10 of chapter 4).
Remark 4. The analogue of Proposition 4.2 for commutative algebras states
that a generic commutative quadratic algebra with dim A\ = m and dim A<i — s is
Koszul iff s ^ m2/4 orO (m^_1) - m (see [57]).
5. Examples with small dim^ and dim^2
In this section we consider examples of the stratification of the variety of
quadratic algebras by the dimension of A3. By a "component" we mean an "irreducible
component". We use the notation m = dimAi, 5 = dim^2, и = dim A3.
1. m ^ 2, 5 = 1. The stratification has the form Qm?i = Qm,i,o U 2тдд,
where the closed sub variety Qm,i,i consists of quadratic algebras isomorphic to В ,
where В = Ik{#i,..., xm}/(x2). All algebras in Qm?i are Koszul. If in addition Ik is
quadratically closed (contains all square roots) then all of them are PBW-algebras.
The possible Hilbert series of algebras in Qm?i are
1 + mz + z2,
1 + mz + z2 + z3 + z4 + zb + ...
For m = 2, any algebra from 62,1,0 is isomorphic to k{x, y}/(xy-Xyx) with Л G P£,
or to k{x, y}/(xy — yx + x2).
xThe argument of Zvyagina (Zapiski Nauchn. Sem. LOMI 155, 1986) concerning this problem
is incorrect.

126 6. FAMILIES OF QUADRATIC ALGEBRAS AND HILBERT SERIES
In the dual case m = 2, 5 = 3, the Hilbert series are
1 + 2z + 3z2 + 4z3 + 5z4 + 6z5 + ...
1 + 2z + 3z2 + 5z3 + 8z4 + Ш5 + ...
2. ra = 2, 5 = 2. We have 62,2 = 62,2,0 L-1 62,2,1 LI 62,2,2- The closed subva-
riety 62,2.2 consists of three 2-dimensional irreducible components. The algebras
corresponding to their generic points are k{x,y}/(x2,y2), k{x,y}/(x2,xy — \yx),
and k{x,y}/(xy,yx), and any algebra from 62,2,2 is isomorphic to one of these.
The first and the third components intersect the second one (at Л = — 1 and A = 1,
respectively), but do not intersect each other. Geometrically, this is a union of three
surfaces P1 xP1, where the lines P1 x {^1} in the second component are identified
with the diagonals in the first and the third components.
The locally closed subvariety 62,2,1 is a disjoint union of two 3-dimensional
components consisting of algebras isomorphic to k{x, y}/(x2,y(x + y)) and to
k{x,y}/(xy,y(x + ?/)), respectively. Their closures contain the first two and the
last two components of 62,2,2, respectively. The open subset 62,2,0 consists of
algebras isomorphic to k{x, y}/(xy, x2 + Xyx + y2). A2 ^ 1.
A quadratic algebra from 62,2 is Koszul iff it belongs to 62,2,2- The Hilbert
series of any such algebra is equal to
1 + 2z + 2z2 + 2z3 + 2z4 + 2z5 + ...
3. m ^ 3, 5 = 2. This case was analyzed in [15]. The stratification
has the same form 6m,2 — 6m,2,o L-l 6m,2,1 L-l 6m,2,2- The closed subvariety
6m,2,2 consists of Koszul algebras of the form i П (I ф У), where A E 62,2,2
and dim V — m — 2, so it has three 2ra — 2-dimensional components (see the
case m = 2). The subvariety 6m,2,1 consists of two intersecting components of
dimensions 3m — 3 and m2 + m — 3. A generic algebra on the first component
is quadratic dual to Ik{#i,... ,хт}/(х1Х2,Х2Хз). Other algebras from this
component form two non-Koszul isomorphism classes: that of the algebra В dual
to Ik{#i,... ,xra\j(x\,x\X2 + #3X1) and of the algebra ЛП(кф V), where A =
k{x, y}/(x2, y{x-\-y)) (A corresponds to the first component of 62,2,1)- The second
component of 6m,2,1 consists of the algebras A such that A- has a relation x2 = 0.
All algebras of this component are Koszul except for those isomorphic to В
(constituting the intersection with the first component) and except for algebras arising
from the second component of 62,2,1 (by the direct sum with кфУ). All algebras on
6m,2,0 are Koszul except for those arising from 62,2,0 and the isomorphism classes
of 6 algebras listed in [15]. The Hilbert series of Koszul algebras with dim Ai = m
and dim A^ = 2 are
1 + mz + 2z2
1 + mz + 2z2 + z3
1 + mz + 2z2 + z3 + z4 + z5 + ...
1 + mz + 2z2 + 2z3 + 2z4 + 2z5 + ...
4. rn = 3, 5 = 3. In this case (and in the next one) we do not know either
a description of the stratification or the list of Hilbert series of Koszul algebras.
The possible values of и are и = 0, ... , 5. For Koszul algebras one has и ф 0.
We can only list the Hilbert series of PBW-algebras (together with a possible set

6. KOSZULNESS IS NOT CONSTRUCTIBLE
127
Sc{l,2,3}2):
l + 3z + 3z2 + z3 (12,23,13)
1 + 3z + 3z2 + 2z3 + 2z4 + 2z5 + 2z6 + ... (12, 22,13)
1 + 3z + 3z2 + 3z3 + 3z4 + 3z5 + 3z6 + ... (11, 22,33)
1 + 3z + 3z2 + 4z3 + 4z4 + 4z5 + 4z6 + ... (12, 22, 23)
1 + 3z + 3z2 + 4z3 + 5z4 + 6z5 + 7z6 + ... (11,12, 22)
1 + 3z + 3z2 + 5z3 + 8z4 + 13z5 + 21z6 + ... (11,12, 21).
5. m = 3, 5 = 4. For quadratic algebras we can have и = 0, ... , 8. There
exist Koszul algebras with и = 4, ... , 8. There are no Koszul algebras with и = 0,
1, or 2. We do not know whether there exists a Koszul algebra with dimAi = 3,
dim A2 = 4 and dim A3 = 3. Here are the Hilbert series of PBW-algebras:
1 + 3z + 4z2 + 4z3 + 4z4 + 4z5 + ...
1 + 3z + 4z2 + 5z3 + 6z4 + 7z5 + ...
1 + 3z + 4z2 + 5z3 + 7z4 + 9z5 + 12z6 + 16z7 + 21z8 + 28z9 + ...
1 + 3z + 4z2 + 6z3 + 8z4 + 10z5 + 12z6 + 14z7 + 16z8 + 18z9 + ..
1 + 3z + 4z2 + 6z3 + 8z4 + 12z5 + 16z6 + 24z7 + 32z8 + 48z9 + ..
1 + 3z + 4z2 + 6z3 + 9z4 + 13z5 + 19z6 + 28z7 + 41z8 + 60z9 + ..
1 + 3z + 4z2 + 6z3 + 9z4 + 14z5 + 22z6 + 35z7 + 56z8 + 90z9 + ..
1 + 3z + 4z2 + 6z3 + 10z4 + 16z5 + 26z6 + 42z7 + 68z8 + 110z9 +
1 + 3z + 4z2 + 7z3 + llz4 + 18z5 + 29z6 + 47z7 + 76z8 + 123z9 +
1 + 3z + 4z2 + 8z3 + 16z4 + 32z5 + 64z6 + 128z7 + 256z8 + ...
6. Koszulness is not constructible
(12,13,23,32)
(11,12,22,33)
(12,23,32,31)
(11,12,22,23)
(12,21,13,31)
(11,12,23,31)
(11,12,21,33)
.(11,12,21,23)
.(11,12,21,13)
(11,12,21,22).
The following examples from the paper [56] demonstrate that the set of all
Koszul algebras with 3 generators and 3 relations is not constructible:
fzy = yz + \y2
or
Лек*
xz = xy
zx = 0, AGlk.
For the first algebra one has хуп+1х — xzynx = Xnxynzx = Anxyn+1x, so the
Hilbert series differs from the generic one (equal to (1 + z)(l - 2z + z3)"1) iff Л
is a root of unity. The Hilbert series of the second algebra is equal to (1 — z)-3
unless A-1 = 1, 2, 3, ... . We claim that in both cases algebras corresponding
to non-exceptional values of Л are Koszul. To prove this note that these algebras
admit a Г-grading (in the sense of section 7 of chapter 4) with values in the free
group with two generators a, /3. Namely, set A1?a = (x) and A\$ = (y, z). Since
dim Ai)a = 1, it is enough to check that the relation lattice in Ai^a<S>Af ^+1(g)i1)a is
distributive. But in the gradings a/3/3, /3/3/3 and /3/3a one has I®Vr\V®I = 0, hence
Koszulness follows from the calculation of the Hilbert series (cf. Proposition 2.3).
Taking Л = — 1 in the second example we obtain a quadratic algebra A over Z
such that A ®z Q is Koszul but A <g>% (Z/pZ) is not Koszul for any prime. Note

128 6. FAMILIES OF QUADRATIC ALGEBRAS AND HILBERT SERIES
that the quadratic dual A] (defined over Z) is a free Z-module. Substituting a new
generator и instead of x in the third relation we obtain a family with the generic
series (1 — 4z + 3z2)-1.
Let us outline a more general approach to such examples following [56].
Suppose we are given a collection of linear automorphisms Mi,..., M^: V —► V of
a vector space V and two subspaces Vj, Vr С V. Consider the quadratic
algebra A with the generator space (#i,..., #&, a/, ar) Ф V and the relations aiVi — 0,
xiv = Mi(v)xi for v G V, and Vrar = 0. We claim that the Hilbert series of A is
given by the formula
oo к
hA{z)-1 = 1 - mz + rz2 - J2 ^П+3 Yl dim VinMh"- МъгУт.
n=Q ii,...,in = l
This can be easily verified if all these intersections are zero. One can reduce to
this case iteratively taking quotients of A by strongly free elements (see the end
of section 5 of chapter 2). Moreover, a straightforward computation shows that
gl dim A ^ 3 and
к
dimExt^3+n(Ik, k) = ^ dim Vt П Mh • • • MinVr.
2i,...,2n = l
Therefore, A is Koszul iff V\ П M^ • • • MinVr = 0 for any гь ..., гп, n ^ 1, and
A3 = 0 iff Vi П Vr = 0. There exists also a construction with the same
properties for which A- is commutative [8]. Furthermore, it is shown in [9] that for any
system F of (quasi-)polynomial Diophantine equations in nonnegative integer
variables z\,..., Zk with coefficients in Ik there exists a construction of linear operators
Mi,..., Mfc on a vector space V D V\, Vr (with dim V bounded by the "size" of
F) such that the right-hand side of the above formula is equal to the number of
solutions of F with z\ Л V z^ = n.
Thus, the set of Koszul algebras can be non-constructible. However, we do not
know whether there exists an irreducible component of some Qm,s?u that contains
a Koszul algebra but does not contain a nonempty Zariski open subset consisting
of Koszul algebras.
7. Families of quadratic algebras over schemes
Our goal in this section is to prove that there is a finite number of Hilbert series
of Koszul algebras with fixed dim A\ over all ground fields uniformly. 2
Proposition 7.1. Let Л be an arbitrary ring, W a K-module, and R\, ... ,
Rn С W a collection of submodules generating a distributive lattice ft of submodules
in W. Assume that for any subset I C [1, n] the K-module W/ ^2ieI Ri is projective.
Then for any 5, T G f£ such that S С Т the K-module T/S is projective. Moreover,
there exists a direct sum decomposition W = 0а€д Wa over К such that each Ri
is the sum of a set of submodules Wa.
Proof: Every submodule T £ ft can be presented as the intersection Ti П • • • flTfc of
submodules of the form Yliei ^* -^ us Prove ^У induction in к that the quotient
2We are grateful to J. Bernstein for posing this question.

7. FAMILIES OF QUADRATIC ALGEBRAS OVER SCHEMES
129
module W/T is projective. Indeed, if the modules W/Tu W/T2, and W/(Ti+T2)
are projective then it follows from the exact sequence
0 > W/(T± П T2) > W/Тг 0 W/T2 > W/(T± + T2) > 0
that the module W/(T\ Г\Т2) is also projective. It remains to use the exact triple
0 -> T/S -> W/S -> ТУ/Т -> 0 to deduce projectivity of T/S. The second assertion
is clear from the proof of Proposition 7.1 of chapter 1. □
Proposition 7.2. Let A be a Noetherian local ring with the residue field к and
let Ri, ... , Rn С W be a collection of submodules in a finitely generated A-module.
Assume that
(1) all of the A-modules W, W/Ri, and W/(Ri + Rj) are free, where i,j G
[l,n];
(2) the collection of subspaces (Ri ®л Ik), i G [l,n], m W ®л к is distributive
(they are indeed subspaces since the modules W/Ri are free).
Then the collection of submodules (Ri) generates a distributive lattice and for any
submodule T С W from this lattice the quotient module W/T is free.
Proof: It suffices to consider the case n = 3. The case n > 3 will follow exactly
as in the proof of Proposition 1.1. Consider the complex of free Л-modules C2 —>
C\ —► C0 given by
W > W/R1®W/R2®W/R3 > W/(R1 + R2)®W/(R1 + R3)@W/(R2 + R3).
We have H0(C.) = W/(Ri + R2 + Д3) and H2(C.) = R1nR2D R3. Furthermore,
the triple (Дь Д2, Дз) is distributive iff Hi(Cm) = 0 (see Proposition 7.2 of
chapter 1). The analogous statements hold for the complex C. <S>\ к and the collection of
subspaces (Ri ®л Ik) since the tensor product commutes with quotients. In
particular, we have Hi(C. <8>a k) = 0. Now consider the hyperhomology spectral sequence
for the functor Tor^(—, Ik) and the complex Cm:
Elq = Tor£(Hq(C.),k) => Hp+q(C.®Ak)
with the differentials dr: E^ —> E^ Jrr_1. This sequence implies that
Tor^(H0(C.),к) С H^C. ®Ak) =0 and therefore the Л-module H0(C.) =
W/(R1 + R2 + Д3) is free. It follows that Tor2(H0(C.),k) = 0 and H^C.) ®л Ik ~
Н\(Сщ <8>a Ik) = 0. Hence, Hi(Cm) = 0 and the collection (Ri) is distributive. It
remains to apply Proposition 7.1. □
Theorem 7.3. Let A be a Noetherian local ring with the residue field к and let
л = лелеЛе-- = ta(Ai)/(i), 1сЛ^л Л
be a quadratic algebra over A with finitely generated A-modules Ai. Then the
following two conditions are equivalent:
(a) the A-modules A\, A2 and A3 are free and the quadratic algebra А®а к
over the field к is Koszul;
(b) for any n the collection of submodules (*4fA*_1 ®А1®А А^АП~1~1), i G
[l,n— 1], generates a distributive lattice in the A-module Afn (n-th tensor
power over A) and all the quotients of this module by submodules from this
lattice are free over A.

130
6. FAMILIES OF QUADRATIC ALGEBRAS AND HILBERT SERIES
If these conditions are satisfied then all the grading components An are free Л-
modules. If in addition A is a domain with the field of quotients K. then the algebra
Л 0л 1К is Koszul. The same statements are true if we consider these properties in
internal degree ^ N.
Proof: This follows easily from Proposition 7.2. □
For a closed point у of a scheme Y we denote by k(?/) the corresponding residue
field. Also, for a coherent sheaf T over Y we set T(y) = T <S>oY k(y).
Lemma 7.4. Let Y be a Noetherian scheme, p: £ —> T a morphism of locally
free sheaves of finite ranks over Y. Let also к be an integer. Assume that for any
closed point у £ Y one has dim^) coker (p)(y) ^ / — k, where f = vkT. Then
the sheaf coker (p) is locally free of rank f — к iff the morphism Afc+1 p : Afc+1£ —>
д/c+i^r is identically zero.
Proof: "Only if": If the sheaf TI p{£) is locally free of rank f — к then the map p
factors through a locally free sheaf p(£) of rank к and therefore Ak+1p = 0.
"If": Since д\тцу^(Т/р(£))(y) ^ / — fc, locally in Y one can choose a set of
к sections si,..., Sk of the sheaf p{£) such that their images in Т(у) are linearly
independent over k(y). Since Afc+1p = 0, it is easy to see that the sections si,..., Sk
generate p(£). The desired statement follows immediately. □
Lemma 7.5. Let X be a Noetherian scheme and let p: £ —► T be a morphism
of locally free sheaves of finite rank. Then for any integer к there exists a locally
closed subscheme X^ С X with the following universal property: a morphism of
Noetherian schemes f: Y —> X factors through X^ iff the pull-back /* coker(p) is
locally free of rank k.
Proof: The condition dim^) coker (p)(x) ^ к defines an open subscheme in X.
Inside this subscheme the condition f\f~k+lp = 0 defines the closed subscheme X^.
The universal property follows from Lemma 7.4. □
In the following proposition for coherent sheaves T and Q on a scheme Y we
denote their tensor product over Oy simply by T <S> Q.
Proposition 7.6. For any integers m,s,u ^ 0 there exists a scheme Q =
2m,s,u of finite type over SpecZ together with a subsheaf Tq С (CJq)®2 of the free
sheaf of rank m2 such that
(i) the sheaf ((9q )®2/Tq is locally free of rank s,
(ii) the sheaf (0^3/(0^ <g> IQ + IQ <g> Og) is locally free of rank u,
and the scheme Q^ s u has the following universal property: for any Noetherian
scheme Y together with a subsheaf Ту С О®2 with the properties similar to (i) and
(ii) there exists a unique morphism f: Y —> Q^ s u such that Ту coincides with
the pull-back ofTQ as a subsheaf in (O™)®2 = f*(6%)®2.
Proof: Note that the scheme Q should be empty unless s ^ m2. Recall that for any
integers 0 ^ r ^ n there exists the Grassmannian scheme Gr?n of finite type over
Spec Z together with a locally free subsheaf Tq С О^ of rank r with a locally free
quotient sheaf Oq/Tq such that the scheme Gr?n is universal in the category of all
schemes Y equipped with subsheaves Ту С Oy with the same properties (see [84]).
It remains to consider the scheme X = Q^?s = Gm2_s?m2 and apply Lemma 7.5
to the morphism p: 0% ®TQ 0lQ <g> 0£ —^ (6>g)03 to obtain the locally closed
subscheme Q^su с Q^ □

7. FAMILIES OF QUADRATIC ALGEBRAS OVER SCHEMES
131
Corollary 7.7. For any integers m,s,u > 0 and any field к there exists a
natural bijective correspondence between the set of quadratic algebras A over к with
the fixed generators space A\ — кш such that dim A2 = s, dim A3 = и and the set
of all k-points of the scheme Q = Q^ s u. More precisely, a point f: Speck —> Q
corresponds to the quadratic algebra A = {km,/} with I = /*2q С Ikm = /*(9q.
Furthermore, n-Koszul algebras A correspond to k-points that belong to an open
subscheme n/C^ s u С Q^ s u- This subscheme has a natural decomposition into a
disjoint union of open subschemes rJ^n^.u ~ Uh(z) п^ъ numbered by power series
h(z) = 1 + mz + sz2 + uz3 H— • considered modulo zn+1 such that k-points in nJCf
correspond to n-Koszul algebras A with Ha{z) mod zn+1 = h(z).
Proof: To prove the first statement apply Proposition 7.6 to Y = Speck. The
remaining assertions follow easily from Theorem 7.3. □
Corollary 7.8. The total number of Hilbert series of Koszul algebras with
fixed dim A\ = m over all fields к is finite.
Proof: The number of Koszul Hilbert series cannot exceed the number of irreducible
components of the schemes Q = Q^ s u. Indeed, if a point /: к —► Q
corresponding to a Koszul algebra A belongs to an irreducible component with the general
point 77: К —> Q then it follows from Corollary 7.7 that the quadratic algebra В
over the field К corresponding to 77 is also Koszul and has the same Hilbert series
hA(z) = hB(z). D

CHAPTER 7
Hilbert series of Koszul algebras and
one-dependent processes
In this chapter we describe a relation (first observed in [100]) between Hilbert
series of Koszul algebras and one-dependent stochastic sequences of O's and l's.
The connection is based on certain polynomial inequalities satisfied by dimensions of
grading components of arbitrary Koszul algebras (see section 2). Our interest in this
relation is due to its potential relevance for the conjecture on rationality of Hilbert
series of Koszul algebras (see sections 1 and 8). It is also interesting that some
of the features of the theory of Koszul algebras have analogues for one-dependent
processes. For example, there is a class of one-dependent processes playing the role
similar to that of PBW-algebras (see section 5). Also, one can define modules over
one-dependent processes (see section 10) and consider operations similar to those
studied in chapter 3 (see section 6).
1. Conjectures on Hilbert series of Koszul algebras
Conjecture 1. The Hilbert series of a Koszul algebra A is a rational function. In
particular, the growth of A is either polynomial or exponential.
This conjecture is true for PBW-algebras and for commutative PBW-algebras
(see sections 6 and 8 of chapter 4). Perhaps the most important philosophical reason
to believe Conjecture 1 is provided by the finiteness of Hilbert series of Koszul
algebras with fixed dim A\ (see sections 2 and 7 of chapter 6) It is well known that
neither this nor Conjecture 1 is true for general quadratic algebras (see [8, 56] and
section 6 of chapter 6). Among partial confirmations of Conjecture 1 let us mention
the result of Piontkovskii [93] stating that algebras admitting a Koszul filtration
have rational Hilbert series.
Remark 1. It is well known that any power series with integral coefficients
defining a meromorphic function on the entire complex plane is rational (see [32]).
Hence, to prove Conjecture 1 it suffices to show that the Hilbert series of a Koszul
algebra admits a meromorphic continuation to C. It is clear that the Hilbert series of
a one-generated algebra is holomorphic in the disk \z\ < m_1, where m = dim A\.
In section 8 we will show that the Hilbert series of a Koszul algebra admits a
meromorphic continuaton to the disk \z\ < 2m~1 (see Corollary 8.3).
Note that Conjecture 1 would also imply that every Koszul module over a
Koszul algebra has a rational Hilbert series (by Corollary 5.5 of chapter 2). One
can hope that the following stronger version of this assertion is true.
Conjecture Ibis. Let A be a Koszul algebra. Then there exists a polynomial q(z)
depending only on A such that for every Koszul A-module M one has Hm(z) =
p(z)/q(z) for some polynomial p(z).
133

134 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
Finally, let us point out that Conjecture 1 would imply rationality of Hilbert
series for a much wider class of graded algebras.
Proposition 1.1. Assume that Conjecture 1 holds. Then the Hilbert series of
any graded algebra of finite rate (see section 3 of chapter 3) is rational.
Proof: Note that if rate (A) ^ d then the algebra A^ is Koszul (by Theorem 3.1).
Consider the decomposition A = 0rZo A^d,r\ where A\ ',r) — Аг+<ц. It suffices
to show that each A^d^ has rational Hilbert series. For r = 0 this follows from
the Koszulness of A^d\ On the other hand, applying Proposition 3 of chapter 3
to the A-module A^ = ®°^0 Ar+i for r > 0 we derive that A^d^ is a Koszul
A(d)-module. Hence, its Hilbert series is also rational. □
We have seen that there are examples of Koszul algebras with Hilbert series
different from that of any PBW-algebra (see section 8 of chapter 4). Nevertheless,
all of the known Hilbert series of Koszul algebras are quotients of polynomials
of degree ^ dim A\. The following particular case of this observation may be of
independent interest.
Conjecture 2. Any Koszul algebra A of finite global homological dimension d has
dim A\ ^ d. By duality this is equivalent to the following statement: for a Koszul
algebra В with Bd+i = 0 and B^ ф 0 one has dimi?i ^ d.
This conjecture holds trivially for d ^ 2. To verify it in the case d = 3 we
just have to rule out the possibility dimAi = 2 which is not difficult since all the
Hilbert series of Koszul algebras with two generators are known (see section 5 of
chapter 6). It is also easy to check that Conjecture 2 holds for PBW-algebras:
the proof reduces to the fact that all vertices in a path of maximal length in an
oriented graph are distinct. The second statement of the conjecture is not true for
non-Koszul quadratic algebras: for the quadratic algebra
B = k{x,y}/(x2, y{x + y))
one has Hb(z) = 1 + 2z + 2z2 + z3. Also, the first statement of Conjecture 2 is not
true for general one-generated graded algebras: the algebra A = {x,y}/(x2y,xy2)
with two generators and two cubic relations has global dimension 3 and the Hilbert
series (1 - z)~2(l - z2)~l [119].
Another simple observation can be formulated as follows.
Conjecture 3. Let Abe a Koszul algebra. If both algebras A and A[ have polynomial
growth then the Hilbert series of A coincides with that of the tensor product of a
symmetric algebra and an exterior algebra:
(1 + zY
Ьа(х) = 7 гг for some a, b ^ 0.
{l-z)b
For a finite-dimensional algebra A Conjecture 3 follows easily from
Conjecture 2. The first nontrivial case to check would be that of the series h(z) =
1 + 3z + 4z2 + 3z3 + z4 = (1 + z + z2)(l + z)2. Both conjectures imply that
there should be no Koszul algebras with h(z) as the Hilbert series. We do not know
whether this is true (see also Example 5 in section 5 of chapter 6).
Here is a proof of Conjecture 3 for PBW-algebras. According to Proposition 6.3
of chapter 4 we should consider a subset S С [1, m]2 such that both graphs Gs and
Gg have no intersecting cycles. Assume that Gs has a cycle of length a. Since Gs
has no intersecting cycles, the only edges between the corresponding a vertices are

2. KOSZUL INEQUALITIES
135
the ones constituting the cycle. Since G$ also has no intersecting cycles, it follows
that a ^ 2. Thus, the only cycles in Gs are loops and 2-cycles (cycles of length 2).
Note that if there are no edges connecting i and j in Gs (we say in this case
that {i,j} is a disconnected pair of vertices) then i and j form a 2-cycle in G§.
Since 2-cycles in Gs cannot intersect, we derive that for every vertex i £ [l,m]
there is at most one other vertex j such that {i,j} is a disconnected pair of vertices
in Gs- Note also that for every disconnected pair {i,j} in Gs we should have loops
at both vertices г and j.
Let us call a one- or two-element subset in [l,m] a quasi-cycle if it is either a
disconnected pair of vertices, or a vertex with a loop that does not belong to any
such pair, or a 2-cycle, or a vertex that does not belong to any cycle. It is clear
that every vertex belongs to exactly one quasi-cycle and the sets of quasi-cycles
for Gs and for G$ are the same. By the construction there is exactly one edge in
Gs between any two vertices from different quasi-cycles. It is not difficult to see
that all the edges joining two fixed quasi-cycles start in one of them, and that this
relation defines a total order on the set of quasi-cycles.
It remains to observe that our set S coincides with the subset of [1, m]2
associated with the natural PBW-basis in the tensor product of the monomial quadratic
algebras with ^ 2 generators corresponding to the quasi-cycles (see section 4 of
chapter 4). Therefore, the Hilbert series of S is equal to the product of those
Hilbert series. □
Remark 2. It is also easy to verify Conjecture 3 for commutative and skew-
commutative algebras A. Indeed, a quadratic algebra A is skew-commutative (resp.,
commutative) iff the quadratic dual algebra A- is the universal enveloping algebra
of a quadratic Lie algebra (resp., Lie super algebra), so this follows from Examples
2 and 3 in section 2 of chapter 2.
Remark 3. Our Conjecture 3 is also reminiscent of the well-known conjecture
on Hilbert series of Artin-Schelter regular algebras [12] (see also [79]). Recall
that a graded algebra is called regular, if it is Gorenstein, of finite homological
dimension and has polynomial growth. It is expected that the Hilbert series of a
regular algebra A coincides with that of a polynomial algebra with homogeneous
generators in appropriate degrees:
hA(z) = f[(l-znT1-
2=1
If A is a Koszul regular algebra this would follow immediately from either
Conjecture 2 or Conjecture 3 (to deduce it from Conjecture 2 observe that roots of
h^(—z) have absolute value 1 and their sum is equal to dimAi).
2. Koszul inequalities
In this section we show that the numbers аг = dim A{ for a Koszul algebra A
satisfy a certain system of polynomial inequalities. In section 4 we will use these
inequalities to associate with A a one-dependent stochastic sequence of O's and l's.
The inequalities that we want to prove generalize inequalities
&i-\-j ^ CLi&j

136 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
that hold for every one-generated algebra. For every collection of indices zi,... , ir ^
1 (where r ^ 1) let us consider the polynomial
$ii,...,ir(a#) = $ii;...,v(ai,a2,... ,ail+...+iJ :=
&ii • ■ • 0>ir ~ Qix+ii&iz • • • ^ir ~ aii #42+13^4 * " • ^ir ~ • • • ~ &ii • ■ • &iT-_2a,iT._i+iT.
~r aii+i2+i3ai4 • • ■ O'ir + • • • + CLii • • • &гг-3агг-2+гг-1+гг
"г aii+12^13+14^15 • • • air ~r • ■ • H~ aii • ■ ■ a-iT-_4a'iT-_3+iT—2a'iT-_i+iT- — ■ • • ± 0,^ + ...+^-
The summation here is taken over all partitions of the segment [1, r] into nonempty
subsegments [1, £i], \t\ + 1, t2], ..., [ts-i + 1, r] and the sign is (—l)r_s.
Theorem 2.1. If A is an m-Koszul algebra then the numbers a* = dirnA^ for
i ^ 1 satisfy inequalities
(2-1) ФП11...,„ГИ):=Ф (a.) ^ 0
for every ni,... , nr ^ 1 sitc/i £/ш£ ni +... + nr ^ m.
Proof: Consider the complex of Proposition 8.3 of chapter 2. Since it is exact, we
get
$ni,...,nr(a.) = dimV^ni,... ,nr) ^ 0.
а
Remark 1. One can get a more direct proof of the above theorem using
Backelin's criterion of m-Koszulness (see section 4 of chapter 2) and the following
elementary observation. Suppose we are given a finite set 5 and a collection of
subsets Si,... , Sm С S. For every J с [1, m] let bj be the number of elements in
Sj := f]jeJ Sj. Then for every 7 С [1, m] one should have
£(-l)"JH%£0.
JDI
Indeed, the set S can be partitioned into a disjoint union of the atomic subsets
7> = (nu)n(p|sa
iei iei
where Sf denotes the complement to Si in S, 7 runs through subsets of [1, m]. The
subsets Sj are recovered from the partition into T/s by the formula Sj = Uj^e/ ^i-
Using the exclusion-inclusion formula one can easily express cardinalities // := \Tj\
in terms of (bj):
(2.2) // = £(-l)|JH%.
JDI
This implies the above inequalities.
Remark 2. Inequalities (2.1) with r = 1 are true for any graded algebra, the
inequalities with r = 2 hold for a one-generated algebra A, and those with r = 3
are satisfied for a quadratic algebra. This observation can be generalized further:
the inequality Ф^.^ДА) ^ 0 holds provided that Ext^(k, k) = 0 for all i ^ r — 1
and for i < j ^ ii + • • • + ir. Moreover, if one has is ^ с for all s, then it suffices
to consider (i — \)c < j — 1 instead of i < j in the condition above. This follows
from the Z-algebra version of Backelin's theorem on the cohomology of Veronese
subalgebras (see [16] and section 3 of chapter 3).
There is a nice determinantal formula for polynomials Ф^,...,^-

2. KOSZUL INEQUALITIES
137
Proposition 2.2. For every zi,... , ir ^ 1 one has
Фг1;...,гДа.) = detX(iu... ,гг),
where X(ii,... , гг) = (xmn) zs t/ie r x r matrix with the entries
(airn>n, m^n, where ^п = Т^=т is;
Xmn = U, ГП = 71+ 1,
I 0, m > n+ 1.
Proof: We use induction in r. If r = 1 then the assertion is trivial. To prove the
induction step we use the following recursion formula:
(2-3) Ф*1 ir(fli) = ЯгхФга, ..,гг(а«) ~ Фг1+г2;г3)... ,гг (<*•) •
The similar formula for det X(ii,... , гг) is immediately obtained by expanding the
determinant in the first column. □
The determinants appearing in the above formula for Ф^,...^ are actually
minors of the infinite Toeplitz matrix X = (о^-Ог.^сь where we set a* = 0 for
г < 0. More precisely, X(i\,... , ir) corresponds to the r x r minor of X with rows
(0, ii, %i + гг,. • • , i\ + . • • + V-1) and columns {i\,i\ + гг,.. • , г\ + ... + гг).
The reason we think this determinantal formula is of interest is because of the
relation with total positivity. Recall that a sequence (an) is called totally positive
(or Poly a frequency sequence) if the matrix X is totally positive, i.e., all its minors
are nonnegative. It is well known (see [71]) that in this case the Hilbert series
h(z) = En>o anZn meromorphically extends to the entire complex plane (and hence
is rational in the case when an are integers). Moreover, in this case poles (resp.,
zeros) of h(z) are positive (resp., negative) real numbers. However, it is not true
that for every Koszul algebra the sequence (an) is totally positive because the above
condition on poles and zeros is not necessarily satisfied (even for PBW-algebras). In
[42] Davydov proved that total positivity holds for a quadratic algebra associated
with an i?-matrix satisfying the Hecke condition (R + l)(R — q) = 0.
It is not true that every sequence a# = (an) of positive integers satisfying
inequalities (2.1) comes from a Koszul algebra. In fact, there are some extra non-
homogeneous inequalities that an should satisfy. Here is the simplest example.
Proposition 2.3. Assume that A is a one-generated algebra and let an =
dimAn. Then
&k ^ CLiO'j
/ormax(i,j) ^ к ^ i + j.
Proof: This is proved using Grobner basis techniques. Let xi,... ,xm be a basis
of A\. Let J\f be the set of normal monomials in x^s. By definition, J\f consists
of all monomials xa E A such that Xе* is not a linear combination (in A) of Xе*
with a' < a in lexicographical order (cf. section 1 of chapter 4). Let J\fn С N be
the set of monomials of degree n in ЛЛ It is clear that J\fn is a basis of An, hence,
|A/*n| = an. Consider the map
Л4 —- Mi x Mj : xa ^ ((х")^г, (х«)>кч),
where {xa)^i (resp., (xa)>/c_^) denotes the truncation of the monomial xa to its
initial segment of length г (resp., its final segment of length j). It is easy to see
that this map is injective for max(z,j) ^ к ^ г + j. Hence,
|A4KIM|-|A/}|. D

138 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
Example. The series h(z) = 1 + 9z + z2 + 2z3 cannot be the Hilbert series of a
Koszul algebra since the inequality аз ^ a\ does not hold for it. However, we claim
that it satisfies all inequalities (2.1). Indeed, this follows from the fact that
h(8z) = l + 72z + 64z2 + 1024z3
is the Hilbert series of the monomial algebra corresponding to the oriented graph
with 72 vertices
{ui,..., u32, v, wi,..., w32, 5i,..., 57}
and 64 edges
{(ui, v),..., (гх32, v), (v, wi),..., (v, w32)}>
3. Koszul duality and inequalities
For a sequence of numbers a# = (ao = l,ai,a2,...) let us define the dual
sequence a9 = (1, aj, a2,...) by the equality
52ап*п = (Еоп(-г)пГ1-
Thus, if a# consists of dimensions of grading components of a Koszul algebra A
then a9 corresponds to the Koszul dual algebra A\
Proposition 3.1. For every n ^ 1 one has
an = ф1П(а.),
where ln = (1,... , 1) (1 repeated n times).
Proof: It suffices to prove the following identity for every n ^ 1:
n
^(-1)гап_г<Ма.) = 0.
г=0
This identity is easily derived from the recursion formula (2.3) applied to the
sequences ln, (2, ln"2), (3, ln"3), etc. П
The next proposition is analogous to Proposition 8.5 of chapter 2. Recall that
(zi,... , ir) 1—> Jii,... лг denotes the correspondence between collections of positive
numbers (zi,... , ir) such that i\ + ... + ir = n and subsets of [1, n — 1] considered
in Lemma 8.2 of ch. 2.
Proposition 3.2. For any sequence of numbers a. = (an) one has
Фгь...,гДаО = Ф^...^(а-),
where a[ is the dual sequence, Jj1,...js is the complement to Jii,...,ir гп [1?^- — 1]-
One can prove this by mimicking the proof of Proposition 8.5 of chapter 2. We
will give another proof using one-dependent sequences in section 4.
Corollary 3.3. For any quadratic m-Koszul algebra A and the quadratic dual
algebra A one has
Фй,...,<гИ) = Фл,...лИ!).
where Jj1,...js = Jfx ir, provided that i\ + • • • + ir = j\ + • • • + js — n ^ m.

4. ONE-DEPENDENT PROCESSES
139
4. One-dependent processes
In this section we show how to associate a one-dependent stationary stochastic
sequence of O's and l's with a Koszul algebra. Such a sequence is determined by
a certain collection of probablities that can be viewed as a function on the free
algebra in two variables. We will give a definition first and then explain its origin
in probablity theory.
Definition. A one-dependent process is a linear functional ф : M{#o,£i} —> К such
that
« Ф(1) = 1,
(ii) ф(т) ^ 0 for every monomial m in xq and #i, and
(ш)
(4.1) Ф(/-(хо+х1)-д)=ф(Л-ф(д)
for all /, g G R{xq,Xi}.
Let us point out some easy corollaries from conditions (i)-(iii).
Proposition 4.1. For every one-dependent process one has
(a) ф{/ • (xo + xi)n • g) = </>(/) • ф(д), where n ^ 1, /, g e Щх0, Хх};
(b) ф — ф о l, where t : R{xo,xi} —> R{#o,£i} is the anti-involution sending
Xi^ . . . Xir ЪО Xir . . . Xi^ .
Proof: (a) This follows from (4.1) by induction.
(b) Let us prove by induction that for every monomial m of length n one has
ф(и(т)) — ф(т). We can use the second induction degXo(m). If m = x™ then the
assertion is clear. Otherwise, we can write m = m'xQXk. Then i(m) = xkXQi{m'),
hence
ф(и(т)) = ф{хк1{х1 + х0)с(т')) - ф(хк+1и{т')) = ф(хк1)ф{т) - </>(m'a;*+1),
where we used the induction assumption for m' and for m'x\+l. Therefore,
ф(т) = ф(т'(х0 + xi)xf) - ф{тхк+1) = ф{хк)ф{т') - ф{т'хк+1) = ф(с(т)).
П
Now let us explain a more standard definition of a one-dependent process in
probablity theory. A 0-1-valued stochastic sequence is a sequence of random 0-1-
valued variables (£i)iez- Such a sequence is determined by the set of probabilities
(4.2) 0 ^ P{£m = £m,£m+l = £m+l? • • • ? £n = enf ^ 17
where m^n, £г g{0,1}. These numbers should satisfy the obvious compatibility
conditions:
* ism+1 = £m+\i •••;Sn =£n/ =
^{sm = 0, Цт+l = £m+l? • • • ? sn = £nf H~ -* \sm = 17 sm+1 = £m+l? • • • > Sn = £71J ?
-* IS^n = &ГП1 - ' - 1 Sn—1 = ^П— 1 J
Pism = £mi • • • 5 Sn—1 = £n—1? Sn = 0) + .r |c;m = £m, . . . , qn_1 = 6n— 1, £n = 1}.
A stochastic sequence (£i) is called one-dependent if the collection of all £; with
г < к is independent from the collection of all ^ with i > к for any /c G Z—
"the future is independent of the past provided that nothing is known about the

140 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
present". Equivalently, probabilities (4.2) should satisfy the following system of
identities:
(4.3) P{£rn = £m? • • • > ffc-1 = £fc-bffc = 0,ffc+l = £fc+b • • • > f n = ^n} +
P{t>rn = ^m? • • • 7 ffc-1 = £fc-bffc = l,ffc+l = £fc+b • • • , ^п = £71} =
P{£,rn — £m, • • • > £fc-l = £k-l} ' P{£k+1 = £fc+b • • • > £n = ^n}7
where m ^ к ^ n. Note that the cases к = m and к — n correspond precisely to
the compatibility conditions between elementary probabilities (4.2).
A stochastic sequence is called stationary if the probabilities are invariant under
the shift £i i—> £i+i- In this case the elementary probablities are invariant under
the simultaneous shift of m and n.
We claim that a one-dependent process can be viewed as a generating function
for the set of probablities associated with a stationary 0-1-valued one-dependent
stochastic sequence. Indeed, if we set
ф(х£1 ...ж£п) = P{£i =£i....,fn = £n}
then we immediately see that the condition of one-dependence (4.3) is equivalent to
(4.1). Note that given a one-dependent process ф : M{#o, xi} ~^ ^ one can realize it
canonically by a stochastic sequence (£n) on a probabilistic space O. Namely, let О
be the space of all infinite sequences (en)n^i of 0's and l's and let £n : Cl —> {0,1} be
the projection onto the n-th component. Let S(ei,..., en) С Ct be the subset of all
sequences starting with (е1?..., en). Then we set P(S(ei,..., en)) = ф(хЕ1 ... x£n)
and extend it by сг-additivity. Correctness of this is guaranteed by the identity
Ф(1х0) + ф(/х1) = ф(/)
that follows from property (i) in the definition of a one-dependent process.
Now the crucial observation is that to an arbitrary sequence of numbers a# =
(an) satisfying Koszul inequalities (2.1) (and hence to any Koszul algebra) one can
assign a one-dependent process. To explain this connection let us note that a one-
dependent process is determined by the sequence of numbers an := </>(x™-1) (i.e.,
by the probabilities of several units in a row). Indeed, we can rewrite (4.1) as
(4.4) ф(т1Хот2) = Ф(ггы)ф(т2) — 0(mixi?7i2)
for a pair of monomials mi, 7712. Using this identity we can express ф(т) for every
monomial m in terms of ф(т!) where degXQ(mf) < degXQ(m). Iterating this
procedure we can express all numbers ф(т) as some universal polynomials of an. A
one-dependent process with prescribed numbers an exists iff all these polynomial
expressions of an's are nonnegative. It is not difficult to check that these
expressions coincide with polynomials Фгь... ,гДаь #25..., an) (where ol\ = 1), so we have
the following result.
Theorem 4.2. Assume that a\ > 0. A sequence of numbers a. = (an)
satisfies (2.1) iff there exists a one-dependent process ф : R{#o,£i} —> ^ such that
(4.5) an = ф{хГг) = ап/а%
for all n ^ 1. For such a one-dependent process one has
(4.6)
Фг1,...,гДаъа2,... ,an)
ф(х£1 ...X£n_J =Фг1,...,гД1,а2,... , <Хп

4. ONE-DEPENDENT PROCESSES
141
where in the notation of Lemma 8.2 of chapter 2, J^,... ,ir = {j € [1, n—1] : Sj = 1}.
Proof: Clearly, it is enough to prove (4.6). Let us denote by т^,...,^ the monomial
of degree n— 1 that has x\ exactly in places corresponding to J = Jii,...,ir- It suffices
to show that the quantities 0(mii,...,ir) satisfy the recursion similar to (2.3). Note
that J = Jii,...,^ = [l?^i — l]U(ii +J'), where J' = Ji2,...,ir С [l,n —ii — 1]. On the
other hand, «/ii+i2,i3,...,ir = «/U {ii}. Hence applying (4.4) for the decomposition
0(raii,...,ir) = aii ' </>(m;2,...,iJ - </>(mil+i2,...,iJ
as required. П
Corollary 4.3. To every Koszul algebra A one can assign a one-dependent
process фл such that
Фа{х^-1) =ап/а1{,
where an — dim An.
Remark 1. More explicitly, given a Koszul algebra A = {V, R} and an integer n,
one can find a realization for the finite part £1, ... , £n_i of the stochastic sequence
corresponding to фл in the following way. Consider a distributing basis fi(n) С V®n
as a finite probabilistic space with the uniform measure P{w} = a±n and define
the random variables & as follows:
(4.7)
&:П(П) .{0Д}, №) = (!' WlS) г = 1,...,п-1.
Both the compatibility and one-dependence conditions follow from the nature of
the subspaces R,\n) = V®*'1 ®R® V71'^1 С V®n.
Remark 2. Equality (4.1) corresponds to the following identity for polynomials
Ф<1,...,<г:
^ii,...,is " ^is-f-i,...,zV ^i\,...,ir \ ^ii,...,is_i,is+zs+i,zs+2,.-4V'
where 1 ^ s < r. For Koszul algebras the exact sequence of Corollary 8.4 of
chapter 2 can be considered as a categorification of this identity.
Remark 3. It was proved in [59] that for every one-dependent sequence with
a = P{£i = 1} > 1/2 one has
P{fi = U2 = lK<*3/2.
3/2
The corresponding inequality аз ^ а2 holds for an arbitrary one-generated algebra
with a2 ^ a?/2 (see Remark 1 in section 4 of chapter 6).
Let us denote by a : R{xo>#i} —> Ш{хо,х\} the automorphism switching xo
and x\. Then for every one-dependent process ф : R{#o,£i} —> К we define the
dual one-dependent process
ф- := ф о а.
Note that if (£n) is a one-dependent stochastic sequence corresponding to ф then
the sequence corresponding to ф' is £n := 1 — £n.
Proposition 4.4. T/ie construction of Theorem 4.2 is compatible with duality.
Namely, if ф is the one-dependent process corresponding to a sequence of numbers
a. then the dual process ф1 corresponds to am.

142 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
Proof: We have for n ^ 2
Since J\n = 0, using (4.6) we obtain
□
Now we can give a proof of Proposition 3.2.
Proof of Proposition 3.2. Let us denote J = Jii,...,^ С [l,n — 1] and let mj be
the monomial of degree n — 1 that has #i exactly in places corresponding to J.
Applying (4.6) together with Proposition 4.4 we obtain
til til
where Jj1,...js = Jc. □
Remark 4. One can define a matrix-valued one-dependent process to be a linear
functional ф : R{#o,£i} —» Matn(R) with values in n x n-matrices over R with
properties similar to (i)-(iii) in the definition of a one-dependent process. We only
have to modify (ii) by requiring ф(т) to have nonnegative entries for every
monomial m in xq and x\. Such processes appear naturally when considering Koszul
algebras with A0 =kn. We leave for the reader to work out the details.
5. PBW-algebras and two-block-factor processes
In the previous section we showed how to assign a one-dependent process to
every Koszul algebra. It turns out that there is a natural class of one-dependent
processes containing all sequences associated with PBW-algebras (equivalently, with
monomial quadratic algebras). Namely, this is the class of two-block-factor
processes. In probability theory, a two-block-factor stochastic sequence is a sequence
of the form £n = f(rjn,rjn+i), where (rjn) is a sequence of independent identically
distributed random R-valued variables, / is a 0-1-valued measurable function of
two variables. It is clear that every two-block-factor sequence is stationary and
one-dependent. Note that without loss of generality one can assume that each rjn
is distributed uniformly on the interval [0,1] (since every random R-valued variable
rj has form ф(г]и), where rju is uniformly distributed on [0,1] and ф is a measurable
function). Then / is the characteristic function of a measurable subset А С [0, l]2.
Thus, we arrive at the following definition.
Definition. Let А С [0, l]2 be a measurable subset. The two-block-factor process
associated with A is the one-dependent process </>д defined by
Фа{х61 •.•^етг_1) =/in(A(£i,...,£n_i)),
where
A(£b...,£n_i) =
{(xb. ..,xn) g [0, l]n I (xi,Xi+i) e A for ei = 1 and (xi,xi+i) £ A for Ei = 0},
where /in is the standard measure on Rn. In particular,
^д(^Г1) = Мп(Д(п)),
■1)=Ф(хГ1)-
,<*n) =
Ф1п(а1,а2,... ,ап)

6. OPERATIONS ON ONE-DEPENDENT PROCESSES
143
where
A(n) = A(ln_1)
- {(xb...,xn) e [0,i]n | (xux2) e A,(x2,x3) e A,...,(xn_bxn) e A}.
Now assume that we have a subset S С {1,..., m}2. We can view S as the set
of edges of an oriented graph Gs with vertices {1,..., m}. Let us associate with S
the subset
A5= U [—.-]*[—Д] С [0,1]2.
(ij)es
Let As be the quadratic monomial algebra associated with S (such that XiXj = 0
inAs for (i,j) 0 5).
Proposition 5.1. The one-dependent process 4>As associated with As coincides
with the two-block-factor process ф&3 associated with A5.
Proof: We have
д(п)= (J (W il]x...x[»n^l »n]c[01]„
w mm mm
(ii,...,in)6S(n)
where 5^n^ was defined in section 1 of chapter 4. It follows that
where an = dim A%. П
Let us equip the space of linear functionals on M{#o,£i} with the topology of
pointwise convergence, i.e., we say that фп —> ф if for every / G M{xq,Xi} one has
Фп(/) -^ Ф{/)- The set of one-dependent processes is closed with respect to this
topology. Now we will show that the set of processes associated with PBW-algebras
is dense in the set of all two-block-factors.
Proposition 5.2. Let ф be a two-block-factor process. Then there exists a
sequence of quadratic monomial algebras (ASi)i^i such that фАБ1 —> ф as i —> 00.
Proof: By definition we have ф = ф& for some measurable subset А с [0, l]2.
We can find a sequence of subsets of the form A^, where Si С {1,..., т^}2, such
that Ast tends to A with respect to the measure. It is clear that this implies that
</>д5. —► ф. It remains to apply the previous proposition. □
Recall that there exist Koszul algebras with Hilbert series different from those
of all PBW-algebras (see section 8 of chapter 4). The first examples of non-two-
block-factor one-dependent sequences were found in [2]. Two-block-factor and non-
two-block-factor sequences are also studied in the monograph [122].
6. Operations on one-dependent processes
Recall that the set of Koszul and PBW-algebras is closed under a large set of
natural operations (see chapter 3). In this section we are going to consider similar
operations for one-dependent processes.
First, let us describe a family of unary operations on one-dependent processes
generalizing Veronese powers. Namely, assume we are given a triple of elements

144 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
/о»/ъ<7 £ R{#o»#i}> all of them positive linear combinations of monomials, such
that
d
/0 + /1 =^2ai(x0+x1y
i=l
for some numbers ai ^ 0 such that J^i a^ = 1. Then we can introduce an operation
ф 1—► 0? * on one-dependent processes by setting
,9 m #*(/))
where deg(/) is the total degree of / and к : R{xo, xi} —> M{xo, x{\ is the following
linear map:
(6.1) k(x£ix£2 ...x£n) = gf£lgfe2g• ■•gfer.g-
Note that this operation is defined on ф only if ф(д) > 0. The fact that фf1 is again
a one-dependent process follows from the identity
«(/•(xo+a:i)-/,) = «(/)(/o + /iM//)
and from Proposition 4.1(a). For fi = x\-i and g = 1 we recover the duality:
ф{ = ф^ Setting ф^ — фхЪ,хц where N > 0 we get analogues of Veronese
powers. One has
6(N), п-1ч _ Ф(х1 П~ )
Hence, for a Koszul algebra A one has
фА(м) = фА .
We can generalize the above construction to define a huge family of n-ary
operations on one-dependent processes. For simplicity we will consider only binary
operations. First, for a pair of one-dependent processes ф\ : IR{xo,xi} —> Ш and
Ф2 : R{ycb2/i} —> К one can consider the tensor product of the corresponding
functionals:
фг 0 ф2 : Щх0, хг} 0r R{y0,2/1} -► К.
Note that (</>i 0 <fe)(l) — 1 and in addition we have the following identity:
(Ф1 0 02)(/ ' (*o + ^1Г(У0 + У1)П^) = (Ф1 0 &)(/) • (Ф1 ® 02)Ы,
where ra ^ 1, n ^ 1. Next, let us fix a triple of positive linear combinations of
monomials /0, /1, <7 £ R{#cb #1} 0r К {2/0» 2/1} sucn that
/0 + /1 = (zo + zi)(2/o+2/i)^(zo + zi,2/o+2/i)
for some polynomial F such that F(l, 1) = 1. Then we can define an operation
h ®/0|/i fc(/) := 0ig|^(g)dag(/)+1>
where « : M{x0,xi} —> M{x0,Xi} 0r R{y0,2/i} is still given by formula (6.1). This
operation is well defined provided (ф\ 0 <fe)(<7) > 0- For # = 1, Д = £12/1 and
/0 = #o2/o + #i2/o + #o2/i we get an analogue of the Segre product ф\ о </>2 that is
characterized by the equality

6. OPERATIONS ON ONE-DEPENDENT PROCESSES
145
This construction is compatible with the Segre product of algebras: if A and В are
Koszul algebras then
ФаоВ = Фа° Фв-
If (fn) and (£fn) are one-dependent stochastic sequences independent from each other
then the stochastic sequence realizing their Segre product is simply min(£n, £fn). On
the other hand, if we take g = 1, Д = xiyo + xqjji and /о = хоУо + #i2/i> we get an
operation realized by the addition of one-dependent sequences (independent from
each other).
Another family of operations is motivated by the direct sum of Koszul algebras.
We are going to check that if we have two sequences of numbers a. = (an)n^i and
fr« = (frn)n^i satisfying inequalities (2.1) then the sequence am + bm = (an + bn) also
satisfies them. This will lead to a family of operations on one-dependent processes.
Lemma 6.1. Let aQm = (a°)n^i and a\ = (ajjn^i be two sets of variables. For
every (ni,..., nr) one has
$nl5..,nr(a2 + ai) =
E E Фп1|...|п4^а;)ФП41+1|...|П4^а$+1)...Фп.в_1+1|...|ПЛа$+--1)
e=0,l l^.t1<t2<---<ts=r
where we set a™ = Cmod2.
Proof: In the case when aen = dimA^ for some Koszul algebras A0 and A1 the
assertion follows immediately from the decomposition of
VAonAi(ni, ...,nr)cW = (A°ni ф<)®-0 (А°Пг ф О
into a direct sum of subspaces, induced by the natural decomposition of W. It
remains to observe that collections of numbers (an) arising from Koszul algebras are
algebraically independent. Indeed, it is enough to consider direct sums of algebras
of polynomials. □
Proposition 6.2. For every number A such that 0 < Л < 1 and every pair
of one-dependent processes ф and ф' there exists a one-dependent sequence ф Г\\ ф'
such that
(ф UA ф'КхГ1) = АЖ"1) + (1 - A)V(хГ1)-
Proof: Let an = </>(x™_1), (3n = (^/(x™-1). Applying Lemma 6.1 to sequences an =
\nan and bn = (l-\)n(3n (where a\ = A, bi = 1-Л) we get ФП1,...,пг(а. + b.) ^ 0.
It remains to apply Theorem 4.2. □
Finally, let us consider analogues of the tensor product. Given a pair of
sequences of numbers a. = (an)n^i and bm = (bn)n^i let us define a sequence
a.®b. = (cn)n^i by setting
c<n = a<n + an-ibi H h aibn_i + bn.
Lemma 6.3. In the above situation ФП1г.)Пг(с) can be written as a universal
polynomial о/(Фг1,...,г8(а)) and (&j1,...jr{b)) with nonnegative integer coefficients.
Proof: The proof is similar to Lemma 6.1: first, one reduces to the case when a and
b come from Koszul algebras A and Б, and then uses a direct sum decomposition
°f ^4<8)0в(пъ • • •»nr)- Note that it is more convenient to use the one-sided product
A 0° В rather than the usual tensor product A® B. □
As before this leads to the following family of operations on one-dependent
processes.

146 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
Proposition 6.4. For every number A such that 0 < Л < 1 and every pair of
one-dependent processes ф and ф' there exists a one-dependent process ф®\ ф' such
that
n
г=0
where o.i = ф(х\~1), а'{ = ф^х1^1) for г ^ 1 and а$ = а'0 = 1. □
7. Hilbert space representations of one-dependent processes
In this section we discuss representations of one-dependent sequences in terms
of operators in Hilbert spaces considered by de Valk in [121].
Let Л be a bounded operator on a Hilbert space H and let x, у £ H be a pair of
vectors such that (x,y) = 1. With these data we can associate a pair of operators
Aq and Ai as follows. Let P* be the operator given by
P^(v) = (v,x)y.
Then we set Ai = A, Aq = P* — A, Assume that for every sequence £i,..., £n of
O's and l's one has
(Aei...Aeny,x) ^0.
Then there exists a one-dependent process ф such that
ф(х£1 ... x£n) = {A£l ... A£ny, x).
Indeed, this follows from the identity
(BP*Cy,x) = (By,x)-(Cy,x)
that holds for an arbitrary pair of operators В, С In this situation we say that
(if, A, x, у) is a Hilbert space representation of a one-dependent process ф. The
following result is due to de Valk [121] (we give a slightly different proof below).
Proposition 7.1. For every one-dependent process ф there exists a Hilbert
space representation (if, A,x,x) of 0.
Proof: Consider a Hilbert space H with an orthonormal basis (en)n^i. Set
x = -ei + Л • ^2 Ф^'^еп/п е Я,
where Л is a positive constant (the sum converges since ф(х^~1) < 1). Let us
choose Л such that (x,x) = 1. Now let A be the unique operator on H such that
A(x) = \e<z and A(en) = 21^en+i for n ^ 2. It is easy to see that A is bounded.
Also, for n ^ 1 one has
{Anx,x) = IL—(еп+ьж) = ф(х^).
This implies that (H, A,x,x) is a representation of ф. □
Some one-dependent processes admit finite-dimensional representations. In
particular, this is true if there exists a 0 — 1 sequence £i,...,£jv with N ^ 1
such that P{£i = £i, • • • ,£jv = £jv} = 0. Such a sequence is called a zero-cylinder
of length N. De Valk proves in [121] that a one-dependent process with a zero-
cylinder of length N admits a representation in the iV-dimensional space. It is
not known how to characterize one-dependent processes with a finite-dimensional

8. HILBERT SERIES OF ONE-DEPENDENT PROCESSES
147
representation. The only result in this direction is Theorem 4.3 of [121]
asserting that if a one-dependent process admits a two-dimensional representation then
it is a two-block-factor process. The structure of minimal zero-cylinders for one-
dependent processes is also not known. The conjecture of de Valk states that the
only possible minimal zero-cylinders are [101], [010], [1N] and [0N].
It is very probable that there exist two-block-fact or processes that do not admit
a finite-dimensional representation. On the other hand, we have the following result.
Proposition 7.2. Every two-block-factor process ф admits a Hilbert space
representation (H, A, x,x), where A is a Hilbert-Schmidt operator.
Proof: Let ф = Фа for some measurable subset А С [0, l]2. Consider H =
L2([0,1], dx) and let A: H —> H be the Hilbert-Schmidt operator with the kernel
1д (characteristic function of A). Take x to be the constant function 1 on [0,1]. It
is easy to see that (H, A, x, x) is a representation of ф. □
8. Hilbert series of one-dependent processes
With every one-dependent process ф we can associate its Hilbert series
oo
M*) := 1 + I>(*rV-
71=1
Clearly, this series converges for \z\ < 1.
Conjecture. The series Нф has a meromorphic continuation to the entire complex
plane.
Note that if Фа is a process associated with a Koszul algebra A then
^A(z) = hA(z/m),
where m = dimAi. Thus, this conjecture would imply Conjecture 1 in section 1
(see Remark 1 of section 1). The above conjecture is true for two-block-factor
processes (see Proposition 8.4 below). For general one-dependent processes we can
prove only the following weaker version (along with some additional information).
Theorem 8.1. The series Нф(г) has a meromorphic continuation to the disk
\z\ < 2. Furthermore, 1 + h<f,(z) has no zeros in this disk.
The proof is based on the following result that can be of independent interest.
Let M С M{xo,xi} be the set of all noncommutative monomials in xq and x\
(including 1).
Proposition 8.2. Consider the noncommutative generating series
F(xq,xi)= ^^ Ф(т)т £ Щхо.х^.
тем
Then one has
F(x0,x1) = [f(x! -xq)'1 -xo]"1,
where f(z) = F(0,z) = (кф(г) - l)/z.
Proof: Equation (4.1) leads to the following identity in R{x,y}:
(8.1) ^^ Ф{т1(хо + x1)m2)m1m2 = F(x0,x1)2.
77li,7712 EM

148 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
Consider the derivations do and d\ of Ж{хо,х\} defined by di(xj) = Sij (and
satisfying the Leibnitz identity di(ab) = di(a)b + adi(b)). Set D = до + д\. Then (8.1)
can be written as
D(F) = F2.
This is equivalent to
D(F~l) = -F~1D(F)F-1 = -1,
or to D(F~1 + xo) = 0. It is clear that for any f(z) 6 1 + 2R[[z]] one has
D(f(Xl-x0)) = 0.
Therefore, for any such f(z) the series [f(xi — xq) — £o]_1 is a unique solution of
(8.1) with the initial condition F(0, z) = f(z)~1. It remains to take / = /_1. П
Proof of Theorem 8.1: From the formula of Proposition 8.2 we get
(8.2) F(-t,t) = [f(2t)-1+t}-1.
Note that F(—t,t) = ^2n>oantn, where
\an\ ^</>((x0+xi)n) = l.
Hence, F(—t,t) converges for \t\ < 1. Now (8.2) defines a meromorphic continuation
for f(z) in \z\ < 2. Moreover, it shows that
is holomorphic in this disk. □
Corollary 8.3. If A is a Koszul algebra with dimAi = m then the Hilbert
series Ha{z) admits a meromorphic continuation to the disk \z\ < 2/ra. Moreover,
1 + Ha(z) has no zeros in this disk.
Note that the result that 1 + Ha(z) has no zeros in \z\ < 2/ra is sharp: for
Ha{z) = 1 + mz a zero appears on the boundary of this disk.
Proposition 8.4. If ф is a two-block-factor process then Нф extends meromor-
phically to the entire complex plane.
Proof: According to Proposition 7.2 there exists a Hilbert space representation
(Я, A,x,x) of О with A Hilbert-Schmidt. Therefore,
oo
Ъ,ф(г) = 1 + ^(An_1x, x)zn = 1 + z((l - zA)~lx, x).
71=1
Since A is compact, the operator-valued function (1 — zA)~l extends meromorphi-
cally to the entire complex plane. Therefore, the same is true for кф(г). □
The above proof indicates that one possible approach to the conjecture on
meromorphic continuation of Нф is to try to prove the existence of a Hilbert space
representation (H,A,x,y) of ф such that the operator A is compact.

9. HERMITIAN CONSTRUCTION OF ONE-DEPENDENT PROCESSES
149
9. Hermitian construction of one-dependent processes
In this section we describe the construction due to B. Tsirelson (unpublished)
of a one-dependent process associated with a quadratic algebra and a Hermitian
form on the space of generators.
Let У be a Hermitian vector space, i.e., a finite-dimensional complex vector
space equipped with a positive-definite Hermitian form (-,-)• Recall that for a pair
of subspaces K,L cV one can define the angle between К and L by the formula
(K,L) = (K,L)v = ti(PKPL),
where Рк and Pl are orthogonal projections onto К and L respectively. The angle
(K, L) is always a nonnegative real number and it is equal to zero iff К and L are
orthogonal. In fact, there is another formula for (K, L) that shows this. To state
it we need the following elementary result.
Lemma 9.1. For every pair of subspaces K, L с V there exist orthonormal
bases (/c;)i^dimK, (Ij)i^j^dimL of К and L such that (h,lj) = 0 for i ф j.
Proof: If KCiL1- ф 0 then the statement easily reduces to the subspaces K/KCiL1-
and L in V/K П lA Therefore, we can assume that К П ZA = L П K1- = 0. In
this case the restriction of the projection Pl to К gives an isomorphism p — Pl\k •
К —► L. Let (ki) be the orthonormal basis of К that diagonalizes the pull-back
of the Hermitian form on L by p. Setting U = p{ki)/\\p{ki)\\ we get the required
bases (ki) and (k) in К and L. □
Let us choose bases (ki) and (lj) of К and L as in the above lemma. Then one
has
dim L min(dim L,dim L)
(K,L) = tT(PKPL) = J2 (h^Kh) = J2 &Л*;Л')*;)
j=l j=l
min(dim if,dim L)
E №л-)12-
This shows that (K, L) is a nonnegative real number. Note that if К П L ф 0 then
the above bases (ki) and (lj) can be chosen in such a way that they both contain
the same basis of К П L. This implies that (K, L) ^ dim KC\L. The equality holds
iff there exists an orthonormal basis of V containing bases for К and L.
We can equip tensor powers of V with positive-definite Hermitian forms in the
standard way. This allows us to consider angles between subspaces in V®71 for all
n^ 1.
Definition. A Hermitian pair is a pair (V, R) consisting of a Hermitian vector space
V and a subspace R С У02.
Let us define the Hilbert series 1iv,r(z) = 1 + Sn>i an(V, R)zn associated with
a Hermitian pair (V, R) by the formulas
a2k+i(V,R) = (R*k®V,V®R*k) V®(2fc + 1),
a2fc(V, R) = (R0k, V ® Я®**"1) ® V)v%w.
Note that ax{V, R) = (V, V) = dimV and a2(V, R) = (R, V®2) = dimД.

150 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
Proposition 9.2. For every Hermitian pair (V, R) there exists a one-dependent
process 4>{v,R) such that
n_i,_ an(V,R)
(9-D ^(*i )-(dimy)«
for all n ^ I, i.e.,
^v,r(z) = hVlR{z / dimV).
Proof: Let us set
(9.2)
ф(х£1 .. .x£n_J = (dim V)"71 • (Я51 0 Я52 0 ..., V 0 R£2 0 RSA 0 .. .)v9n,
where R1 = R and R° = RL (if n is odd then the last factor of the first tensor
product is V; if n is even then the last factor of the second tensor product is V).
Clearly, this formula is compatible with (9.1), so we just have to check identity (4.3).
Let us denote by P/ and P? the orthogonal projections to V®^-1) 0 R0 yn~i-1
and V®^-1) 0 R± 0 vn~i_1, respectively. Note that for \j - i\ ^ 2 the operators
Pf and Pj commute, so we can rewrite (9.2) as
0(xei...xen_1) = (dim^)-».tr( JJ Р^Г- П P2?)'
1^2i+l^n-l 1^2j<n-l
Now using the fact that P? + P} = 1 we obtain
ф{хе1...х£к_1(х0-\-х1)х£к+1 ...x£n) = (dimF)"71-1 x
tr( П Р^' П P2?).
l<2i+l^n-l,2i+l^fe 1^2j<n-l,2j^fe
It remains to observe that the right-hand side is equal to
(dim У)"71-1 x
M П ВД- П p2?)-tr( п р^г- П p2?)
which leads to the required identity. □
Note that the above construction is compatible with duality: formula (9.2)
immediately implies that
Фу^ = ftv,R-
It would be interesting to characterize one-dependent processes associated with
Hermitian pairs. It is easy to see that all processes associated with PBW-algebras
are in this set. Namely, if S С [1,га]2 is a subset then the process associated with
the quadratic monomial algebra As coincides with фсп,я3) where Cn is equipped
with the standard Hermitian structure and Rs = ©^ j\eS С • (e$ 0ej). Indeed, this
follows from the fact that (K, L) = dim К П L for a pair of subspaces that can be
distributed by an orthonormal basis.
Another interesting problem is to check whether for an arbitrary Hermitian
pair the Hilbert series hv,R extends meromorphically to the entire complex plane.
It seems that this case should be more accessible than the similar conjecture for
general one-dependent processes.

10. MODULES OVER ONE-DEPENDENT PROCESSES
151
10. Modules over one-dependent processes
A (left) module over a one-dependent process ф is a linear map ф : R{xo> x\} —> R
taking nonnegative values on all monomials, such that
(Ю.1) гР(/-(х0 + х1)-д) = ф(Л-ф(д)
for all /, g G R{xo,xi}.
Example. For every monomial m in xq and x\ we have a module фш over ф
defined by iprn(f) — (/>(/ • ra). Also, if ф\ and -02 are modules over ф then for any
constants c\ ^ 0 and C2 ^ 0 the map ci^i + С2Ф2 is also a module over ф.
The above definition is motivated by the following construction: given a Koszul
module M over a Koszul algebra A we can associate with M a module over the
one-dependent process Фа using the infinitesimal Hopf module Va,m (see the end
of section 8 of chapter 2). Namely, we set
(ЦщУа,м(у1ь--- ,nr_i,nr - 1)
^M(mJni). >nJ- (dimill)ni+-+nr-i
where ni,..., nr ^ 1, the correspondence (n\... , nr) 1—> Jni,... ,nr С [1, n — 1] was
defined in Lemma 8.2 of chapter 2 (where n — n\ + ... + тгг)5 and mj denotes the
monomial of degree n — 1 in x0 and xi, having xi on places corresponding to J.
Let us define the Hilbert series of a module ф by
лг*(*) = 2>(*i)zn-
For example, h^od(z) — (Нф(г) — l)/z. Note that for the module фм associated
with a Koszul module M we get the usual Hilbert series of M up to rescaling:
/CdW = hM(z/m),
where m — dimAi. The next result shows that ф is completely determined by its
Hilbert series. We use the notation from Proposition 8.2.
Proposition 10.1. Let ф be a left module over a one-dependent process ф.
Consider the noncommutative generating function
G(xo,xi)= ^ ф(т)т e R{x0,xi}.
тем
Then one has
G(x0, xi) = F(x0, xi) • д{хг - x0),
where g(z) = G(0, z)/F(0, z) = h™d(z)z/(^(z) - 1).
Proof: Equation (10.1) leads to the equality
D(G) = FG.
Since D(F) = F2 this implies that D(F~lG) = 0. Hence, F_1G is of the form
g{xi-x0). □
Note that the fact that G{xq,x\) has nonnegative coefficients is equivalent
to certain polynomial inequalities on coefficients of W2od and Нф that are module
analogues of Koszul inequalities considered in section 2.
As in section 8 we can derive the following.

152 7. HILBERT SERIES OF KOSZUL ALGEBRAS AND ONE-DEPENDENT PROCESSES
Corollary 10.2. The series h7V,od(z) has a meromorphic continuation to the
disk \z\ < 2. Moreover, the ratio
1 + h<p(z)
is holomorphic in this disk.
Proof: Since ф((хо +Xi)n) — ф(1) for n > 1, we obtain that the series G(—t: t) has
bounded coefficients. Hence, it converges for \t\ < 1 (as does F(—t,t)). It remains
to use the identity G(-t, t) = F(-t, t)g(2t) together with (8.2). □
Corollary 10.3. For a Koszul module M over a Koszul algebra A the Hilbert
series Iim{z) admits a meromorphic continuation to the disk \z\ < 2/m, where
m — dim A\. Moreover, the ratio
hM(z)
l + hA(z)
is holomorphic in this disk.

APPENDIX A
DG-algebras and Massey products
Definition. A DG-algebra is a graded algebra A — 0neZ An equipped with a
differential dA : A —> A such that dA(An) С An+i, d2A — 0, and the Leibnitz identity is
satisfied:
dA{x • y) = dA{x) • у + (-l)*z • dA(y),
where x G A%.
A DG-module M over a DG-algebra Л is a graded A-module M — 0neZ Mn
equipped with a differential dM • M —> M such that dM(Mn) С Mn+i, d^ = 0,
and the Leibnitz identity is satisfied:
^m(g • m) = d^(a) • m + (—l)aa • с?м(^)5
where a G A^, m G M.
Observe that in section 4 of chapter 5 we consider only nonnegatively graded
DG-algebras.
The cohomology H*(A) — H^A(A) of a DG-algebra A has a natural structure
of a graded algebra, and the cohomology Н%м (M) of a DG-module M over A has
a natural structure of a graded if*(A)-module.
A morphism of DG-algebras f : A —> В is a homomorphism of graded
algebras such that fdA — dsf • Such a morphism induces a homomorphism H*(f) :
H*(A) —> H*{B). If H*(f) is an isomorphism then we say that / is a quasi-
isomorphism.
Let M (resp., N) be a DG-module over a DG-algebra A (resp., B). Assume
we are given a morphism of a DG-algebras f : A —> B. Then a morphism of
DG-modules g : M —> N compatible with / is a homomorphism of graded A-
modules such that д&м — d^g. It induces a homomorphism of if*(A)-modules
H*{g) : H*{M) —> H*(N). We say that g is a quasi-isomorphism if H*(g) is an
isomorphism.
We leave for the reader to give formally dual definitions of a DG-coalgebra and
a DG-comodule (and morphisms between them).
Let Л be a DG-algebra. Massey products are certain natural multivalued
partially defined operations on H*(A) preserved under quasi-isomorphisms. The
simplest example is a triple Massey product тз(х1,Х2,х3), where x\ G Нг(А),
x2 G Hj(A), x3 G Hk(A) are cohomology classes satisfying
xix2 = x2x3 = 0.
Let us choose cycles x[ G Ai, x'2 G Aj and х'ъ G A^ representing xi, x2 and £3,
respectively. Since x[xf2 is a coboundary, there exists x12 G Ai+j-i such that
dA{xu) =x[x'2.
153

154
A. DG-ALGEBRAS AND MASSEY PRODUCTS
Similarly, we can choose #23 £ Aj+k-i such that
(1а(х2з) = x'2x'3.
Now we set
(хьх2,х3) = Х12Х3 - (-1)гх[х2з mod \m(dA).
The obtained element in Нг^+к~1(А) depends on the choices made. However, its
coset with respect to the subspace x\ • H^k~1(A) + #г+^-1 . x3 is well denned.
One can also define more general Massey products by replacing the
decomposable tensor x\ ® £2 ® X3 with more general tensors. The definition of тг-ary Massey
products for n > 3 is similar but is more involved. They appear as differentials in
the spectral sequence associated with the natural nitration on the bar-complex of
A (we considered the dual spectral sequence in section 7 of chapter 5; see also [82];
Stasheff [114] defined similar notions in a more general context of Aoo-algebras).
In the case when A is the cobar-complex of a graded algebra the above spectral
sequence shows that an algebra is Koszul iff all the higher Massey operations on its
cohomology are trivial (see [104]).

Bibliography
J. Aaronson, D. Gilat, M. Keane. On the structure of 1-dependent Markov chains. J. Theoret.
Probab. 5, #3, 545-561, 1992.
J. Aaronson, D. Gilat, M. Keane, V. de Valk. An algebraic construction of a class of one-
dependent processes. Ann. Probab. 17, #1, 128-143, 1989.
M. Aguiar. Infinitesimal Hopf algebras. New trends in Hopf algebra theory (La Falda, 1999),
1-29. Contemp. Math. 267, AMS, Providence, RI, 2000.
M. Aguiar. Infinitesimal bialgebras, pre-Lie and dendriform algebras. Hopf Algebras, 1-33.
Lect. Notes in Pure and Appl. Math. 237, 2004.
R. Aharoni. A problem in rearrangements of (0, 1) matrices. Discrete Math. 30, #3, 191-
201, 1980.
R. Ahlswede, G. О. Н. Katona. Graphs with maximal number of adjacent pairs of edges.
Acta Math. Acad. Sci. Hungar. 32, #1-2, 97-120, 1978.
D. J. Anick. Noncommutative graded algebras and their Hilbert series. J. Algebra 78, #1,
120-140, 1982.
D. J. Anick. Generic algebras and CW-complexes. Algebra, topology and algebraic K-theory
(Princeton, NJ, 1983), 247-321, Ann. of Math. Stud. 113, Princeton Univ. Press, 1987.
D. J. Anick. Diophantine equations, Hilbert series, and undecidable spaces. Annals of Math.
122, #1, 87-112, 1985.
E. Arbarello, M. Cornalba, P. Griffiths, J. Harris. Geometry of algebraic curves I, Springer-
Verlag, New York/Berlin, 1985.
S. Arkhipov. Koszul duality for quadratic algebras over a complex. Fund. Anal. Appl. 28,
#3, 202-204, 1994.
M. Artin, W. Schelter. Graded algebras of global dimension 3. Advances in Math. 6, #2,
171-216, 1987.
M. Artin, J. Tate, M. Van den Bergh. Some algebras associated to automorphisms of elliptic
curves. The Grothendieck Festschrift, vol. 1, 33-85, 1990.
L. Avramov, D. Eisenbud. Regularity of modules over a Koszul algebra. J. Algebra 153,
#1, 85-90, 1992.
J. Backelin. A distributiveness property of augmented algebras and some related
homological results. Ph. D. Thesis, Stockholm, 1981. Available at http://www.
matematik.su.se/~joeb/avh/
J. Backelin. On the rates of growth of the homologies of Veronese subrings. Algebra, topology
and their interactions (Stockholm, 1983), 79-100, Lecture Notes in Math. 1183, Springer,
Berlin, 1986.
J. Backelin. Some homological properties of "high" Veronese subrings. J. Algebra 146, #1,
1-17, 1992.
J. Backelin. Private communication, 1991.
J. Backelin. Koszul duality for parabolic and singular category O. Represent. Theory 3,
139-152, 1999.
J. Backelin, R. Froberg. Koszul algebras, Veronese subrings and rings with linear resolutions.
Revue Roumaine Math. Pures Appl. 30, #2, 85-97, 1985.
D. Bayer, M. Stillman. A criterion for detecting m-regularity. Inventiones Math. 87, #1,
1-11, 1987.
D. Bayer, M. Stillman. On the complexity of computing syzygies. J. Symbolic Comput. 6,
#2-3, 135-147, 1988.
A. Beilinson, V. Ginzburg, V. Schechtman. Koszul duality. J. Geom. Phys. 5, #3, 317-350,
1988.
155

156 BIBLIOGRAPHY
[24] A. Beilinson, V. Ginzburg, W. Soergel. Koszul duality patterns in representation theory. J.
Amer. Math. Soc. 9, #2, 473-527, 1996.
[25] A. Beilinson, R. MacPherson, V. Schechtman. Notes on motivic cohomology. Duke Math.
J. 54, #2, 679-710, 1987.
[26] G. Bergman. The diamond lemma for ring theory. Advances in Math. 29, #2, 178-218,
1978.
[27] I. Bernstein, I. Gelfand, S. Gelfand. Algebraic vector bundles on Pn and problems of linear
algebra. (Russian) Funktsional. Anal, i Prilozhen. 12, # 3, 66-67, 1978.
[28] R. Bezrukavnikov. Koszul DG-algebras arising from configuration spaces. Geom. and Fund.
Anal. 4, #2, 119-135, 1994.
[29] R. Bezrukavnikov. Koszul property and Frobenius splitting of Schubert varieties. Preprint
alg-geom/9502021.
[30] A. I. Bondal. Helices, representations of quivers and Koszul algebras. Helices and vector
bundles, 75-95, London Math. Soc. Lecture Note Ser., 148, Cambridge Univ. Press, Cambridge,
1990.
[31] A. I. Bondal, A. E. Polishchuk. Homological properties of associative algebras: the method
of helices. Russian Acad. Sci. Izvestiya Math. 42, #2, 219-260, 1994.
[32] E. Borel. Sur une application d'un theoreme de M. Hadamard. Bull. Sc. math., 2e serie,
t.18, 22-25, 1894.
[33] A. Braverman, D. Gaitsgory. The Poincare-Birkhoff-Witt theorem for quadratic algebras
of Koszul type. J. Algebra 181, #2, 315-328, 1996.
[34] W. Bruns, J. Gubeladze, Ngo Viet Trung. Normal polytopes, triangulations, and Koszul
algebras. J. Reine Angew. Math. 485, 123-160, 1997.
[35] B. Buchberger. Grobner bases: An algorithmic method in polynomial ideal theory.
Multidimensional Systems Theory (N. K. Bose ed.), 184-232, Reidel, Dordrecht, 1985.
[36] B. Buchberger, F. Winkler, eds., Grobner bases and applications, Cambridge Univ. Press,
1998.
[37] D. С Butler. Normal generation of vector bundles over a curve. J. Diff. Geom. 39, #1,
1-34, 1994.
[38] A. Conca. Grobner bases for spaces of quadrics of low codimension. Adv. in Appl. Math.
24, #2, 111-124, 2000.
[39] A. Conca, M. E. Rossi, G. Valla. Grobner flags and Gorenstein algebras. Compositio Math.
129, #1, 95-121, 2001.
[40] A. Conca, N. V. Trung, G. Valla. Koszul property for points in projective spaces. Math.
Scand. 89, #2, 201-216, 2001.
[41] A. Connes. Noncommutative differential geometry. Publ. Math. IHES 62, 257-360, 1985.
[42] A. A. Davydov. Totally positive sequences and Я-matric quadratic algebras. Journal of
Math. Sciences 100, #1, 1871-1876, 2000.
[43] V. G. Drinfeld. On quadratic quasi-commutational relations in quasi-classical limit. Mat.
Fizika, Funkc. Analiz, 25-34, "Naukova Dumka", Kiev, 1986. English translation in: Selecta
Math. Sovietica 11, #4, 317-326, 1992.
[44] L. Ein, R. Lazarsfeld. Syzygies and Koszul cohomology of smooth projective varieties of
arbitrary dimension. Inventiones Math. Ill, #1, 51-67, 1993.
[45] D. Eisenbud, S. Goto. Linear free resolutions and minimal multiplicity. J. Algebra 88, #1,
89-133, 1984.
[46] D. Eisenbud, A. Reeves, B. Totaro. Initial ideals, Veronese subrings, and rates of algebras.
Advances in Math. 109, #2, 168-187, 1994.
[47] P. Etingof, V. Ostrik. Module categories over representations of SLq{2) and graphs. Preprint
math.QA/0302130.
[48] B. Feigin, A. Odesskii. Sklyanin's elliptic algebras. Funct. Anal. Appl 23, #3, 207-214,
1990.
[49] B. Feigin, B. Tsygan. Cyclic homology of algebras with quadratic relations, universal
enveloping algebras, and group algebras. K-theory, arithmetic and geometry (Moscow, 1984~
1986), 210-239, Lecture Notes in Math. 1289, Springer, Berlin, 1987.
[50] M. Finkelberg, A. Vishik. The coordinate ring of general curve of genus ^ 5 is Koszul. J.
Algebra 162, 535-539, 1993.
[51] G. Floystad. Koszul duality and equivalences of categories, preprint math.RA/0012264.
[52] R. Froberg. Determination of a class of Poincare series. Math. Scand. 37, #1, 29-39, 1975.

BIBLIOGRAPHY
157
R. Froberg. A study of graded extremal rings and of monomial rings. Math. Scand. 51, #1,
22-34, 1982.
R. Froberg. Rings with monomial relations having linear resolutions. J. Pure Appl. Algebra
38, #2-3, 235-241, 1985.
R. Froberg. Koszul algebras. Advances in commutative ring theory (Fez, 1997), 337-350.
Dekker, New York, 1999.
R. Froberg, T. Gulliksen, C. Lofwall. Flat one-parameter family of Artinian algebras with
infinite number of Poincare series. Algebra, topology and their interactions (Stockholm, 1983),
170-191, Lecture Notes in Math. 1183, Springer, Berlin, 1986.
R. Froberg, C. Lofwall. Koszul homology and Lie algebras with appplication to generic forms
and points. Homology Homotopy Appl. 4, no. 2, 227-258, 2002.
W. Fulton, Intersection theory, Springer, 1998.
A. Gandolfi, M. Keane, V. de Valk. Extremal two-correlations of two-valued stationary
one-dependent processes. Probab. Theory Related Fields 80, #3, 475-480, 1989.
I. M. Gelfand, V. A. Ponomarev. Model algebras and representations of graphs. Fund. Anal
Appl. 13, 157-166, 1979.
V. Ginzburg, M. Kapranov. Koszul duality for operads. Duke Math. J. 76, #1, 203-272,
1994. Erratum in Duke Math. J. 80, #1, 293, 1995.
E. S. Golod. Standard bases and homology. Algebra—some current trends (Varna, 1986),
88-95, Lecture Notes in Math. 1352, Springer, Berlin, 1988.
E. S. Golod, I. R. Shafarevich. On the tower of class fields. Izvestiya Akad. Nauk. SSSR,
Ser. Mat. 28, #2, 261-272, 1964 (in Russian).
G. Gratzer. General lattice theory. Birkhauser, Basel, 1978.
M. Green, R. Lazarsfeld. A simple proof of Petri's theorem on canonical curves, Geometry
Today (Rome, 1984), 129-142, Progress in Math. 60, Birkhauser, Boston, 1985.
R. Hartshorne. Algebraic geometry. Springer, New York, 1977.
S. P. Inamdar, V. B. Mehta. Frobenius splitting of Schubert varieties and linear syzygies.
Amer. Joum. of Math. 116, #6, 1569-1586, 1994.
S. A. Joni, G. C. Rota. Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61,
#2, 93-139, 1979.
B. Jonsson. Distributive sublattices of a modular lattice. Proc. Amer. Math. Soc. 6, #5,
682-688, 1955.
P. Jorgensen, Linear free resolutions over non-commutative algebras. Compositio Math. 140,
#4, 1054-1058, 2004.
S. Karlin. Total positivity. Stanford Univ. Press, 1968.
G. Kempf. Some wonderful rings in algebraic geometry. J. Algebra 134, #1, 222-224, 1990.
G. Kempf. Syzygies for points in projective space. J. Algebra 145, #1, 219-223, 1992.
G. Kempf. Projective coordinate rings of abelian varieties. Algebra analysis, geometry, and
number theory (Baltimore, MD, 1988), 225-235, Johns Hopkins Univ. Press, Baltimore,
MD, 1989.
C. Lofwall. On the subagebra generated by the one-dimensional elements in the Yoneda
Ext-algebra. Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture
Notes in Math. 1183, 291-338, Springer, Berlin, 1986.
A. Malkin, V. Ostrik, M. Vybornov, Quiver varieties and Lusztig's algebra, preprint
math.RT/0403222.
Yu. I. Manin. Some remarks on Koszul algebras and quantum groups. Ann. Inst. Fourier
37, #4, 191-205, 1987.
Yu. I. Manin. Quantum groups and non-commutative geometry. CRM, Universite de
Montreal, 1988.
Yu. I. Manin. Topics in noncommutative geometry. Princeton Univ. Press, Princeton, 1991.
Yu. I. Manin. Notes on quantum groups and quantum de Rham complexes. Theoret. and
Math. Physics 92, #3, 997-1019, 1992.
R. Martinez-Villa. Applications of Koszul algebras: the preprojective algebra.
Representation Theory of Algebras (Cocoyoc, 1994), 487-504, CMS Conf. Proc, vol. 18, AMS,
Providence, RI, 1996.
J. P. May. The cohomology of augmented algebras and generalized Massey products for
DGA-algebras. Trans. Amer. Math. Soc. 122, 334-340, 1966.

158 BIBLIOGRAPHY
[83] Т. Mora. An introduction to commutative and noncommutative Grobner bases. Theoret.
Comput. Science 134, #1, 131-173, 1994.
[84] D. Mumford. Lectures on curves on an algebraic surface. Princeton Univ. Press, Princeton,
1966.
[85] D. Mumford. Varieties defined by quadratic relations, Questions on Algebraic Varieties
(C.I.M.E., III Ciclo, Varenna, 1969), 29-100, Edizioni Cremonese, Rome, 1970.
[86] R. Musti, E. Buttafuoco. Sui subreticoli distributivi dei reticoli modulari. Boll. Unione Mat.
Ital. (3) 11, #4, 584-587, 1956.
[87] P. Orlik, H. Terao. Arrangements of hyperplanes. Springer-Verlag, Berlin, 1992.
[88] S. Papadima, S. Yuzvinsky. On rational K[ir, 1] spaces and Koszul algebras. J. Pure Appl.
Algebra 144, #2, 157-167, 1999.
[89] G. Pareschi. Koszul algebras associated to adjunction bundles. J. Algebra 157, #1, 161-169,
1993.
[90] G. Pareschi, B. Purnaprajna. Canonical ring of a curve is Koszul: A simple proof. Illinois
J. of Math. 41, #2, 266-271, 1997.
[91] I. Peeva, V. Reiner, B. Sturmfels. How to shell a monoid. Math. Ann. 310, #2, 379-393,
1998.
[92] D. Piontkovskii. On the Hilbert series of Koszul algebras. Fund. Anal. Appl. 35, #2, 133-
137, 2001.
[93] D. Piontkovskii. Noncommutative Koszul filtrations. Preprint math.RA/0301233.
[94] D. Piontkovskii. Sets of Hilbert series and their applications. Preprint math.RA/0502149.
Fundam. Prikl. Mat. 10, 143-156, 2004.
[95] D. Piontkovskii. Algebras associated to pseudo-roots of noncommutative polynomials are
Koszul. Preprint math.RA/0405375.
[96] A. Polishchuk. On the Koszul property of the homogeneous coordinate ring of a curve. J.
Algebra 178, #1, 122-135, 1995.
[97] A. Polishchuk. Perverse sheaves on a triangulated space. Math. Res. Letters 4, #2-3, 191-
199, 1997.
[98] A. Polishchuk. Koszul configurations of points in projective spaces. Preprint math.AG/
0412441.
[99] L. Positselski. Nonhomogeneous quadratic duality and curvature. Fund. Anal. Appl. 27,
#3, 197-204, 1993.
[100] L. Positselski. Koszul inequalities and stochastic sequences. M. A. Thesis, Moscow State
University, 1993.
[101] L. Positselski. The correspondence between the Hilbert series of dual quadratic algebras
does not imply their having the Koszul property. Fund. Anal. Appl. 29, #3, 1995.
[102] L. Positselski. Koszul property and Bogomolov conjecture. Harvard Ph. D. Thesis, 1998,
available at http://www.math.uiuc.edu/K-theory/0296.
[103] L. Positselski, A. Vishik. Koszul duality and Galois cohomology. Math. Research Letters 2,
#6, 771-781, 1995.
[104] S. Priddy. Koszul resolutions. Trans. Amer. Math. Soc. 152, 39-60, 1970.
[105] J.-E. Roos. Homology of free loop spaces, cyclic homology, and non-rational Poncare-Betti
series in commutative algebra. Algebra—some current trends (Varna, 1986), 173-189,
Lecture Notes in Math. 1352, Springer, Berlin, 1988.
[106] J.-E. Roos. On the characterization of Koszul algebras. Four counter-examples. Comptes
Rendus Acad. Sci. Paris, ser. I, 321, #1, 15-20, 1995.
[107] J.-E. Roos. A description of the homological behavior of families of quadratic forms in four
variables. Syzygies and Geometry (Boston, 1995), 86-95, Northeastern Univ., Boston, MA,
1995.
[108] M. Rosso. An analogue of P.B.W. theorem and the universal Я-matrix for Uh,sl(N + 1).
Commun. Math. Phys. 124, 307-318, 1989.
[109] P. Schenzel. Uber die freien Auflosungen extremaler Cohen-Macauley-Ringe. J. Algebra 64,
93-101, 1980.
[110] A. Schwarz. Noncommutative supergeometry and duality. Lett. Math. Phys. 50, #4, 309-
321, 1999.
[Ill] S. Serconek, R. Wilson. The quadratic algebras associated with pseudo-roots of
noncommutative polynomials are Koszul algebras. J. Algebra 278, 473-493, 2004.

BIBLIOGRAPHY
159
В. Shelton, С. Tingey. On Koszul algebras and a new construction of Artin-Schelter regular
algebras. J. Algebra 241, 789-798, 2001.
B. Shelton, S. Yuzvinsky. Koszul algebras from graphs and hyperplane arrangements. J.
London Math. Soc. (2) 56, #3, 477-490, 1997.
J. Stasheff. Homotopy associativity of if-spaces, II. Trans. Amer. Math. Soc. 108, 293-312,
1963.
B. Sturmfels. Grobner bases and convex polytopes. University Lecture Series, 8. AMS,
Providence, RI, 1996.
B. Sturmfels. Four counterexamples in combinatorial algebraic geometry. J. Algebra 230,
#1, 282-294, 2000.
J. Tate, M. van den Bergh. Homological properties of Sklyanin algebras. Inventiones Math.
124, #1-3, 619-647, 1996.
B. Tsirelson. A new framework for old Bell inequalities. Helv. Phys. Acta 66, #7-8, 858-874,
1993.
V. A. Ufnarovski. Combinatorial and asymptotic methods in algebra. Algebra, VI, 1-196.
Encyclopaedia Math. Sci. 57, Springer, Berlin, 1995.
V. de Valk. A problem on 0-1 matrices. Compositio Math. 71, #2, 139-179, 1989.
V. de Valk. Hubert space representations of m-dependent processes. Ann. Probab. 21, #3,
1550-1570, 1993.
V. de Valk. One-dependent processes: two-block factors and non-two-block factors. Stichting
Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1994.
M. Van den Bergh. Noncommutative homology of some three-dimensional quantum spaces.
K-theory, 8, 213-230, 1994.
M. Van den Bergh. A relation between Hochschild homology and cohomology for Gorenstein
rings. Proc. AMS, 126. #4, 1345-1348, 1998.
A. M. Vershik. Algebras with quadratic relations. Spektr. teoriya operatorov i beskonech-
nomern. analiz, 32-57, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1984. English translation
in: Selecta Math. Sovietica 11, #4, 293-316, 1992.
M. Vybornov. Mixed algebras and quivers related to cell complexes and Koszul duality,
Math. Res. Letters 5, 675-683, 1998.
M. Vybornov. Sheaves on triangulated spaces and Koszul duality. Preprint math.AT/
9910150.