{A∞-bimodules A-infinity-bimodules and Serre {A∞-functors [A-infinity-functors] [Elektronische Ressource] / Oleksandr Manzyuk
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{A∞-bimodules A-infinity-bimodules and Serre {A∞-functors [A-infinity-functors] [Elektronische Ressource] / Oleksandr Manzyuk

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A -bimodules and Serre A -functors∞ ∞Oleksandr ManzyukVom Fachbereich Mathematik der Technischen Universit¨at Kaiserslauternzur Verleihung des akademischen Grades Doktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation1. Gutachter: Prof. Dr. Gert-Martin Greuel2. Gutachter: Prof. Dr. Bernhard KellerVollzug der Promotion 25.10.2007D 386AbstractThis dissertation is intended totransport the theory ofSerre functors into the contextof A -categories. We begin with an introduction to multicategories and closed multi-∞categories, which form a framework in which the theory of A -categories is developed.∞We prove that (unital) A -categories constitute a closed symmetric multicategory. We∞define the notion of A -bimodule similarly to Tradler and show that it is equivalent to∞anA -functor of two arguments which takes values in the differential graded category of∞complexes ofk-modules, wherek is a commutative ground ring. Serre A -functors are∞defined via A -bimodules following ideas of Kontsevich and Soibelman. We prove that∞a unital closed under shifts A -category A over a fieldk admits a Serre A -functor if∞ ∞0and only if its homotopy categoryH A admits an ordinary Serre functor. The proof usescategories enriched in K, the homotopy category of complexes ofk-modules, and SerreK-functors. Another important ingredient is an A -version of the Yoneda Lemma.

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Published 01 January 2007
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A -bimodules and Serre A -functors∞ ∞
Oleksandr Manzyuk
Vom Fachbereich Mathematik der Technischen Universit¨at Kaiserslautern
zur Verleihung des akademischen Grades Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation
1. Gutachter: Prof. Dr. Gert-Martin Greuel
2. Gutachter: Prof. Dr. Bernhard Keller
Vollzug der Promotion 25.10.2007
D 386Abstract
This dissertation is intended totransport the theory ofSerre functors into the context
of A -categories. We begin with an introduction to multicategories and closed multi-∞
categories, which form a framework in which the theory of A -categories is developed.∞
We prove that (unital) A -categories constitute a closed symmetric multicategory. We∞
define the notion of A -bimodule similarly to Tradler and show that it is equivalent to∞
anA -functor of two arguments which takes values in the differential graded category of∞
complexes ofk-modules, wherek is a commutative ground ring. Serre A -functors are∞
defined via A -bimodules following ideas of Kontsevich and Soibelman. We prove that∞
a unital closed under shifts A -category A over a fieldk admits a Serre A -functor if∞ ∞
0and only if its homotopy categoryH A admits an ordinary Serre functor. The proof uses
categories enriched in K, the homotopy category of complexes ofk-modules, and Serre
K-functors. Another important ingredient is an A -version of the Yoneda Lemma.∞Acknowledgements
I have been fortunate to have two advisors of this dissertation. I would like to thank
Volodymyr Lyubashenko, who taught me everything I know about A -categories and∞
guided my research. I am grateful to Gert–Martin Greuel for his continued enthusiasm,
encouragement, and support. Both advisors helped writing immensely, and I amthankful
for their patience and wisdom.
My research benefited fromnumerous discussions with the participants oftheA -cat-∞
egory seminar held atthe Institute ofMathematics in Kyiv, Ukraine, especially with Yuri
Bespalov and Sergiy Ovsienko.Contents
Chapter 0. Introduction 1
0.1. Motivation 1
0.2. Notation and conventions 3
0.3. Chapter synopsis 4
Chapter 1. Tools 7
1.1. Lax Monoidal categories 8
1.2. Multicategories 22
1.3. Closed multicategories 47
Chapter 2. Serre functors for enriched categories 71
2.1. Preliminaries on enriched categories 71
2.2. Basic properties of Serre functors 75
2.3. Serre functors and base change 86
Chapter 3. A -categories 93∞
3.1. Graded modules and complexes 94
3.2. A -categories andA -functors 98∞ ∞
3.3. A -categories of A -functors 112∞ ∞
3.4. Unital A -categories 118∞
3.5. Opposite A -categories 135∞
Chapter 4. A -bimodules 141∞
4.1. Definitions 142
4.2. Regular A -bimodule 158∞
4.3. Operations onA -bimodules 161∞
4.4. Unital A -bimodules 166∞
4.5. A -modules 169∞
Chapter 5. Serre A -functors 173∞
5.1. Basic properties of Serre A -functors 173∞
5.2. The strict case of Serre A -functors 176∞
Appendix A. The Yoneda Lemma 179
Bibliography 217
List of symbols 221
Index 223
vCHAPTER 0
Introduction
0.1. Motivation
It is widely accepted that Verdier’s notion of triangulated category is not quite satis-
factory. It suffers from numerous deficiencies, starting from non-functorial cone up to the
failure of descent for the derived category of quasi-coherent sheaves on a scheme, see e.g.
[58]. As a remedy, Bondal and Kapranov [5] introduced the notion of pretriangulated
differential graded (dg) category that enhances the notion of triangulated category. In
thparticular, the 0 cohomology (called also the homotopy category) of a pretriangulated
dg category admits a natural triangulated structure. Remarkably, triangulated categories
arising in algebraic geometry and representation theory are of that kind. It is therefore
desirable to develop the relevant homological algebra at the level of pretriangulated dg
categories rather than at the level of triangulated categories. Drinfeld gave in [13] an ex-
plicit construction (implicitly present in Keller’s paper [25]) ofa quotient ofdg categories
and proved that it is compatible with Verdier’s quotient of triangulated categories.
The notion of dg category is a particular case of a more general and more flexible
notion ofA -category. The notion of pretriangulated dg category generalizes toA -cat-∞ ∞
egories. It is being developed independently by Kontsevich and Soibelman [32] and by
Bespalov, Lyubashenko, and the author [3]. Consequently, there are attempts to rewrite
homological algebra using A -categories. Lyubashenko and Ovsienko extended in [42]∞
Drinfeld’s construction of quotients to unital A -categories. In [39], the author jointly∞
with Lyubashenko constructed another kind of a quotient and proved that it enjoys a
certain universal property. We also proved that both constructions of quotients agree,
i.e., produce A -equivalent unital A -categories. In [3], it is proven that the homotopy∞ ∞
category of a pretriangulated A -category is triangulated. Furthermore, the quotient of∞
a pretriangulated A -category over a pretriangulated A -subcategory is again pretrian-∞ ∞
gulated.
The reasons towork withA -categories ratherthan with dg categories are the follow-∞
ing. On the one hand, it is the mirror symmetry conjecture of Kontsevich. In one of its
versions, it asserts that the homotopy category of the Fukaya A -category, constructed∞
from the symplectic structure of a Calabi–Yau manifold, is equivalent to the derived cat-
egory of coherent sheaves on a dual complex algebraic variety. The construction of the
Fukaya A -category is not a settled question yet, it is a subject of current research, see∞
e.g. Seidel [48]. However, it is clear that the Fukaya A -category is in general not a dg∞
category. On the other hand, the supply of dg functors between dg categories is not suf-
ficient for the purposes of theory. Instead of extending the class of dg functors to a wider
class of (unital) A -functors, some authors prefer to equip the category of dg categories∞
with a suitable model structure and to work in the homotopy category of dg categories.
This approach is being developed, e.g., by Tabuada [54, 53] and To¨en [57]. There are
evidences that both approaches may be equivalent, at least if the ground ring is a field.
However, the precise relation is not yet clear to the author.
12 0. INTRODUCTION
The goal of this dissertation is to transport the theory of Serre functors into the
context of A -categories.∞
The notion of Serre functor was introduced by Bondal and Kapranov [4], who used
it to reformulate Serre-Grothendieck duality for coherent sheaves on a smooth projective
variety as follows. Let X be a smooth projective variety of dimension n over a field k.
bDenote by D (X) the bounded derived category of coherent sheaves on X. Then there
exists an isomorphism of k-vector spaces
• • • • ∗∼Hom b (F ,G ⊗ω [n]) = Hom b (G ,F ) ,XD (X) D (X)
• • bnatural in F ,G ∈ D (X), where ω is the canonical sheaf, and ∗ denotes the dualX
k-vector space. IfF andG are sheaves concentrated in degrees i and 0 respectively, the
above isomorphism specializes to the familiar form of Serre duality:
n−i i ∗∼Ext (F,G ⊗ω ) = Ext (G,F) .X
In general, a k-linear functor S from a k-linear categoryC to itself is a Serre functor if
there exists an isomorphism of k-vector spaces
∗∼C(X,SY) =C(Y,X) ,
natural inX,Y ∈ ObC. A Serre functor, if it exists, is determined uniquely up to natural
bisomorphism. In the above example, C =D (X) and S =−⊗ω [n]. Being an abstractX
categorytheorynotion,Serrefunctorshavebeendiscoveredinothercontexts,forexample,
in Kapranov’s studies of constructible sheaves on stratified spaces [24]. The existence of
a Serre functor imposes strong restrictions on the category. For example, Reiten and van
den Bergh have shown that Serre functors in abelian categories of modules are related to
Auslander–Reiten sequences and triangles, and they classified the noetherian hereditary
Ext-finite abelian categories with Serre duality [47]. Serre functors play an important
role in reconstruction of a variety from its derived category of coherent sheaves [6]. An-
other rapidly developing area where Serre functors find applications is non-commutative
geometry.
The idea of non-commutative geometry that goes back to Grothendieck is that cate-
gories should be thought of as non-commutative counterparts of geometric objects. For
example, the derived category of quasi-coherent sheaves on a scheme X reflects a great
deal of geometric properties of X. The general philosophy suggests to forget about the
scheme itself and to work with its derived category of quasi-coherent sheaves. Thus, in-
stead ofdefining what a non-commutative scheme is, non-commutative geometry declares
an arbitrary (sufficiently nice) triangulated category to be the derived category of quasi-
coherent sheaves on a non-commutative scheme. In the spirit of the agenda explained
above, Keller in the talk at ICM 2006and Kontsevich and Soibelman in [31] suggested to
consider pretriangulated dg categories resp. A -categories as non-commutative schemes.∞
Then, to a (commutative) scheme X its derived dg category D (X) is associated. Bydg
definition,D (X)is Drinfeld’squotient ofthedgcategoryofcomplexes ofquasi-coherentdg
sheaves on X modulo the dg subcategory of acyclic complexes. Its homotopy category is
the ordinary derived category of X. Geometric properties of the scheme X (smoothness,
properness etc.) correspond to certain algebraic properties of its derived dg category.
Abstracting these yields a definition of smooth, proper etc. non-commutative schemes.
This approach is being actively developed by To¨en, see e.g. [57] and [56].
In non-commutative geometry, triangulated categories admitting a Serre functor (and
satisfying some further conditions) are considered as non-commutative analogs of smooth
projective varieties. Modern homological algebra insists on working with pretriangulated