Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild Cohomology [PhD thesis]

178 Pages·2015·0.878 MB·English

by  Cris Negron

(cid:13)c Copyright 2015
Cris Negron
Alternate Approaches to the Cup Product and Gerstenhaber
Bracket on Hochschild Cohomology
Cris Negron
A dissertation
submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
University of Washington
2015
Reading Committee:
James Zhang, Chair
Max Lieblich
John Palmieri
Program Authorized to Offer Degree:
Mathematics
University of Washington
Abstract
Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild
Cohomology
Cris Negron
Chair of the Supervisory Committee:
Professor of Mathematics James Zhang
Mathematics Department
The Hochschild cohomology HH•(A) of an algebra A is a derived invariant of the algebra
which admits both a graded ring structure (called the cup product) and a compatible graded
Lie algebra structure (called the Gerstenhaber bracket). The Lie structure is particularly
important as it provides a means of addressing the deformation theory of the algebra A.
In this thesis we produce some new methods for analyzing the cup product and Gersten-
haber bracket on Hochschild cohomology. For the cup product we produce a number of new,
and rather fundamental, relations between the theories of twisting cochains and Hochschild
cohomology. In the case of a Koszul algebra A, our results imply that the Hochschild coho-
mology ring of A is a subquotient of the tensor product algebra A⊗A! of A with its Koszul
dual A!.
We also investigate the Hochschild cohomology of smash product algebras A∗G. (Here A
is an algebra equipped with an action of a Hopf algebra G.) In this setting, we produce new
methods for computing both the cup product and Gerstenhaber bracket. For the Gersten-
haber bracket in particular, we show that there is an intermediate cohomology H• (A∗G)
Int
which is a braided commutative algebra in the category of Yetter-Drinfeld modules over G,
admits a braided anti-commutative bracket [,] , and can be used to recover both the cup
YD
product and Gerstenhaber bracket on the standard Hochschild cohomology of A∗G.
TABLE OF CONTENTS
Page
0.1 An introduction to Hochschild cohomology and what it’s good for . . . . . . 9
0.2 More background and the specific contents of this thesis . . . . . . . . . . . 12
0.2.1 More on the cup product and the contents of Chapter 1: twisting
cochains and the cup product on Hochshcild cohomology for Koszul
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
0.2.2 Chapter 2: spectral sequences for the cohomology rings of a smash
product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
0.2.3 Chapter 3: braided structures and Hochschild cohomology . . . . . . 21
0.3 The Hochschild cochain complex, dg Lie algebras, and formal deformations . 24
0.4 A list of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 1: Twisting Cochains, the Cup Product on Hochschild Cohomology, and
Koszul Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.1.1 Our cup product versus Gerstenhabers cup product and the Gersten-
haber bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.2 Reminders on dg algebras, dg coalgebras, and filtered Koszul algebras . . . . 38
1.2.1 Dg algebras and coalgebras . . . . . . . . . . . . . . . . . . . . . . . 38
1.2.2 Graded Koszul duality with signs . . . . . . . . . . . . . . . . . . . . 39
1.2.3 (Augmented) filtered Koszul algebras . . . . . . . . . . . . . . . . . . 42
1.2.4 Nonaugmented filtered Koszul algebras and curved dg structures . . . 43
1.2.5 Example: The nth Weyl algebra . . . . . . . . . . . . . . . . . . . . . 46
1.2.6 Example: PBW deformations of skew polynomial rings . . . . . . . . 46
1.3 Twisting cochains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.1 Twisting cochains in the curved/nonaugmented case . . . . . . . . . . 52
1.4 Koszul resolutions via twisting cochains . . . . . . . . . . . . . . . . . . . . . 54
1.5 A complex calculating HH•(A,M) . . . . . . . . . . . . . . . . . . . . . . . 56
1.5.1 Example continued: PBW deformations of skew polynomials . . . . . 60
i
1.6 The cup product on HH•(A) . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.7 A Remark on A -Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
∞
1.8 Hochschild cohomology of dg algebras and the complete proof of Theorem 1.32 67
1.9 Example: the Heisenberg Lie algebra . . . . . . . . . . . . . . . . . . . . . . 71
1.9.1 The dg algebra U!⊗U and HH0(U) . . . . . . . . . . . . . . . . . . . 73
(cid:101)
1.9.2 Calculating HH1(U) . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
1.9.3 Calculating HH2(U) and HH3(U) . . . . . . . . . . . . . . . . . . . 77
1.9.4 A multiplication table for HH•(U) . . . . . . . . . . . . . . . . . . . 79
Chapter 2: Spectral Sequences for the Cohomology Rings of a Smash Product . . 82
2.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.2 A Hopf bimodule resolution of G . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.3 Bimodule resolutions of A∗G via a smash product construction . . . . . . . 89
2.4 Hochschild cochains as derived invariants . . . . . . . . . . . . . . . . . . . . 93
2.5 ReminderofthecupproductsonHochschildcohomologyandderivedinvariant
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2.6 Hochschild cohomology as a derived invariant algebra . . . . . . . . . . . . . 100
2.7 Algebras of extensions as derived invariant algebras . . . . . . . . . . . . . . 108
2.8 Two Examples With Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.8.1 Example 1: Zn acting on An by translation . . . . . . . . . . . . . . 115
k
2.8.2 Example 2: The Untwisted Skew Polynomial Ring . . . . . . . . . . . 122
Chapter 3: Braided Structures and Hochschild Cohomology . . . . . . . . . . . . . 131
3.1 Algebras over a separable base and the intermediate cohomology . . . . . . . 132
3.1.1 When S = G and R = A∗G: the complex C• (A∗G) and its G-coaction133
Int
3.2 The Yetter-Drinfeld structure on C• (A∗G), naive bracket, and braided com-
Int
mutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.2.1 A reconsideration of the complex C• (A∗G) . . . . . . . . . . . . . . 139
Int
3.2.2 Identities for the naive bracket: braided commutativity of H• (A∗G) 146
Int
3.3 (In)finiteness of the square braiding on YDG . . . . . . . . . . . . . . . . . . 149
G
3.3.1 The Drinfeld double . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.3.2 The exponent of a Hopf algebra and the order of L: results from
Etingof and Gelaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.4 The Yetter-Drinfeld bracket [,] on C• (A∗G) . . . . . . . . . . . . . . . . 153
YD Int
ii
3.4.1 A remark on characteristic . . . . . . . . . . . . . . . . . . . . . . . . 154
3.4.2 The Yetter-Drinfeld bracket . . . . . . . . . . . . . . . . . . . . . . . 155
3.5 Interpretations of the intermediate cohomology in low degree . . . . . . . . . 160
3.5.1 Braided derivations and H1 (A∗G) . . . . . . . . . . . . . . . . . . . 160
Int
3.5.2 Quantum deformations and H2 (A∗G) . . . . . . . . . . . . . . . . . 165
Int
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
—– acknowledgments
iii
ACKNOWLEDGMENTS
I should thank, first and foremost, David Negron and Faye Negron. I find it difficult to
explain how essential their many sacrifices were to my existence here as a mathematician. I
would also like to thank Lance Negron, because he is my brother!
As far as the production of this thesis is concerned, I am happily indebted to my advisor
James Zhang. His dedication to his graduate students, the international noncommutative
community, and his family is both admirable and inspirational. I would like to thank Chelsea
Walton, Max Lieblich, and Sarah Witherspoon, who’s advice and assistance have played a
fundamental role in my development as a fledgling mathematician. Thanks also to Guangbin
Zhuang and Xingting Wang for listening to me talk for hours, and hours, and hours about
any number of homological gadgets. Many of the topics I’ve written about here were first
introduced to me either by Guangbin or Xingting, and without them I would not have
developed nearly the same breadth of understanding that I have today (any perceived lack
of depth is, of course, my own fault :P). In this same regard, I owe a thanks to Bharathwaj
Palvannan.
I also have to thank Marcel Purnell, Gregory E. Whiting, Kristina Clark, Danielle Fumia,
Iris Viveros Avendan˜o, Isolynn Dean, and Jamaal Trey Songs Jackson. They have all made
Seattle, a naturally enraging and perplexing local, tolerable and even enjoyable. Thanks
finally to Daraka Gardner, who has informed me that dg categories are actually named after
him!
iv
DEDICATION
To Aiyana Jones, Rekia Boyd, and Freddie Gray.
v
9
INTRODUCTION AND THE SECRETS WITHIN
0.1 An introduction to Hochschild cohomology and what it’s good for
Let k be a field of arbitrary characteristic. The Hochschild cohomology of a (k-)algebra is
a derived invariant A (cid:55)→ HH•(A) which produces the center in degree 0, classifies (outer)
derivations on A in degree 1, infinitesimal deformations of A in degree 2, and obstructions
to lifting deformations in degree 3. This cohomology was first introduced by Hochschild [30]
and then popularized in the many works of Gerstenhaber, including the foundational papers
[18, 19]. Formally, we can define the cohomology as the graded extension group
HH•(A) = Ext• (A,A)
A-bimod
with its standard Yoneda product. The product on HH•(A) is often referred to as the cup
product. (It is shown that the cup product and Yoneda product agree at [9, Proposition
1.1].)
Hochschild cohomology also carries a, somewhat mysterious, graded Lie structure which
is compatible with the cup product. The Hochschild cohomology, or more specifically the
Hochschild cochain complex, along with its Lie structure controls the deformation theory of
A(inthegeneralsenseofdeformationtheoryviadgLiealgebras). Thispointiselaboratedon
in Section 0.3. The graded Lie bracket on Hochschild cohomology is called the Gerstenhaber
bracket, and the compatibility condition between the cup product and Gerstenhaber bracket
requires that for each f ∈ HHi(A) the operation [f,−] is a graded degree (i−1) derivation
on the graded ring HH•(A) (see Section 1.1.1).
One can also consider the Hochschild cohomology HH•(A,M) with “coefficients” in
a bimodule M. This cohomology is, again, given by bimodule extensions HH•(A,M) =
Much of this material is based upon work supported by the National Science Foundation Graduate
Research Fellowship under Grant No. DGE-1256082.
10
Ext• (A,M), and when M = B is an algebra extension of A the associated Hochschild
A-bimod
cohomology has a canonical graded ring structure. The cohomology rings HH•(A,B), when
B is something other than A, can be of some use, and in fact will be a principle object
of study in the latter two chapters of this dissertation. There is also an obvious notion
of Hochschild homology, which is defined using Tor groups. In the case of a d-Calabi-Yau
algebraAtheHochschildcohomologyandhomologyarerelatedbya(VandenBergh)duality
HH•(A) →∼= HH (A) [87].
d−•
The simplest example of Hochschild cohomology is provided by the famed Hochschild-
Kostant-Rosenberg theorem [32]. They state that for a smooth affine scheme X, with global
functions k[X], we have an isomorphism
(cid:94)•
HH (A) = Ω , where Ω is the module of global Kahler differentials
• X X
k[X]
and a cohomological version of the theorem provides an isomorphism
(cid:94)•
HH•(A) = T , where T is the module of global vector fields.
X X
k[X]
Here the ring structure is, as the notation suggests, just the exterior algebra generated by
T , and the Lie structure is induced by the Lie bracket on vector fields. As a slightly more
X
interesting example, when X is complex affine space and G is a finite group acting on X,
then the Hochschild cohomology of the skew group ring k[X]∗G is given by
HH•(k[X]∗G) = (cid:0)⊕ (cid:94)•−codim(Xg)T (cid:1)G. (0.1.1)
g∈G Xg
k[Xg]
Here Xg is the subscheme fixed by g ∈ G [16]. The same formula holds when X is a
symplectic manifold and G acts by symplectic automorphisms, and the result was originally
provided in this setting [22, proof of Proposition 6.2].
Hochschild cohomology is related to a number of other invariants and important math-
ematical structures. For a rather deep example one can consider cyclic (co)homology, both
the standard version HC and negative version HC−. This (co)homology plays a principle
role in Connes’ theory of noncommutative geometry [13].