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Secondary Hochschild and Cyclic (Co)Homologies SECONDARY HOCHSCHILD AND CYCLIC (CO)HOMOLOGIES Jacob Laubacher A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2017 Committee: Mihai D. Staic, Advisor John Laird, Graduate Faculty Representative Xiangdong Xie Juan Bes` Copyright c May 2017 Jacob Laubacher All rights reserved iii ABSTRACT Mihai D. Staic, Advisor Hochschild cohomology was originally introduced in 1945. Much more recently in 2013 a gen- eralization of this theory, the secondary Hochschild cohomology, was brought to light. In this dis- sertation we provide the details behind the simplicial structure for the chain complexes associated to the (secondary) Hochschild (co)homology. For this we introduce the notion of simplicial alge- bras and simplicial modules. The key results are two lemmas (3.4.1 and 3.4.2) that can be thought of as analogues of the Tor and Ext functors in the context of simplicial modules. It was a pleasant surprise that the higher order Hochschild homology over the 2-sphere can also be described using simplicial structures. We study some other related concepts like the secondary Hochschild and cyclic homologies associated to the triple (A; B; "), as well as some of their properties. iv ACKNOWLEDGMENTS I would first like to thank my parents, KC and Laura, for distilling in me the notion to do what I love. Without them I would be lost and miserable, albeit a bit richer. I would like to thank my siblings Rachael, Gretchen, and Seth for their patience and tolerance for a brother who was inter- minably in school, as well as Kathy for putting up with me during this final stretch. My colleagues also deserve a heartfelt thanks for their support and collaboration: Dave Walmsley, Mark Medwid, Sam Carolus, John Haman, Todd Romutis, and Rob Kelvey. I would also like to thank my commit- tee members Xiangdong Xie, Juan Bes,` and John Laird for their helpful comments, feedback, and suggestions. Finally, a special thanks to my advisor Mihai Staic. The task of mathematical research seemed daunting, yet with his careful guidance and instruction, he navigated me to a dissertation worthy of the title that I have pursued for so long. v TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION . 1 CHAPTER 2 PRELIMINARIES . 4 2.1 Notations and Conventions . 4 2.2 Basic Homological Algebra . 4 2.3 The Simplicial Category . 7 2.4 Bicomplexes . 9 2.5 Hochschild (Co)homology . 13 2.6 Cyclic (Co)homology . 17 2.7 Secondary Hochschild Cohomology . 18 2.8 Higher Order Hochschild (Co)homology . 21 CHAPTER 3 SIMPLICIAL STRUCTURES . 23 3.1 Simplicial Algebras . 23 3.1.1 Definition . 23 3.1.2 Examples of Simplicial Algebras . 24 3.2 Simplicial Modules . 29 3.2.1 Definition . 29 3.2.2 Examples of Simplicial Modules . 29 3.3 Co-simplicial Modules . 54 3.3.1 Definition . 54 3.3.2 Examples of Co-simplicial Modules . 55 3.4 The Hom and Tensor Lemmas . 62 vi CHAPTER 4 APPLICATIONS OF THE HOM AND TENSOR LEMMAS . 68 4.1 Hochschild (Co)homology Examples . 68 4.2 Higher Order Hochschild Homology Examples . 75 4.2.1 Higher Order Hochschild Homology Over the Circle . 76 4.2.2 Higher Order Hochschild Homology Over the Sphere . 79 4.2.3 A Special Case Over the Sphere . 113 CHAPTER 5 FOUNDATIONS OF THE SECONDARY CYCLIC HOMOLOGY . 117 5.1 Definition and Construction . 117 5.1.1 The Secondary Connes’ Complex . 122 5.1.2 The Secondary Cyclic Bicomplex . 129 5.1.3 The Secondary Diagonal Bicomplex . 132 5.2 Proofs of the Equivalences . 137 5.3 The Connes’ Long Exact Sequence for the Secondary Case . 142 5.3.1 Proof 1 . 142 5.3.2 Proof 2 . 145 5.3.3 Proof 3 . 147 CHAPTER 6 PROBLEMS FOR FUTURE RESEARCH . 149 BIBLIOGRAPHY . 150 1 CHAPTER 1 INTRODUCTION Hochschild introduced the cohomology bearing his name in 1945 in [21]. His main motivation was to study extensions of associative algebras over a field k. Employing the bar resolution B(A), one can show that the Hochschild cohomology of an associative k-algebra A with coefficients in the A-bimodule M is n n n H (A; M) := H (HomA⊗Aop (B(A);M)) = ExtA⊗Aop (A; M): In 1964, Gerstenhaber made a connection between Hochschild cohomology and deformation theory. The main result from [16] (Theorem 2.5.7) determines the obstruction for the multiplication law 2 3 mt(a ⊗ b) = ab + c1(a ⊗ b)t + c2(a ⊗ b)t + c3(a ⊗ b)t + ··· to be associative. The Hochschild homology can be defined in a similar manner. More precisely, A⊗Aop Hn(A; M) := Hn(M ⊗A⊗Aop B(A)) = Torn (A; M): Notice again the use of the bar resolution. A special case is obtained by taking M = A, and this is denoted HHn(A) := Hn(A; A). The Hochschild homology associated to A has many nice properties, and under certain conditions it is a good substitute for the module of differential forms n ΩAjk. In 1962 this fact was made clear through the celebrated Hochschild-Kostant-Rosenberg Theorem (see [22]). Connes first introduced cyclic homology in 1981 in [8] (for detailed construction and further reading we refer to [24], [25], [28], and [42]). Cyclic homology has close ties to the de Rham cohomology and differential forms. For more on this, one can see [1], [28], and [32]. Most 2 importantly, the Hochschild and cyclic homologies were united in Connes’ long exact sequence: B∗ I∗ S∗ B∗ I∗ ::: −−! HHn(A) −−! HCn(A) −−! HCn−2(A) −−! HHn−1(A) −−! ::: There is an analogous long exact sequence for the cohomology, and both have proved useful for computations. In 2013 Staic introduced the secondary Hochschild cohomology of the triple (A; B; ") with coefficients in M in [39]. Here A is an associative k-algebra, B a commutative k-algebra, and " : B −! A a morphism of k-algebras such that "(B) ⊆ Z(A) (i.e., A is a B-algebra). We now let M be an A-bimodule which is B-symmetric (that is, m"(α) = "(α)m). We define n ⊗n ⊗ n(n−1) C ((A; B; "); M) = Homk(A ⊗ B 2 ;M), and in [39] it was shown that there exists a • " n complex (C ((A; B; "); M); δn). Its cohomology is denoted H ((A; B; "); M) and is called the secondary Hochschild cohomology of the triple (A; B; ") with coefficients in M. The secondary Hochschild cohomology is used to study deformations A[[t]] for a B-algebra A. More properties of this new cohomology were seen in [11] and [40], including relations to the higher order Hochschild cohomology, cup and bracket products (establishing a Gerstenhaber algebra structure), as well as a Hodge-type decomposition (for characteristic zero). In [27] we introduce the notion of simplicial algebras and simplicial modules. In particular, we showed how they can be used to describe the secondary Hochschild cohomology. Using simpli- cial modules allows one to vary the module of coefficients, which was the main impediment for describing Hn((A; B; "); M) as an Ext functor. The main ingredient in our construction is the so-called secondary bar resolution (i.e., the simplicial left module B(A; B; ")). On top of its original use for the secondary cohomology, B(A; B; ") was employed to define a secondary homology, as well as cyclic counterparts. The working force behind these descriptions are Lemma 3.4.1 and Lemma 3.4.2. In the case where B = k, our results are essentially equivalent to the classical description of the Hochschild coho- mology. 3 Here is a short summary of the results in this dissertation. Chapter 2 is dedicated to introducing the necessary material so as to keep this dissertation as self-contained as possible. Chapter 3 focuses on the results in [27]. Whereas in a paper the details can be lacking, here we give all sufficient reasoning and justification. In [27] we studied the simplicial structure of the complex C•((A; B; "); M), which is associated to the secondary Hochschild cohomology of the triple (A; B; ") with coefficients in M. We were able to define a secondary cyclic (co)homology as well as study some of its properties. In Chapter 4 we present a few more applications to the above-mentioned lemmas. Besides the Hochschild and secondary Hochschild examples that were presented in [27] (Section 4.1), we also give a presentation of the higher order Hochschild homology (Section 4.2). Chapter 5 is devoted to establishing the secondary cyclic homology associated to the triple (A; B; "). This was introduced in [27], where we also proved the existence of a Connes-like long exact sequence concerning the secondary Hochschild and cyclic homology: B∗ I∗ S∗ B∗ I∗ ::: −−! HHn(A; B; ") −! HCn(A; B; ") −−! HCn−2(A; B; ") −−! HHn−1(A; B; ") −! ::: Here we prove this in three different ways, one of which is done without using bicomplexes (under the condition that k contains Q). Chapter 6 lists a few problems and possible research directions that give avenues for future work. 4 CHAPTER 2 PRELIMINARIES In this chapter we define and present all material deemed necessary for the remaining chapters. The aim is to keep this dissertation as self-contained as possible. 2.1 Notations and Conventions For this dissertation we fix k to be a field. Furthermore, all tensor products are over k unless otherwise stated (that is, ⊗ = ⊗k). We fix the following: A is an associative k-algebra and M is an A-bimodule. We fix B to be a commutative k-algebra and " : B −! A a morphism of k-algebras such that "(B) ⊆ Z(A).
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