An Introduction to Hochschild Cohomology

Description

This book gives a thorough and self-contained introduction to the theory of Hochschild cohomology for algebras and includes many examples and exercises. The book then explores Hochschild cohomology as a Gerstenhaber algebra in detail, the notions of smoothness and duality, algebraic deformation theory, infinity structures, support varieties, and connections to Hopf algebra cohomology. Useful homological algebra background is provided in an appendix. The book is designed both as an introduction for advanced graduate students and as a resource for mathematicians who use Hochschild cohomology in their work. Cover 1 Title page 2 Introduction 8 Chapter 1. Historical Definitions and Basic Properties 14 1.1. Definitions of Hochschild homology and cohomology 14 1.2. Interpretation in low degrees 22 1.3. Cup product 26 1.4. Gerstenhaber bracket 29 1.5. Cap product and shuffle product 33 1.6. Harrison cohomology and Hodge decomposition 35 Chapter 2. Cup Product and Actions 38 2.1. From cocycles to chain maps 38 2.2. Yoneda product 40 2.3. Tensor product of complexes 45 2.4. Yoneda composition and tensor product of extensions 48 2.5. Actions of Hochschild cohomology 51 Chapter 3. Examples 58 3.1. Tensor product of algebras 58 3.2. Twisted tensor product of algebras 65 3.3. Koszul complexes and the HKR Theorem 71 3.4. Koszul algebras 75 3.5. Skew group algebras 83 3.6. Path algebras and monomial algebras 87 Chapter 4. Smooth Algebras and Van den Bergh Duality 92 4.1. Dimension and smoothness 92 4.2. Noncommutative differential forms 96 4.3. Van den Bergh duality and Calabi-Yau algebras 101 4.4. Skew group algebras 104 4.5. Connes differential and Batalin-Vilkovisky structure 107 Chapter 5. Algebraic Deformation Theory 112 5.1. Formal deformations 112 5.2. Infinitesimal deformations and rigidity 117 5.3. Maurer-Cartan equation and Poisson bracket 121 5.4. Graded deformations 123 5.5. Braverman-Gaitsgory theory and the PBW Theorem 125 Chapter 6. Gerstenhaber Bracket 130 6.1. Coderivations 131 6.2. Derivation operators 134 6.3. Homotopy liftings 138 6.4. Differential graded coalgebras 144 6.5. Extensions 149 Chapter 7. Infinity Algebras 154 7.1. 𝐴_{∞}-algebras 154 7.2. Minimal models 158 7.3. Formality and Koszul algebras 161 7.4. 𝐴_{∞}-center 162 7.5. 𝐿_{∞}-algebras 165 7.6. Formality and algebraic deformations 168 Chapter 8. Support Varieties for Finite-Dimensional Algebras 172 8.1. Affine varieties 173 8.2. Finiteness properties 175 8.3. Support varieties 180 8.4. Self-injective algebras and realization 183 8.5. Self-injective algebras and indecomposable modules 186 Chapter 9. Hopf Algebras 194 9.1. Hopf algebras and actions on rings 194 9.2. Modules for Hopf algebras 198 9.3. Hopf algebra cohomology and actions 203 9.4. Bimodules and Hochschild cohomology 209 9.5. Finite group algebras 215 9.6. Spectral sequences for Hopf algebras 218 Appendix A. Homological Algebra Background 224 A.1. Complexes 224 A.2. Resolutions and dimensions 227 A.3. Ext and Tor 231 A.4. Long exact sequences 234 A.5. Double complexes 237 A.6. Categories, functors, derived functors 239 A.7. Spectral sequences 243 Bibliography 248 Index 260 Back Cover 265