| Introduction |
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vii | |
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Chapter 1 Historical Definitions and Basic Properties |
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1 | (24) |
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1.1 Definitions of Hochschild homology and cohomology |
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1 | (8) |
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1.2 Interpretation in low degrees |
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9 | (4) |
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13 | (3) |
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16 | (4) |
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1.5 Cap product and shuffle product |
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20 | (2) |
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1.6 Harrison cohomology and Hodge decomposition |
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22 | (3) |
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Chapter 2 Cup Product and Actions |
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25 | (20) |
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2.1 From cocycles to chain maps |
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25 | (2) |
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27 | (5) |
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2.3 Tensor product of complexes |
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32 | (3) |
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2.4 Yoneda composition and tensor product of extensions |
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35 | (3) |
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2.5 Actions of Hochschild cohomology |
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38 | (7) |
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45 | (34) |
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3.1 Tensor product of algebras |
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45 | (7) |
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3.2 Twisted tensor product of algebras |
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52 | (6) |
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3.3 Koszul complexes and the HKR Theorem |
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58 | (4) |
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62 | (8) |
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70 | (4) |
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3.6 Path algebras and monomial algebras |
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74 | (5) |
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Chapter 4 Smooth Algebras and Van den Bergh Duality |
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79 | (20) |
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4.1 Dimension and smoothness |
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79 | (4) |
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4.2 Noncommutative differential forms |
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83 | (5) |
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4.3 Van den Bergh duality and Calabi-Yau algebras |
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88 | (3) |
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91 | (3) |
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4.5 Connes differential and Batalin-Vilkovisky structure |
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94 | (5) |
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Chapter 5 Algebraic Deformation Theory |
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99 | (18) |
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99 | (5) |
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5.2 Infinitesimal deformations and rigidity |
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104 | (4) |
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5.3 Maurer-Cartan equation and Poisson bracket |
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108 | (2) |
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110 | (2) |
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5.5 Braverman-Gaitsgory theory and the PBW Theorem |
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112 | (5) |
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Chapter 6 Gerstenhaber Bracket |
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117 | (24) |
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118 | (3) |
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121 | (4) |
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125 | (6) |
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6.4 Differential graded coalgebras |
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131 | (5) |
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136 | (5) |
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Chapter 7 Infinity Algebras |
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141 | (18) |
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141 | (4) |
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145 | (3) |
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7.3 Formality and Koszul algebras |
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148 | (1) |
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149 | (3) |
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152 | (3) |
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7.6 Formality and algebraic deformations |
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155 | (4) |
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Chapter 8 Support Varieties for Finite-Dimensional Algebras |
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159 | (22) |
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160 | (2) |
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8.2 Finiteness properties |
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162 | (5) |
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167 | (3) |
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8.4 Self-injective algebras and realization |
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170 | (3) |
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8.5 Self-injective algebras and indecomposable modules |
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173 | (8) |
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181 | (30) |
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9.1 Hopf algebras and actions on rings |
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181 | (4) |
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9.2 Modules for Hopf algebras |
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185 | (5) |
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9.3 Hopf algebra cohomology and actions |
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190 | (6) |
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9.4 Bimodules and Hochschild cohomology |
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196 | (6) |
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9.5 Finite group algebras |
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202 | (3) |
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9.6 Spectral sequences for Hopf algebras |
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205 | (6) |
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Appendix A Homological Algebra Background |
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211 | (24) |
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211 | (3) |
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A.2 Resolutions and dimensions |
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214 | (4) |
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218 | (3) |
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221 | (3) |
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224 | (2) |
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A.6 Categories, functors, derived functors |
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226 | (4) |
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230 | (5) |
| Bibliography |
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235 | (12) |
| Index |
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247 | |
