Preface ....................................................... vii
Introduction .................................................... 1
Chapter I. Noncommutative projective geometry .................. 13
Introduction ................................................... 13
1 Review of basic background and the Diamond Lemma ............ 14
2 Artin-Schelter regular algebras ............................. 30
3 Point modules ............................................... 42
4 Noncommutative projective schemes ........................... 51
5 Classification of noncommutative curves and surfaces ........ 62
Chapter II. Deformations of algebras in noncommutative
geometry ....................................................... 71
Introduction ................................................... 71
1 Motivating examples ......................................... 75
2 Formal deformation theory and Kontsevich's theorem ......... 104
3 Hochschild cohomology and infinitesimal deformations ....... 124
4 Dglas, the Maurer-Cartan formalism, and proof of
formality theorems ......................................... 136
5 Calabi-Yau algebras and isolated hypersurface
singularities .............................................. 157
Chapter III. Symplectic reflection algebras ................... 167
Introduction .................................................. 167
1 Symplectic reflection algebras ............................. 171
2 Rational Cherednik algebras at t = 1 ....................... 184
3 The symmetric group ........................................ 197
4 The KZ functor ............................................. 211
5 Symplectic reflection algebras at t = 0 .................... 224
Chapter IV Noncommutative resolutions ........................ 239
Introduction .................................................. 239
Acknowledgments ............................................... 240
1 Motivation and first examples .............................. 240
2 NCCRs and uniqueness issues ................................ 250
3 From algebra to geometry: quiver GIT ....................... 261
4 Into derived categories .................................... 270
5 McKay and beyond ........................................... 285
6 Appendix: Quiver representations ........................... 297
Solutions to the exercises .................................... 307
I Noncommutative projective geometry ........................ 307
II Deformations of algebras in noncommutative geometry ....... 316
III Symplectic reflection algebras ............................ 332
IV Noncommutative resolutions ................................ 337
Bibliography .................................................. 343
Index ......................................................... 353
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