Quadratic algebras

Quadratic Algebras......Page 3
Contents......Page 5
Introduction......Page 7
0. Conventions and notation......Page 13
1. Bar constructions......Page 14
2. Quadratic algebras and modules......Page 18
4. Minimal resolutions......Page 19
5. Low-dimensional cohomology......Page 21
6. Lattices and distributivity......Page 23
7. Lattices of vector spaces......Page 27
1. Koszulness......Page 30
2. Hilbert series......Page 32
3. Koszul complexes......Page 36
4. Distributivity and n-Koszulness......Page 40
5. Homomorphisms of algebras and Koszulness I......Page 43
6. Homomorphisms of algebras and Koszulness II......Page 48
7. Koszul algebras in algebraic geometry......Page 51
8. Infinitesimal Hopf algebra associated with a Koszul algebra......Page 56
9. Koszul algebras and monoidal functors......Page 60
10. Relative Koszulness of modules......Page 64
1. Direct sums, free products and tensor products......Page 66
2. Segre products and Veronese powers I......Page 70
3. Segre products and Veronese powers II......Page 74
4. Internal cohomomorphism......Page 79
5. Koszulness cannot be checked using Hilbert series......Page 88
1. PBW-bases......Page 92
2. PBW-theorem......Page 93
3. PBW-bases and Koszulness......Page 95
4. PBW-bases and operations on quadratic algebras......Page 96
5. PBW-bases and distributing bases......Page 97
6. Hilbert series of PBW-algebras......Page 98
7. Filtrations on quadratic algebras......Page 99
8. Commutative PBW-bases......Page 102
9. Z-algebras......Page 106
10. Z-PBW-bases......Page 107
11. Three-dimensional Sklyanin algebras......Page 109
1. Jacobi identity......Page 111
2. Nonhomogeneous PBW-theorem......Page 113
3. Nonhomogeneous quadratic modules......Page 114
4. Nonhomogeneous quadratic duality......Page 115
5. Examples......Page 118
6. Nonhomogeneous duality and cohomology......Page 121
7. Bar construction for CDG-algebras and modules......Page 122
8. Homology of completed cobar-complexes......Page 127
1. Openness of distributivity......Page 129
2. Deformations of Koszul algebras......Page 130
3. Upper bound for the number of Koszul Hilbert series......Page 132
4. Generic quadratic algebras......Page 133
5. Examples with small dim A1 and dim A2......Page 135
6. Koszulness is not constructible......Page 137
7. Families of quadratic algebras over schemes......Page 138
1. Conjectures on Hilbert series of Koszul algebras......Page 142
2. Koszul inequalities......Page 144
3. Koszul duality and inequalities......Page 147
4. One-dependent processes......Page 148
5. PBW-algebras and two-block-factor processes......Page 151
6. Operations on one-dependent processes......Page 152
7. Hilbert space representations of one-dependent processes......Page 155
8. Hilbert series of one-dependent processes......Page 156
9. Hermitian construction of one-dependent processes......Page 158
10. Modules over one-dependent processes......Page 160
Appendix A: DG-algebras and Massey products......Page 162
Bibliography......Page 164