Homological Algebra of Semimodules and Semicontramodules (eBook)

Contents 8
Preface 12
Introduction 16
0 Preliminaries and Summary 26
0.1 Unbounded Tor and Ext 26
0.2 Coalgebras over fields Cotor and Coext
0.3 Semialgebras over coalgebras over fields 36
0.4 Nonhomogeneous Koszul duality over a base ring 43
1 Semialgebras and Semitensor Product 50
1.1 Corings and comodules 50
1.2 Cotensor product 52
1.3 Semialgebras and semimodules 57
1.4 Semitensor product 60
2 Derived Functor SemiTor 64
2.1 Coderived categories 64
2.2 Coflat complexes 65
2.3 Semiderived categories 66
2.5 Main theorem for comodules 68
2.6 Main theorem for semimodules 70
2.7 Derived functor SemiTor 73
2.8 Relatively semiflat complexes 76
2.9 Remarks on derived semitensor product of bisemimodules 78
3 Semicontramodules and Semihomomorphisms 81
3.1 Contramodules 81
3.2 Cohomomorphisms 83
3.3 Semicontramodules 89
3.4 Semihomomorphisms 95
4 Derived Functor SemiExt 100
4.1 Contraderived categories 100
4.2 Coprojective and coinjective complexes 100
4.3 Semiderived categories 101
4.5 Main theorem for comodules and contramodules 102
4.6 Main theorem for semimodules and semicontramodules 104
4.7 Derived functor SemiExt 106
4.8 Relatively semiprojective and semiinjective complexes 108
4.9 Remarks on derived semihomomorphisms from bisemimodules 110
5 Comodule-Contramodule Correspondence 112
5.1 Contratensor product and comodule/contramodule homomorphisms 112
5.2 Associativity isomorphisms 114
5.3 Relatively injective comodules and relatively projective contramodules 118
5.4 Comodule-contramodule correspondence 120
5.5 Derived functor Ctrtor 124
5.6 Coext and Ext, Cotor and Ctrtor 127
6 Semimodule-Semicontramodule Correspondence 129
6.1 Contratensor product and semimodule/semicontramoduleho momorphisms 129
6.2 Associativity isomorphisms 132
6.3 Semimodule-semicontramodule correspondence 139
6.4 Birelatively contraflat, projective, and injective complexes 140
6.5 Derived functor CtrTor 142
6.6 SemiExt and Ext, SemiTor and CtrTor 145
7 Functoriality in the Coring 147
7.1 Compatible morphisms 147
7.2 Properties of the pull-back and push-forward functors 151
7.3 Derived functors of pull-back and push-forward 154
7.4 Faithfully flat/projective base ring change 156
7.5 Remarks on Morita morphisms 159
8 Functoriality in the Semialgebra 164
8.1 Compatible morphisms 164
8.2 Complexes, adjusted to pull-backs and push-forwards 171
8.3 Derived functors of pull-back and push-forward 174
8.4 Remarks on Morita morphisms 181
9 Closed Model Category Structures 189
9.1 Complexes of comodules and contramodules 189
9.2 Complexes of semimodules and semicontramodules 193
10 A Construction of Semialgebras 203
10.1 Construction of comodules and contramodules 203
10.2 Construction of semialgebras 205
10.3 Entwining structures 208
10.4 Semiproduct and semimorphisms 211
11 Relative Nonhomogeneous Koszul Duality 213
11.1 Graded semialgebras 213
11.2 Differential semialgebras 214
11.3 One-sided SemiTor 218
11.4 Koszul semialgebras and corings 219
11.5 Central element theorem 225
11.6 Poincar´e–Birkhoff–Witt theorem 228
11.7 Quasi-differential comodules and contramodules 233
11.8 Koszul duality 237
11.9 SemiTor and Cotor, SemiExt and Coext 241
Appendices 246
A Contramodules over Coalgebras over Fields 247
A.1 Counterexamples 247
A.2 Nakayama’s Lemma 250
A.3 Contraflat contramodules 252
B Comparison with Arkhipov’s Ext8/2+* and Sevostyanov’s Tor8/2+* 255
B.1 Algebras R and R# 255
B.2 Finite-dimensional case 258
B.3 Semijective complexes 259
B.4 Explicit resolutions 261
B.5 Explicit resolutions for a finite-dimensional subalgebra 262
C Semialgebras Associated to Harish-Chandra Pairs 265
C.1 Two semialgebras 265
C.2 Morita equivalence 268
C.3 Semitensor product and semihomomorphisms, SemiTor and SemiExt 272
C.4 Harish-Chandra pairs 275
C.5 Semiinvariants and semicontrainvariants 278
D Tate Harish-Chandra Pairs and Tate Lie Algebras 282
D.1 Continuous coactions 282
D.2 Construction of semialgebra 288
D.3 Isomorphism of semialgebras 298
D.4 Semiinvariants and semicontrainvariants 307
D.5 Semi-infinite homology and cohomology 311
D.6 Comparison theorem 318
E Groups with Open Profinite Subgroups 325
E.1 Morita equivalent semialgebras 325
E.2 Semiinvariants and semicontrainvariants 328
E.3 SemiTor and SemiExt 332
E.4 Remarks on the Gaitsgory–Kazhdan construction 334
F Algebraic Groupoids with Closed Subgroupoids 338
F.1 Coring associated to affine groupoid 338
F.2 Canonical Morita autoequivalence 339
F.3 Distributions and generalized sections 340
F.4 Lie algebroid of a groupoid 341
F.5 Two Morita equivalent semialgebras 343
F.6 Compatibility verifications 345
Bibliography 347
Notation 353
Index 356