Preface ........................................................ xi
Introduction ................................................... xv
0 Preliminaries and Summary
0.1 Unbounded Tor and Ext ................................... 1
0.2 Coalgebras over fields; Cotor and Coext ................. 3
0.3 Semialgebras over coalgebras over fields ............... 11
0.4 Nonhomogeneous Koszul duality over a base ring ......... 18
1 Semialgebras and Semitensor Product
1.1 Corings and comodules .................................. 25
1.2 Cotensor product ....................................... 27
1.3 Semialgebras and semimodules ........................... 32
1.4 Semitensor product ..................................... 35
2 Derived Functor SemiTor
2.1 Coderived categories ................................... 39
2.2 Coflat complexes ....................................... 40
2.3 Semiderived categories ................................. 41
2.4 Semiflat complexes ..................................... 41
2.5 Main theorem for comodules ............................. 43
2.6 Main theorem for semimodules ........................... 45
2.7 Derived functor SemiTor ................................ 48
2.8 Relatively semiflat complexes .......................... 51
2.9 Remarks on derived semitensor product of
bisemimodules .......................................... 53
3 Semicontramodules and Semihomomorphisms
3.1 Contramodules .......................................... 57
3.2 Cohomomorphisms ........................................ 59
3.3 Semicontramodules ...................................... 65
3.4 Semihomomorphisms ...................................... 71
4 Derived Functor SemiExt
4.1 Contraderived categories ............................... 77
4.2 Coprojective and coinjective complexes ................. 77
4.3 Semiderived categories ................................. 78
4.4 Semiprojective and semiinjective complexes ............. 78
4.5 Main theorem for comodules and contramodules ........... 79
4.6 Main theorem for semimodules and semicontramodules ..... 81
4.7 Derived functor SemiExt ................................ 83
4.8 Relatively semiprojective and semiinjective
complexes .............................................. 85
4.9 Remarks on derived semihomomorphisms from
bisemimodules .......................................... 87
5 Comodule-Contramodule Correspondence
5.1 Contratensor product and comodule/contramodule
homomorphisms .......................................... 89
5.2 Associativity isomorphisms ............................. 91
5.3 Relatively injective comodules and relatively
projective contramodules ............................... 95
5.4 Comodule-contramodule correspondence ................... 97
5.5 Derived functor Ctrtor ................................ 101
5.6 Coext and Ext, Cotor and Ctrtor ....................... 104
6 Semimodule-Semicontrainpdule Correspondence
6.1 Contratensor product and semimodule/semicontramodule
homomorphisms ......................................... 107
6.2 Associativity isomorphisms ............................ 110
6.3 Semimodule-semicontramodule correspondence ............ 117
6.4 Birelatively contraflat, projective, and injective
complexes ............................................. 118
6.5 Derived functor CtrTor ................................ 120
6.6 SemiExt and Ext, SemiTor and CtrTor ................... 123
7 Functoriality in the Coring
7.1 Compatible morphisms .................................. 125
7.2 Properties of the pull-back and push-forward
functors .............................................. 129
7.3 Derived functors of pull-back and push-forward ........ 132
7.4 Faithfully Hat/projective base ring change ............ 134
7.5 Remarks on Morita morphisms ........................... 137
8 Functoriality in the Semialgebra
8.1 Compatible morphisms .................................. 143
8.2 Complexes, adjusted to pull-backs and push-forwards ... 150
8.3 Derived functors of pull-back and push-forward ........ 153
8.4 Remarks on Morita morphisms ........................... 160
9 Closed Model Category Structures
9.1 Complexes of comodules and contramodules .............. 169
9.2 Complexes of semimodules and semicontramodules ........ 173
10 A Construction of Semialgebras
10.1 Construction of comodules and contramodules ........... 183
10.2 Construction of semialgebras .......................... 185
10.3 Entwining structures .................................. 188
10.4 Semiproduct and semimorphisms ......................... 191
11 Relative Nonhomogeneous Koszul Duality
11.1 Graded semialgebras ................................... 193
11.2 Differential semialgebras ............................. 194
11.3 One-sided SemiTor ..................................... 198
11.4 Koszul semialgebras and corings ....................... 199
11.5 Central element theorem ............................... 205
11.6 Poincare-Birkhoff-Witt theorem ........................ 208
11.7 Quasi-differential comodules and contramodules ........ 213
11.8 Koszul duality ........................................ 217
11.9 SemiTor and Cotor, SemiExt and Coext .................. 221
Appendices
A Contramodules over Coalgebras over Fields
A.l Counterexamples ....................................... 229
A.2 Nakayama's Lemma ...................................... 232
A.3 Contraflat contramodules .............................. 234
В Comparison with Arkhipov's Ext∞/2+* and Sevostyanov's
Tor∞/2+*
B.l Algebras R and R# ..................................... 237
B.2 Finite-dimensional case ............................... 240
B.3 Semijective complexes ................................. 241
B.4 Explicit resolutions .................................. 243
B.5 Explicit resolutions for a finite-dimensional
subalgebra ............................................ 244
С Semialgebras Associated to Harish-Chandra Pairs
by Leonid Positselski and Dmitriy Rumynin
C.l Two semialgebras ...................................... 247
C.2 Morita equivalence .................................... 250
C.3 Semitensor product and semihomomorphisms, SemiTor
and SemiExt ........................................... 254
C.4 Harish-Chandra pairs .................................. 257
C.5 Semiinvariants and semicontrainvariants ............... 260
D Tate Harish-Chandra Pairs and Tate Lie Algebras
by Sergey Arkhipov and Leonid Positselski
D.l Continuous coactions .................................. 265
D.2 Construction of semialgebra ........................... 271
D.3 Isomorphism of semialgebras ........................... 281
D.4 Semiinvariants and semicontrainvariants ............... 290
D.5 Semi-infinite homology and cohomology ................. 294
D.6 Comparison theorem .................................... 301
E Groups with Open Profinite Subgroups
E.l Morita equivalent semialgebras ........................ 309
E.2 Semiinvariants and semicontrainvariants ............... 312
E.3 SemiTor and SemiExt ................................... 316
E.4 Remarks on the Gaitsgory-Kazhdan construction ......... 318
F Algebraic Groupoids with Closed Subgroupoids
F.l Coring associated to affine groupoid .................. 323
F.2 Canonical Morita Autoequivalence ...................... 324
F.3 Distributions and generalized sections ................ 325
F.4 Lie algebroid of a groupoid ........................... 326
F.5 Two Morita equivalent semialgebras .................... 328
F.6 Compatibility verifications ........................... 330
Bibliography .................................................. 333
Notation ...................................................... 339
Index ......................................................... 343
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