Homological Algebra of Semimodules and Semicontramodules

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ISBN: 9783034604352 | 3034604351
Cover: Hardcover
Copyright: 9/8/2010

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This monograph deals with semi-infinite homological algebra. Intended as the definitive treatment of the subject of semi-infinite homology and cohomology of associative algebraic structures, it also contains material on the semi-infinite (co)homology of Lie algebras and topological groups, the derived comodule-contramodule correspondence, its application to the duality between representations of infinite-dimensional Lie algebras with complementary central charges, and relative non-homogeneous Koszul duality.

Preface	p. xi
Introduction	p. xv
Preliminaries and Summary
Unbounded Tor and Ext	p. 1
Coalgebras over fields; Cotor and Coext	p. 3
Semialgebras over coalgebras over fields	p. 11
Nonhomogeneous Koszul duality over a base ring	p. 18
Semialgebras and Semitensor Product
Corings and comodules	p. 25
Cotensor product	p. 27
Semialgebras and semimodules	p. 32
Semitensor product	p. 35
Derived Functor SemiTor
Coderived categories	p. 39
Coflat complexes	p. 40
Semiderived categories	p. 41
Semiflat complexes	p. 41
Main theorem for comodules	p. 43
Main theorem for semimodules	p. 45
Derived functor SemiTor	p. 48
Relatively semiflat complexes	p. 51
Remarks on derived semitensor product of bisemimodules	p. 53
Semicontramodules and Semihomomorphisms
Contramodules	p. 57
Cohomomorphisms	p. 59
Semicontramodules	p. 65
Semihomomorphisms	p. 71
Derived Functor SemiExt
Contraderived categories	p. 77
Coprojective and coinjective complexes	p. 77
Semiderived categories	p. 78
Semiprojective and semiinjective complexes	p. 78
Main theorem for comodules and contramodules	p. 79
Main theorem for semimodules and semicontramodules	p. 81
Derived functor SemiExt	p. 83
Relatively semiprojective and semiinjective complexes	p. 85
Remarks on derived semihomomorphisms from bisemimodules	p. 87
Comodule-Contramodule Correspondence
Contratensor product and comodule/contramodule homomorphisms	p. 89
Associativity isomorphisms	p. 91
Relatively injective comodules and relatively projective contramodules	p. 95
Comodule-contramodule correspondence	p. 97
Derived functor Ctrtor	p. 101
Coext and Ext, Cotor and Ctrtor	p. 104
Semimodule-Semicontramodule Correspondence
Contratensor product and semimodule/semicontramodule homomorphisms	p. 107
Associativity isomorphisms	p. 110
Semimodule-semicontramodule correspondence	p. 117
Birelatively contraflat, projective, and injective complexes	p. 118
Derived functor CtrTor	p. 120
SemiExt and Ext, SemiTor and CtrTor	p. 123
Functoriality in the Coring
Compatible morphisms	p. 125
Properties of the pull-back and push-forward functors	p. 129
Derived functors of pull-back and push-forward	p. 132
Faithfully flat/projective base ring change	p. 134
Remarks on Morita morphisms	p. 137
Functoriality in the Semialgebra
Compatible morphisms	p. 143
Complexes, adjusted to pull-backs and push-forwards	p. 150
Derived functors of pull-back and push-forward	p. 153
Remarks on Morita morphisms	p. 160
Closed Model Category Structures
Complexes of comodules and contramodules	p. 169
Complexes of semimodules and semicontramodules	p. 173
A Construction of Semialgebras
Construction of comodules and contramodules	p. 183
Construction of semialgebras	p. 185
Entwining structures	p. 188
Semiproduct and semimorphisms	p. 191
Relative Nonhomogeneous Koszul Duality
Graded semialgebras	p. 193
Differential semialgebras	p. 194
One-sided SemiTor	p. 198
Koszul semialgebras and corings	p. 199
Central element theorem	p. 205
Poincaré-Birkhoff-Witt theorem	p. 208
Quasi-differential comodules and contramodules	p. 213
Koszul duality	p. 217
SemiTor and Cotor, SemiExt and Coext	p. 221
Appendices
Contramodules over Coalgebras over Fields
Counterexamples	p. 229
Nakayama's Lemma	p. 232
Contraflat contramodules	p. 234
Comparison with Arkhipov's Ext<$$$> and Sevostyanov's Tor<$$$>
Algebras R and R^#	p. 237
Finite-dimensional case	p. 240
Semijective complexes	p. 241
Explicit resolutions	p. 243
Explicit resolutions for a finite-dimensional subalgebra	p. 244
Semialgebras Associated to Harish-Chandra Pairs
Two semialgebras	p. 247
Morita equivalence	p. 250
Semitensor product and semihomomorphisms, SemiTor and SemiExt	p. 254
Harish-Chandra pairs	p. 257
Semiinvariants and semicontrainvariants	p. 260
Tate Harish-Chandra Pairs and Tate Lie Algebras
Continuous coactions	p. 265
Construction of semialgebra	p. 271
Isomorphism of semialgebras	p. 281
Semiinvariants and semicontrainvariants	p. 290
Semi-infinite homology and cohomology	p. 294
Comparison theorem	p. 301
Groups with Open Profinite Subgroups
Morita equivalent semialgebras	p. 309
Semiinvariants and semicontrainvariants	p. 312
SemiTor and SemiExt	p. 316
Remarks on the Gaitsgory-Kazhdan construction	p. 318
Algebraic Groupoids with Closed Subgroupoids
Coring associated to affine groupoid	p. 323
Canonical Morita autoequivalence	p. 324
Distributions and generalized sections	p. 325
Lie algebroid of a groupoid	p. 326
Two Morita equivalent semialgebras	p. 328
Compatibility verifications	p. 330
Bibliography	p. 333
Notation	p. 339
Index	p. 343
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Homological Algebra of Semimodules and Semicontramodules

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Homological Algebra of Semimodules and Semicontramodules

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