Homological algebra of semimodules and semicontramodules: semi-infinite homological algebra of associative algebraic structures
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Basel
Birkhäuser
2010
|
| Series: | Monografie Matematyczne
70 |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 349 S. graph. Darst. |
| ISBN: | 9783034604352 |
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MARC
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| 245 | 1 | 0 | |a Homological algebra of semimodules and semicontramodules |b semi-infinite homological algebra of associative algebraic structures |c Leonid Positselski |
| 264 | 1 | |a Basel |b Birkhäuser |c 2010 | |
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| adam_text | Titel: Homological algebra of semimodules and semicontramodules
Autor: Positselski, Leonid
Jahr: 2010
Contents
Préface...................................... 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 catégories......................... 39
2.2 Coflat complexes........................... 40
2.3 Semiderived catégories........................ 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............................
3.2 Cohomomorphisms..........................
3.3 Semicontramodules..........................
71
3.4 Semihomomorphisms.........................
4 Derived Functor SemiExt
77
4-1 Contraderived catégories.......................
77
4.2 Coprojective and coinjective complexes...............
4-3 Semiderived catégories........................
4-4 Semiprojective and senriinjective complexes............
4.5 Main theorem for comodules and contramodules.......... 79
. . Contents
vin
4.6 Main theorem for semimodules and semicontramodules...... 81
4.7 Derived functor SemiExt....................... °°
4.8 Relatively semiprojective and semiinjective complexes...... 85
4.9 Remarks on derived semihomomorphisms
from bisemimodules ......................... °
5 Comodule-Contramodule Correspondence
5.1 Contratensor product and comodule/contramodule
homomorphisms............................ °9
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-Semicontramodule Correspondence
6.1 Contratensor product and semimodule/semicontramodule
homomorphisms............................ 107
6.2 Associativity isomorphisms ..................... 110
6.3 Semimodule-semicontramodule correspondence.......... 117
6.4 Birelatively contrafiat, 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 fiat/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
Contents ix
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 Senhproduct 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 clément theorem....................... 205
11.6 Poincaré-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 Counteroxamples........................... ^
A.2 Nakayama s Lenima ......................... 232
A.3 Contrariât contramodules ...................... 234
B Comparison with Arkhipov s Ext°°/2+* and Sevostyanov s Toroo/2+*
B.l Algebras R and R*..........................237
B.2 Finitc-dimonsional case........................
B.3 Semijeetive complexes........................
B.4 Explicit resolutions..........................
B.5 Explicit résolutions for a finite-dimensional subalgebra......24
C Semialgebras Associated to Harish-Chandra Pairs
by Leonid Po.sitsel.ski and Dmitriy Rumynin
Cl Two semialgebras...........................
C-2 Morita équivalence.........................
C-3 Semitensor product and somihomomorphisms,
SemiTor and SemiExt ...................... ?
C-4 Harish-Chandra pairs....................... ocn
n r ? ... 260
Co Semiinvariants and seinicontrainvanants.........
x Contents
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-infinité homology and cohomology.............. 294
D.6 Comparison theorem......................... 301
E Groups with Open Profinite Subgroups
E.l Morita équivalent 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 équivalent semialgebras ................ 328
F.6 Compatibility vérifications...................... 330
Bibliography................................... 333
Notation...................................... 339
Index....................................... 343
|
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| indexdate | 2024-07-09T22:37:15Z |
| institution | BVB |
| isbn | 9783034604352 |
| language | English |
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| physical | XXIV, 349 S. graph. Darst. |
| publishDate | 2010 |
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| spelling | Positselski, Leonid Verfasser aut Homological algebra of semimodules and semicontramodules semi-infinite homological algebra of associative algebraic structures Leonid Positselski Basel Birkhäuser 2010 XXIV, 349 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Monografie Matematyczne 70 Monografie Matematyczne 70 (DE-604)BV000003532 70 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020195386&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Positselski, Leonid Homological algebra of semimodules and semicontramodules semi-infinite homological algebra of associative algebraic structures Monografie Matematyczne |
| title | Homological algebra of semimodules and semicontramodules semi-infinite homological algebra of associative algebraic structures |
| title_auth | Homological algebra of semimodules and semicontramodules semi-infinite homological algebra of associative algebraic structures |
| title_exact_search | Homological algebra of semimodules and semicontramodules semi-infinite homological algebra of associative algebraic structures |
| title_full | Homological algebra of semimodules and semicontramodules semi-infinite homological algebra of associative algebraic structures Leonid Positselski |
| title_fullStr | Homological algebra of semimodules and semicontramodules semi-infinite homological algebra of associative algebraic structures Leonid Positselski |
| title_full_unstemmed | Homological algebra of semimodules and semicontramodules semi-infinite homological algebra of associative algebraic structures Leonid Positselski |
| title_short | Homological algebra of semimodules and semicontramodules |
| title_sort | homological algebra of semimodules and semicontramodules semi infinite homological algebra of associative algebraic structures |
| title_sub | semi-infinite homological algebra of associative algebraic structures |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020195386&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003532 |
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