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Gelfand S.I., Manin Yu.I. — Methods of Homological Algebra
Gelfand S.I., Manin Yu.I. — Methods of Homological Algebra

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Название: Methods of Homological Algebra

Авторы: Gelfand S.I., Manin Yu.I.

Аннотация:

Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn about a modern approach to homological algebra and to use it in their work.


Язык: en

Рубрика: Математика/Алгебра/Гомология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1989

Количество страниц: 372

Добавлена в каталог: 22.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelian group filtered      117
Abelian group filtered, simplicial      46
Abelian group filtered, topological      118
Additivity of the derived category      162
Affine group scheme      83
Bialgebra      83
Bisimplicial set      18
Boundary      29
Canonical decomposition of a morphism in an abelian category      112
Cap lower      240
Cap lower, upper      240
Cartier duality      85
Category      57
Category of an ordered set      60
Category of an ordered set, categories      67
Category of an ordered set, coindices      196
Category of an ordered set, complexes      58 168
Category of an ordered set, functors      63
Category of an ordered set, indices      86
Category of an ordered set, inductive limits      197
Category of an ordered set, projective limits      197
Category, abelian      113
Category, additive      113
Category, closed model      293
Category, cyclic      67
Category, derived      141
Category, dual      61
Category, exact      275
Category, examples of      58 59
Category, homotopic      159 246 332 354
Category, Probenius      276
Category, quotient      77
Category, simplicial      146
Category, stable      276
Category, triangulated      239
Chain      23
Chaindegenerate      53
Class complementary      291
Class complementaryof morphisms localizing      147
Class complementaryof morphisms localizing in a triangulated category      251
Class complementaryof morphisms, objects adapted to a functor      187
Classified space (of a group)      8 61
Coboundary      24
Cochain      24
Coefficient system      28
Coequalizer      88
Cofibration      293 331 335 350
Cohomology      47
Cohomology of a cochain complex      25 123
Cohomology of a cochain group      30
Cohomology of a cochain Lie algebra      48
Cohomology of a cochain sheaf      219
Cohomology of a cochain sheaf with the compact support      227
Cohomology, Cech      30 47
Cohomology, cyclic      54
Cohomology, Hochschild      54
Cokernel      111
Colimit      88
Complex of Cech      46
Complex, acyclic      123
Complex, bounded (from the left, from the right, from both sides)      147
Complex, chain      25
Complex, cochain      25 123
Complex, de Rham      48
Complex, de Rham, singular chains      47
Complex, diagonal      208
Complex, double      207 208
Complex, dualizing      234
Complex, K-injective      199
Complex, K-projective      199
Complex, Koszul      54 108
Complex, rigid      269
Complex, rigid, finite      269
Complex, twisted      262
Cone      26 154 244
Convolution of a complex      258
Core of a triangulated category      278
Coskeleton      16 93
CYCLE      25
Cylinder of a morphism      154
Cylinder of a morphism an object      332
Degeneration      14
Diagonal of a bisimplicial set      15
Differential graded algebra (DG-algebra)      338
Differential graded algebra, minimal      343
DIMENSION      227
Dimension, homological      171
Dimension, infective      172
Dimension, projective      172
Eilenberg — Zilber theorem      19
Element of an object of an abelian category      118
Epimorphism      114
Equilizer      87
Equivalence homotopic      301
Equivalence homotopic of categories      71
Equivalence homotopic of categories, examples      71—76
Equivalence, weak      293 331 334 350
Exact pair      215
Exact pair, sequence      40
Exact pair, sequence of cohomology      40
Exact pair, triple      40 42
Extension      257
Extension and $Ext^1$      184
Extension by Yoneda      166
Face      14
Fiber of a sheaf in a point      34
Fibration      293 331 334 350
Fibration, locally trivial      321
Fibration, minimal      317
Fibration, universal      325
Filtration , regular      202
Filtration and $Ext^2$      184
Filtration of a complex      203
Filtration of a complex, canonical      203
Filtration of a complex, stupid      203
Five-lemma      120
functor      60
Functor, additive      122
Functor, adjoint      89
Functor, cohomological      243
Functor, contravariant      61
Functor, covariant      61
Functor, derived      188
Functor, derived and $Ext^i$      194
Functor, derived and $Ext^i$, $Tor_i$      195
Functor, derived, classical      194
Functor, derived, of a composition      200
Functor, derived, weak      197
Functor, diagonal      86
Functor, exact      125 185 285
Functor, exact, from the left      125
Functor, exact, from the right      125
Functor, examples      60—62
Functor, faithful      64
Functor, forgetful      63 91
Functor, full      64
Functor, representable      78
Functor, representable in a category of sheaves      229
Geometrical realization      6
Geometrical realization of a simplicial topological space      18
Germ of a section      34
Gluing      1 288
Gluing data      1
Grassmann algebra      267
Grothendieck topology      100
Group cosimplicial      46
Group cosimplicial in a category      82
Group cosimplicial, simplicial      46
Hilbert theorem      175 176
Homology      47
Homology of a chain complex      25 30
Homology of a group      30
Homology of a Lie algebra      48
Homology, cyclic      54
Homology, Hochschild      54
Homology, singular      47 50
Homotopy      49 53 107 301 333 351 353
Homotopy of a morphism of complexes      49 140
Homotopy of simplicial mappings      23
Homotopy of simplicial sets      301
Homotopy, group      356
Homotopy, group of a DG-algebra      353
Horn      294
Hypercohomology      213
Hypersimplex      260
Image      113
Image, direct      133
Image, direct with compact support      224
Image, direct with compact support, higher      227
Image, direct, higher      228
Image, inverse      134—136 223
Image, inverse, with compact support      228
Injectivity diagram      125
Isomorphism of triangles      157
Isomorphism of triangles, functors      69
Jordan — Holder theorem      122
Kan set      299
Kernel      111
Lie superalgebra      344
Limit, inductive      88 197
Limit, projective      86
localization      145
Manifolds      93
Massey product      266
Meier — Vietoris theorem      236
Minimal model of a de Rham algebra      352
Minimal model of a de Rham algebra DG-algebra      347
Mittag — Lefler condition      198
Module flat      128 217
Monomorphisms      114
Morita equivalence      77
Morphism      57
Morphism of coefficient systems      44
Morphism of complexes      41
Morphism of functors      63
Morphism of functors, examples      63
Morphism of sheaves      32
Morphism of spectral sequences      202
Morphism of triangles      157
Morphism, diagonal      80
Morphism, flat      128 224
Morphism, inner      105 108
Morphism, proper      224
Nerve of a category      104
Nerve of a category, covering      7
Object      57
Object, A-acyclic      133
Object, cofibrant      332
Object, F-acyclic (for functor F)      195
Object, fibrant      332
Object, final      65
Object, initial      65
Object, injective      125 275
Object, projective      125 275
Object, representing      78
Object, simple      121
Object, zero      110
Octahedron diagram      240
Poincare duality      234
Polycategory      65
Postnikov system      257
Presheaf      32 100
Product direct      80
Product direct, fiber      80
Projectivity diagram      128
Quadratic algebras      108
Quasi-isomorphism      143
quiver      288
Quotient category      77
Quotient category, (a la Serre)      121
Resolution, Cartan — Eilenberg      207
Resolution, flat      217
Resolution, free      139
Resolution, injective      143 179 199
Resolution, projective      141 199
Retraction      292
Ringed space      97
Roof      151
Scheme      98
Scheme, affine      95 97
Section of a sheaf      32
Section of a sheaf with compact support      225
Section of a sheaf with compact support in a subset      234
Sheaf      32
Sheaf of morphisms      38
Sheaf on a site      100
Sheaf, associated to a presheaf      35 114
Sheaf, constant      35
Sheaf, fine      38
Sheaf, flabby      37 219
Sheaf, flat      222
Sheaf, injective      218
Sheaf, locally constant      36
Sheaf, quasicoherent      40
Sheaf, soft      38 226
Sieve      100
Sieve, covering      100
Simplex n-dimensional      2 6
Simplex n-dimensional, non-degenerate      9
Simplex n-dimensional, singular      7
Simplices g-connected      316
Simplicial set      6
Simplicial set, N-truncated      16
Simplicial set, singular      60
Simplicial topological space      17
Site      100
Skeleton      3 13 92
Snake-lemma      120
Spectral sequence      201
Spectral sequence of a composition of functors      207
Spectral sequence of a double complex      207
Spectral sequence of a filtered complex      203 217
Spectral sequence of Cech cohomology      214
Spectral sequence of Eilenberg — Moore      263
Spectral sequence of hypercohomology      213
Spectral sequence of Serre — Hochschild      214
Spectral sequence, degenerate      202
Sphere simplicial      2 295
Square cartesian      81
Square cartesian, cocartesian      81
Subcategory      64
Subcategory, full      64 76
Subcategory, localizing      153
Subcategory, thick      266
Sum amalgamated      81
Sum, direct      138
Superspace      98
Suspension      276
Syzygies      139
t-structure      276 286
Tensor product of algebras      82
Translation functor      157 239
triangle      156
Triangle, distinguished      157 239
Triangulated space      2
Triangulation      2
Triangulation, canonical of the product of two simplices      4
Truncation functor      279
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