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Levine M. — Mixed motives
Levine M. — Mixed motives

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Название: Mixed motives

Автор: Levine M.


The author constructs and describes a triangulated category of mixed motives over an arbitrary base scheme. The resulting cohomology theory satisfies the Bloch-Ogus axioms; if the base scheme is a smooth scheme of dimension at most one over a field, this cohomology theory agrees with Bloch's higher Chow groups. Most of the classical constructions of cohomology can be made in the motivic setting, including Chern classes from higher X-theory, push-forward for proper maps, Riemann-Roch, duality, as well as an associated motivic homology, Borel-Moore homology and cohomology with compact supports. The motivic category admits a realization functor for each Bloch-Ogus cohomology theory which satisfies certain axioms; as examples the author constructs Betti, etale, and Hodge realizations over smooth base schemes.
This book is a combination of foundational constructions in the theory of motives, together with results relating motivic cohomology with more explicit constructions, such as Bloch's higher Chow groups. It is aimed at research mathematicians interested in algebraic cycles, motives and X-theory, starting at the graduate level. It presupposes a basic background in algebraic geometry
and commutative algebra.

Язык: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 515

Добавлена в каталог: 23.03.2005

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Предметный указатель
$\mathbb{Z}/2$-set      435
$\mathbb{Z}/2$-set, equivalence relations      436
$\mathbb{Z}/2$-set, graded      435
$\mathcal{D}$-module, holonomic      282
$\mathcal{V}$-category      377
$\mathcal{V}$-functor      377
$\mathcal{V}$-natural transformation      378
2-category      379
2-resolution      239
2-resolution, strict      240
2-resolution, weak      247
Absolute Hodge complexes      273—275
Absolute Hodge complexes, enlarged diagrams      274—275
Adams degree      258
Additive category      381
Additive category, free      384
Adjoining a base-point      361
Adjoining morphisms to a category      384
Adjoining morphisms to a DG tensor category      422—424
Adjoining morphisms to a tensor category      393
Alexander — Whitney map      463 478
Bi-products      380
Bloch’s formula      93
Blow-up distinguished triangle      237—239
Canonical cochains      258
Canonical filtration      277
Canonical topology      482
Canonical truncation      461
Categorical cochain operations      464—466
Category of pairs      385
Cech resolution      53 56
Chern character      169—171
Chern character for diagrams      170
Chern character for higher K-theory      164
Chern character, isomorphism      179 188
Chern classes for $K_0$      163
Chern classes for higher K-theory      119 122 164
Chern classes for higher K-theory of a diagram      122—126
Chern classes for higher K-theory with support      123
Chern classes for higher K-theory, mod n      124 126
Chern classes for higher K-theory, total      165
Chern classes for line bundles      112
Chern classes for line bundles, properties      112
Chern classes for relative higher K-theory      124
Chern classes for relative higher K-theory with support      124
Chern classes for vector bundles      116
Chern classes, compatibility with localization and relativization      126
Chern classes, naturality      117
Chern classes, properties      116
Chern classes, total      116 164
Chern classes, universal      121
Chow motives      214—215
Chow motives, definition      214
Chow motives, embedding into $\mathcal{D}\mathcal{M}$      215
Chow’s Lemma      237
Classifying scheme      357—358
Classifying scheme, bundles      358
Classifying space      357
Closed simplicial model category      475
Cocontinuous      255
Codegeneracy map      449
Coface map      449
Cohomology over a category      466 467
Compactifiable embedding      225
Compactification      277
Complexes over a DG category      409 411—414
Complexes over a DG category, cone sequence      416
Complexes over a DG category, homotopy category      411
Complexes over a DG category, homotopy category, distinguished triangles      416
Complexes over a DG category, homotopy category, triangulated structure      416—420
Complexes over a DG category, tensor structure      414
Cone for complexes over a DG category      411
Cone for complexes over an additive category      410
Cone for Pre-Tr      411
Cone sequence for complexes over a DG category      416
Cone sequence for complexes over an additive category      410
Connected by a subset      335
Connected in codimension one      335
Connectivity      335
Construction      38—40
Correspondences      210 314
Cosimplicial scheme, motive of a      26
Cosimplicial scheme, very smooth      27
Coskeleton      482
Cup product      453
Cycle class map      47—51
Cycle class map for K-theory      183—185
Cycle class map for units      294
Cycle class map for varieties      48
Cycle class map with support      48
Cycle class map with support for varieties      48
Cycle class map, compatibility with Gysin morphism      138
Cycle class map, motivic      76—78 82
Cycle class map, motivic, injectivity      84
Cycle class map, motivic, isomorphism      88
Cycle class map, motivic, surjectivity      78 82
Cycle class map, naive      71 76
Cycle class map, properties      48—51
Cycle class map, relative      218
Cycle class map, relative, push-forward for      218
Cycle complex, Bloch’s      65 68
Cycle complex, comparison isomorphism      68
Cycle complex, motivic      68 69
Cycle complex, motivic for varieties      77
Cycle map      47—51
Cycle map for varieties      48
Cycle map with support      17
Cycle map with support for varieties      48
Cycles and cycle classes      47
Cycles for $\mathcal{L}(\mathcal{V})$      10
Cycles functor      13 38
Cycles of relative dimension d      332
Cycles on simplicial schemes      107 110
Cycles on simplicial schemes, products      109
Cycles, basic definition      10
Cycles, effective      332
Cycles, equi-dimensional, properties of      352—356
Cycles, equi-dimensional, pull-back for      346
Cycles, functoriality      10
Cycles, intersection multiplicity      333
Cycles, intersection multiplicity over a normal base      340
Cycles, intersection multiplicity over a reduced base      349
Cycles, relative      181 331
de Rham functor      282
Decalage      278
Decomposition into type      445
Deformation diagram      131
Differential graded category      381
Differential graded category, homotopy equivalence      420
DIMENSION      331
Dimension over a scheme      331
Discriminant      239
Distinguished octahedra      429
Dold — Kan equivalence      362
Dual of a morphism      193
Dual of an object      192
Dual, canonical      194
Duality criterion for a tensor category      195—197
Duality criterion for a triangulated tensor category      204
Duality in tensor categories      191
Duality in triangulated tensor categories      198
Duality involution for a graded tensor category      195
Duality involution for a tensor category      194
Duality involution for a triangulated tensor category      201 204
Duality involution for smooth projective schemes      205—206
Duality involution for the triangulated motivic category      206 207
Duality involution for the triangulated Tate category      235
Duality involution, explicit formulae      210—214
Effective motives, category of      311
Effective motives, tensor product for      312
Eilenberg — maclane map      477
Equi-dimensional, cycle      332
Equi-dimensional, scheme      331
Etale site      482
Exact category      358
Excision isomorphism      18
Extended total complex      455
External product      391
External product, categorical      391
External product, universal      392
Fiber functor      483
Fiber functor, associated pro-object      485
Fiber functor, stalk      484
Fibrant, functor      475
Fibrant, simplicial set      475
Finite category      468
Flat, inverse system      271
Flat, presheaf      256
Flat, sheaf      256
Flatness      497
General linear group      358
Generated by a set of objects      425
Generic projection      95
Geometric cohomology theory      255 257
Geometric motives, category of      313
Geometric point      269 331
Gluing cycles      347
Godement resolution      486 490—499
Godement resolution and cohomology with support      490 492
Godement resolution and flatness      499
Godement resolution and sheaf cohomology      490
Godement resolution, associated complex      490
Godement resolution, products      493 496
Good compactifications      224
Good compactifications, duality for      224
Graded category      381
Graded homomorphism      383
Graded symmetric monoidal category      436
Graded symmetric monoidal category, punctual      436
Grothendieck group      359
Grothendieck pre-topology      481
Grothendieck site      481
Grothendieck site, presheaf on a      482
Grothendieck site, sheaf on a      482
Grothendieck topology      481
Grothendieck topology, covering families      481
Group homology      357
Gysin distinguished triangle      132
Gysin isomorphism      18
Gysin morphism      20 130 141
Gysin morphism for a closed embedding      131—132
Gysin morphism for a split embedding      130—131
Gysin morphism, compatibility with cycle classes      138
Gysin morphism, compatibility with products      142
Gysin morphism, functoriality      132
Gysin morphism, properties      132
Gysin sequence      132
Higher Chow groups and motivic cohomology      103 105
Higher Chow, Bloch’s      65 66
Higher Chow, comparison isomorphism      71\
Higher Chow, motivic      75—77
Higher Chow, motivic and hypercohomology      77
Higher Chow, motivic and X-theory      179
Higher Chow, motivic for varieties      77
Higher Chow, naive      70
Higher Chow, naive for varieties      72
Homotopy category of a DG category      409
Homotopy category of the category of complexes      411
Homotopy commutative product      463
Homotopy commutativity      322 441 445 455
Homotopy equivalence of DG categories      420
Homotopy fiber      127
Homotopy limit for simplicial sets      474—475
Homotopy limit, additive      467 468 471
Homotopy limit, additive and cohomology      473
Homotopy limit, additive and hypercohomology      474
Homotopy limit, additive, distinguished triangle      472
Homotopy limit, additive, functoriality      472
Homotopy limit, additive, non-degenerate      471
Homotopy one point category      13 435 440—441
Homotopy one point category, universal mapping property      441
Homotopy property      17
Homotopy property for homological motives      216
Homotopy property for motives of diagrams      35
Homotopy property for motives of schemes      19
Hurewicz map      362—363
Hurewicz map for diagrams      363—364
Hurewicz map, compatibility with products      364—369
Hypercohomology      62
Hypercohomology for motives      63 64
Hypercover      482
Hypercover of a sheaf      483
Hypercover of an object      483
Hyperresolutions for motives      59
Hyperresolutions for motives, maps of      59
Hyperresolutions for motives, the category of      59 61
Hyperresolutions, cubical      237 240
Hyperresolutions, cubical, category of      240
Hyperresolutions, cubical, strict      240
Hyperresolutions, cubical, weak      247
Index of inseparability      338
Intersection multiplicity on a regular scheme      333
Intersection multiplicity over a normal base      340
Intersection multiplicity over a reduced base      349
Inverse systems of sheaves      269
Inverse systems of Tate sheaves      270
Inverse systems, category of      269—270
Inverse systems, category of internal Horn      270
Inverse systems, category of tensor structure      270
Inverse systems, continuous hypercohomology for      270
Inverse systems, strongly acyclic      271
Inverse systems, strongly normalized      271
K-group with support      123
K-group, relative      123
K-group, relative with support      124
K-theory and homology of GL      362
K-theory for diagrams      360
K-theory for schemes      358
Kunneth isomorphism      18
Kunneth isomorphism for compactly supported motives      217
Kunneth isomorphism for diagrams      36
Kunneth isomorphism for homological motives      216
L-functions of motives      292
Lambda ring      161
Lambda ring, Adams operations      180
Lambda ring, Adams operations for higher K-theory      181
Lambda ring, augmented      180
Lambda ring, gamma filtration      180
Lambda ring, special      162—163
Lambda ring, structure for $K_0$      163
Lambda ring, structure for higher K-theory      179—180
Leibnitz rule      295
Localization of a triangulated category      425—426
Localization of a triangulated tensor category      426
Localization, connecting homomorphism      299—302
Localization, distinguished triangle      22
Localization, distinguished triangle for homological motives      216
Localization, distinguished triangle for motives with support      22
Localization, distinguished triangle for relative motives      34
Localization, sequence      126 129
Localization, sequence, compatibility with Chern classes      128—130
MAH complex      285
Mayer — Vietoris for homological motives      216
Mayer — Vietoris for motives      21
Mayer — Vietoris for motives with support      22
Milnor K-groups      293 298—303
Milnor K-groups and motivic cohomology      298
Milnor K-groups, tame symbol      302
Milnor K-sheaf      303
Mixed absolute Hodge complex      285
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