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Levine M. — Mixed motives
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Название: Mixed motivesАвтор: Levine M. Аннотация: 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.
Язык: Рубрика: Математика /Алгебра /Алгебраическая геометрия /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1998Количество страниц: 515Добавлена в каталог: 23.03.2005Операции: Положить на полку |
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Предметный указатель
Mixed Hodge modules 282 Monoidal category 375 Motive of a cosimplicial scheme 27—28 Motive of a cubical hyperresolution 241 Motive of a cubical hyperresolution, blow-up diagram for 241 Motive of a diagram 34—35 Motive of a diagram, distinguished triangle 35—36 Motive of a diagram, products 36 Motive of a diagram, properties 156 Motive of a k-scheme 241 Motive of a k-scheme, Borel — Moore 251—253 Motive of a k-scheme, cohomological 246—249 Motive of a k-scheme, compactly supported 251—253 Motive of a k-scheme, comparisons 250 Motive of a k-scheme, homological 249—250 Motive of a k-scheme, products 248 Motive of a non-degenerate simplicial scheme 30 Motive of a simplicial scheme 28—31 Motive of a variety 21 Motive of an n-cube 31—32 Motive with support 21 Motive with support of a diagram 156 Motive with support of a simplicial scheme 109 Motive, Borel — Moore 153 Motive, Borel — Moore for singular schemes 225 227—231 Motive, Borel — Moore with support 153 226 Motive, Borel — Moore, cap products 217 Motive, Borel — Moore, functoriality 153 Motive, Borel — Moore, properties 217 Motive, Cech resolution 54 58 Motive, compactly supported 215 Motive, compactly supported for singular schemes 227—231 Motive, compactly supported, cup products 217 Motive, compactly supported, properties 217 Motive, fundamental properties 19 Motive, homological 215 Motive, homological, cap products 217 Motive, homological, projection formula 217 Motive, homological, properties 216—217 Motive, homological, relative 216 Motive, homological, Thorn isomorphism 216 Motive, Lefschetz 23 Motive, open cover 54 58 Motive, open cover, push-forward 56 Motive, relative 31 33—34 123 Motive, relative with support 34 124 Motive, relative, duality for 219—224 Motivic, Borel — Moore functor 216 Motivic, Borel — Moore homology 215 Motivic, cohomology 21 22 Motivic, cohomology and higher Chow groups 103 Motivic, cohomology and homology of GL 121—122 Motivic, cohomology as Zariski hypercohomology 89 Motivic, cohomology for simplicial schemes 30—31 Motivic, cohomology of a diagram 36 Motivic, cohomology with compact support 215 Motivic, cohomology, mod n 21 Motivic, cohomology, relative 34 123 218 225 Motivic, cohomology, semi-purity 140 Motivic, cohomology, special properties 89 Motivic, cycle complex 67 Motivic, cycles functor 38 Motivic, DG category 12 36 Motivic, DG category, construction 12—16 Motivic, DG category, structure 36—38 Motivic, functor with compact support 216 Motivic, Gersten complex 91 Motivic, Gersten conjecture 92 Motivic, Gersten resolution 93 Motivic, homological functor 215 Motivic, homology 215 Motivic, homology, relative 225 Motivic, homotopy category 40 Motivic, homotopy category, structure 40—44 Motivic, hypercohomology 62 Motivic, local to global spectral sequence 90 Motivic, polylogarithm 303 Motivic, polylogarithm as a diagram of schemes 303—304 Motivic, polylogarithm as a motive 309 Motivic, polylogarithm, distinguished triangle 309 Motivic, polylogarithm, spectral sequence 305—306 Motivic, pull-back 24 Motivic, Quillen spectral sequence 91 Motivic, suspension 68 70 76 187 Motivic, triangulated category 16 44 Motivic, triangulated category, definition 17—19 Motivic, triangulated category, structure 44—47 Motivic, triangulated Tate category 234 235 Moving Lemma 20 94 96 102 Moving lemma for diagrams 36 Moving lemma, isomorphism 18 Multiplication for a cosimplicial functor 464 Multiplication in a symmetric monoidal category 453 Multiplication of cosimplicial objects 452 Multiplication, commutative 453 Multiplicative system 424—425 N-cubes 32 N-cubes, distinguished triangle 33 N-cubes, lifting 32 N-cubes, relative motives 31 Negligible complexes 270 Nerve of a category 468 Nisnevic sheaf with transfers 311 Nisnevic sheaf with transfers, homotopy invariant 311 Nisnevic topology 311 Normal crossing subschemes 208 Normalized complex of a cosimplicial abelian group 275 Normalized complex of an inverse system 269 Octahedra 429 Operad 444 Pentagonal identity 375 Permutative bi-module 402—403 Perverse sheaf 282 Plus construction and Q-construction 359 Plus construction for diagrams 361—362 Plus construction for diagrams, H-group structure 365 Plus construction for rings 359 Plus construction for schemes 360 Point 483 Point, associated pro-object 485 Point, conservative family 484 Point, neighborhood of a 485 Point, stalk 484 Pre-additive category 378 Pre-differential graded category 379 Pre-graded category 378 Pre-tensor category 382 Pre-Tr 410—411 Presheaf of sets 482 Presheaf with values in a category 482 Products and holim 476—479 Products for cosimplicial objects 452 Products for equi-dimensional cycles 355 Products for motives of diagrams 36 Products for motives of simplicial schemes 31 Products for motivic Borel — Moore homology 217 Products for motivic cohomology 22 Products for motivic cohomology with compact support 217 Products for motivic homology 217 Products, Alexander — Whitney 452 Projection formula for a projection 144 Projection formula for a projective morphism 150 Projection formula for an embedding 136 Projective bundle formula 113 114 Properties 66 Pseudo-abelian hull 427 Pseudo-abelian hull of a tensor category 427 Pseudo-abelian hull of a triangulated category 427—433 Pseudo-tensor functor 382—383 Push-forward for a projection 142 Push-forward for a projective morphism 146 Push-forward for a projective morphism, compatibility with cycle classes 152 Push-forward for a projective morphism, compatibility with products 151 Push-forward for a projective morphism, compatibility with pull-back 150 Push-forward for a projective morphism, definition 146—147 Push-forward for a projective morphism, functoriality 149 Push-forward for a projective morphism, naturality 151 Push-forward for diagrams 153 159—161 Push-forward for K-theory 171—172 Push-forward, additional properties 151 Q-construction 358 Real Frobenius 268 Realization for geometric cohomology theories 261 Realization, Betti 267 Realization, Betti, complex 267—268 Realization, Betti, real 268 Realization, Chow 79 80 Realization, etale 268 271—272 Realization, etale, 272 Realization, etale, mod n 272 Realization, functor 260 266 Realization, functor for cohomology 267 Realization, Hodge 273 280 281 Realization, Hodge over a smooth base 282 284 Realization, Hodge, real 281 282 Realization, motivic 284 291 Reduction 240 Relative cycles 331 Relativization, distinguished triangle 33 Relativization, sequence 126 129 Relativization, sequence, compatibility with Chern classes 128—130 Resolution of singularities 237 Riemann — Roch for singular schemes 231 233 Riemann — Roch for smooth schemes 176 177 Riemann — Roch without denominators 171 174 S-smooth stratification 225 Semi-monoidal category 375—376 Semi-monoidal category, strictly associative 375 Sheaf of -R-modules 486 Sheaf of -R-modules, flat 497 Sheaf of sets 482 Sheaf with values in a category 482 Sheafification of a presheaf 482 Simplicial and cosimplicial objects 449 Simplicial closed subset 109—110 Simplicial schemes, lifting 29 Simplicial schemes, line bundles on 112 Simplicial schemes, vector bundles on 110 Smoothly decomposable scheme 225 Splitting idempotents 433 Splitting principle for 164 Splitting principle for the motive of a diagram 161 Splitting principle for the motive of a simplicial scheme 116 Stabilization map 358 Stalk 484 Stalk of a presheaf 486 Stalk of a sheaf 486 Steinberg relation 295 296 Structured category 377 Support of a cycle 332 Suspension 67 Symmetric monoidal 380 Symmetric monoidal 384 Symmetric monoidal category 375 380 Symmetric monoidal category, constructions 383 385 386 Symmetric monoidal category, examples 376 Symmetric monoidal category, graded 436 Symmetric semi-monoidal category 375 Tate mixed Hodge structure 279 Tate twist 20 Tensor category 381 Tensor category, DG 382 Tensor category, DG without unit 382 Tensor category, graded 381 Tensor category, graded without unit 382 Tensor category, triangulated 416 Thick subcategory 424 Thom — Sullivan cochains 275—277 Thom — Sullivan cochains, tempered 287—289 Todd character 176 Todd class 176 Topos 483 Topos as a site 483 Topos with enough points 484 Topos, fiber functor 483 Topos, point of a 483 Total Chern class 116 Total complex functor 412—413 Translation structure 403 Translation structure for complexes over an additive category 409 Translation structure for Pre-Tr 411 Translation structure, free 406—409 Translation structure, free compatible 408 Translation structure, tensor compatible 404 Translation structures 401 Transverse cartesian square 150 Triangulated category 414 Triangulated category of complexes over a DG category 416 Triangulated category, axioms 414—415 Triangulated category, exact functor 415 Triangulated category, five lemma 415 Triangulated category, localization 424—426 Triangulated category, multiplicative system 424—425 Triangulated category, pseudo-abelianization 427 Triangulated category, thick subcategory 424 Triangulated tensor category 416 Triangulated tensor category, localization 426 Triangulation 477 Twisted duality theory 258 Unit isomorphism 18 Unit motive 20 Universal vector bundle 358 Whitney product formula 117
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