Triangulated Categories of Mixed Motives

Contents......Page 6
A.1 The conjectural theory described by Beilinson......Page 11
A.2 Voevodsky’s motivic complexes......Page 12
A.3 Morel and Voevodsky’s homotopy theory......Page 13
A.4 Voevodsky’s cross functors and Ayoub’s thesis......Page 14
A.5 The Grothendieck six functors formalism......Page 15
B Voevodsky’s motivic complexes......Page 18
C.1 Definition and fundamental properties......Page 20
C.2 Constructible Beilinson motives......Page 22
C.3 Comparison theorems......Page 23
C.4 Realizations......Page 26
D.1 The Grothendieck six functors formalism (Part I)......Page 28
D.2 The constructive part (Part II)......Page 32
D.3 Motivic complexes (Part III)......Page 33
D.4 Beilinson motives (Part IV)......Page 36
E.1 Nisnevich motives with integral coefficients......Page 37
E.2 Étale motives with integral coefficients and ℓ-adic realization......Page 38
E.3 Motivic stable homotopy theory with rational coefficients......Page 39
E.5 Enriched realizations......Page 40
Notations and conventions......Page 41
Part I Fibred categories and the six functors formalism......Page 43
1.1.a Definitions......Page 44
1.1.b Monoidal structures......Page 51
1.1.c Geometric sections......Page 54
1.1.d Twists......Page 56
1.2.a The general case......Page 58
1.2.b The monoidal case......Page 59
1.3.a Abstract definition......Page 61
1.3.b The abelian case......Page 63
1.3.c The triangulated case......Page 64
1.3.d The model category case......Page 67
1.4 Premotivic categories......Page 69
2 Triangulated P-fibred categories in algebraic geometry......Page 72
2.1 Elementary properties......Page 73
2.2.a The support axiom......Page 77
2.2.b Exceptional direct image......Page 79
2.2.c Further properties......Page 84
2.3.a Definition......Page 87
2.3.b First consequences of localization......Page 89
2.3.c Localization and exchange properties......Page 91
2.3.d Localization and monoidal structure......Page 94
2.4.a The stability property......Page 98
2.4.b The purity property......Page 103
2.4.c Duality, purity and orientation......Page 109
2.4.d Motivic categories......Page 119
3.1.a The general case......Page 123
3.1.b The model category case......Page 126
3.2 Hypercovers, descent, and derived global sections......Page 139
3.3 Descent over schemes......Page 150
3.3.a Localization and Nisnevich descent......Page 151
3.3.b Proper base change isomorphism and descent by blow-ups......Page 153
3.3.c Proper descent with rational coefficients I: Galois excision......Page 155
3.3.d Proper descent with rational coefficients II: separation......Page 166
4.1 Resolution of singularities......Page 171
4.2 Finiteness theorems......Page 174
4.3 Continuity......Page 186
4.4 Duality......Page 194
Part II Construction of fibred categories......Page 207
5.1.a Abelian premotives: recall and examples......Page 208
5.1.b The t-descent model category structure......Page 210
5.1.c Constructible premotivic complexes......Page 220
5.2.a Localization of triangulated premotivic categories......Page 225
5.2.b The homotopy relation......Page 231
5.2.c Explicit A1-resolution......Page 236
5.2.d Constructible A1-local premotives......Page 240
5.3.a Modules......Page 242
5.3.b Symmetric sequences......Page 244
5.3.c Symmetric Tate spectra......Page 246
5.3.d Symmetric Tate Ω-spectra......Page 249
5.3.e Constructible premotivic spectra......Page 255
6.1 Generalized derived premotivic categories......Page 258
6.2 The fundamental example......Page 262
6.3 Nearly Nisnevich sheaves......Page 263
6.3.a Support property (effective case)......Page 264
6.3.b Support property (stable case)......Page 267
6.3.c Localization for smooth schemes......Page 268
7.1 Rings......Page 269
7.2 Modules......Page 275
Part III Motivic complexes and relative cycles......Page 283
8.1.a The category of cycles......Page 284
8.1.b Hilbert cycles......Page 286
8.1.c Specialization......Page 289
8.1.d Pullback......Page 293
8.2.a Commutativity......Page 301
8.2.b Associativity......Page 302
8.2.c Projection formulas......Page 304
8.3.a Constructibility......Page 305
8.3.b Samuel multiplicities......Page 309
9.1 Definition and composition......Page 316
9.2 Monoidal structure......Page 322
9.3.a Base change......Page 323
9.3.b Restriction......Page 324
9.3.c A finiteness property......Page 325
9.4 The fibred category of correspondences......Page 326
10 Sheaves with transfers......Page 327
10.1 Presheaves with transfers......Page 328
10.2 Sheaves with transfers......Page 329
10.3 Associated sheaf with transfers......Page 331
10.4 Examples......Page 339
10.5.a Change of coefficients......Page 341
10.5.c qfh-sheaves and transfers......Page 342
11.1.a Premotivic categories......Page 345
11.1.b Constructible and geometric motives......Page 347
11.1.c Enlargement, descent and continuity......Page 349
11.2.a Definition and functoriality......Page 352
11.2.b Effective motivic cohomology in weight 0 and 1......Page 354
11.2.c The motivic cohomology ring spectrum......Page 359
11.3 Orientation and purity......Page 361
11.4 The six functors......Page 365
Part IV Beilinson motives and algebraic K-theory......Page 370
12.2 Orientation......Page 371
13.1 The K-theory spectrum......Page 374
13.2 Periodicity......Page 375
13.3 Modules over algebraic K-theory......Page 376
13.4 K-theory with support......Page 377
13.5 The fundamental class......Page 379
13.6 Absolute purity for K-theory......Page 380
13.7 Trace maps......Page 382
14.1 The γ-filtration......Page 386
14.2 Definition......Page 388
14.3 Motivic proper descent......Page 393
14.4 Motivic absolute purity......Page 395
15.1 Definition and basic properties......Page 396
15.2 The Grothendieck six functors formalism and duality......Page 398
16.1 Comparison with Voevodsky motives......Page 399
16.2 Comparison with Morel motives......Page 403
17.1 Tilting......Page 411
17.2 Mixed Weil cohomologies......Page 416
References......Page 427
Index......Page 434
Notation......Page 441
Index of properties of P-fibred triangulated categories......Page 442