Triangulated Categories of Mixed Motives (Springer Monographs in Mathematics)

The primary aim of this monograph is to achieve part of Beilinson’s program on mixed motives using Voevodsky’s theories of **A**1-homotopy and motivic complexes. Historically, this book is the first to give a complete construction of a triangulated category of mixed motives with rational coefficients satisfying the full Grothendieck six functors formalism as well as fulfilling Beilinson’s program, in particular the interpretation of rational higher Chow groups as extension groups. Apart from Voevodsky’s entire work and Grothendieck’s SGA4, our main sources are Gabber’s work on étale cohomology and Ayoub’s solution to Voevodsky’s cross functors theory. We also thoroughly develop the theory of motivic complexes with integral coefficients over general bases, along the lines of Suslin and Voevodsky. Besides this achievement, this volume provides a complete toolkit for the study of systems of coefficients satisfying Grothendieck’ six functors formalism, including Grothendieck-Verdier duality. It gives a systematic account of cohomological descent theory with an emphasis on h-descent. It formalizes morphisms of coefficient systems with a view towards realization functors and comparison results. The latter allows to understand the polymorphic nature of rational mixed motives. They can be characterized by one of the following properties: existence of transfers, universality of rational algebraic K-theory, h-descent, étale descent, orientation theory. This monograph is a longstanding research work of the two authors. The first three parts are written in a self-contained manner and could be accessible to graduate students with a background in algebraic geometry and homotopy theory. It is designed to be a reference work and could also be useful outside motivic homotopy theory. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given. 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 "The primary aim of this monograph is to achieve part of Beilinsons program on mixed motives using Voevodskys theories of A1-homotopy and motivic complexes. Historically, this book is the first to give a complete construction of a triangulated category of mixed motives with rational coefficients satisfying the full Grothendieck six functors formalism as well as fulfilling Beilinsons program, in particular the interpretation of rational higher Chow groups as extension groups. Apart from Voevodskys entire work and Grothendiecks SGA4, our main sources are Gabbers work on étale cohomology and Ayoubs solution to Voevodskys cross functors theory. We also thoroughly develop the theory of motivic complexes with integral coefficients over general bases, along the lines of Suslin and Voevodsky. Besides this achievement, this volume provides a complete toolkit for the study of systems of coefficients satisfying Grothendieck six functors formalism, including Grothendieck-Verdier duality. It gives a systematic account of cohomological descent theory with an emphasis on h-descent. It formalizes morphisms of coefficient systems with a view towards realization functors and comparison results. The latter allows to understand the polymorphic nature of rational mixed motives. They can be characterized by one of the following properties: existence of transfers, universality of rational algebraic K-theory, h-descent, étale descent, orientation theory. This monograph is a longstanding research work of the two authors. The first three parts are written in a self-contained manner and could be accessible to graduate students with a background in algebraic geometry and homotopy theory. It is designed to be a reference work and could also be useful outside motivic homotopy theory. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given"--Publisher's description The primary aim of this monograph is to achieve part of Beilinson's program on mixed motives using Voevodsky's theories of $\mathbb{A}^1$-homotopy and motivic complexes. Historically, this book is the first to give a complete construction of a triangulated category of mixed motives with rational coefficients satisfying the full Grothendieck six functors formalism as well as fulfilling Beilinson's program, in particular the interpretation of rational higher Chow groups as extension groups. Apart from Voevodsky's entire work and Grothendieck's SGA4, our main sources are Gabber's work on étale cohomology and Ayoub's solution to Voevodsky's cross functors theory. We also thoroughly develop the theory of motivic complexes with integral coefficients over general bases, along the lines of Suslin and Voevodsky. Besides this achievement, this volume provides a complete toolkit for the study of systems of coefficients satisfying Grothendieck' six functors formalism, including Grothendieck-Verdier duality. It gives a systematic account of cohomological descent theory with an emphasis on h-descent. It formalizes morphisms of coefficient systems with a view towards realization functors and comparison results. The latter allows to understand the polymorphic nature of rational mixed motives. They can be characterized by one of the following properties: existence of transfers, universality of rational algebraic K-theory, h-descent, étale descent, orientation theory. This monograph is a longstanding research work of the two authors. The first three parts are written in a self-contained manner and could be accessible to graduate students with a background in algebraic geometry and homotopy theory. It is designed to be a reference work and could also be useful outside motivic homotopy theory. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given The primary aim of this monograph is to achieve part of Beilinson's program on mixed motives using Voevodsky's theories of A 1-homotopy and motivic complexes. Historically, this book is the first to give a complete construction of a triangulated category of mixed motives with rational coefficients satisfying the full Grothendieck six functors formalism as well as fulfilling Beilinson's program, in particular the interpretation of rational higher Chow groups as extension groups. Apart from Voevodsky's entire work and Grothendieck's SGA4, our main sources are Gabber's work on étale cohomology and Ayoub's solution to Voevodsky's cross functors theory. We also thoroughly develop the theory of motivic complexes with integral coefficients over general bases, along the lines of Suslin and Voevodsky. Besides this achievement, this volume provides a complete toolkit for the study of systems of coefficients satisfying Grothendieck' six functors formalism, including Grothendieck-Verdier duality. It gives a systematic account of cohomological descent theory with an emphasis on h-descent. It formalizes morphisms of coefficient systems with a view towards realization functors and comparison results. The latter allows to understand the polymorphic nature of rational mixed motives. They can be characterized by one of the following properties: existence of transfers, universality of rational algebraic K-theory, h-descent, étale descent, orientation theory. This monograph is a longstanding research work of the two authors. The first three parts are written in a self-contained manner and could be accessible to graduate students with a background in algebraic geometry and homotopy theory. It is designed to be a reference work and could also be useful outside motivic homotopy theory. The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given.