Triangulated categories of mixed motives

Abstract

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, Röndigs, Østvær, and others. In particular, the relation of motives with $K$-theory is addressed in full, and we prove the absolute purity theorem with rational coefficients, using Quillen's localization theorem in algebraic $K$-theory together with a variation on the Grothendieck-Riemann-Roch theorem. Using resolution of singularities via alterations of de Jong-Gabber, this leads to a version of Grothendieck-Verdier duality for constructible motivic sheaves with rational coefficients over rather general base schemes. We also study versions with integral coefficients, constructed via sheaves with transfers, for which we obtain partial results. Finally, we associate to any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors, establishing a correspondence between mixed Weil cohomologies and suitable systems of coefficients. The results of this book have already served as ground reference in many subsequent works on motivic sheaves and their realizations, and pointers to the most recent developments of the theory are given in the introduction.

This is the final version. To appear in the series Springer Monographs in Mathematics

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• The motive of a classifying space
• Enhanced six operations and base change theorem for higher Artin stacks
• Weights for relative motives; relation with mixed complexes of sheaves
• Rigidity in etale motivic stable homotopy theory
• The weight complex functor is symmetric monoidal
• Künneth formulas for motives and additivity of traces
• Positive model structures for abstract symmetric spectra
• Preorientations of the derived motivic multiplicative group
• Gluing pseudo functors via $n$-fold categories
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• Borel isomorphism and absolute purity
• Triangulated categories of motives over fs log schemes
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Cited by in corpus (136)

• The six operations in equivariant motivic homotopy theory
• From algebraic cobordism to motivic cohomology
• A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
• Fundamental classes in motivic homotopy theory
• Motivic and Real Etale Stable Homotopy Theory
• The motive of a classifying space
• Weights for relative motives; relation with mixed complexes of sheaves
• Perfection in motivic homotopy theory
• Donaldson-Thomas theory for categories of homological dimension one with potential
• Arakelov motivic cohomology I
• On the effectivity of spectra representing motivic cohomology theories
• On higher regulators of Siegel threefolds II: the connection to the special value
• The intersection motive of the moduli stack of shtukas
• MW-motivic complexes
• The six-functor formalism for rigid analytic motives
• The Milnor-Witt motivic ring spectrum and its associated theories
• A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections
• Virtual fundamental classes of derived stacks I
• Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes
• Intermediate extension of Chow motives of Abelian type
• Gersten weight structures for motivic homotopy categories; direct summands of cohomology of function fields and coniveau spectral sequences
• K-theory and G-theory of derived algebraic stacks
• The rigid syntomic ring spectrum
• Triangulated categories of motives in positive characteristic
• Rigidity in etale motivic stable homotopy theory
• On Modules Over Motivic Ring Spectra
• Bivariant theories in motivic stable homotopy
• The localization theorem for framed motivic spaces
• On the rational motivic homotopy category
• Polylogarithm for families of commutative group schemes
• On relative $K$-motives, weights for them, and negative $K$-groups
• The motivic Satake equivalence
• The intrinsic stable normal cone
• Rigid cohomology via the tilting equivalence
• Motivic Realizations of Singularity Categories and Vanishing Cycles
• Gluing restricted nerves of $\infty$-categories
• Framed transfers and motivic fundamental classes
• Nakayama categories and groupoid quantization
• Effective motives with and without transfers in characteristic $p$
• A formula for the Voevodsky motive of the moduli stack of vector bundles on a curve
• Stable operations and cooperations in derived Witt theory with rational coefficients
• Anticyclotomic Euler systems for unitary groups
• Mixed Artin-Tate motives over number rings
• Motivic complexes and special values of zeta functions
• The tame site of a scheme
• A characterization of finite étale morphisms in tensor triangular geometry
• Gersten weight structures for motivic homotopy categories; retracts of cohomology of function fields, motivic dimensions, and coniveau spectral sequences
• Towards the dual motivic Steenrod algebra in positive characteristic
• The Grothendieck ring of varieties and algebraic K-theory of spaces
• On constructing weight structures and extending them to idempotent extensions
• Adjoint functors on the derived category of motives
• The p-cohomology of algebraic varieties and special values of zeta functions
• Weights and t-structures: in general triangulated categories, for 1-motives, mixed motives, and for mixed Hodge complexes and modules
• Gluing pseudo functors via $n$-fold categories
• Algebraic Cobordism and Étale Cohomology
• On the Nori and Hodge realisations of Voevodsky motives
• On the intersection motive of certain Shimura varieties: the case of Siegel threefolds
• Intersecting the dimension filtration with the slice one for (relative) motivic categories
• The slice spectral sequence for singular schemes and applications
• Motivic Homotopy Theory of Algebraic Stacks
• A better comparison of cdh- and ldh-cohomologies
• Tate motives on Witt vector affine flag varieties
• Motivic twisted K-theory
• Positive model structures for abstract symmetric spectra
• Thom isomorphisms in triangulated motivic categories
• MW-homotopy sheaves and Morel generalized transfers
• Quaternionic Projective Bundle Theorem and Gysin Triangle in MW-Motivic Cohomology
• On the Northcott property for special values of L-functions
• Overconvergent global analytic geometry
• Tensor structure on perverse Nori motives
• Tensor structures on fibered categories
• Derivator Six Functor Formalisms -- Definition and Construction I
• Artin-Tate motivic sheaves with finite coefficients over an algebraic variety
• Derivator Six-Functor-Formalisms -- Construction II
• Stable maps and singular curves on K3 surfaces
• Cohomological Hall algebras and vertex algebras
• K-Motives and Koszul Duality
• On perverse homotopy $t$-structures, coniveau spectral sequences, cycle modules, and relative Gersten weight structures
• Category Of C-Motives Over Finite Fields
• On weights for relative motives with integral coefficients
• Detecting effectivity of motives, their weights, connectivity, and dimension via Chow-weight (co)homology: a "mixed motivic decomposition of the diagonal"
• A note on Tannakian categories and mixed motives
• The Berkovich realization for rigid analytic motives
• Motivic Birational Covers and Finite Filtrations on Chow Groups
• The geometric Satake equivalence for integral motives
• Projective Bundle Theorem in MW-Motivic Cohomology
• Triangulated categories of motives over fs log schemes
• Borel isomorphism and absolute purity
• Mahler's measure and elliptic curves with potential complex multiplication
• K-theory Soergel Bimodules
• Derived categories and birationality
• On some finiteness results in real étale cohomology
• Motivic interpretation of Albanese varieties of smooth varieties
• A Motivic Snaith Decomposition
• General Motivic Cohomology and Symplectic Orientation
• On uniqueness of $p$-adic period morphisms, II
• Description of mixed motives
• Chow motives without projectivity, II
• Constructing monoidal structures on fibered categories via factorizations
• Universality of Barwick's unfurling construction
• Graded and Geometric Parabolic Induction for Category $\mathcal{O}$
• Dimensional homotopy t-structure in motivic homotopy theory
• On the Riemann-Roch formula without projective hypothesis
• Galois equivariance and stable motivic homotopy theory
• Motivic homology of semiabelian varieties
• Orientable homotopy modules
• Stable motivic pi_1 of low-dimensional fields
• Motivic splitting principle
• Parabolic induction and perverse sheaves on h/W
• $T$-equivariant motives of flag varieties
• Motivic Steenrod operations in characteristic $p$
• Grothendieck Duality theories -- abstract and concrete, I: pseudo-coherent finite maps
• Milnor-Witt motivic cohomology of complements of hyperplane arrangements
• Absolute intersection motive
• Motivic homological stability of configuration spaces
• Integration of Voevodsky motives
• A Gersten complex on real schemes
• On the Orientability of the Slice Filtration
• Motivic Intersection Complex of Certain Shimura varieties
• The localization spectral sequence in the motivic setting
• Motivic mirror symmetry and $\chi$-independence for Higgs bundles in arbitrary characteristic
• Modules over the de Rham cohomology spectrum
• The stable Galois correspondence for real closed fields
• Trace formalism for motivic cohomology
• Deligne-Beilinson cycle maps for Lichtenbaum cohomology
• Zariski-local framed ${\mathbb{A}}^1$-homotopy theory
• Trace maps in motivic homotopy and local terms
• From Six Functors Formalisms to Derived Motivic Measures
• Embedding of category of twisted Chow-Witt correspondences into geometric stable $\A^1$-derived category over a field
• The integral identity conjecture in motivic homotopy theory
• Chow groups and equivariant geometry
• Rational and $p$-local Motivic Homotopy Theory
• On the higher Riemann-Roch without denominators
• E-motives and motivic stable homotopy
• Motives for an elliptic curve