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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Cited by in corpus (137)

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