On the Nori and Hodge realisations of Voevodsky motives

Abstract

We show that the derived category of perverse Nori motives and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$-categorical lifts of the six operations and therefore to obtain realisation functors from the category of Voevodsky étale motives to the derived categories of perverse Nori motives and mixed Hodge modules that commute with the operations. We give a proof that the realisation induces an equivalence of categories between Artin motives in the category of étale motives and Artin motives in the derived category of Nori motives. We also prove that if a motivic $t$-structure exists then Voevodsky étale motives and the derived category of perverse Nori motives are equivalent. Finally we give a presentation of the indization of the derived category of perverse Nori motives as a category of modules in Voevodsky étale motives that gives a continuity result for perverse Nori motives.

arXiv comment58 pages, accepted version

References in corpus (11)

• Triangulated categories of mixed motives
• Étale motives
• Structures de poids à la Bondarko sur les motifs de Beilinson
• The intersection motive of the moduli stack of shtukas
• The six-functor formalism for rigid analytic motives
• The universal six-functor formalism
• An isomorphism of motivic Galois groups
• Adjoint functor theorems for $\infty$-categories
• Tensor triangular geometry of filtered objects and sheaves
• Tannaka duality and stable infinity-categories
• Mixed motivic sheaves (and weights for them) exist if 'ordinary' mixed motives do

Cited by in corpus (3)

• Tensor structure on perverse Nori motives
• Malcev Completions, Hodge Theory, and Motives
• The integral identity conjecture in motivic homotopy theory