On the Nori and Hodge realisations of Voevodsky motives
Swann Tubach
TL;DR
The paper constructs ∞-categorical realisation functors from Voevodsky’s étale motives to the derived categories of perverse Nori motives and mixed Hodge modules, lifting the six operations to a coherent ∞-categorical framework. It shows that perverse Nori motives arise as the hearts of motivic t-structures in étale motives and realisations factor compatibly with Betti and ℓ-adic theories, with Nori motives expressed as modules over an algebra object in DM^{et}. A key result is that Artin motives in étale motives correspond to Artin motives in the Nori setting, and, under a conjectural motivic t-structure, DM^{et}_c(X) would be equivalent to D^b(M_{perv}(X)). The work unifies motivic realisations, ∞-categorical enhancements, and module presentations, enabling continuity and comparison across motivic contexts and suggesting a path toward an integral, weight-aware motivic framework. These results provide a robust pathway to compare and transport cohomological theories through motivic realisations with stability under the six operations, impacting the study of motives, Hodge theory, and their interrelations.
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.