Nori motives (and mixed Hodge modules) with integral coefficients
Raphaël Ruimy, Swann Tubach
TL;DR
The paper constructs an integral theory of Nori motivic sheaves over characteristic-zero schemes by passing from Nori motives over fields to a six-functor formalism inside étale motives, realized via a universal Nori algebra. Relative and geometric Nori motives are realized as modules over the Nori algebra, with ordinary and perverse t-structures whose hearts yield abelian categories of integral Nori motives, and with motivic Leray spectral sequences and arc-descent results. The theory extends to integral mixed Hodge modules over real and complex schemes, providing a parallel six-functor framework and connections to variations of mixed Hodge structures with integral lattices. Collectively, the work unifies motivic and Hodge-theoretic perspectives at the integral level, showing deep compatibilities with Betti/ℓ-adic realizations and offering a robust foundation for integral motivic t-structures and descent properties.
Abstract
We construct abelian categories of integral Nori motivic sheaves over a scheme of characteristic zero. The first step is to study the presentable derived category of Nori motives over a field. Next we construct an algebra in étale motives such that modules over it afford a t-structure that restricts to constructible objects. This category of integral Nori motives has the six operations and arc-descent. We finish by providing analogous constructions and results for mixed Hodge modules on schemes over the reals.