Local and stable homological algebra in Grothendieck abelian categories

TL;DR

This paper develops model category structures on chain complexes over Grothendieck abelian categories, facilitating the study of motives and mixed motives over schemes through homological algebra tools.

Contribution

It introduces new model structures depending on generating families, enhancing the understanding of triangulated categories of motives in algebraic geometry.

Findings

Model structures depend on generating families

Tools for constructing triangulated categories of motives

Application to mixed motives over schemes

Abstract

We define model category structures on the category of chain complexes over a Grothendieck abelian category depending on the choice of a generating family, and we study their behaviour with respect to tensor products and stabilization. This gives convenient tools to construct and understand triangulated categories of motives and we consider here the case of mixed motives over a regular base scheme.