Cartesian modules in small categories

TL;DR

This paper extends homological algebra results to cartesian modules over flat presheaves on small categories, enabling new applications to quasi-coherent sheaves on stacks, broadening the theoretical framework.

Contribution

It generalizes previous results to a broader setting of cartesian modules over flat presheaves on small categories, with applications to algebraic stacks.

Findings

Extended homological algebra results to new categories

Applied theory to quasi-coherent sheaves on stacks

Provided a framework for further research in algebraic geometry

Abstract

In this note we extend the main results of [E. Enochs and S. Estrada, Relative homological algebra in the category of quasi-coherent sheaves. Adv. in Math. 194(2005), 284-295] to the category of cartesian modules over a flat presheaf of rings and on an arbitrary small category. This provides with new applications of that paper to the categories of quasi-coherent sheaves on an Artin stack or on a Deligne-Mumford stack.