Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves

TL;DR

This paper develops a comprehensive framework for constructing compatible model structures on exact categories, linking approximation theory and cotorsion pairs, with applications to derived categories of quasi-coherent sheaves.

Contribution

It generalizes existing results on model structures in exact categories and applies these to the derived category of quasi-coherent sheaves over schemes.

Findings

Constructed compatible model structures on exact categories.

Established connections between model structures, approximation theory, and cotorsion pairs.

Applied the theory to monoidal model structures for derived categories of sheaves.

Abstract

Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close connection of this theory to approximation theory and cotorsion pairs. We also discuss the motivating applications with the emphasis on constructing monoidal model structures for the derived category of quasi-coherent sheaves of modules over a scheme.