On exact categories and applications to triangulated adjoints and model structures

TL;DR

This paper extends Quillen's small object argument to exact categories, enabling new insights into cotorsion pairs, triangulated adjoints, and model structures, especially in the context of complexes of quasi-coherent sheaves.

Contribution

It generalizes the small object argument to exact categories and applies it to derive new results on cotorsion pairs and model structures in algebraic geometry.

Findings

Extended Quillen's small object argument to exact categories

Established new links between cotorsion pairs and triangulated functors

Streamlined proofs of recent results in complexes of quasi-coherent sheaves

Abstract

We show that Quillen's small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, J{\o}rgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.