Cotorsion Pairs and Quillen Adjunctions

TL;DR

This paper establishes conditions under which adjunctions between abelian categories induce Quillen adjunctions on their chain complex categories, facilitating computations like the derived tensor product using model structures.

Contribution

It provides new criteria for when adjunctions induce Quillen adjunctions on chain complexes, based on properties of cotorsion pairs in abelian categories.

Findings

Criteria for Quillen adjunctions depend on cotorsion pairs.

Model structures can be checked at the abelian category level.

Applications include computing derived tensor products.

Abstract

Let $F:\mathcal{A}\to \mathcal{B}$ be a left adjoint between abelian categories and let $Ch(F)$ be the induced left adjoint on chain complexes. If the abelian categories $\mathcal{A}$ and $\mathcal{B}$ are equipped with sufficiently nice cotorsion pairs, then we find model structures on their categories of chain complexes. This paper gives novel criteria under which the induced adjunction becomes a Quillen adjunction. In particular, our criteria can be checked on the level of the underlying abelian categories instead of chain complexes. We do the same for adjunctions of $n$ variables and we also show how our results imply that the flat model structure can be used to compute the derived tensor product.