Approximations and Hovey triples by objects of finite homological dimensions: Applications to sheaves

TL;DR

This paper develops a framework for approximating objects in abelian categories using finite homological dimensions, leading to new model structures for sheaves on schemes.

Contribution

It introduces weaker assumptions for Hovey triples, constructs hereditary Hovey triples from resolution dimensions, and applies these to sheaf categories.

Findings

Established that $ ext{Q}_n$ forms the first class of a hereditary Hovey triple.

Proved $ ext{Q}_n$ is part of a complete hereditary cotorsion pair.

Constructed an abelian model structure on quasi-coherent sheaves with Gorenstein dimensions.

Abstract

Let $\mathcal{Q}$ be a class of objects in an abelian category $\mathcal{A}$ which need not have enough projective or injective objects. In this paper, we prove that if $\mathcal{Q}$ is the first class of a Hovey triple $(\mathcal{Q},\mathcal{W},\mathcal{R})$ in $\mathcal{A}$ satisfying certain assumptions-weaker than those required in the recent literature-then $\mathcal{Q}_n$, the class of objects with $\mathcal{Q}$-resolution dimension at most an integer $n\ge 0$, forms the first class of a hereditary Hovey triple $\mathcal{M}_n=(\mathcal{Q}_n,\mathcal{W}_{\mathcal{Q},n},\mathcal{R}_{\mathcal{Q},n})$, where $\mathcal{W}_{\mathcal{Q},n}$ and $\mathcal{R}_{\mathcal{Q},n}$ are described explicitly. Consequently, $\mathcal{Q}_n$ is the left-hand side of a complete hereditary cotorsion pair and hence a special precovering class. The dual statement is also established. As a main application, we construct an abelian model structure on $Qcoh(X)$, the category of quasi-coherent sheaves over a semi-separated Noetherian scheme $X$, in which the cofibrant (resp. fibrant) objects are precisely the sheaves with Gorenstein flat (resp. Gorenstein injective) dimension at most $n$.