Models for homotopy categories of injectives and Gorenstein injectives

TL;DR

This paper introduces locally type $FP_{ abla}$ categories, generalizes the concept of absolutely clean objects, and explores their derived categories, embedding properties, and Gorenstein AC-injectives within a model category framework.

Contribution

It generalizes the framework of absolutely clean objects and Gorenstein AC-injectives to locally type $FP_{ abla}$ categories, including sheaves over schemes, and establishes new categorical and homotopical structures.

Findings

The derived category of absolutely clean objects is always compactly generated.

$ ext{D}( ext{AC})$ embeds fully into the homotopy category of injectives.

Gorenstein AC-injectives form the fibrant objects in a cofibrantly generated model structure.

Abstract

A natural generalization of locally noetherian and locally coherent categories leads us to define locally type $FP_{\infty}$ categories. They include not just all categories of modules over a ring, but also the category of sheaves over any concentrated scheme. In this setting we generalize and study the absolutely clean objects recently introduced in a paper of Bravo-Gillespie-Hovey. We show that $\mathcal{D}(\mathcal{AC})$, the derived category of absolutely clean objects, is always compactly generated and that it is embedded in $K(Inj)$, the chain homotopy category of injectives, as a full subcategory containing the DG-injectives. Assuming the ground category $\mathcal{G}$ has a set of generators satisfying a certain vanishing property, we also show that there is a recollement relating $\mathcal{D}(\mathcal{AC})$ to the (also compactly generated) derived category $\mathcal{D}(\mathcal{G})$. Finally, we generalize the Gorenstein AC-injectives of Bravo-Gillespie-Hovey, showing that they are the fibrant objects of a cofibrantly generated model structure on $\mathcal{G}$.