Models for homotopy categories of injectives and Gorenstein injectives

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 $\mathcal{G}$.