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A natural generalization of locally noetherian and locally coherentcategories leads us to define locally type $FP_{\infty}$ categories. Theyinclude not just all categories of modules over a ring, but also the categoryof sheaves over any concentrated scheme. In this setting we generalize andstudy the absolutely clean objects recently introduced in a paper ofBravo-Gillespie-Hovey. We show that $\mathcal{D}(\mathcal{AC})$, the derivedcategory of absolutely clean objects, is always compactly generated and that itis embedded in $K(Inj)$, the chain homotopy category of injectives, as a fullsubcategory 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 modelstructure on $\mathcal{G}$.

Models for homotopy categories of injectives and Gorenstein injectives