A note on homotopy categories of FP-Injectives

TL;DR

This paper studies a specific subcategory of the homotopy category of FP-injective objects in a Grothendieck category, showing it is compactly generated and identifying it with the derived category in locally coherent cases.

Contribution

It introduces a new subcategory of the homotopy category of FP-injectives, proves it is compactly generated, and relates it to the derived category in certain categories, extending previous dual results.

Findings

The subcategory of FP-injective homotopy category is compactly generated.

In locally coherent categories, this subcategory equals the derived category of FP-injectives.

The results extend duality principles similar to Neeman's work on flat modules.

Abstract

For a locally finitely presented Grothendieck category $\mathcal{A}$, we consider a certain subcategory of the homotopy category of FP-injective objects in $\mathcal{A}$ which we show is compactly generated. In the case where $\mathcal{A}$ is locally coherent, we identify this subcategory with the derived category of FP-injective objects in $\mathcal{A}$. Our results are, in a sense, dual to the ones obtained by Neeman on the homotopy category of flat modules. Our proof is based on extending a characterization of the pure acyclic complexes which is due to Emmanouil.