A note on homotopy categories of FP-Injectives

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.