A note on homotopy categories of FP-injectives

For a locally finitely presented Grothendieck category $\mathcal{A}$, we consider a certain subcategory of the homotopy category of FP-injectives 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-injectives 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.

Keywords

FP-injective, purity, locally coherent category, compactly generated triangulated category

2010 Mathematics Subject Classification

16E35, 18E30, 18G25

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The author is supported by the Fundación Séneca of Murcia 19880/GERM/15.

Received 5 March 2018

Received revised 20 June 2018

Accepted 28 June 2018

Published 29 August 2018