On purity and applications to coderived and singularity categories

TL;DR

This paper proves that the homotopy category of injective complexes in a locally coherent Grothendieck category is compactly generated, with the subcategory of compact objects identified as the bounded derived category of finitely presentable objects.

Contribution

It establishes the compact generation of coderived and derived categories in locally coherent Grothendieck categories using model theory and pure derived categories.

Findings

Homotopy category of injectives is compactly generated.

Compact objects correspond to bounded derived category of finitely presentable objects.

Conditions for the existence of Krause's recollement are identified.

Abstract

Given a locally coherent Grothendieck category G, we prove that the homotopy category of complexes of injective objects (also known as the coderived category of G) is compactly generated triangulated. Moreover, the full subcategory of compact objects is none other than D^b(fp G). If G admits a generating set of finitely presentable objects of finite projective dimension, then also the derived category of G is compactly generated and Krause's recollement exists. Our main tools are (a) model theoretic techniques and (b) a systematic study of the pure derived category of an additive finitely accessible category.