The homotopy category of acyclic complexes of pure-projective modules

TL;DR

This paper demonstrates that the homotopy category of acyclic complexes of pure-projective modules over any ring is compactly generated and establishes a new model for the derived category through abelian model structures.

Contribution

It constructs abelian model structures that relate the homotopy category of acyclic pure-projective complexes to the derived and pure derived categories, providing a new perspective.

Findings

The homotopy category is compactly generated.

Recollement links to derived and pure derived categories.

New model for the derived category introduced.

Abstract

Let $R$ be any ring with identity. We show that the homotopy category of all acyclic chain complexes of pure-projective $R$-modules is a compactly generated triangulated category. We do this by constructing abelian model structures that put this homotopy category into a recollement with two other compactly generated triangulated categories: The usual derived category of $R$ and the pure derived category of $R$. This also gives a new model for the derived category.