On Pure Derived Categories

TL;DR

This paper explores the properties of pure derived categories of module categories, demonstrating their similarities to classical derived categories and introducing new concepts like pure projective and injective dimensions.

Contribution

It introduces the pure projective and injective dimensions of complexes, providing criteria for their computation and characterizing the pure global dimension of rings.

Findings

Pure derived categories share many properties with classical derived categories.

Bounded pure derived categories can be realized as certain homotopy categories.

Criteria for computing pure projective and injective dimensions are established.

Abstract

We investigate the properties of pure derived categories of module categories, and show that pure derived categories share many nice properties of classical derived categories. In particular, we show that bounded pure derived categories can be realized as certain homotopy categories. We introduce the pure projective (resp. injective) dimension of complexes in pure derived categories, and give some criteria for computing these dimensions in terms of the properties of pure projective (resp. injective) resolutions and pure derived functors. As a consequence, we get some equivalent characterizations for the finiteness of the pure global dimension of rings. Finally, pure projective (resp. injective) resolutions of unbounded complexes are considered.