# On pure derived and pure singularity categories

## TL;DR

This paper investigates pure derived and pure singularity categories, comparing them with classical derived categories, and explores their properties, invariance, and relation to pure-global dimension of rings.

## Contribution

It introduces the concept of pure singularity categories and analyzes their invariance and relation to pure-global dimension, extending the understanding of pure-derived categories.

## Key findings

- Pure singularity category reflects pure-global dimension finiteness.
- Comparison between pure and usual derived categories under strong assumptions.
- Invariance of pure singularity in recollements of pure derived categories.

## Abstract

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient of bounded pure derived category modulo the bounded homotopy category of pure-projective modules, which is called a pure singularity category since we show that it reflects the finiteness of pure-global dimension of rings. Moreover, invariance of pure singularity in a recollement of bounded pure derived categories is studied.