The Pure derived category of quasi-coherent sheaves

TL;DR

This paper investigates the pure derived category of quasi-coherent sheaves on schemes, focusing on geometrical purity and proposing new approaches for its categorical construction.

Contribution

It introduces replacements for geometrical pure derived categories of schemes, advancing the understanding of pure derived categories in algebraic geometry.

Findings

New constructions for geometrical pure derived categories

Enhanced understanding of categorical pure derived categories

Proposed methods applicable to non-semiseparated schemes

Abstract

Let X be a quasi-compact and quasi-separated (not necessarily semiseparated) scheme. The category QcoX of all quasi-coherent sheaves of OX-modules has several diferent pure derived categories. Recently, categorical pure derived categories of X have been studied in more details. In this work, we focus on the geometrical purity and find replacements for geometrical pure derived categories of X.