Pure injective and absolutely pure sheaves

TL;DR

This paper explores two notions of purity in sheaf categories, proves the existence of pure injective envelopes under broad conditions, and characterizes certain subschemes using locally absolutely pure sheaves.

Contribution

It introduces the concept of locally absolutely pure sheaves and characterizes locally Noetherian subschemes via this new class.

Findings

Pure injective envelopes exist under general assumptions.

Introduction of locally absolutely pure sheaves.

Characterization of locally Noetherian subschemes.

Abstract

We study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally we introduce the class of locally absolutely pure (quasi--coherent) sheaves, with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.