Purity in categories of sheaves

TL;DR

This paper explores different notions of purity in categories of sheaves over schemes, analyzing their relationships and providing explicit examples including the Ziegler spectra for quasicoherent sheaves on the projective line.

Contribution

It introduces a detailed comparison of categorical and geometric purity in sheaf categories and computes explicit examples, enriching the understanding of purity in algebraic geometry.

Findings

Relations between categorical and geometric purity are clarified.

Explicit descriptions of Ziegler spectra for quasicoherent sheaves are provided.

Examples over the projective line illustrate the theoretical concepts.

Abstract

We consider categorical and geometric purity for sheaves of modules over a scheme satisfying some mild conditions, both for the category of all sheaves and for the category of quasicoherent sheaves. We investigate the relations between these four purities and compute a number of examples, in particular describing both the geometric and categorical Ziegler spectra for the category of quasicoherent sheaves over the projective line over a field.