The Ziegler spectrum for enriched ringoids and schemes

TL;DR

This paper generalizes the Ziegler spectrum to enriched categories like schemes, connecting classical ring theory with algebraic geometry and introducing new topological and categorical insights.

Contribution

It defines the Ziegler spectrum for enriched ringoids and schemes, extending classical concepts and establishing relationships with quasi-coherent sheaves and scheme spectra.

Findings

Ziegler spectrum for enriched categories is well-defined and studied.

Connections between the Ziegler spectrum and scheme spectra are established.

Recollement relates quasi-coherent and generalized quasi-coherent sheaves.

Abstract

The Ziegler spectrum for categories enriched in closed symmetric monoidal Grothendieck categories is defined and studied in this paper. It recovers the classical Ziegler spectrum of a ring. As an application, the Ziegler spectrum as well as the category of generalised quasi-coherent sheaves of a reasonable scheme is introduced and studied. It is shown that there is a closed embedding of the injective spectrum of a coherent scheme endowed with the tensor fl-topology (respectively of a noetherian scheme endowed with the dual Zariski topology) into its Ziegler spectrum. It is also shown that quasi-coherent sheaves and generalised quasi-coherent sheaves are related to each other by a recollement.