The Ziegler spectrum for enriched ringoids and schemes
Grigory Garkusha
TL;DR
The paper develops Ziegler spectra in an enriched setting by introducing the enriched Ziegler spectrum ${}_{\mathcal{A}}\mathsf{Zg}$ for enriched ringoids $\mathcal{A}$ and relating it to the injective spectrum with tensor fl-topology. It builds a robust framework using enriched category theory (including Day convolution, enriched Yoneda, and modules over enriched ringoids) and Grothendieck $\mathcal{V}$-categories to study both ordinary and generalized modules, with a recollement connecting $\mathop{Mod}\mathcal{A}$ and ${}_{\mathcal{A}}\mathcal{C}$. The approach recovers classical Ziegler spectra when $\mathcal{V}=\mathrm{Ab}$ and $\mathcal{A}$ is a ring, and extends to schemes by defining a Ziegler spectrum for a scheme $X$ via generalized quasi-coherent sheaves, including embeddings from the injective spectrum and dual Zariski considerations for coherent and noetherian cases. Enriched localizing subcategories and quotients underpin these results, enabling a cohesive link between quasi-coherent and generalized quasi-coherent sheaves. Overall, the work unifies model-theoretic spectrum techniques with enriched categorical methods to analyze schemes and their module categories.
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.