The Ziegler spectrum for enriched ringoids and schemes

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.