Tuesday, April 16, 15:15

Kevin Schlegel, Exact Structures and Purity

Abstract:
The notion of a purity category and a Ziegler spectrum is, in its most general form, defined for a locally finitely presented category A. Given an exact structure E on the full subcategory of finitely presented objects in A, we define a relative purity category P such that A is a full subcategory of P closed under extensions. This induces an exact structure on A with enough injective objects and we show that the indecomposable injective objects form a closed set of the Ziegler spectrum of A. Moreover, we specify to the case that A is a module category over an Artin algebra. In this case, the closed set of the Ziegler spectrum obtained from the exact structure E gives rise to an fp-idempotent ideal. This is used to prove generalized Auslander-Reiten formulas for the Ext-groups relative to the exact structure.