The Positselski-Štovíček Correspondence for the Recollements of Purity

Abstract: The theory of purity for modules over a ring R has been studied using the covariant as well as the contravariant functor categories. In their work on quiver Grassmanians, Crawley-Boevey and Sauter introduced a third such category, the projective quotient functor category. We will explain how these three functor categories are related to the three equivalent definitions of purity. Ironically, the projective quotient functor category is closest in spirit to Prüfer's original definition.

Each of these functor categories may be regarded as the middle term of a recollement of abelian categories whose localization/colocalization is given by the category R-Mod of R-modules. We will describe the basic theory of recollements of functor categories and indicate how it reveals the common features of the three functor categories. Each of these functor categories is related to the other two by a triangle of Positselski-Štovíček correspondences, which allows a detailed analysis of its homological properties.

This is joint work with Xianhui Fu.

rings and algebrasrepresentation theory

Audience: researchers in the topic

ONCAS Online Noncommutative Algebra Seminar

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