Abstract

In a 2006 paper of Angeleri Hugel and Saorin, it was shown that any module with a perfect decomposition is Sigma-coperfect over its endomorphism ring, and a question was asked whether the converse implication holds. In this talk, I will present a topological algebra approach to this problem. The ring of endomorphisms of any module is endowed with the so-called finite topology, making it a complete, separated topological ring with a base of neighborhoods of zero formed by open right ideals. Both the existence of a perfect decomposition and the endo-Sigma-coperfectness properties of a module are shown to be equivalent to certain properties of the topological ring of endomorphisms. The equivalence of these two properties of a topological ring is an assertion extending Bass' famous Theorem P to the realm of topological rings. One can prove it for commutative topological rings and for topological rings with a countable base of neighborhoods of zero. It follows that the answer to the question of Angeleri Hugel and Saorin is positive for modules with a commutative endomorphism ring and for countably generated modules.

This talk is based on a joint work of Jan Stovicek and the speaker.

Abstract

In this talk, we consider a specific class of finite dimensional algebras of infinite representation type, called “tubular algebras”. Pure-injective modules over tubular algebras have been partially classified by Angeleri Hügel and Kussin, in 2016, and we want to give a contribution to the classification of the ones of “irrational slope”. First, we move to a more geometrical framework, i.e. we work in the category of quasi-coherent sheaves over a tubular curve, and we approach our classification problem from the point of view of tilting/cotilting theory. More precisely, we consider the Happel-Reiten-Smalø heart of torsion pairs cogenerated by infinite dimensional cotilting sheaves. These hearts are locally coherent Grothendieck categories in which the pure-injective sheaves over the tubular curve become injective objects. In order to study injective objects in a Grothendieck category is fundamental to know the classification of the simple objects. We will use techniques coming from the continued fractions and universal extensions to provide a method to construct an infinite dimensional sheaf of a prescribed irrational slope that becomes simple in the Grothendieck category given as the heart of a precise t-structure.