Alessandro Rapa, Università di Verona

Title: Simple objects in the heart of a t-structure

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.