# Sigma-pure-injective modules for string algebras via the theory of linear relations

**Date**: Wednesday 18 October 2017, 12:00 – 13:00**Location**: EC Stoner SR (8.90)**Type**: Pure Mathematics seminars, Algebra seminars, Seminar series**Cost**: Free

#### Part of the pure mathematics algebra seminar series.

Let R be a fixed ring. By a theorem of Baur and Monk every formula in the language of R-modules is equivalent to a Boolean combination of positive primitive formulas. In a fixed R-module M, the set of solutions to a formula of this type defines a subgroup. M is called Sigma-pure-injective provided any descending chain of such subgroups must terminate.

In this talk I shall introduce a class of rings called string algebras which include infinite dimensional algebras such as k[x,y]/(xy) (where k is any field). I shall then present a classification of the Sigma-pure-injective modules over these algebras. If time permits I will say something about the proof, which uses the classification of finite dimensional representations of the Kronecker quiver.

This is based on joint work with Bill Crawley-Boevey (arxiv: 1705.10145).