Using linear relations to study Σ-algebraically-compact modules for string algebras.

Raphael Bennett-Tennenhaus, University of Bielefeld. Part of the algebra, logic and algorithms seminar series.

I will give an informal introduction to some (quite combinatorial) model theoretic algebra. Formal definitions will be avoided, and replaced with a focus on examples. Consequently the talk is self contained and a general audience is welcome.

Positive-primitive definable subgroups (of an R-module M) are given by finite systems of R-linear equations in M. Codirected families of such subgroups define finitely solvable systems of equations. M is called algebraically-compact if each such system has a simultaneous solution. M is Σ-algebraically-compact provided it has the descending chain condition on positive-primitive definable subgroups.

A string algebra is defined using a quiver (=finite oriented graph). The finite dimensional modules over these algebras have been classified by Butler and Ringel. They may be defined using combinatorics involving finite walks in the quiver. These combinatorics were adapted by Ringel to provide examples of algebraically-compact modules which may be infinite-dimensional. These modules are given by infinite walks, and Ringel conjectured there were no others. Under some restrictions on R, Prest and Puninski gave a proof of this conjecture.

This talk is on joint work with Bill Crawley-Boevey (arXiv:1705.10145). Here we use the same method used by Butler and Ringel to classify Σ-algebraically-compact modules over (possibly infinite-dimensional) string algebras. This method involves studying linear relations.

All are very welcome. Tea/coffee in the Staff Common Room before hand at 3.45pm. 

Organised by Olaf Beyersdorff and Vincenzo Mantova.

Raphael Bennett-Tennenhaus, University of Bielefeld