String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them. But not all. To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.
This is joint work with Rosie Laking and Gena Puninski:
G. Puninski and M. Prest, Ringel's conjecture for domestic string algebras, arXiv:1407.7470;
R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.
- Algebra Seminar