REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: There are two classical settings, where the existence of almost split sequences was established by Auslander: the category of finite dimensional modules over finite dimensional algebras, and the category of finitely generated Cohen- Macaulay modules over an isolated singularity. There is another classical result, also due to Auslander: if M is a finitely presented module with a local endomorphism ring over a ring R, then M is a sink of an almost split sequence in the category of all R-modules. We will show that an analogue of this Auslander's result holds true for any definable category closed with respect to extensions. The proof uses a model theoretic approach developed by Ivo Herzog.