Zengqiang Lin (Huaqiao University, Quanzhou), The category of short exact sequences, Auslander-Reiten sequences and Auslander algebras.

Abstract:

Let C be an abelian category. Denote by S(C) the category of short exact sequences in C and SS(C) the full subcategory of S(C) consisting of split short exact sequences. It is well known that S(C) is an exact category, but it is not abelian in general. I will show that the quotient category S(C)/SS(C) is abelian and the kernels and cokernels are given by pushout and pullback diagrams. Assume that A is an Artin algebra and modA is the category of finitely generated right A-modules. I will prove that there is a bijection between the class of isomorphism classes of simple objects in S(modA)/SS(modA) and the class of isomorphism classes of Auslander-Reiten sequences in modA. In particular, if A is a representation-finite Artin algebra, I will prove that the category S(modA)/SS(modA) is isomorphic to modB where B is the stable Auslander algebra of A. For the category of triangles from a triangulated category, there are some similar results.

This talk is based on ongoing joint discussions with Steffen Koenig; related earlier work has been done by Ronald Gentle, Erik Backelin and Omar Jaramillo.