Reinhard Knörr
Auslander--Reiten sequences and a certain ideal in mod--$FG$
The paper is published: Rostocker Mathematische Kolloquium, Rostock. Math. Kolloq. 49, 89-97 (1995)
20C20 Modular representations and characters
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
Abstract: In [1], Auslander and Reiten defined an almost split sequence as a
certain type of non--split short exact sequence of modules over artin
algebras (the definition is repeated here
in (19) below). Since then, this concept has been extensively used in the general
representation theory of finite dimensional algebras, but also --- for example by Erdmann
[2] --- in the much more restricted case of group algebras. In this situation, the most
basic property of Auslander--Reiten sequences, namely their existence, turns out to be
closely related to an old acquaintance, the so called projective maps. The purpose of
the present note is to exhibit this relation, thereby giving a rather explicit construction
of Auslander--Reiten sequences for group algebras. In fact, the construction can be
relativized up to a point (Theorem 17). The reader not interested in this
can always
take $ H = 1 $ in what follows. For another treatment of the topic, see
Greens paper [3].
Notes: Abstract contains the first few lines of text of the paper.