Auslander and the Brauer-Thrall conjectures
The first Brauer-Thrall conjecture for algebras was solved by Roiter in 1968,
but his proof did not extend to artinian rings. In 1974, Auslander
published a proof for the general case. This proof (partially
modified by Yamagata) is now considered as the standard one and it
may be considered as the basis of what is called the
Auslander-Reiten theory. But it seems to be worthwhile to draw attention
also to the original proof of Roiter and the influence it had.
After all, it is clear that Auslander was strongly impressed by Roiter's
approach, and he extended the scope of the methods of Roiter (and of the
interpretation of these methods by Gabriel) considerably: let us mention
his construction of indecomposables of infinite length, as well as his
joint work with Smalo on preprojective and preinjective modules.
The lecture will focus the attention to these developments in the
representation theory of algebras. In particular, we also want to draw
attention to Auslander's interest on the second Brauer-Thrall conjecture
and the trichotomy of finite, tame, and wild representation type:
in 1993 a workshop was held - on his request - at the Bielefeld SFB dealing
with these questions...
Ringel
Last modified: Wed Sep 22 19:49:16 CEST 2004