- Helmut Lenzing (University of Paderborn), Hereditary noetherian categories and
singularities
Abstract: A remarkable result of Maurice
Auslander establishes existence
of almost split sequences for (a suitable category of modules over) an
isolated singularity (and also a converse). We are going to link this
question with the topic of hereditary noetherian categories which, due
to the work of a number of authors, has attracted a lot of attention in
recent years. In the case of a, possibly graded, surface singularity we
investigate the category of coherent sheaves on the punctured (graded)
spectrum of the singularity. Each such category is a hereditary, in
particular abelian, category whose objects are noetherian and which
satisfies Serre duality, in a sense to be specified.
In a related context we take
up former studies of Auslander and Reiten on the completion from graded
Cohen Macaulay modules over suitable graded rings to Cohen Macaulay
modules over complete local rings and discuss properties of the
completion functor, with a focus on factoriality.
- Idun Reiten (NTNU, Trondheim), Connections between work of Maurice and my
recent work
Abstract: The emphasis of my lecture will be
some work by (and with) Maurice, which has connections with some of my
work (with various coauthors) during the last years, in particular on
hereditary categories with Serre duality and cluster-tilted algebras.
In the process we include some recent results with Buan and Marsh.
- Claus Michael Ringel (University of Bielefeld), Auslander and the Brauer-Thrall conjectures
Abstract: 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 Smalø on preprojective and
preinjective modules.
The lecture will focus the
attention on these developments in representation theory of
algebras. In particular, we also want to draw attention to Auslander's
interest in 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...
