ALGEBRAIC COMPACTNESS, DIRECT SUM DECOMPOSITIONS, AND
REPRESENTATION TYPE
Birge Huisgen-Zimmermann
Department of Mathematics
University of California
Santa Barbara, California 93106
USA
LECTURE I. Algebraic compactness, product-preserving
functors, and direct sum decompositions.
LECTURE II. Generic modules and representation type.
The first lecture will start by tracing the historical
origins of algebraic compactness (= pure injectivity), a concept
which already arose in various contexts during the 1950's.
The red thread through the core of the lecture will be
provided by the following problem which goes back to Koethe,
Cohen, and Kaplansky: Characterize the rings for which each
(left) module is a direct sum of finitely generated components.
This decomposition property obviously constitutes the strongest
possible tie one can ask for between finitely and non-finitely
generated modules. First headway towards a solution was made by
Koethe, Cohen-Kaplansky, Chase, Griffith, Warfield, and Ringel-
Tachikawa.
It was this problem that triggered new, important
descriptions of algebraically compact and \Sigma-algebraically
compact modules in terms of finite matrix subgroups (= subgroups
of finite definition) which were simultaneously discovered by
Gruson-Jensen and W. Zimmermann in the mid-70's. Finite matrix
subgroups of a module will be introduced from a functorial point
of view emphasizing one of their pivotal properties, namely
compatibility with direct products. The most important therorems
in this context will be stated, and proofs of a few crucial links
will be sketched. On this occasion, it will become apparent how,
in various guises, pure injectivity lies at the heart of many
arguments in classical module theory.
Combined with an idea of Chase, the theory of algebraic
compactness will then lead us to a series of decomposition
theorems for direct products of modules, due to Gruson-Jensen, W.
Zimmermann, and the speaker. These, in turn, will take us back to
our original problem, allowing us to at least satisfactorily
characterize the rings for which all left AND right modules have
the stipulated decomposition properties. Aiming at a full
solution of the problem - unfortunately, not available to date
- we will further outline results due to Auslander, Simson,
Prest, W. Zimmermann, Herzog, and the speaker.
The second lecture will start with a discussion of a duality
between the lattices of finite matrix subgroups of left modules
and their character modules relative to an injective cogenerator.
This duality will then be invested into a sketch of an argument
backing one of the culminating theorems of Lecture I. We will
follow with a brief outline of other dualities that have arisen in
the context of algebraic compactness and modules of finite
endolength. (The topic of duality is to be addressed in greater
depth by other speakers.)
This will guide us to the main subject of Lecture II, namely
generic modules over Artin algebras in the sense of Crawley-
Boevey. We will closely follow work of Crawley-Boevey in
outlining how the generic modules 'govern' families of finitely
generated indecomposable modules and in deducing their impact on
the representation type of the underlying algebra. In particular,
we will address the coincidence of classical tameness with generic
tameness in the case of a finite dimensional algebra over an
algebraically closed base field.
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Last modified: 22-06-1998