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.

REFERENCES

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Last modified: 22-06-1998