Strong preinjective partitions and almost split morphisms
by
Nguyen Viet Dung
Hanoi Institute of Mathematics, Vietnam and Ohio University, Athens, USA

A ring R is of left pure global dimension zero if and only if every left R-module is a direct sum of finitely generated modules. It is well known that rings of left and right pure global dimension zero are precisely rings of finite representation type, i.e., Artinian rings with finitely many isomorphism classes of finitely generated indecomposable modules. However, it is still an open problem whether rings of left pure global dimension zero are always of finite representation type. Huisgen-Zimmermann and Zimmermann [3] and Prest [4] showed that rings of left pure global dimension zero come "close" to having finite representation type, namely that for such a ring R and any natural number n, there are only finitely many isomorphism classes of finitely presented indecomposable left or right R-modules of length at most n. Moreover, Huisgen-Zimmermann [2] showed that if R is a ring of left pure global dimension zero, then every family of finitely presented indecomposable left (right) R-modules has a unique strong preinjective (resp. preprojective) partition of countable length. The notions of strong preinjective and preprojective partitions introduced in [2] are generalizations of those introduced by Auslander and Smalø [1] in the context of Artin algebras.

In this talk, we present a different method based on the concepts of almost split morphisms in subcategories (due to Auslander, Reiten and Smalø) allowing us to recover and generalize the results in [2], [3], [4] (which were proved in these works by duality techniques). For example, we show that if R is any ring and C is a family of indecomposable R-modules of finite length satisfying the property that all submodules of arbitrary direct sums of modules in C are direct sums of finitely generated submodules, then every subfamily of C has right almost split morphisms. Moreover, if C contains a finite cogenerating set, then for each natural number n there are only finitely many non-isomorphic modules of length at most n in C, and every subfamily of C has a unique strong preinjective partition of countable length. Our methods can also be used to prove similar results for Grothendieck categories of pure global dimension zero, or more generally, Grothendieck categories that satisfy the Kulikov property.

REFERENCES.

[1] M. Auslander and S.O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61-122.

[2] B. Huisgen-Zimmermann, Strong preinjective partitions and representation type of artinian rings, Proc. Amer. Math. Soc. 109 (1990), 309-322.

[3] B. Huisgen-Zimmermann and W. Zimmermann, On the sparsity of representations of rings of pure global dimension zero, Trans. Amer. Math. Soc. 320 (1990), 695 -711.

[4] M. Prest, Duality and pure-semisimple rings. J. London Math. Soc. 38 (1988), 403-409.

Date received: February 3, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-68.