On the existence of modules which are neither preprojectives nor preinjectives
type: Item
instance of: scholarly article
title: On the existence of modules which are neither preprojectives nor preinjectives
zbMATH Open document ID: 0820.16007
author: R. Bautista
author: Flávio U. Coelho
published in: Journal of Algebra
publication date: 1994-09-27T00:00:00Z
review text: Let \(A\) be a finite dimensional algebra over a field, \(\text{mod }A\) the category of finitely generated left \(A\)-modules, and \(\text{ind }A\) a full subcategory of \(\text{mod }A\) of chosen representatives of isoclasses of indecomposable \(A\)-modules. [In J. Algebra 66, 61-122 (1980; Zbl 0477.16013)] \textit{M. Auslander} and \textit{S. O. Smalø} have shown that \(\text{ind }A\) admits a unique partition \({\mathcal P}_ 0, {\mathcal P}_ 1,\dots,{\mathcal P}_ \infty\) of \(\text{ind }A\), called preprojective partition, such that \(\text{ind }A = \bigcup_{i \leq \infty} {\mathcal P}_ i\), \({\mathcal P}_ i \cap {\mathcal P}_ j = \emptyset\) if \(i \neq j\), and for each \(i < \infty\), \({\mathcal P}_ i\) is a finite minimal cover of \(\bigcup_{j \geq i} {\mathcal P}_ j\). The modules in \(\bigcup_{i < \infty} {\mathcal P}_ i\) are called preprojective. Dually, one defines a preinjective partition of \(\text{ind }A\) and preinjective modules. More than 16 years ago C. M. Ringel raised the question whether \(A\) is of finite representation type if any indecomposable module from \(\text{ind }A\) is preprojective or preinjective. It is shown in the paper that this is the case for finite dimensional algebras over infinite perfect fields. In the proof the validity of the second Brauer-Thrall conjecture for such algebras is used. Recently, the above problem has been solved in the affirmative for arbitrary (even Artin) algebras by the reviewer and \textit{S. O. Smalø} [in Arch. Math. 64, 8-10 (1995; Zbl 0813.16007)].
reviewed by: Andrzej Skowroński
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 645088
zbMATH Keywords: category of finitely generated left \(A\)-modules
zbMATH Keywords: indecomposable \(A\)- modules
zbMATH Keywords: preprojective partition
zbMATH Keywords: preinjective partition
zbMATH Keywords: preinjective modules
zbMATH Keywords: finite representation type
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: second Brauer-Thrall conjecture
zbMATH Keywords: Artin algebras
MaRDI profile type: Publication
full work available at URL: jabr.1994.1237
OpenAlex ID: W1972539805
DOI: 10.1006/JABR.1994.1237
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description: scientific article; zbMATH DE number 645088