Auslander-Reiten sequences over Artinian rings
type: Item
instance of: scholarly article
title: Auslander-Reiten sequences over Artinian rings
zbMATH Open document ID: 0661.16024
DOI: 10.1016/0021-8693(88)90066-X
author: Wolfgang Zimmermann
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: In the middle part of the paper under review the author presents the following example: Let k be some field of characteristic 0, \(F=k((X))\) the field of formal Laurent series in X, \({}_ FN\) be a two-dimensional left vector-space with basis x, y which is made a right vector space by \(x\alpha =\alpha x\), \(y\alpha =\alpha y+\alpha 'x\) (\(\alpha\in F\), ' the formal derivation) and let \(R=\left( \begin{matrix} F\\ 0\end{matrix} \begin{matrix}_ FN_ F\\ F\end{matrix} \right)\) be the bimodule algebra. By \textit{C. M. Ringel} [J. Algebra 41, 269-302 (1976; Zbl 0338.16011), Theorem 4 and {\S}{\S}6, 7] the indecomposable finitely presented modules are classified; if \(C\in mod R\) is indecomposable then either \(\partial (C)\neq 0\) (\(\partial\) the defect) which implies \underbar{dim} C\(=(m,n)\) with \(| m-n| =1\) or C is in the abelian category \({\mathfrak m}\times {\mathfrak u}\) of regular modules. The surprising result (Theorem 13) of section 2 is the following: For each finitely presented, indecomposable, non projective right R-module C there exists an Auslander-Reiten sequence \(0\to A\to B\to C\to 0\) in mod R. This is also an Auslander-Reiten sequence in Mod R if and only if C is not in \({\mathfrak m}.\) For the proof the author uses his general results on Auslander-Reiten sequences of section 1, f.e. ``Theorem 1: Let R be a semiperfect ring and \(C_ R\) a finitely presented, non-projective module with a local endomorphism ring. There exists an Auslander-Reiten sequence \(0\to A\to B\to C\to 0\) in mod R if and only if (Tr C)\({}^ o\) contains a finitely presented, pure submodule with a local endomorphism ring. Up to isomorphism, the latter is uniquely determined''. For the application of these results to the described example the author needs results on systems of differential equations over the field F. These results are proved in section 3 of the paper.
reviewed by: Otto Kerner
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 4079672
zbMATH Keywords: formal Laurent series
zbMATH Keywords: bimodule algebra
zbMATH Keywords: indecomposable finitely presented modules
zbMATH Keywords: abelian category
zbMATH Keywords: regular modules
zbMATH Keywords: Auslander-Reiten sequences
zbMATH Keywords: local endomorphism ring
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4079672