Decompositions of degenerations
type: Item
instance of: scholarly article
title: Decompositions of degenerations
zbMATH Open document ID: 0964.16010
author: R. Aehle
published in: Manuscripta Mathematica
publication date: 2001-04-19T00:00:00Z
review text: Let \(k\) be an algebraically closed field and let \(\Lambda\) be a finite dimensional associative \(k\)-algebra with unit. By definition, a \(d\)-dimensional \(\Lambda\)-module is the vector space \(k^d\) endowed with a multiplication by \(\Lambda\). Making use of a purely algebraic description of the partial order \(\leq_{\text{deg}}\) [\textit{G. Zwara}, Compos. Math. 121, No. 2, 205-218 (2000; Zbl 0957.16007)], the author develops a method of decomposition of a given degeneration of \(\Lambda\)-modules into finer ones. He then applies this method to the case when \(\Lambda\) has a directed Auslander-Reiten quiver and gives a new proof of the equivalence of the orders \(\leq_{\text{deg}}\) and \(\leq_{\text{ext}}\) for such algebras. In fact, this result has been earlier obtained in a different context by \textit{K. Bongartz} [in Algebras and modules I. Workshop on representations of algebras and related topics, Trondheim, Norway, 1996. CMS Conf. Proc. 23, 1-27 (1998; Zbl 0915.16008)].
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 1491674
zbMATH Keywords: degenerations of modules
zbMATH Keywords: partial orders
zbMATH Keywords: directed Auslander-Reiten quivers
zbMATH Keywords: indecomposable modules
zbMATH Keywords: almost split sequences
zbMATH Keywords: vector bundles
zbMATH Keywords: finite-dimensional algebras
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: s002291020169
OpenAlex ID: W2069558449
DOI: 10.1007/S002291020169
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description: scientific article; zbMATH DE number 1491674