A degeneration-like order for modules
type: Item
instance of: scholarly article
title: A degeneration-like order for modules
zbMATH Open document ID: 0964.16008
author: G. Zwara
published in: Archiv der Mathematik
publication date: 2001-01-31T00:00:00Z
review text: Let \(A\) be a finite dimensional associative \(K\)-algebra with an identity over an algebraically closed field \(K\). It is known [see \textit{Ch. Riedtmann}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 275-301 (1986; Zbl 0603.16025)] that the existence of the exact sequence \(0\to N\to M\oplus Z\to Z\to 0\) of finite dimensional \(A\)-modules implies that the module \(M\) degenerates to \(N\). The author proves that the corresponding relation \(M\leq_R N\) is a partial order on the set of isomorphism classes of finite dimensional \(A\)-modules, and this relation is in fact equivalent to the existence of a short exact sequence \(0\to Z\to Z\oplus M\to N\to 0\).
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 1292434
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: representations
zbMATH Keywords: degenerations of modules
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: Dynkin quivers
zbMATH Keywords: indecomposable modules
zbMATH Keywords: tilted algebras
zbMATH Keywords: exact sequences
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: s000130050288
OpenAlex ID: W2071022949
DOI: 10.1007/S000130050288
description: scientific article; zbMATH DE number 1292434