On wild tilted algebras which are squids
type: Item
instance of: scholarly article
title: On wild tilted algebras which are squids
zbMATH Open document ID: 0727.16010
DOI: 10.1007/BF01191689
author: Luise Unger
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: The author calls a finite-dimensional algebra B over an algebraically closed field an r-squid, if B is a one-point extension \(B=D[R]\), where D is a hereditary algebra with exactly one simple injective module S, and R is an indecomposable D-module with top R\(=S^ r\). She proves as main result: Let C be a wild hereditary algebra with at least three simple modules. For all \(N\in {\mathbb{N}}\) there is a tilted algebra B of C which is an r-squid for some \(r\geq N\). Moreover B has exactly one preprojective component, namely the preprojective component of D and exactly one preinjective component, which is the preinjective component of the generalized Kronecker-algebra \(K_ r.\) The idea of the proof is the following: One may assume that the algebra C is a one-point co-extension of a representation-infinite algebra A. Choosing a suitable indecomposable module U in A-mod with \(Ext^ 1(U,U)=0\), consider the minimal projective generator M of U in A-mod. Then there exists exactly one indecomposable C-module N such that \(T=U\oplus M\oplus N\) is a tilting module in C-mod and \(B=End(T)\) has the desired properties. A consequence of this concept is the result that almost all preprojective A-modules are quasi-simple regular C-modules. In the short first part the author proves some interesting lemmas, which seem to be useful for many purposes.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 4200411
zbMATH Keywords: finite-dimensional algebra
zbMATH Keywords: r-squid
zbMATH Keywords: one-point extension
zbMATH Keywords: wild hereditary algebra
zbMATH Keywords: simple modules
zbMATH Keywords: tilted algebra
zbMATH Keywords: preprojective component
zbMATH Keywords: preinjective component
zbMATH Keywords: generalized Kronecker-algebra
zbMATH Keywords: representation- infinite algebra
zbMATH Keywords: tilting module
Wikidata QID: Q113355365
reviewed by: Otto Kerner
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01191689
OpenAlex ID: W2089034072
description: scientific article; zbMATH DE number 4200411