A generalization of rejection lemma of Drozd-Kirichenko
type: Item
instance of: scholarly article
title: A generalization of rejection lemma of Drozd-Kirichenko
zbMATH Open document ID: 0916.16007
author: Osamu Iyama
published in: Journal of the Mathematical Society of Japan
publication date: 1999-07-12T00:00:00Z
review text: The paper provides a new efficient tool for the investigation of (not only) representation-finite orders and artinian algebras. For an order \(\Lambda\) over a complete discrete valuation domain, let \(\text{Ind }\Lambda\) denote the set of isomorphism classes of indecomposable \(\Lambda\)-lattices. The rejection lemma of \textit{Yu. A. Drozd} and \textit{V. V. Kirichenko} [Izv. Akad. Nauk SSSR, Ser. Mat. 36, 328-370 (1972; Zbl 0234.13009)] says that for a bijective (i.e. projective and injective in the category \(\Lambda\)-\(\mathbf{lat}\) of \(\Lambda\)-lattices) \(B\in\text{Ind }\Lambda\) with \(B\not\cong\text{Rad }B\), the set \(\text{Ind }\Lambda\setminus\{B\}\) is of the form \(\text{Ind }\Gamma\) for an overorder \(\Gamma\) of \(\Lambda\). In view of this result, the author studies the sets \({\mathcal S}=\text{Ind }\Lambda\setminus\text{Ind }\Gamma\) with \(\Gamma\supset\Lambda\) which he calls rejectable. Note that the rejection lemma extends to the case \(B\cong\text{Rad }B\) if generalized overorders \(\Gamma\) in the sense of Drozd and Kirichenko are allowed, that is, orders \(\Gamma\) connected with \(\Lambda\) by a ring homomorphism \(\Lambda\to\Gamma\) with torsion cokernel. Then every proper subset \({\mathcal S}\subset\text{Ind }\Lambda\) determines the smallest generalized overorder \(\Gamma\) with \({\mathcal S}\subset\text{Ind }\Lambda \setminus\text{Ind }\Gamma\). The author calls \(\mathcal S\) cofaithful whenever this \(\Gamma\) is an overorder of \(\Lambda\). Now assume that the ambient algebra of \(\Lambda\) is semisimple. Generalizing Auslander's proof of the Brauer-Thrall-I conjecture (= Rojter's theorem), the author shows that a rejectable set \(\mathcal S\) with bounded rational rank for all \(E\in{\mathcal S}\) is necessarily finite. The second starting point of his investigation is an observation on Auslander-Reiten sequences for overorders \(\Gamma\supset\Lambda\) given at the end of the paper, namely: if \(M\hookrightarrow E\twohead…
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 1248134
zbMATH Keywords: representation-finite orders
zbMATH Keywords: indecomposable lattices
zbMATH Keywords: rejection lemma
zbMATH Keywords: generalized overorders
zbMATH Keywords: Auslander-Reiten sequences
zbMATH Keywords: rationally exact sequences
zbMATH Keywords: cochain complexes
zbMATH Keywords: finite rejectable sets
zbMATH Keywords: Auslander-Reiten quivers
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05030697
OpenAlex ID: W2110628335
Wikidata QID: Q124995225
DOI: 10.2969/JMSJ/05030697
description: scientific article; zbMATH DE number 1248134