On the finitistic dimension conjecture of Artin algebras.
type: Item
instance of: scholarly article
title: On the finitistic dimension conjecture of Artin algebras.
zbMATH Open document ID: 1176.16014
author: Aiping Zhang
author: Shunhua Zhang
published in: Journal of Algebra
publication date: 2008-10-06T00:00:00Z
review text: Let \(A\) be an Artin algebra. The finitistic dimension of \(A\) is defined as the supremum of the projective dimensions of finitely generated left \(R\)-modules of finite projective dimension. The finitistic dimension conjecture, due to H. Bass in 1960, asserts that the finitistic dimension of any Artin algebra is finite. The representation dimension of \(A\) is defined as the smallest number \(d\) that is equal to the global dimension of the endomorphism ring of a left \(R\)-module which is both a generator and a cogenerator in the category of left \(A\)-modules. In this paper it is proved that if an Artin algebra \(A\) has representation dimension at most three, and \(e\) is an idempotent of \(A\), then the finitistic dimension of \(eAe\) is finite. As a corollary, it is deduced that if every quasi-hereditary algebra has representation dimension at most three, then the finitistic dimension conjecture holds.
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 5349051
zbMATH Keywords: Artin algebras
zbMATH Keywords: finitistic dimension
zbMATH Keywords: representation dimension
zbMATH Keywords: projective dimension
zbMATH Keywords: global dimension
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.12.011
OpenAlex ID: W2023237954
Wikidata QID: Q123231164
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DOI: 10.1016/J.JALGEBRA.2007.12.011
description: scientific article; zbMATH DE number 5349051