Homological domino effects and the first finitistic dimension conjecture
type: Item
instance of: scholarly article
title: Homological domino effects and the first finitistic dimension conjecture
zbMATH Open document ID: 0792.16011
DOI: 10.1007/BF02100610
published in: Inventiones Mathematicae
publication date: 1993-04-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143999
review text: For any field \(k\) and any integer \(m\geq 2\), it is shown that there exist finite dimensional \(k\)-algebras \(\Lambda\) such that the little finitistic dimension of \(\Lambda\) is \(m\), while the big finitistic dimension is \(m+1\). This provides a negative solution to the first of the following Finitistic Dimension Conjectures. For a finite dimensional algebra \(\Lambda\), (1) \(\text{fin}\dim \Lambda = \text{Fin}\dim \Lambda\) and (2) \(\text{fin}\dim \Lambda < \infty\), where \(\text{fin}\dim\Lambda = \sup\{\text{p}\dim M\mid M\) a finitely generated left \(\Lambda\)-module with \(\text{p}\dim M < \infty\}\) is the (left) little finitistic dimension of \(\Lambda\), and \(\text{Fin}\dim \Lambda = \sup\{\text{p}\dim M \mid M\) an arbitrary left \(\Lambda\)-module with \(\text{p}\dim M < \infty\}\) is the (left) big finitistic dimension of \(\Lambda\).
reviewed by: Yoshiki Kurata
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 149678
zbMATH Keywords: finitistic dimension conjectures
zbMATH Keywords: finite dimensional \(k\)-algebras
zbMATH Keywords: little finitistic dimension
zbMATH Keywords: big finitistic dimension
Wikidata QID: Q123255770
author: Birge Zimmermann Huisgen
MaRDI profile type: Publication
cites work: Diagrams for modules
cites work: On the Dimension of Modules and Algebras (III): Global Dimension
cites work: HOMOLOGICAL DIMENSION IN NOETHERIAN RINGS
cites work: Homological Dimension in Local Rings
cites work: Applications of contravariantly finite subcategories
cites work: Finitistic Dimension and a Homological Generalization of Semi-Primary Rings
cites work: Injective Dimension in Noetherian Rings
cites work: Q3240881
cites work: Algebras from diagrams
cites work: Representation theory of graded Artin algebras
cites work: Finitistic dimensions of finite dimensional monomial algebras
cites work: Finitistic dimension of Artinian rings with vanishing radical cube
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Bounds on global and finitistic dimension for finite dimensional algebras with vanishing radical cube
cites work: Predicting syzygies over monomial relations algebras
cites work: Field-dependent homological behavior of finite dimensional algebras
cites work: Syzygy Pairs in a Monomial Algebra
cites work: Q3999266
cites work: Finitistic global dimension for rings
cites work: A Change of Rings Theorem
cites work: Quasi-Frobenius rings and generalizations. QF-3 and QF-1 rings. Notes by Claus Michael Ringel
description: scientific article; zbMATH DE number 149678