Reduction techniques for homological conjectures
type: Item
instance of: scholarly article
title: Reduction techniques for homological conjectures
zbMATH Open document ID: 0809.16021
author: Dieter Happel
published in: Tsukuba Journal of Mathematics
publication date: 1995-04-03T00:00:00Z
review text: The aim of this article is to show that using the language of triangulated categories certain reduction techniques can be obtained which apply to the Finitistic Dimension Conjecture, the Vanishing Conjecture, and the Generalized Nakayama Conjecture. More precisely, it is shown that if \(A\) is a finite dimensional algebra, \(T\) a tilting module, and \(B = \text{End}_ A T\), then \(fd(A) < \infty\) if and only if \(fd(B) < \infty\). It is also shown that if \(D^ b (A)\), the derived category of bounded complexes, has recollement relative to \(D^ b (A')\) and \(D^ b(A'')\) for some finite dimensional algebras \(A'\), \(A''\), then \(fd(A) < \infty\) if and only if \(fd(A')\) and \(fd(A'') < \infty\).
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 515577
zbMATH Keywords: finitistic dimension conjecture
zbMATH Keywords: vanishing conjecture
zbMATH Keywords: generalized Nakayama conjecture
zbMATH Keywords: triangulated categories
zbMATH Keywords: tilting modules
zbMATH Keywords: derived category of bounded complexes
zbMATH Keywords: recollement
zbMATH Keywords: finite dimensional algebras
reviewed by: Ann K. Boyle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162134
OpenAlex ID: W1951788695
Wikidata QID: Q123166372
DOI: 10.21099/TKBJM/1496162134
description: scientific article; zbMATH DE number 515577