Derived and stable equivalence classification of twisted multifold extensions of piecewise hereditary algebras of tree type
type: Item
instance of: scholarly article
title: Derived and stable equivalence classification of twisted multifold extensions of piecewise hereditary algebras of tree type
zbMATH Open document ID: 1024.16006
author: Hideto Asashiba
published in: Journal of Algebra
publication date: 2002-07-15T00:00:00Z
review text: Let \(k\) be an algebraically closed field and let \(\Lambda\) and \(\Pi\) be basic connected selfinjective finite-dimensional \(k\)-algebras. If \(\Lambda\) and \(\Pi\) are derived equivalent, then they are stably equivalent [see \textit{B. Keller} and \textit{D. Vossieck}, C. R. Acad. Sci., Paris, Sér. I 305, 225-228 (1987; Zbl 0628.18003) or \textit{J. Rickard}, J. Pure Appl. Algebra 61, No. 3, 303-317 (1989; Zbl 0685.16016)]. In an earlier paper the author showed that the converse holds if both \(\Lambda\) and \(\Pi\) are of finite representation type [\textit{H. Asashiba}, J. Algebra 214, No. 1, 182-221 (1999; Zbl 0949.16013)]. The question of when the converse holds is related to Broué's conjecture [\textit{M. Broué}, Astérisque 181-182, 61-92 (1990; Zbl 0704.20010)] stating that if \(G\) is a finite group, then a block of the group algebra \(kG\) with Abelian defect is derived equivalent to its Brauer correspondent. The relationship is that in some important cases it is known that the block and its Brauer correspondent are stably equivalent. In the present paper the author shows that the converse holds for a certain class of algebras that need not be of finite representation type. The main result is that if \(\Lambda\) and \(\Pi\) are twisted multifold extensions of piecewise hereditary algebras of tree type, then \(\Lambda\) and \(\Pi\) are derived equivalent if and only if they are stably equivalent.
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1750359
zbMATH Keywords: selfinjective algebras
zbMATH Keywords: derived equivalences
zbMATH Keywords: stable equivalences
zbMATH Keywords: twisted multifold extensions
zbMATH Keywords: piecewise hereditary algebras
reviewed by: Mark Kleiner
MaRDI profile type: Publication
OpenAlex ID: W2066527449
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DOI: 10.1006/JABR.2001.9063
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description: scientific article; zbMATH DE number 1750359