Folded tilting complexes for Brauer tree algebras.
type: Item
instance of: scholarly article
title: Folded tilting complexes for Brauer tree algebras.
zbMATH Open document ID: 1040.20003
author: Jeremy Rickard
author: Mary Schaps
published in: Advances in Mathematics
publication date: 2003-03-26T00:00:00Z
review text: Let \(A\) and \(B\) be two \(k\)-algebras, projective as \(k\)-modules for an integral domain \(k\). Then, the derived categories of \(A\)-modules and the derived category of \(B\)-modules are equivalent if and only if there is a so-called tilting complex \(T\) of \(A\)-modules so that \(B\) is the endomorphism ring of \(T\). The complex \(T\) is the image of the rank one free \(B\)-module under the equivalence. Among the very first examples studied for this theory were Brauer tree algebras. These are algebras associated to a finite graph with additional combinatorial data, a Brauer tree. Using an idea of \textit{S. König} and the reviewer [Commun. Algebra 24, No. 6, 1897--1913 (1996; Zbl 0851.16012)], \textit{M. Schaps} and \textit{E. Zakay-Illouz} constructed a combinatorially defined complex \(T\) over the Brauer tree algebra \(A\) associated to a star with endomorphism ring being any of the algebras amongst those which are derived equivalent to the given one [J. Algebra 246, No. 2, 647--672 (2001; Zbl 0994.16011)], so that \(T=F(A)\) for an equivalence \(F:D^b(B)\rightarrow D^b(A)\). In the paper under review a complex \(S\) is constructed corresponding to \(G(A)\) for a quasi-inverse \(G\) of \(F\).
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E05
zbMATH DE Number: 1888321
zbMATH Keywords: derived equivalences
zbMATH Keywords: Brauer tree algebras
zbMATH Keywords: derived categories
MaRDI profile type: Publication
full work available at URL: aima.2001.2064
OpenAlex ID: W1970838262
cites work: Q3722697
cites work: Walking around the Brauer tree
cites work: Derived categories and stable equivalence
cites work: A modular version of Maschke's theorem for groups with cyclic \(p\)-Sylow subgroups
cites work: Pointed Brauer trees
cites work: Combinatorial partial tilting complexes for the Brauer star algebras
cites work: Two sided tilting complexes for Green orders and Brauer tree algebras
DOI: 10.1006/AIMA.2001.2064
description: scientific article; zbMATH DE number 1888321