Two sided tilting complexes for Green orders and Brauer tree algebras
type: Item
instance of: scholarly article
title: Two sided tilting complexes for Green orders and Brauer tree algebras
zbMATH Open document ID: 0871.20003
author: Alexander Zimmermann
published in: Journal of Algebra
publication date: 1997-08-13T00:00:00Z
review text: Derived equivalences became popular in representation theory of finite groups because of results of \textit{J. Rickard}. First of all, his fundamental theorem [J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989; Zbl 0672.16034)] characterizes derived equivalences between rings by the existence of certain tilting complexes. Moreover, \textit{J. Rickard} provided examples of non-trivial derived equivalences by constructing some tilting complexes [J. Pure Appl. Algebra 61, No. 3, 303-317 (1989; Zbl 0685.16016)] inducing derived equivalences between Brauer tree algebras. This covers blocks of group algebras (over fields) having cyclic defect. Later on [J. Lond. Math. Soc., II. Ser. 43, No. 1, 37-48 (1991; Zbl 0683.16030)] \textit{J. Rickard} has shown that under assumptions satisfied for group algebras, if there are any derived equivalences, then there are as well derived equivalences which are just derived tensor functors, and the object to be tensored with is a so-called twosided tilting complex. The article under review gives a rather explicit combinatorial construction of twosided tilting complexes (whose existence had been predicted by Rickard's result) for both Brauer tree algebras and Green orders, the latter covering blocks of cyclic defect of group algebras over discrete valuation rings. The article also contains a detailed exposition of the results and concepts mentioned above.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 1014019
zbMATH Keywords: derived equivalences
zbMATH Keywords: tilting complexes
zbMATH Keywords: Brauer tree algebras
zbMATH Keywords: blocks of group algebras
zbMATH Keywords: derived tensor functors
zbMATH Keywords: Green orders
zbMATH Keywords: blocks of cyclic defect
reviewed by: Steffen Koenig
MaRDI profile type: Publication
full work available at URL: jabr.1996.6799
OpenAlex ID: W2058426608
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DOI: 10.1006/JABR.1996.6799
description: scientific article; zbMATH DE number 1014019