Derived equivalences of triangular matrix rings arising from extensions of tilting modules
type: Item
instance of: scholarly article
title: Derived equivalences of triangular matrix rings arising from extensions of tilting modules
zbMATH Open document ID: 1228.18010
author: Sefi Ladkani
published in: Algebras and Representation Theory
publication date: 2011-06-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0702060
review text: Let \(R\) and \(S\) be rings and let \(_RM_S\) be an \(R\)-\(S\)-bimodule. Then one may form a ring as \(2\times 2\)-matrices, where in the upper right corner one has entries in \(M\), the upper diagonal entry are entries in \(R\) and the lower diagonal have entries in \(S\). Denote this ring by \((R,S,M)\). The main result in the paper shows that if \(T\) is a tilting module over \(S\) and if \(M\) has finite projective dimension as \(S\)-module and if \(\text{Ext}^n_S(M,T)=0\) for all \(n>0\), then there is a tilting complex over \((R,S,M)\) with endomorphism ring \((\text{End}_S(T),R,\Hom_S(M,T))\). In particular these two algebras are derived equivalent. The proof uses a construction similar to a recollement construction. The author gives examples showing that the hypotheses are really necessary. Numerous interesting applications are given.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 5903795
zbMATH Keywords: triangular matrix ring
zbMATH Keywords: derived equivalence
zbMATH Keywords: tilting complex
MaRDI profile type: Publication
OpenAlex ID: W3104885544
arXiv ID: math/0702060
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DOI: 10.1007/S10468-009-9175-0
description: scientific article