Algebras and Representation Theory

, Volume 14, Issue 1, pp 57–74

Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules

Open Access
Article

Abstract

A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and RMS is an S-R-bimodule. In the main theorem of this paper we show that if TS is a tilting S-module, then under certain homological conditions on the S-module MS, one can extend TS to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and TS, and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and MS is finitely generated. In this case, (S′,R,M′) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, MS has a finite projective resolution and ExtSn(MS, S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, HomS(M, S)).

Keywords

Triangular matrix ring Derived equivalence Tilting complex 

Mathematics Subject Classifications (2000)

18E30 16S50 18A25 

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Max-Planck-Institut für MathematikBonnGermany

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