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

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 ExtS(MS , S) = 0 for all n > 0. In this case, (S′, R, M ′) = (S, R,HomS(M, S)).