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

A triangular matrix ring I > is defined by a triplet (R,S,M) where R and S are rings and (R) M (S) is an S-R-bimodule. In the main theorem of this paper we show that if T (S) is a tilting S-module, then under certain homological conditions on the S-module M (S) , one can extend T (S) to a tilting complex over I > inducing a derived equivalence between I > 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 T (S) , 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 M (S) 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, M (S) has a finite projective resolution and Ext (S) (n) (M (S) , S) = 0 for all n > 0. In this case, (S',R,M') = (S, R, Hom (S) (M, S)).