On the vanishing of Ext over formal triangular matrix rings

Abstract

Let A and B be two rings, M be a left B, right A bimodule and T = ( MA B0) be the formal triangular matrix ring. It is known that the category of right T-modules is equivalent to the category Ω of triples (X, Y)f, where X is a right A-module, Y is a right B-module and f : Y ⊗B M → X is a right A-homomorphism. Using this alternative description of right T-modules, in the first part of this paper, we study the vanishing of the extension functor 'Ext' over T. To this end, we first describe explicitly the structure of (right) T-modules of finite projective (respectively, injective) dimension. Using these results, we shall characterize respectively modules in Auslander's G-class, Gorenstein injective modules, cotorsion modules and tilting and cotilting modules over T. As another application we investigate the structure of the flat covers of right T-modules. © de Gruyter 2006.