The balance of relative Ext groups defined by semidualizing modules

Abstract

Let [Formula: see text] be a commutative Noetherian ring with identity and [Formula: see text] a semidualizing module for [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote, respectively, the classes of [Formula: see text]-projective and [Formula: see text]-injective [Formula: see text]-modules. We show that their induced Ext bifunctors Ext[Formula: see text] and Ext[Formula: see text] coincide for all [Formula: see text] if and only if [Formula: see text] is projective. Also, we provide some other criteria for [Formula: see text] to be projective by using some special cotorsion theories.