Quasi-homological dimensions with respect to semidualizing modules

Abstract

Gheibi, Jorgensen and Takahashi recently introduced the quasi-projective dimension of a module over commutative Noetherian rings, a homological invariant extending the classic projective dimension of a module, and Gheibi later developed the dual notion of quasi-injective dimension. Takahashi and White in 2010 introduced the projective and injective dimension of a module with respect to a semidualizing module, which likewise generalize their classic counterparts. In this paper we unify and extend these theories by defining and studying the quasi-projective and quasi-injective dimension of a module with respect to a semidualizing module. We establish several results generalizing classic formulae such as the Auslander-Buchsbaum formula, Bass' formula, Ischebeck's formula, Auslander's depth formula and Jorgensen's dependency formula. Furthermore, we prove a special case of the Auslander-Reiten conjecture and investigate rigidity properties of Ext and Tor.