Quasi-homological dimensions with respect to semidualizing modules

TL;DR

This paper introduces and studies the quasi-homological dimensions with respect to semidualizing modules, unifying and extending existing homological invariants and formulas in commutative algebra.

Contribution

It defines quasi-projective and quasi-injective dimensions relative to semidualizing modules, extending classic homological invariants and formulas.

Findings

Generalized Auslander-Buchsbaum and Bass' formulas

Proved a special case of the Auslander-Reiten conjecture

Investigated rigidity properties of Ext and Tor

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.