Quasi-homological dimensions with respect to semidualizing modules
Souvik Dey, Luigi Ferraro, Mohsen Gheibi
TL;DR
The paper defines $C$-quasi-projective and $C$-quasi-injective dimensions relative to a semidualizing module $C$, unifying existing quasi-homological notions with the $C$-projective/$C$-injective framework. It develops transfer formulae linking these dimensions to classical ones, including a $C$-version of the Auslander–Buchsbaum, Bass, and Ischebeck formulas, as well as depth and dependency results in the $C$-context. It also proves a special case of the Auslander–Reiten conjecture and establishes rigidity results for Ext and Tor, with significant consequences for when rings are Cohen–Macaulay or Gorenstein, particularly in the presence of a dualizing module or complex. The work thus provides a cohesive theory that both generalizes and connects quasi-homological dimensions with semidualizing-module theory, yielding practical criteria for Cohen–Macaulayness, dualizing conditions, and related homological behavior.
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.