On a Chouinard's formula for $C$-quasi-injective dimension

TL;DR

This paper extends classical homological formulas, including Bass' and Chouinard's formulas, to modules with finite $C$-quasi-injective dimension, unifying previous notions of injective dimensions relative to a semidualizing module.

Contribution

It provides new formulas for $C$-quasi-injective dimension, broadening the understanding of homological invariants in module theory.

Findings

Extended Bass' formula to $C$-quasi-injective dimension.

Derived a version of Chouinard's formula for modules with finite $C$-quasi-injective dimension.

Unified and generalized previous results on injective dimensions relative to semidualizing modules.

Abstract

The $C$-quasi-injective dimension is a recently introduced homological invariant that unifies and extends the notions of quasi-injective dimension and of injective dimension with respect to a semidualizing module, previously studied by Gheibi and by Takahashi and White, respectively. In the main results of this paper, we provide extensions of the Bass' formula and a version of the Chouinard's formula for modules of finite $C$-quasi-injective dimension over an arbitatry ring.