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

TL;DR

This work extends Chouinard's and Bass' formulas to the setting of $C$-quasi-injective dimension over a commutative Noetherian ring. Under the vanishing condition $\text{Tor}_{>0}^R(C,M)=0$ and finite finite $C$-qid, the authors establish a Chouinard-type formula: $\text{ extit{C}-qid}_R M = \sup\{ \operatorname{depth} R_{\mathfrak p} - \operatorname{width}_{R_{\mathfrak p}} M_{\mathfrak p} \mid \mathfrak p \in \operatorname{Spec} R \}$, with a finitely generated specialization to $\sup\{ \operatorname{depth} R_{\mathfrak p} \mid \mathfrak p \in \operatorname{Supp} M \}$. The paper also furnishes a dual finiteness criterion for $\text{ extit{C}-id}_R M$ and a corollary dual to ferraro, showing finiteness of $\text{ extit{C}-id}_R M$ from finite $\text{ extit{C}-qid}_R M$ together with Ext-vanishing, thereby recovering known results when $C=R$ and connecting to Tri's and Gheibi's frameworks. Overall, the results unify Bass-type and Chouinard-type formulas in the semidualizing setting and illuminate how $C$-id, $C$-qid, width, and depth interact across spectra.

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.

Theorems & Definitions (20)

• Theorem 1.1: See Theorem \ref{['Cchouinard']}
• Theorem 1.2: See Theorem \ref{['theoremfg']}
• Corollary 1.3: See Corollary \ref{['crl']}
• Definition 2.6
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Corollary 3.4