On $\operatorname{Ext}$-finite modules, quasi-injective dimension and width of modules

TL;DR

This work extends Bass-type formulas to Ext-finite modules that are not necessarily finitely generated by providing a homological characterization of the quasi-injective dimension $qid_R M$ and establishing vanishing criteria via Ext against quotients by maximal regular sequences. It proves that for a local ring $(R,\mathfrak{m},k)$ and nonzero Ext-finite $M$ with finite $qid$, we have $qid_R M \le n$ iff certain Ext-vanishing conditions hold, in particular $qid_R M=0$ iff $\Ext_R^{i>0}(R/(\boldsymbol{x}),M)=0$ for a maximal $R$-sequence $\boldsymbol{x}$, and if $qid_R M>0$ then $qid_R M = \sup\{ i : \Ext_R^i(R/(\boldsymbol{x}),M) \neq 0\}$. In the Cohen–Macaulay setting, the authors establish a key formula $qid_R M + \operatorname{width}_R M = \operatorname{depth} R$ for Ext-finite $M$ with finite width, generalizing Bass-type equalities to a broader class of modules. These results connect depth, width, and quasi-injective dimension in local/Cohen–Macaulay contexts and extend classical invariants to non-finitely generated Ext-finite modules.

Abstract

Let $(R,\mathfrak{m},k)$ be a commutative Noetherian local ring. It is well-known that if $M$ is a finitely generated $R$-module of finite quasi-injective dimension, then $\operatorname{qid}_RM = \operatorname{depth} R$. In this paper, we demonstrate that under the weaker condition that $M$ is $\operatorname{Ext}$-finite and has finite quasi-injective dimension, the equality $\operatorname{qid}_R M =0$ holds if and only if $\operatorname{Ext}_R^{i>0}(R/(\boldsymbol{x}),M)=0$, where $\boldsymbol{x} \in \mathfrak{m}$ is a maximal $R$-sequence and if $\operatorname{qid}_R M \neq 0$, we show then that $\operatorname{qid}_R M = \sup \lbrace i : \operatorname{Ext}_R^i(R/(\boldsymbol{x}),M) \neq 0 \rbrace$. Also, we show that if $R$ is a Cohen-Macaulay local ring and $M$ is an $\operatorname{Ext}$-finite $R$-module of finite quasi-injective dimension, then $\operatorname{depth} R = \operatorname{qid}_R M + \inf \lbrace i : \operatorname{Tor}_i^R(k,M) \neq 0 \rbrace$, provided that $\inf \lbrace i : \operatorname{Tor}_i^R(k,M) \neq 0 \rbrace< \infty$.

Theorems & Definitions (28)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.2
• Lemma 2.5
• Theorem 2.6
• Proposition 2.7
• Lemma 2.8
• Theorem 3.1
• proof