On $\operatorname{Ext}$-finite modules, quasi-injective dimension and width of 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 $\inf \lbrace i : \operatorname{Tor}_i^R(k,M) \neq 0 \rbrace< \infty$.