On depth of modules in an ideal

Abstract

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$-module with $\dim_R(M)=d$. Denote by $\depth_R(I,M)$ the depth of $M$ in $I$. In \cite{HT}, C. Huneke and V. Trivedi proved that if $R$ is a quotient of a regular ring then there exists a finite subset $\Lambda_M $ of $\Spec(R)$ such that $$\depth_R(I,M)=\underset{\p\in \Lambda_M}{\min} \big\{ \depth_{R_{\p}}(M_{\p})+ \docao\big((I+\p)/\p\big) \big\}.$$ Denote by $\Psupp^i_R(M)=\{\frak p\in\Spec(R)\mid H^{i-\dim(R/\frak p)}_{\frak p R_{\frak p}}(M_{\frak p})\neq 0\}$ the $i$-th pseudo support of $M$ defined by M. Brodmann and R. Y. Sharp \cite{BS1}. In this paper, we prove that if $\Psupp^i_R(M)$ is closed for all $i\leq d$ then the above formula of $\depth_R(I,M)$ holds true, where $\Lambda_M =\underset{0\leq i\leq d}{\bigcup} \min \Psupp^i_R(M)$. In particular, if $R$ is a quotient of a Cohen-Macaulay local ring then $\Lambda_M =\underset{0\leq i\leq d}{\bigcup}\min\Var\big(\Ann_R(H_{\m}^i(M))\big)$. We also give some examples to clarify the results.