Czechoslovak Mathematical Journal, first online, pp. 1-10


$S$-depth on $ZD$-modules and local cohomology

Morteza Lotfi Parsa

Received February 27, 2020.   Published online October 29, 2020.

Abstract:  Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-${\rm depth}(I, M)$ and ${\rm depth}(I, J, M)$, the $S$-${\rm depth}$ of $(I, J)$ on $M$ is defined as $S$-${\rm depth}(I, J, M)=\inf\{S$-${\rm depth}(\frak{a}, M) \colon\frak{a}\in\widetilde{\rm W}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-${\rm depth}(I, J, M)$ and $H^i_{I,J}(M)$ are studied, and it is proved that $S$-${\rm depth}(I, J, M)=\inf\{i \colon H^i_{I,J}(M)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let ${\rm Supp}_R H^i_{I,J}(M)$ be a finite subset of ${\rm Max}(R)$ for all $i<t$, where $M$ is an arbitrary $R$-module and $t$ is an integer. It is shown that there are distinct maximal ideals $\frak m_1, \frak m_2,\ldots,\frak m_k\in{\rm W}(I, J)$ such that $H^i_{I,J}(M)\cong H^i_{\frak m_1}(M)\oplus H^i_{\frak m_2}(M)\oplus\cdots \oplus H^i_{\frak m_k}(M)$ for all $i<t$.
Keywords:  depth; local cohomology; Serre subcategory; $ZD$-module
Classification MSC:  13C15, 13C60, 13D45
DOI:  10.21136/CMJ.2020.0088-20

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Affiliations:  \!\! Morteza Lotfi Parsa, Sayyed Jamaleddin Asadabadi University, Asadabad, 6541861841, Iran, e-mail:  lotfi.parsa@sjau.ac.ir, lotfi.parsa@yahoo.com


 
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