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

Lotfi Parsa, Morteza. "$S$-depth on $ZD$-modules and local cohomology." Czechoslovak Mathematical Journal 71.3 (2021): 755-764. <http://eudml.org/doc/297665>.

@article{LotfiParsa2021,
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 $S$-$\{\rm depth\}(I, J, M)=\inf \lbrace S$-$\{\rm depth\}(\mathfrak \{a\}, M) \colon \mathfrak \{a\}\in \widetilde\{\rm W\}(I,J)\rbrace $, 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 \lbrace i \colon H^\{i\}_\{I,J\}(M)\notin S\rbrace $, 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 $\mathfrak \{m\}_1, \mathfrak \{m\}_2,\ldots ,\mathfrak \{m\}_k\in \{\rm W\}(I, J)$ such that $H^\{i\}_\{I,J\}(M)\cong H^\{i\}_\{\mathfrak \{m\}_1\}(M)\oplus H^\{i\}_\{\mathfrak \{m\}_2\}(M)\oplus \cdots \oplus H^\{i\}_\{\mathfrak \{m\}_k\}(M)$ for all $i<t$.},
author = {Lotfi Parsa, Morteza},
journal = {Czechoslovak Mathematical Journal},
keywords = {depth; local cohomology; Serre subcategory; $ZD$-module},
language = {eng},
number = {3},
pages = {755-764},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$S$-depth on $ZD$-modules and local cohomology},
url = {http://eudml.org/doc/297665},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Lotfi Parsa, Morteza
TI - $S$-depth on $ZD$-modules and local cohomology
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 755
EP - 764
AB - 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 $S$-${\rm depth}(I, J, M)=\inf \lbrace S$-${\rm depth}(\mathfrak {a}, M) \colon \mathfrak {a}\in \widetilde{\rm W}(I,J)\rbrace $, 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 \lbrace i \colon H^{i}_{I,J}(M)\notin S\rbrace $, 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 $\mathfrak {m}_1, \mathfrak {m}_2,\ldots ,\mathfrak {m}_k\in {\rm W}(I, J)$ such that $H^{i}_{I,J}(M)\cong H^{i}_{\mathfrak {m}_1}(M)\oplus H^{i}_{\mathfrak {m}_2}(M)\oplus \cdots \oplus H^{i}_{\mathfrak {m}_k}(M)$ for all $i<t$.
LA - eng
KW - depth; local cohomology; Serre subcategory; $ZD$-module
UR - http://eudml.org/doc/297665
ER -