| Title:
|
$S$-depth on $ZD$-modules and local cohomology (English) |
| Author:
|
Lotfi Parsa, Morteza |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
755-764 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $i<t$. (English) |
| Keyword:
|
depth |
| Keyword:
|
local cohomology |
| Keyword:
|
Serre subcategory |
| Keyword:
|
$ZD$-module |
| MSC:
|
13C15 |
| MSC:
|
13C60 |
| MSC:
|
13D45 |
| idZBL:
|
07396195 |
| idMR:
|
MR4295243 |
| DOI:
|
10.21136/CMJ.2020.0088-20 |
| . |
| Date available:
|
2021-08-02T08:05:30Z |
| Last updated:
|
2023-10-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149054 |
| . |
| Reference:
|
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