| Title:
|
A generalization of the finiteness problem of the local cohomology modules (English) |
| Author:
|
Abbasi, Ahmad |
| Author:
|
Roshan-Shekalgourabi, Hajar |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
69-78 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative Noetherian ring and ${\mathfrak a}$ an ideal of $R$. We introduce the concept of ${\mathfrak a}$-weakly Laskerian $R$-modules, and we show that if $M$ is an ${\mathfrak a}$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\rm Ext}^j_R(R/{\mathfrak a}, H^i_{{\mathfrak a}}(M))$ is ${\mathfrak a}$-weakly Laskerian for all $i<s$ and all $j$, then for any ${\mathfrak a}$-weakly Laskerian submodule $X$ of $H^s_{{\mathfrak a}}(M)$, the $R$-module ${\rm Hom}_R(R/{\mathfrak a},H^s_{{\mathfrak a}}(M)/X)$ is ${\mathfrak a}$-weakly Laskerian. In particular, the set of associated primes of $H^s_{\mathfrak a}(M)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an ${\mathfrak a}$-weakly Laskerian $R$-module such that $ H^i_{{\mathfrak a}}(N)$ is ${\mathfrak a}$-weakly Laskerian for all $i<s$, then the set of associated primes of $H^s_{\mathfrak a}(M, N)$ is finite. This generalizes the main result of S. Sohrabi Laleh, M. Y. Sadeghi, and M. Hanifi Mostaghim (2012). (English) |
| Keyword:
|
local cohomology module |
| Keyword:
|
weakly Laskerian module |
| Keyword:
|
${\mathfrak a}$-weakly Laskerian module |
| Keyword:
|
associated prime |
| MSC:
|
13C05 |
| MSC:
|
13D45 |
| MSC:
|
13E10 |
| idZBL:
|
Zbl 06391477 |
| idMR:
|
MR3247445 |
| DOI:
|
10.1007/s10587-014-0084-y |
| . |
| Date available:
|
2014-09-29T09:35:27Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143950 |
| . |
| Reference:
|
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| Reference:
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