| Title:
|
On the minimaxness and coatomicness of local cohomology modules (English) |
| Author:
|
Hatamkhani, Marzieh |
| Author:
|
Roshan-Shekalgourabi, Hajar |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
72 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
177-190 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $\mathcal {C}$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that $(H^i_I(M))_{\frak m}$ is a minimax $R_{\frak m}$-module for all $\frak m \in {\rm Max} (R)$ and for all $i < n$, then the set ${\rm Ass}_R(H^n_I(M))$ is finite. Also, if $H^i_I(M)$ is minimax for all $i \geq n \geq 1$, then $H^i_I(M)$ is Artinian for $i \geq n$. It is shown that if $M$ is a $\mathcal {C}$-minimax module over a local ring such that $H^i_I(M)$ are $\mathcal {C}$-minimax modules for all $i < n$ (or $i\geq n$), where $n\geq 1$, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules. (English) |
| Keyword:
|
local cohomology module |
| Keyword:
|
minimax module |
| Keyword:
|
coatomic module |
| Keyword:
|
Artinian module |
| Keyword:
|
local-global principle |
| MSC:
|
13C05 |
| MSC:
|
13D45 |
| MSC:
|
13E05 |
| idZBL:
|
Zbl 07511560 |
| idMR:
|
MR4389113 |
| DOI:
|
10.21136/CMJ.2021.0383-20 |
| . |
| Date available:
|
2022-03-25T08:29:29Z |
| Last updated:
|
2024-04-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149580 |
| . |
| Reference:
|
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