| Title:
|
Artinian cofinite modules over complete Noetherian local rings (English) |
| Author:
|
Sadeghi, Behrouz |
| Author:
|
Bahmanpour, Kamal |
| Author:
|
A'zami, Jafar |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
877-885 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(R,\mathfrak {m})$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module. In this paper it is shown that if $\mathfrak p$ is a prime ideal of $R$ such that $\dim R/\mathfrak p=1$ and $(0:_M\mathfrak p)$ is not finitely generated and for each $i\geq 2$ the $R$-module ${\rm Ext}^i_R(M,R/\mathfrak p)$ is of finite length, then the $R$-module ${\rm Ext}^1_R(M,R/\mathfrak p)$ is not of finite length. Using this result, it is shown that for all finitely generated $R$-modules $N$ with $\operatorname {Supp}(N)\subseteq V(I)$ and for all integers $i\geq 0$, the $R$-modules ${\rm Ext}^i_R(N,M)$ are of finite length, if and only if, for all finitely generated $R$-modules $N$ with $\operatorname {Supp}(N)\subseteq V(I)$ and for all integers $i\geq 0$, the $R$-modules ${\rm Ext}^i_R(M,N)$ are of finite length. (English) |
| Keyword:
|
Artinian module |
| Keyword:
|
cofinite module |
| Keyword:
|
Krull dimension |
| Keyword:
|
local cohomology |
| MSC:
|
13D45 |
| MSC:
|
13E10 |
| MSC:
|
14B15 |
| idZBL:
|
Zbl 06282116 |
| idMR:
|
MR3165502 |
| DOI:
|
10.1007/s10587-013-0059-4 |
| . |
| Date available:
|
2014-01-28T14:02:59Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143604 |
| . |
| Reference:
|
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