| Title:
|
Finiteness of local homology modules (English) |
| Author:
|
Rezaei, Shahram |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
56 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
31-41 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $I$ be an ideal of Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper, we introduce the concept of weakly colaskerian modules and by using this concept, we give some vanishing and finiteness results for local homology modules. Let $I_{M}:=\operatorname{Ann}_{R}(M/IM)$, we will prove that for any integer $n$ \begin{enumerate} \item[(i)] If $N$ is a weakly colaskerian linearly compact $R$-module such that $(0:_N {I_M})\neq 0$ then $$ \operatorname{width}_{I_M}(N)= \inf\{i\mid \operatorname{H}_i^{I_M}(N)\neq 0 \} =\inf\{i \mid \operatorname{H}_i^I(M,N)\neq 0 \}\,. $$ \item[(ii)] If $(R,\frak{m})$ is a Noetherian local ring and $N$ is an artinian $R$-module then \begin{multline*} \cup_{i<n}\operatorname{Cos}_R\big(\operatorname{H}_i^{I_M}(N)\big)=\cup_{i<n}\operatorname{Cos}_R\big(\operatorname{H}_i^I(M,N)\big)=\\ \cup_{i<n}\operatorname{Cos}_R\big(\operatorname{Tor}_i^R(M/IM,N)\big)\,, \end{multline*} \begin{multline*} \inf\{i \mid \operatorname{H}_i^{I_M}(N) \text{ is not Noetherian $R$-module\,} \}=\\ \inf\{i \mid \operatorname{H}_i^I(M,N) \mbox{\ is not Noetherian $R$-module\,}\}\,. \end{multline*} \end{enumerate} (English) |
| Keyword:
|
coregular sequence |
| Keyword:
|
local homology |
| Keyword:
|
weakly colaskerian |
| MSC:
|
13D45 |
| MSC:
|
16E30 |
| idZBL:
|
Zbl 07177878 |
| idMR:
|
MR4075886 |
| DOI:
|
10.5817/AM2020-1-31 |
| . |
| Date available:
|
2020-03-02T09:06:50Z |
| Last updated:
|
2020-08-26 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148034 |
| . |
| Reference:
|
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