Volume 9, Issue 3 (12-2023)                   mmr 2023, 9(3): 136-146 | Back to browse issues page

XML Persian Abstract Print


Download citation:
BibTeX | RIS | EndNote | Medlars | ProCite | Reference Manager | RefWorks
Send citation to:

Rezaei S. Attached primes of top local cohomology modules. mmr 2023; 9 (3) :136-146
URL: http://mmr.khu.ac.ir/article-1-3200-en.html
PNU , sha.Rezaei@gmail.com
Abstract:   (331 Views)
Introduction
        Let a  be an ideal of Noetherian ring R  and M,N  be finitely generated R -modules. Recall that, for each i≥0 , i-th generalized local cohomology module M, N with respect to a  is defined by
HaiM,Nlimn Hai(M/anM,N).
Also, recall that  cda,M,N,  the cohomological dimension of R-modules M and N with respect to an ideal a  of a commutative Noetherian ring R  is
supiN0: Hai M,N≠0.
An important problem concerning local cohomology is determining the set of attached prime ideals of the top local cohomology modules. This problem has been studied by several authors. In this paper, we study attached prime ideals of top local cohomology modules.
 Material and methods
   In this paper, we first obtain some subsets of the set of attached prime ideals of top local cohomology module. By using these, we obtain a result about finiteness of  top local cohomology modules.

Results and discussion
    Let R  be a Noetherian ring and a  be an ideal of  R . Let M  and N  be non-zero finitely generated R -modules.  Assume that  pdM=d<∞ , cda,N=c<∞ . We will prove that 
i) pSuppRNcda,R/p=c, dimRp=cAttR(HacN),
ii)  AttR(Had+cM,N)pSuppRNcda,M,R/p=d+c,
iii) pSuppRN :cda,M,Rp=d+c,dimRp=c AttR(Had+cM,N).
Conclusion
    Let M  and N  be non-zero finitely generated R -modules.  Assume that  pdM=d<∞ , cda,N=c<∞ The following conclusions were drawn from this research.
 •  If  (R,m)   is a Noetherian local ring such that   Had+cM,N≠0  then Had+cM,N  is not of finite length.      
  •  If R is a Noetherian domain, then under certain conditions we have 
AttR(Had+cM,N)=AttR(HacN) .
 
Full-Text [PDF 298 kb]   (122 Downloads)    
Type of Study: Original Manuscript | Subject: alg
Received: 2021/05/26 | Revised: 2024/02/19 | Accepted: 2022/04/22 | Published: 2023/12/31 | ePublished: 2023/12/31

Add your comments about this article : Your username or Email:
CAPTCHA

Send email to the article author


Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

© 2024 CC BY-NC 4.0 | Mathematical Researches

Designed & Developed by : Yektaweb