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

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Attached primes of top local cohomology modules
Shahram Rezaei *
PNU , sha.Rezaei@gmail.com
Abstract:   (1431 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,N≔limn 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
supi∈N0: 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) p∈SuppRN:  cda,R/p=c, dimRp=c⊆AttR(HacN),
ii)  AttR(Had+cM,N)⊆p∈SuppRN:  cda,M,R/p=d+c,
iii) p∈SuppRN :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) .
 
Keywords: local cohomology modules, attached primes
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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
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