Annihilators and Attached Primes of Top Local Cohomology Modules
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Czechoslovak Mathematical Journal
7 pp
Online first
ANNIHILATORS AND ATTACHED PRIMES OF TOP LOCAL COHOMOLOGY MODULES Shahram Rezaei, Tehran Received November 5, 2019. Published online September 30, 2020.
Abstract. Let a be an ideal of Noetherian ring R and M a finitely generated R-module. cd(a,M ) cd(a,M ) In this paper we determine AnnR (Ha (M )) and AttR (Ha (M )), which are two cd(a,M ) important problems concerning the last nonzero local cohomology module Ha (M ). cd(a,M ) We show that AnnR (Ha (M )) = AnnR (M/TR (a, M )), where TR (a, M ) is the largest submodule of M such that cd(a, TR (a, M )) < cd(a, M ). Using the above result we determine cd(a,M ) the attached primes of the top local cohomology module AttR (Ha (M )). In fact, cd(a,M ) we show that AttR (Ha (M )) = {p ∈ SuppR M : cd(a, R/p) = cd(a, M )}. Then by using these, we obtain some main results of A. Atazadeh, M. Sedghi, R. Naghipour (2014), K. Bahmanpour, J. A’zami, G. Ghasemi (2012) and K. Divaani-Aazar (2009). Keywords: annihilator; attached prime; local cohomology MSC 2020 : 13D45, 14B15, 13E05
1. Introduction Throughout this paper, R is a commutative Noetherian ring with identity, a is an ideal of R and M is an R-module. An important problem concerning local cohomology is determining the annihilators of the ith local cohomology module Hia (M ). This problem has been studied by several authors, see for example [2], [3], [4], [5] and [9]. In [5], Bahmanpour et al. proved an intersting result about the dim(M ) annihilator AnnR (Hm (M )) in the case when (R, m) is a complete local ring. In fact, in [5], Theorem 2.6, they proved that if (R, m) is a complete local ring, S dim(M ) then AnnR (Hm (M )) = AnnR (M/TR (M )), where TR (M ) = {N : N 6 M and dim N < dim M }. More recently, Atazadeh et al. in [2] generalized this main result by determindim(M ) ing AnnR (Ha (M )) for an arbitrary Noetherian ring R. In [2], Theorem 2.3 DOI: 10.21136/CMJ.2020.0479-19
1
dim(M )
by using the above main result, they showed that if Ha (M ) 6= 0, then S dim(M ) AnnR (Ha (M )) = AnnR (M/TR (a, M )), where TR (a, M ) = {N : N 6 M and cd(a, N ) < cd(a, M )}. cd(a,M ) In [2], there is a question about AnnR (Ha (M )). Is it possible to determine cd(a,M ) AnnR (Ha (M ))? In this paper, by an easy and short proof, we answer this question and determine it. Clearly, it is a generalization of the above main results. The following theorem is the first our main result: Theorem 1.1. Let R be a Noetherian ring and a be an ideal of R. Let M be cd(a,M ) a finitely generated R-module. Then AnnR (Ha (M )) = AnnR (M/TR (a, M )), S where TR (a, M ) = {N : N 6 M and cd(a, N ) < cd(a, M )}. By using the above theorem, we obtain the main result concerning attached prime ideals of top local cohomology modules. In [2], Theorem 3.3 it is shown that cd(a,M )
AttR (Ha
(M )) ⊆ {p ∈ SuppR M : cd(a, R/p) = cd(a, M )}.
Is the above containment an equality? This is the second question in [2]. Here we answer this question by the following main result: Theorem 1.2. Let R be a Noetherian ri
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