On the annihilators of local cohomology modules

“…One of the important and hard problems in commutative algebra is determining the annihilator of the local cohomology module H i I (M ). This problem has been studied by several authors; see for example [1], [2], [9], [10], [11], [15], [16] and [18]. Recall that, for an R-module M , the cohomological dimension of M with respect to I is defined as cd(I, M ) := max{i ∈ Z : H i I (M ) = 0}.…”

Section: Introductionmentioning

confidence: 99%

Exactness of Ideal Transforms and Annihilators of Top Local Cohomology Modules

Bahmanpour¹

2015

Journal of the Korean Mathematical Society

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Abstract. Let (R, m) be a commutative Noetherian local domain, M a non-zero finitely generated R-module of dimension n > 0 and I be an ideal of R. In this paper it is shown that if x 1 , . . . , xt (1 ≤ t ≤ n) be a subset of a system of parameters for M , then the R-module H t (x 1 ,...,xt) (R) is faithful, i.e., Ann H t (x 1 ,...,xt) (R) = 0. Also, it is shown that, if H i I (R) = 0 for all i > dim R − dim R/I, then the R-module H dim R−dim R/I I (R) is faithful. These results provide some partially affirmative answers to the Lynch's conjecture in [10]. Moreover, for an ideal I of an arbitrary Noetherian ring R, we calculate the annihilator of the top local cohomology module H 1 I (M ), when H i I (M ) = 0 for all integers i > 1. Also, for such ideals we show that the finitely generated R-algebra D I (R) is a flat Ralgebra.

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“…This problem for ordinary local cohomology modules has been studied by several authors, see [6,7,8,9,11], and has led to some interesting results. In particular, Bahmanpour et al in [2] proved an interesting result about…”

Section: Introductionmentioning

confidence: 98%

Annihilators of Top Local Cohomology Modules Defined by a Pair of Ideals

Karimi¹,

Payrovi²

2019

International Electronic Journal of Algebra

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Let R be a commutative Noetherian ring, I, J two proper ideals of R and let M be a non-zero finitely generated R-module with c = cd(I, J, M). In this paper, we first introduce T R (I, J, M) as the largest submodule of M with the property that cd(I, J, T R (I, J, M)) < c and we describe it in terms of the reduced primary decomposition of zero submodule of M. It is shown that Ann R (H d I,J (M)) = Ann R (M/T R (I, J, M)) and Ann R (H d I (M)) = Ann R (H d I,J (M)), whenever R is a local ring, M has dimension d with H d I,J (M) = 0 and J t M ⊆ T R (I, M) for some positive integer t.

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On the annihilators of local cohomology modules

Cited by 19 publications

References 2 publications

Cofiniteness and finiteness of local cohomology modules over regular local rings

Cofiniteness and finiteness of local cohomology modules over regular local rings

Exactness of Ideal Transforms and Annihilators of Top Local Cohomology Modules

Annihilators of Top Local Cohomology Modules Defined by a Pair of Ideals

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