Finiteness of local cohomology modules over rings of small dimension (original) (raw)
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On the finiteness properties of local cohomology modules
2006
Let a denote an ideal in a regular local (Noetherian) ring R and let N be a finitely generated R-module with support in V (a). The purpose of this paper is to show that all homomorphic images of the R-modules Ext j R (N, H i a (R)) have only finitely many associated primes, for all i, j ≥ 0, whenever dim R ≤ 4 or dim R/a ≤ 3 and R contains a field. In addition, we show that if dim R = 5 and R contains a field, then the R-modules Ext j R (N, H i a (R)) have only finitely many associated primes, for all i, j ≥ 0.
On the cofiniteness properties of certain general local cohomology modules
Acta Scientiarum Mathematicarum
Let R be a commutative Noetherian ring, Φ a system of ideals of R, and M a finitely generated R-module. Suppose that a∈Φ and t is a non-negative integer. It is shown that if Ext R i (R/a,H Φ j (M)) is finitely generated for all i and all j<t, then Ext R i (R/a,H Φ t (M)) is finitely generated for i=0,1. In particular, if R is a local ring of dimension at most 2, then Ext R i (R/a,H Φ j (M)) is finitely generated for all i, j.
On the associated primes of generalized local cohomology modules
Communications in Algebra®, 2006
Let be an ideal of a commutative Noetherian ring R with identity and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that Ass R H t M N is contained in the union of the sets Ass R Ext i R M H t−i N , where 0 ≤ i ≤ t. As an immediate consequence, it follows that if either H i N is finitely generated for all i < t or Supp R H i N is finite for all i < t, then Ass R H t M N is finite. Also, we prove that if d = pd M and n = dim N are finite, then H d+n M N is Artinian. In particular, Ass R H d+n M N is a finite set consisting of maximal ideals.
On the finiteness of associated primes of local cohomology modules
Proceedings of the American Mathematical Society, 2010
Let R be a Noetherian ring, a be an ideal of R and M be a finitely generated R-module. The aim of this paper is to show that if t is the least integer such that neither H t a (M ) nor supp(H t a (M)) is non-finite, then H t a (M ) has finitely many associated primes. This combines the main results of Brodmann and Faghani and independently Khashyarmanesh and Salarian.
On the Attached Prime Ideals of Local Cohomology Modules
Communications in Algebra, 2013
Let I and J be two ideals of a commutative Noetherian ring R and M be an R-module. For a non-negative integer n it is shown that, if the sets Ass R (Ext n R (R/I, M)) and Supp R (Ext i R (R/I, H j I,J (M))) are finite for all i ≤ n + 1 and all j < n, then so is Ass R (Hom R (R/I, H n I,J (M))). We also study the finiteness of Ass R (Ext i R (R/I, H n I,J (M))) for i = 1, 2.
Communications in Algebra, 2013
1. Let (R, m) be a Noetherian local ring, I an ideal of R and N a finitely generated R-module. Let k≥ − 1 be an integer and r = depth k (I, N) the length of a maximal N-sequence in dimension > k in I defined by M. Brodmann and L. T. Nhan (Comm. Algebra, 36 (2008), 1527-1536). For a subset S ⊆ Spec R we set S ≥k = {p ∈ S | dim(R/p)≥k}. We first prove in this paper that Ass R (H j I (N)) ≥k is a finite set for all j≤r. Let N = ⊕ n≥0 N n be a finitely generated graded R-module, where R is a finitely generated standard graded algebra over R 0 = R. Let r be the eventual value of depth k (I, N n). Then our second result says that for all l≤r the sets j≤l Ass R (H j I (N n)) ≥k are stable for large n.
Some results on local cohomology modules
Archiv der Mathematik, 2006
Let R be a commutative Noetherian ring, a an ideal of R, and let M be a finitely generated R-module. For a non-negative integer t, we prove that H t a (M) is a-cofinite whenever H t a (M) is Artinian and H i a (M) is a-cofinite for all i < t. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a local ring (R, m), f − depth(a, M) is the least integer i such that H i a (M) ∼ = H i m (M). This result in conjunction with the first one, yields some interesting consequences. Finally, we extend Grothendieck's non-vanishing Theorem to a-cofinite modules.
A generalization of the finiteness problem in local cohomology modules
Proceedings Mathematical Sciences, 2009
Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H i a (N) is a weakly Laskerian R-module for all i < t, then Hom R (R/a, H t a (M, N)) is weakly Laskerian R-module. Also, we prove that Ext i R (R/a, H t a (N)) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp R (H i a (N)) is a finite set for all i < t, then Ext i R (R/a, H t a (N)) is weakly Laskerian R-module for all i = 0, 1.
On the Finiteness Dimension of Local Cohomology Modules
Algebra Colloquium, 2014
Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. Moreover, we prove that [Formula: see text] for all i.
Cofinite Modules and Generalized Local Cohomology
Houston journal of mathematics
Let R be a commutative Noetherian ring, a an ideal of R, and M , N two finitely generated R-modules. We prove that the generalized local cohomology modules H t a (M, N) are a-cofinite; that is, Ext i R (R/a, H t a (M, N)) is finitely generated for all i, t ≥ 0, in the following cases: (i) cd(a) = 1, where cd is the cohomological dimension of a in R. (ii) dim R ≤ 2. Additionally, we show that if cd(a) = 1 then Ext i R (M, H t a (N)) is a-cofinite for all i, t ≥ 0.