On the primefulness of 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.

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.

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.

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.

Journal of Algebra, 2014

Let (R, m) be a Noetherian local ring and M a finitely generated R-module. It is well known that the local cohomology module H i m (M) is Artinian for all i 0. Following I.G. Macdonald [8], denote by Att R H i m (M) the set of attached primes of H i m (M). This paper is concerned with clarifying the structure of the base ring R via a relation between Att R H i m (M) and Att R H i m (M). Some characterizations for R being universally catenary with all Cohen-Macaulay formal fibers are given.

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.

2020

The first part of the paper is concerned to relationship between the sets of associated primes of the generalized d-local cohomology modules and the ordinary generalized local cohomology modules. Assume that R is a commutative Noetherian local ring, M and N are finitely generated R-modules and d, t are two integers. We prove that AssH d(M,N) = ∪ I∈Φ AssH t I(M,N) whenever H i d(M,N) = 0 for all i &lt; t and Φ = {I : I is an ideal of R with dimR/I ≤ d}. In the second part of the paper, we give some information about the non-vanishing of the generalized d-local cohomology modules. To be more precise, we prove that H d(M,R) ̸= 0 if and only if i = n − d whenever R is a Gorenstein ring of dimension n and pdR(M) &lt; ∞. This result leads to an example which shows that AssHn−d d (M,R) is not necessarily a finite set.

Kodai Mathematical Journal, 2006

Let M be a semi-discrete linearly compact module over a commutative noetherian ring R and i a non-negative integer. We show that the set of co-associated primes of the local homology R-module H I i ðMÞ is finite in either of the following cases: (i) The R-modules H I j ðMÞ are finite for all j < i; (ii) I J RadðAnn R ðH I j ðMÞÞÞ for all j < i. By Matlis duality we extend some results for the finiteness of associated primes of local cohomology modules H i I ðMÞ: 383 2000 Mathematics subject classification. Local cohomology 13D45, Homological functors on modules 16E30, Topological rings and modules 13J99.

Arxiv preprint math/0406357, 2004

The notion of weakly Laskerian modules was introduced recently by the authors. Let R be a commutative Noetherian ring with identity, an ideal of R, and M a weakly Laskerian module. It is shown that if is principal, then the set of associated primes of the local cohomology module H i M is finite for all i ≥ 0. We also prove that when R is local, then Ass R H i M is finite for all i ≥ 0 in the following cases: (1) dim R ≤ 3, (2) dim R/ ≤ 1, (3) M is Cohen-Macaulay, and for any ideal , with l = grade M , Hom R R/ H l+1 M is weakly Laskerian.

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.