Top Local Cohomology Research Articles

A Note on Local Cohomology

A certain set of associated primes of the Matlis duality of any top local cohomology module of a complete filter local ring is characterized. Also, it is proved that the set of associated primes of a finitely generated module over a four-dimensional local ring is finite.

Matlis duals of top Local cohomology modules

In the first section of this paper we present generalizations of known results on the set of associated primes of Matlis duals of local cohomology modules; we prove these generalizations by using a new technique. In section 2 we compute the set of associated primes of the Matlis dual of H d-1 J (R), where R is a d-dimensional local ring and J ⊆ R an ideal such that dim(R/J) = 1 and H d J (R) = 0.

Top Local Cohomology Modules

For a finitely generated module M over a commutative Noetherian local ring (R,𝔪), it is shown that there exist only a finite number of non-isomorphic top local cohomology modules [Formula: see text] for all ideals 𝔞 of R. It is also shown that for a given integer r ≥ 0, if [Formula: see text] is zero for all 𝔭 in Supp (M), then [Formula: see text] for all i ≥ r.

Top Local Cohomology and the Catenaricity of the Unmixed Support of a Finitely Generated Module

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module with dim M = d. This paper is concerned with the following property for the top local cohomology module : In this article we will show that this property is equivalent to the catenaricity of the unmixed support USupp M of M which is defined by Usupp M = Supp M/UM(0), where UM(0) is the largest submodule of M of dimension less than d. Some characterizations of this property in terms of system of parameters as well as the relation between the unmixed supports of M and of the 𝔪-adic completion are given.

Matlis Duals of Top Local Cohomology Modules and the Arithmetic Rank of an Ideal

Let I be an ideal of a local ring (R, 𝔪). Using local cohomology, we present new criteria (see 1.4, respectively 1.5) for the conditions ara (I) ≤ 1 respectively ara (I) ≤ 2, where ara (I) stands for the number of generators of I up to radical. Though this works equally well for the local and for the graded case, we show some subtle differences between the local and the graded situation in Section 2. Finally, in Section 3, we show that the Matlis dual of certain local cohomology modules, though not finite, is well behaved in some sense.

Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure

This paper is concerned with the tight closure of an ideal $\mathfrak {a}$ in a commutative Noetherian local ring $R$ of prime characteristic $p$. Several authors, including R. Fedder, K-i. Watanabe, K. E. Smith, N. Hara and F. Enescu, have used the natural Frobenius action on the top local cohomology module of such an $R$ to good effect in the study of tight closure, and this paper uses that device. The main part of the paper develops a theory of what are here called ‘special annihilator submodules’ of a left module over the Frobenius skew polynomial ring associated to $R$; this theory is then applied in the later sections of the paper to the top local cohomology module of $R$ and used to show that, if $R$ is Cohen–Macaulay, then it must have a weak parameter test element, even if it is not excellent.

Quasihomogeneity of isolated hypersurface singularities and logarithmic cohomology

We characterize quasihomogeneity of isolated hypersurface singularities by the injectivity of the map induced by the first differential of the logarithmic differential complex in the top local cohomology supported in the singular point.

On the Associated Primes of Matlis Duals of Top Local Cohomology Modules

After motivating the question, we prove various results about the set of associated primes of Matlis duals of top local cohomology modules. In some cases, we can calculate this set. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules.

Attached primes of the top local cohomology modules with respect to an ideal

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $\mathfrak{a}$ be an ideal of R. Let M be an R-module of dimension n. In this paper we study the attached primes of the top local cohomology module ${\text{H}}_\mathfrak{a}^n (M).$

Bounds on the a-invariant and reduction numbers of ideals

Let R be a d-dimensional standard graded ring over an Artinian local ring. Let M be the unique maximal homogeneous ideal of R. Let h i ( R) n denote the length of the nth graded component of the local cohomology module H i M (R) . Define the Eisenbud–Goto invariant EG( R) of R to be the number ∑ q=0 d−1 d−1 q h q M (R) 1−q. We prove that the a-invariant a( R) of the top local cohomology module H M d(R) satisfies the inequality: a( R)⩽ e( R)−ℓ( R 1)+( d−1)(ℓ( R 0)−1)+ EG( R). This bound is used to get upper bounds for the reduction number of an m -primary ideal I of a Cohen–Macaulay local ring (R, m) , when the associated graded ring of I has depth at least d−1.