Local Cohomology Modules Research Articles

On top local cohomology modules, Matlis duality and tensor products

Let [Formula: see text] be an ideal of a local ring [Formula: see text] with [Formula: see text] the cohomological dimension of [Formula: see text] in [Formula: see text]. In the case that [Formula: see text], we first give a bound for [Formula: see text], where [Formula: see text] and [Formula: see text] is complete. Later, [Formula: see text], [Formula: see text] and [Formula: see text] are examined. In the case [Formula: see text], the set [Formula: see text] is considered.

Generalization of a connectedness result to cohomologically complete intersections

It is a well-known result from Hartshorne that, in projective space over a field, every set-theoretical complete intersection of positive dimension is connected in codimension one. Another important connectedness result (also from Hartshorne) is that a local ring with disconnected punctured spectrum has depth at most 1 1 . The two results are related; Hartshorne calls the latter “the keystone to the proof” of the former. In this short note we show how the latter result generalizes smoothly from set-theoretical to cohomologically complete intersections, i.e., to ideals for which there is in terms of local cohomology no obstruction to be a complete intersection. The proof is based on the fact that for cohomologically complete intersections over a complete local ring, the endomorphism ring of the (only) local cohomology module is the ring itself and hence is indecomposable as a module.

The derived category analogues of Faltings Local-global Principle and Annihilator Theorems

Let [Formula: see text] be a specialization closed subset of Spec R and X a homologically left-bounded complex with finitely generated homologies. We establish Faltings’ Local-global Principle and Annihilator Theorems for the local cohomology modules [Formula: see text] Our versions contain variations of results already known on these theorems.

On asymptotic vanishing behavior of local cohomology

Let R be a standard graded algebra over a field k, with irrelevant maximal ideal $$\mathfrak {m}$$, and I a homogeneous R-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules $$\{{\text {H}}^{i}_{\mathfrak {m}}(R/I^n)\}_{n\in {\mathbb {N}}}$$ for $$i<\dim R/I$$. We show that, when $${{\,\mathrm{{{char}}}\,}}k= 0$$, R / I is Cohen–Macaulay, and I is a complete intersection locally on $${{\,\mathrm{{Spec}}\,}}R {\setminus }\{\mathfrak {m}\}$$, the lowest degrees of the modules $$\{{\text {H}}^{i}_{\mathfrak {m}}(R/I^n)\}_{n\in {\mathbb {N}}}$$ are bounded by a linear function whose slope is controlled by the generating degrees of the dual of $$I/I^2$$. Our result is a direct consequence of a related bound for symmetric powers of locally free modules. If no assumptions are made on the ideal or the field k, we show that the complexity of the sequence of lowest degrees is at most polynomial, provided they are finite. Our methods also provide a result on stabilization of maps between local cohomology of consecutive powers of ideals.

On coLaskerian module and local cohomology

The notion of generalized Laskerian modules was introduced recently by the authors. In this paper, we continue our investigation of such modules. Moreover, we are going to generalize some results on finiteness of associated primes of local cohomology modules to the category of generalized Laskerian modules.

Notes on linkage of modules

AbstractLet R be a Cohen–Macaulay local ring. It is shown that under some mild conditions, the Cohen–Macaulay property is preserved under linkage. We also study the connection of the (Sn) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module.

A sufficient condition for the finiteness of Frobenius test exponents

The Frobenius test exponent Fte ⁡ ( R ) \operatorname {Fte}(R) of a local ring ( R , m ) (R,\mathfrak {m}) of prime characteristic p &gt; 0 p&gt;0 is the smallest e 0 ∈ N e_0 \in \mathbb {N} such that for every ideal q \mathfrak {q} generated by a (full) system of parameters, the Frobenius closure q F \mathfrak {q}^F has ( q F ) [ p e 0 ] = q [ p e 0 ] (\mathfrak {q}^F)^{\left [p^{e_0}\right ]}=\mathfrak {q}^{\left [ p^{e_0}\right ]} . We establish a sufficient condition for Fte ⁡ ( R ) &gt; ∞ \operatorname {Fte}(R)&gt;\infty and use it to show that if R R is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e., H m j ( R ) / 0 H m j ( R ) F H^j_\mathfrak {m}(R)/0^F_{H^j_\mathfrak {m}(R)} is finite length for 0 ≤ j &gt; dim ⁡ ( R ) 0 \le j &gt; \dim (R) , then Fte ⁡ ( R ) &gt; ∞ \operatorname {Fte}(R)&gt;\infty .

English

Let $${\cal I}$$ be a set of ideals of a commutative Noetherian ring R. We use the notion of $${\cal I}$$-closure operation which is a semiprime closure operation on submodules of modules to introduce the class of $${\cal I}$$-Laskerian modules. This enables us to investigate the set of associated prime ideals of certain $${\cal I}$$-closed submodules of local cohomology modules.

Top local cohomology modules over local rings and the weak going-up property

Let (R,m) be a Noetherian local ring and let ˆR denote the m-adic completion of R. In this paper, we introduce the concept of the weak going-up property for the extension R⊆ˆR and we give some characterizations of this property. In particular, we show that this property is equivalent to the strong form of the Lichtenbaum–Hartshorne Vanishing Theorem. Also, when R satisfies the weak going-up property, we show that for a finitely generated R-module M of dimension d, and ideals a and b of R, we have AttR(Hda(M))=AttR(Hdb(M)) if and only if da(M)≅Hd(M), and we find a criterion for the cofiniteness of Artinian top local cohomology modules.

Cofiniteness over Noetherian complete local rings

In this article, we prove the following generalization of a result of Hartshorne: Let be a regular local ring of dimension 4. Assume that is a regular system of parameters for S and . Then for each finitely generated S-module N with the socle of is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring , we characterize the class of all ideals I of R with the property that, for every finitely generated R-module M, the local cohomology modules are I-cofinite for all .

Weakly cofiniteness of local cohomology modules

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] a system of ideals of [Formula: see text] and [Formula: see text]. Let [Formula: see text] be an [Formula: see text]-module (not necessary [Formula: see text]-torsion) such that [Formula: see text], then the [Formula: see text]-module [Formula: see text] is weakly Laskerian, for all [Formula: see text], if and only if the [Formula: see text]-module [Formula: see text] is weakly Laskerian for [Formula: see text]. Let [Formula: see text] be an integer and [Formula: see text] an [Formula: see text]-module such that [Formula: see text] is weakly Laskerian for all [Formula: see text]. We prove that if the [Formula: see text]-module [Formula: see text] is [Formula: see text] for all [Formula: see text], then [Formula: see text] is [Formula: see text]-weakly cofinite for all [Formula: see text], and for any [Formula: see text] (or minimax) submodule [Formula: see text] of [Formula: see text], the [Formula: see text]-modules [Formula: see text] and [Formula: see text] are weakly Laskerian. Let [Formula: see text] be a finitely generated [Formula: see text]-module. We also prove that [Formula: see text] and [Formula: see text] are [Formula: see text]-weakly cofinite for all [Formula: see text] and [Formula: see text] whenever [Formula: see text] is weakly Laskerian and [Formula: see text] is [Formula: see text] for all [Formula: see text]. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.

Finiteness properties and numerical behavior of local cohomology

This manuscript discusses many problems connected with finiteness properties of local cohomology modules, e.g. the associated primes, the minimal primes, Bass numbers, as well as the behavior of injective dimension, Lyubeznik numbers, and content.This article is dedicated to Gennady Lyubeznik on the occasion of his 60th birthday, and celebrates his many contributions to commutative algebra.

Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals

The study of Frobenius actions on local cohomology modules over a local ring of prime characteristic has interesting connections with the theory of tight closure. This paper establishes new connections by developing the notion of relative Frobenius actions on local cohomology. As an application, we show that a ring has F-nilpotent singularities if and only if the tight closure of every parameter ideal is equal to its Frobenius closure.

Lyubeznik numbers and injective dimension in mixed characteristic

We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated $\mathbb{Z}$-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for almost all reductions to positive characteristic. Additionally, we address an open question of Lyubeznik that asks whether the injective dimension of a local cohomology module over a regular ring is bounded above by the dimension of its support. Although we show that the answer is affirmative for several families of $\mathbb{Z}$-algebras, we also exhibit an example where this bound fails to hold. This example settles Lyubeznik's question, and illustrates one way that the behavior of local cohomology modules of regular rings of equal characteristic and of mixed characteristic can differ.

A short note on annihilators of local cohomology modules

Let [Formula: see text] be a commutative Noetherian domain, [Formula: see text] a nonzero [Formula: see text]-module of finite injective dimension, and [Formula: see text] be a nonzero ideal of [Formula: see text]. In this paper, we prove that whenever [Formula: see text], then the annihilator of [Formula: see text] is zero. Also, we calculate the annihilator of [Formula: see text] for finitely generated [Formula: see text]-modules [Formula: see text] and [Formula: see text] with conditions [Formula: see text] and [Formula: see text]. Moreover, if [Formula: see text] is a regular Noetherian local ring and [Formula: see text] such that [Formula: see text], then we show that there exists an ideal [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text].

On a question of D. Rees on classical integral closure and integral closure relative to an Artinian module

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] and [Formula: see text] ideals of [Formula: see text]. Motivated by a question of Rees, we study the relationship between [Formula: see text], the classical Northcott–Rees integral closure of [Formula: see text], and [Formula: see text], the integral closure of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text] (also called here ST-closure of [Formula: see text] on [Formula: see text]), in order to study a relation between [Formula: see text], the multiplicity of [Formula: see text], and [Formula: see text], the multiplicity of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text]. We conclude [Formula: see text] when every minimal prime ideal of [Formula: see text] belongs to the set of attached primes of [Formula: see text]. As an application, we show what happens when [Formula: see text] is a generalized local cohomology module.

Faltings’ local–global principle and annihilator theorem for the finiteness dimensions

Let R be a commutative Noetherian ring, M be a finitely generated R-module and n be a non-negative integer. In this article, it is shown that for a positive integer t, there is a finitely generated submodule Ni of such that for all i < t if and only if there is a finitely generated submodule of such that for all i < t and all p ∈ Spec(R). This generalizes Faltings’ Local–global Principle for the finiteness of local cohomology modules (Faltings’ in Math. Ann. 255:45–56, 1981). Also, it is shown that whenever R is a homomorphic image of a Gorenstein local ring, then the invariants and are equal, for every finitely generated R-module M and for all ideals of R with . As a consequence, we determine the least integer i where the local cohomology module is not minimax (resp. weakly Laskerian).

Finiteness of extension functors of generalized local cohomology modules

Let R be a commutative Noetherian ring with non-zero identity, an ideal of R, M and X finite R–modules, and t a non-negative integer such that is –cofinite for all i < t. In this article, we show that and are finite, and is finite if and only if is finite. As a consequence, we observe that is a finite set.

Strong F-regularity and generating morphisms of local cohomology modules

We establish a criterion for the strong F-regularity of a (non-Gorenstein) Cohen–Macaulay reduced complete local ring of dimension at least 2 and prime characteristic p. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an n×(n−1) matrix X of indeterminates. For a perfect field k of characteristic p≥5, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring over k given by the maximal minors of X is strongly F-regular.

Bass numbers of derived local cohomology with respect to a specialization closed subset

Let [Formula: see text] be a specialization closed subset of Spec [Formula: see text] and [Formula: see text] a left homologically bounded complex with finitely generated homologies. We provide some inequalities between the Bass numbers of [Formula: see text] and its local cohomology modules with respect to [Formula: see text]. As an application of these inequalities, we provide a comparison between the injective dimensions of [Formula: see text] and its nonzero local cohomology module [Formula: see text]. Our versions contain variations of some results already known in these areas.