Top Local Cohomology Research Articles

On the attached primes of top local cohomology modules

Let a be an ideal of Noetherian ring R and M a finitely generated R-module such that cd ( a , M ) = c . In this paper, we investigate Att R ( H a c ( M ) ) . Among other things, it is shown that Max { p ∈ Supp R M | cd ( a , R / p ) = c } ⊆ Att R ( H a c ( M ) ) . We also show that Att R ( H a c ( M ) ) = { p ∈ Supp R M | cd ( a , R / p ) = c , p = Ann R ( H a c ( R / p ) ) } and { p ∈ Supp R M | cd ( a , R / p ) = c , dim R / p − 1 ≤ cd ( a , R / p ) ≤ dim R / p } ⊆ Att R ( H a c ( M ) ) . Finally, we prove that if ( R , m ) is a local ring and dim R / a = 1 then Att R ( H a c ( M ) ) = { p ∈ Supp R M | cd ( a , R / p ) = cd ( a , M ) } . Then by using this, it is shown that if ( R , m ) is a local ring then { p ∈ Supp R M | cd ( a , R / p ) = c , dim R / ( a + p ) = 1 } ⊆ Att R ( H a c ( M ) ) .

Annihilator of top local cohomology and Lynch’s conjecture

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] a proper ideal of [Formula: see text] and [Formula: see text] a nonzero finitely generated [Formula: see text]-module with [Formula: see text]. Let [Formula: see text] (respectively, [Formula: see text]) be the smallest (respectively, greatest) non-negative integer [Formula: see text] such that the local cohomology [Formula: see text] is nonzero. In this paper, we provide sharp bounds under inclusion for the annihilators of the local cohomology modules [Formula: see text], [Formula: see text] and these annihilators are computed in certain cases. Also, we construct a counterexample to Lynch’s conjecture.

On shifted principles for attached primes of the Artinian cofinite top local cohomology modules

Let be a Noetherian local ring, I be an ideal of R and M be a finitely generated R-module such that is Artinian and I-cofinite, where In this paper, we establish shifted principles under localization and completion for attached primes of the top local cohomology module

Indecomposability of top local cohomology modules and connectedness of the prime divisors graphs

Let [Formula: see text] be an ideal of a Noetherian local ring [Formula: see text] with [Formula: see text] and [Formula: see text] be a positive integer. In this paper, it is shown that the top local cohomology module [Formula: see text] (equivalently, its Matlis dual [Formula: see text]) can be written as a direct sum of [Formula: see text] indecomposable summands if and only if the endomorphism ring [Formula: see text] can be written as a direct product of [Formula: see text] local endomorphism rings if and only if the set of minimal primes [Formula: see text] of [Formula: see text] with [Formula: see text] can be written as disjoint union of [Formula: see text] non-empty subsets [Formula: see text] such that for all distinct [Formula: see text] and all [Formula: see text] and all [Formula: see text], we have [Formula: see text]. This generalizes Theorem 3.6 of Hochster and Huneke [Contemp. Math. 159 (1994) 197–208].

Lynch’s conjecture and attached primes of the top local cohomology modules

Lynch’s conjecture and attached primes of the top local cohomology modules

On Shifted Principles for Attached Primes of the Top Local Cohomology Modules

On Shifted Principles for Attached Primes of the Top Local Cohomology Modules

Characterization of some special rings via Linkage

Providing a description of linked ideals in a commutative Noetherian ring in terms of some associated prime ideals, we make a characterization of Cohen-Macaulay, Gorenstein and regular local rings in terms of their linked ideals. More precisely, it is shown that the local ring $(R,fm)$ is Cohen-Macaulay if and only if any linked ideal is unmixed. Also, $(R,fm)$ is Gorenstein if and only if any unmixed ideal $fa$ is linked by every maximal regular sequence in $fa$. We also compute the annihilator of top local cohomology modules in some special cases.

Attached primes of local cohomology modules with respect to a pair of ideals

The paper shows an extension of the Lichtenbaum–Hartshorne Vanishing theorem for local cohomology modules with respect to a pair of ideals. We also study the attached primes of top local cohomology module [Formula: see text] where [Formula: see text] In the case, where [Formula: see text] we show that [Formula: see text]

English

Let $$\mathfrak{a}$$ be an ideal of Noetherian ring R and M a finitely generated R-module. In this paper we determine $${\rm{An}}{{\rm{n}}_R}({\rm{H}}_{\mathfrak{a}}^{{\rm{cd}}(\mathfrak{a},M)}(M))$$ and $${\rm{At}}{{\rm{t}}_R}({\rm{H}}_{\mathfrak{a}}^{{\rm{cd}}(\mathfrak{a},M)}(M))$$ , which are two important problems concerning the last nonzero local cohomology module $${\rm{H}}_{\mathfrak{a}}^{{\rm{cd}}(\mathfrak{a},M)}(M)$$ . We show that $${\rm{An}}{{\rm{n}}_R}({\rm{H}}_{\mathfrak{a}}^{{\rm{cd}}(\mathfrak{a},M)}(M)) = {\rm{An}}{{\rm{n}}_R}(M/{T_R}(\mathfrak{a},M))$$ , where $${T_R}(\mathfrak{a},M)$$ is the largest submodule of M such that $${\rm{cd}}(\mathfrak{a},{T_R}(\mathfrak{a},M)) < {\rm{cd}}(\mathfrak{a},M)$$ . Using the above result we determine the attached primes of the top local cohomology module $${\rm{At}}{{\rm{t}}_R}({\rm{H}}_{\mathfrak{a}}^{{\rm{cd}}(\mathfrak{a},M)}(M))$$ . In fact, we show that $${\rm{At}}{{\rm{t}}_R}({\rm{H}}_{\mathfrak{a}}^{{\rm{cd}}(\mathfrak{a},M)}(M)) = \left\{ {\mathfrak{p} \in {\rm p }{{\rm{p}}_R}M:\;{\rm{cd}}(\mathfrak{a},\;R/\mathfrak{p}) = {\rm{cd}}(\mathfrak{a},M)} \right\}$$ . Then by using these, we obtain some main results of A. Atazadeh, M. Sedghi, R. Naghipour (2014), K. Bahmanpour, J. A’zami, G. Ghasemi (2012) and K. Divaani-Aazar (2009).

Cominimaxness with respect to ideals of dimension two and local cohomology

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] an [Formula: see text]-module with [Formula: see text]. We get equivalent conditions for top local cohomology module [Formula: see text] to be Artinian and [Formula: see text]-cofinite Artinian separately. In addition, we prove that if [Formula: see text] is a local ring such that [Formula: see text] is minimax, for each [Formula: see text], then [Formula: see text] is minimax [Formula: see text]-module for each [Formula: see text] and for each finitely generated [Formula: see text]-module [Formula: see text] with [Formula: see text] and [Formula: see text]. As a consequence we prove that if [Formula: see text] and [Formula: see text], then [Formula: see text] is [Formula: see text]-cominimax if (and only if) [Formula: see text], [Formula: see text] and [Formula: see text] are minimax. We also prove that if [Formula: see text] and [Formula: see text] such that [Formula: see text] is minimax for all [Formula: see text], then [Formula: see text] is [Formula: see text]-cominimax for all [Formula: see text] if (and only if) [Formula: see text] is minimax for all [Formula: see text].

Attached primes and annihilators of top local cohomology modules defined by a pair of ideals

Assume that \(R\) is a complete Noetherian local ring and \(M\) is a non-zero finitely generated \(R\)-module of dimension \(n=\dim(M)\geq 1\). It is shown that any non-empty subset \(T\) of \(\mathrm{Assh}(M)\) can be expressed as the set of attached primes of the top local cohomology modules \(H_{I,J}^n(M)\) for some proper ideals \(I,J\) of \(R\). Moreover, for ideals \(I, J=\bigcap_ {\mathfrak p\in \mathrm{Att}_R(H_{I}^n(M))}\mathfrak p\) and \(J'\) of \(R\) it is proved that \(T=\mathrm{Att}_R(H_{I,J}^n(M))=\mathrm{Att}_R(H_{I,J'}^n(M))\) if and only if \(J'\subseteq J\). Let \(H_{I,J}^n(M)\neq 0\). It is shown that there exists \(Q\in \mathrm{Supp}(M)\) such that \(\dim(R/Q)=1\) and \(H_Q^n(R/{\mathfrak p})\neq 0\), for each \(\mathfrak p \in \mathrm{Att}_R(H_{I,J}^n(M))\). In addition, we prove that if \(I\) and \(J\) are two proper ideals of a Noetherian local ring \(R\), then \(\mathrm{Ann}_R(H_{I,J}^{n}(M))=\mathrm{Ann}_R(M/{T_R(I,J,M)})\), where \(T_R(I,J,M)\) is the largest submodule of \(M\) with \(\mathrm{cd}(I,J,T_R(I,J,M))<\mathrm{cd}(I,J,M)\), here \(\mathrm{cd}(I,J,M)\) is the cohomological dimension of \(M\) with respect to \(I\) and \(J\). This result is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6].

English

english

Cohomological dimension and top local cohomology modules

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this paper, we first determine a condition under which a given integer $t$ is a lower bound for the cohomological dimension $\operatorname {cd}(I,M)$, and use this to conclude that non-catenary Noetherian domains contain prime ideals that are not set-theoretic complete intersection. We also show the existence of a descending chain of ideals with successive diminishing cohomological dimensions. We then resolve the Artinianness of top local cohomology modules over local unique factorization domains of Krull dimension at most three, and obtain several related results on the top local cohomology modules for much more general cases.

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.

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.

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.

English

Let $$\mathfrak{a}$$ , I, J be ideals of a Noetherian local ring $$(R,\mathfrak{m},k)$$ . Let M and N be finitely generated R-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $$H^t_{I,J}$$ (M) and $$H^t_{I,J}$$ (M)), where t is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and D(−):= HomR(−, Er(k)) is the Matlis dual functor. We show that if R is a d-dimensional complete Cohen-Macaulay ring and $$H^i_{I,J}$$ (R) = 0 for all i ≠ t, the natural homomorphism R → Homr( $$H^t_{I,J}$$ (KR), $$H^t_{I,J}$$ (KR)) is an isomorphism, where KR denotes the canonical module of R. Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.

Duality and de Rham cohomology for graded [formula omitted]-modules

We consider the (graded) Matlis dual D(M) of a graded D-module M over the polynomial ring R=k[x1,…,xn] (k is a field of characteristic zero), and show that it can be given a structure of D-module in such a way that, whenever dimk⁡HdRi(M) is finite, then HdRi(M) is k-dual to HdRn−i(D(M)). As a consequence, we show that if M is a graded D-module such that HdRn(M) is a finite-dimensional k-space, then dimk⁡(HdRn(M)) is the maximal integer s for which there exists a surjective D-linear homomorphism M→Es, where E is the top local cohomology module H(x1,…,xn)n(R). This extends a recent result of Hartshorne and Polini on formal power series rings to the case of polynomial rings; we also apply the same circle of ideas to provide an alternate proof of their result. When M is a finitely generated graded D-module such that dimk⁡HdRi(M) is finite, we generalize the above result further, showing that HdRn−i(M) is k-dual to ExtDi(M,E).

Content of local cohomology, parameter ideals, and robust algebras

This paper continues the investigation of quasilength, of content of local cohomology with respect to generators of the support ideal, and of robust algebras begun in joint work of Hochster and Huneke. We settle several questions raised by Hochster and Huneke. In particular, we give a family of examples of top local cohomology modules both in equal characteristic 0 and in positive prime characteristic that are nonzero but have content 0. We use the notion of a robust forcing algebra (the condition turns out to be strictly stronger than the notion of a solid forcing algebra in, for example, equal characteristic 0) to define a new closure operation on ideals. We prove that this new notion of closure coincides with tight closure for ideals in complete local domains of positive characteristic, which requires proving that forcing algebras for instances of tight closure are robust, and study several related problems. This gives, in effect, a new characterization of tight closure in complete local domains of positive characteristic. As a byproduct, we also answer a question of Lyubeznik in the negative.

On the structure of $S_2$-ifications of complete local rings

Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $Spec (R)-V(J)$, and the (finite simple) graph $\Gamma _R$ with vertex set $Min (R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of connected components of $\Gamma _R$ is equal to the maximum number of connected components of $Spec (R)-V(J)$ for all $J$ of height $2$. We further investigate this graph by exhibiting a technique for showing that certain graphs $G$ can be realized in the form $\Gamma _R$.