Top Local Cohomology Research Articles

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 $\Gamma _R$.

F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p > 0

The main aim of this article is to study the relation between F-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many other topics.(1)We prove that if every parameter ideal of a Noetherian local ring of prime characteristic p>0 is Frobenius closed, then it is F-injective.(2)We prove a necessary and sufficient condition for the injectivity of the Frobenius action on Hmi(R) for all i≤fm(R), where fm(R) is the finiteness dimension of R. As applications, we prove the following results. (a) If the ring is F-injective, then every ideal generated by a filter regular sequence, whose length is equal to the finiteness dimension of the ring, is Frobenius closed. It is a generalization of a recent result of Ma and which is stated for generalized Cohen–Macaulay local rings. (b) Let (R,m,k) be a generalized Cohen–Macaulay ring of characteristic p>0. If the Frobenius action is injective on the local cohomology Hmi(R) for all i<dim⁡R, then R is Buchsbaum. This gives an answer to a question of Takagi.(3)We consider the problem when the union of two F-injective closed subschemes of a Noetherian Fp-scheme is F-injective. Using this idea, we construct an F-injective local ring R such that R has a parameter ideal that is not Frobenius closed. This result adds a new member to the family of F-singularities.(4)We give the first ideal-theoretic characterization of F-injectivity in terms the Frobenius closure and the limit closure. We also give an answer to the question about when the Frobenius action on the top local cohomology is injective.

Some results on top local cohomology and top formal local cohomology modules

ABSTRACTLet (R,𝔪) be a complete noetherian local ring and M a finitely generated R-module of dimension n and 𝔞 an ideal of R.At first, we find a relation between top local cohomology and top formal local cohomology modules. In fact, we show that there exists an ideal 𝔟 such that and there exists an ideal 𝔠 such that . Several applications of this result are given. Among other things, it is shown that and .

A note on injectivity of Frobenius on local cohomology of global complete intersections

Given a graded complete intersection ideal J=(f1,…,fc)⊆k[x0,…,xn]=S, where k is a field of characteristic p>0 such that [k:kp]<∞, we show that if S/J has an isolated non-F-pure point then the Frobenius action on top local cohomology Hmn+1−c(S/J) is injective in sufficiently negative degrees, and we compute the least degree of any kernel element. If S/J has an isolated singularity, we are also able to give an effective bound on p ensuring the Frobenius action on Hmn+1−c(S/J) is injective in all negative degrees, extending a result of Bhatt and Singh in the hypersurface case.

Annihilators of Local Cohomology Modules

Let (R, 𝔪) be a commutative Noetherian complete local ring, M a nonzero finitely generated R-module of dimension n, and I be an ideal of R. In this paper we calculate the annihilator of the top local cohomology module . Also, if (R, 𝔪) is a Noetherian local Cohen–Macaulay ring of dimension d and I is a nonzero proper ideal of R, then we calculate the annihilator of the first nonzero local cohomology module . Finally, we show that if R is an arbitrary Noetherian ring, I an ideal of R, and M is a nonzero finitely generated R-module with cd(I, M) = t ≥ 0, then there exists a submodule N of M such that . This is a generalization of the main result of Bahmanpour, A'zami, and Ghasemi [1] for all ideals of an arbitrary Noetherian ring R.

Cohomological dimension filtration and annihilators of top local cohomology modules

Let $\mathfrak a$ denote an ideal in a Noetherian ring $R$, and $M$ a finitely generated $R$-module. We introduce the concept of the cohomological dimension filtration $\mathscr {M} =\{M_i\}_{i=0}^c$, where $ c=\mathop {\rm cd}\nolimits (\mathfrak a,M)$ a

Applications of n-Gorenstein projective and injective modules

Over a commutative noetherian ring, we introduce a generalization of Gorenstein projective and injective modules, which we call, respectively, n-Gorenstein projective and injective modules. These last two classes of modules give us a new characterization of Gorenstein rings in terms of top local cohomology modules of flat modules. We also utilize the n-Gorenstein injective dimension to study an open question of Takahashi. Furthermore, we prove that a nonzero finite module with finite n-Gorenstein projective dimension satisfies the Auslander-Bridger formula.

On the annihilators and attached primes of top local cohomology modules

Let \({\mathfrak{a}}\) be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that \({{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))= {\rm Ann}_R(M/T_R(\mathfrak{a}, M))}\) , where \({T_R(\mathfrak{a}, M)}\) is the largest submodule of M such that \({{\rm cd}(\mathfrak{a}, T_R(\mathfrak{a}, M)) < {\rm cd}(\mathfrak{a}, M)}\) . Several applications of this result are given. Among other things, it is shown that there exists an ideal \({\mathfrak{b}}\) of R such that \({{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))={\rm Ann}_R(M/H_{\mathfrak{b}}^{0}(M))}\) . Using this, we show that if \({ H_{\mathfrak{a}}^{{\rm dim} R}(R)=0}\) , then \({{{\rm Att}_R} H^{{\rm dim} R-1}_{\mathfrak a}(R)= \{\mathfrak{p} \in {\rm Spec} R | \,{\rm cd}(\mathfrak{a}, R/\mathfrak{p}) = {\rm dim} R-1\}.}\) These generalize the main results of Bahmanpour et al. (see [2, Theorem 2.6]), Hellus (see [7, Theorem 2.3]), and Lynch (see [10, Theorem 2.4]).

Erratum to: Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

This note makes a correction to the paper “Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module”.

On the annihilators of local cohomology modules

Let (R,m) be a commutative Noetherian complete local ring, M a non-zero finitely generated R-module of dimension d⩾1, and TR(M):=⋃{N:N⩽M and dimN<dimM}. In this paper we calculate the annihilator of the top local cohomology module Hmd(M). More precisely, we show that 0:RHmd(M)=0:RM/TR(M). Moreover, for every positive integer n, we calculate the radical of the annihilator of Hmn(M). More precisely, we prove that if Hmn(M) is not finitely generated then Rad(0:RHmn(M))=⋂p∈Sp, whereS={p∈SpecR:(Hpn−1(M))p≠0 and dimR/p=1}.

Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

Let \({(R, \mathfrak{m})}\) be a commutative Noetherian local ring of Krull dimension d, and let C be a semidualizing R-module. In this paper, it is shown that if R is complete, then C is a dualizing module if and only if the top local cohomology module of \({R, H _{\mathfrak{m}} ^{d} (R)}\), has finite GC-injective dimension. This generalizes a recent result due to Yoshizawa, where the ring is assumed to be complete Cohen-Macaulay.

Annihilators of Top Local Cohomology

In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider We examine these annihilators for an arbitrary local ring when i is the cohomological dimension of I. When the cohomological dimension coincides with the dimension of the ring, we provide an explicit description of In general, we examine its dimension over R and conjecture that for i > 0.

On Gorenstein injectivity of top local cohomology modules

R. Sazeedeh showed that top local cohomology modules are Gorenstein injective in a Gorenstein local ring with at most two dimensions. In this paper, it is proved that the condition of dimension in his result cannot be relaxed and the conclusion in his result holds for complete local hypersurface rings with any dimension.

A finiteness condition on local cohomology in positive characteristic

In this paper, we present a condition on a local Cohen–Macaulay F-injective ring of positive characteristic p > 2 , which implies that its top local cohomology module with support in the maximal ideal has finitely many Frobenius compatible submodules.

Top local cohomology modules with respect to a pair of ideals

Let (R, m) be a commutative Noetherian local ring, and let I and J be two proper ideals of R. Let M be a non-zero finitely generated R-module. We investigate the top local cohomology module H dim M I, J (M). We get some results about attached prime ideals of the local cohomology module H dim M I, J (M). As a consequence, we find that there exists a quotient L of M such that H dim M I, J (M) ≅ H dim M I (L). Also, we give the generalized version of the Lichtenbaum-Hartshorne Vanishing Theorem for local cohomology modules of a finitely generated module with respect to a pair of ideals.

Cofiniteness and coassociated primes of local cohomology modules

Let $R$ be a noetherian ring, $\mathfrak a$ an ideal of $R$ such that $\dim R/{\mathfrak a}=1$ and $M$ a finite $R$-module. We will study cofiniteness and some other properties of the local cohomology modules $H^{i}_{{\mathfrak a}}(M)$. For an arbitrary ideal $\mathfrak a$ and an $R$-module $M$ (not necessarily finite), we will characterize $\mathfrak a$-cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized.

Some criteria for the Cohen-Macaulay property and local cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Had (M) is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let \( \mathfrak{a} \) be an ideal of R and \( \mathfrak{M} \) denote the intersection of all prime ideals \( \mathfrak{p} \in Supp_R H_\mathfrak{a}^d (M) \). It is shown that $$ H_\mathfrak{a}^d (M) \simeq H_\mathfrak{M}^d (M)/\sum\limits_{n \in \mathbb{N}} {\langle \mathfrak{M}\rangle } (0:_{H_\mathfrak{M}^d (M)} \mathfrak{a}^n ), $$ where for an Artinian R-module A we put \( \langle \mathfrak{M}\rangle A = \cap _{n \in \mathbb{N}} \mathfrak{M}^n \)A. As a consequence, it is proved that for all ideals \( \mathfrak{a} \) of R, there are only finitely many non-isomorphic top local cohomology modules \( H_\mathfrak{a}^d (M) \) having the same support. In addition, we establish an analogue of the Lichtenbaum-Hartshorne vanishing theorem over rings that need not be local.

Parameter-test-ideals of Cohen–Macaulay rings

AbstractWe describe an algorithm for computing parameter-test-ideals in certain local Cohen–Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp’s notion of ‘special ideals’. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article Generators of D-modules in positive characteristic (J. Alvarez-Montaner, M. Blickle and G. Lyubeznik, Math. Res. Lett. 12 (2005), 459–473).