Local cohomology modules via certain flat extension rings

Let (R,𝔪) be a Noetherian local ring, M a finitely generated R-module with finite projective dimension n, N an arbitrary R-module, and 𝔞 be an ideal of R which is generated by s elements. In this article, we provide a surjective homomorphism from ordinary local cohomology module to top generalized local cohomology module , where P n, M is an nth syzygy of a projective resolution of M. Also, by using this epimorphism, we prove some results about the attached primes, coassociated primes, the Betti numbers, and Artinian properties of certain generalized local cohomology modules.

Constructing arbitrary torsion elements for a local cohomology module

On Buchsbaum rings

Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an antinilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the ascending chain condition.

Linkage of modules over Cohen–Macaulay rings

In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing.

Let $$(R,\mathfrak {m})$$ be a commutative Noetherian local ring, M be a finitely generated R-module and $$\mathfrak {a}$$ , I and J are ideals of R. We investigate the structure of formal local cohomology modules of $$\mathfrak {F}^i_{\mathfrak {a},I,J}(M)$$ and $$\check{\mathfrak {F}}^i_{\mathfrak {a},I,J}(M)$$ with respect to a pair of ideals, for all $$i\ge 0$$ . The main subject of the paper is to study the finiteness properties and artinianness of $$\mathfrak {F}^i_{\mathfrak {a},I,J}(M)$$ and $$\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)$$ . We study the maximum and minimum integer $$i\in \mathbb {N}$$ such that $$\mathfrak {F}^i_{\mathfrak {a},\mathfrak {m},J}(M)$$ and $$\check{\mathfrak {F}}^i_{\mathfrak {a},\mathfrak {m},J}(M)$$ are not Artinian and we obtain some results involving cosupport, coassociated and attached primes for formal local cohomology modules with respect to a pair of ideals. Also, we give an criterion involving the concepts of finiteness and vanishing of formal local cohomology modules and Cech-formal local cohomology modules with respect to a pair of ideals.

Abstract. Let (R,m) be a Noetherian local ring and M a finitely gener-ated R-module. For an integer s>−1, we say that Mis Cohen-Macaulayin dimension >s if every system of parameters of M is an M-sequencein dimension > s introduced by Brodmann-Nhan [1]. In this paper, wegive some characterizations for Cohen-Macaulay modules in dimension>sin terms of the Noetherian dimension of the local cohomology mod-ules H im (M), the polynomial type of M introduced by Cuong [5] and themultiplicity e(x;M) of M with respect to a system of parameters x. 1. IntroductionThroughoutthis paper, let (R,m) be a Noetherian local ring and Ma finitelygenerated R-module with dimM= d.It is well known that the Cohen-Macaulay modules play an important rolein the theory of Noetherian rings and finitely generated modules. Recall thatM is called Cohen-Macaulay if every system of parameters (s.o.p. for short)of Mis an M-sequence. The structure of Cohen-Macaulay modules are well-known in the multiplicity, local cohomology, m-adic completion, localization,etc (see [3]). There are some extensions of the concepts of M-sequence andCohen-Macaulay modules, among which are the notions of M-sequence in di-mension >sintroduced by Brodmann-Nhan [1] and Cohen-Macaulay modulesin dimension >sdefined by Zamani [21].Definition. Let s> −1 be an integer. A sequence (x

We prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

The Canonical Element Conjecture holds for those 3 3 -dimensional local rings of depth 2 whose second local cohomology module is a finite direct sum of cyclic modules, and for those 3 3 -dimensional local rings whose second local cohomology module is cyclic.

In this note, we first obtain some bounds for Noetherian dimension of Artinian local cohomology modules with respect to an ideal, in the cases of small levels. Secondly, in the case of top local cohomology modules, some bounds of Noetherian dimension for such modules are given. Received: April 24, 2023Revised: June 16, 2023Accepted: July 6, 2023

Let [Formula: see text] be an ideal of a commutative Noetherian local ring [Formula: see text] and M, N be two finitely generated R-modules such that M is of finite projective dimension n. Let t be a positive integer. We show that if there exists a regular sequence [Formula: see text] with [Formula: see text] and the i-th local cohomology module [Formula: see text] of N with respect to [Formula: see text] is zero for all i > t, then [Formula: see text], where D(-):= Hom R(-,E). Also, we prove that if N is a Cohen-Macaulay R-module of dimension d, then the generalized local cohomology module [Formula: see text] is co-Cohen-Macaulay of Noetherian dimension d. Finally, with an elementary proof, we show that [Formula: see text] is finite.

Let $$(R,\mathfrak {m})$$ be a Noetherian local ring of prime characteristic p and Q be an $$\mathfrak {m}$$ -primary parameter ideal. We give criteria for F-rationality of R using the tight Hilbert function $$H^*_Q(n)=\ell (R/(Q^n)^*)$$ and the coefficient $$e_1^*(Q)$$ of the tight Hilbert polynomial $$P^*_Q(n)=\sum _{i=0}^d(-1)^ie_i^*(Q)\left( {\begin{array}{c}n+d-1-i\\ d-i\end{array}}\right) .$$ We obtain a lower bound for the tight Hilbert function of Q for equidimensional excellent local rings that generalizes a result of Goto and Nakamura. We show that if $$\dim R=2 $$ , the Hochster–Huneke graph of R is connected and this lower bound is achieved, then R is F-rational. Craig Huneke asked if the F-rationality of unmixed local rings may be characterized by the vanishing of $$e_1^*(Q).$$ We construct examples to show that without additional conditions, this is not possible. Let R be an excellent, reduced, equidimensional Noetherian local ring and Q be generated by parameter test elements. We find formulas for $$e_1^*(Q), e_2^*(Q), \ldots , e_d^*(Q)$$ in terms of Hilbert coefficients of Q, lengths of local cohomology modules of R, and the length of the tight closure of the zero submodule of $$H^d_{\mathfrak {m}}(R).$$ Using these, we prove: R is F-rational $$\iff e_1^*(Q)=e_1(Q) \iff {\text {depth}}R\ge 2$$ and $$e_1^*(Q)=0.$$

Local cohomology based on a nonclosed support defined by a pair of ideals

Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties