English

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

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

Let R be a Noetherian ring, M a finitely generated R-module and N an arbitrary R-module. We consider the generalized local cohomology modules, with respect to an arbitrary ideal I of R, and prove that, for all nonnegative integers r, t and all p ? Spec(R) the Bass number ?r(p,HtI (M,N)) is bounded above by ?tj=0?r(p, t?jExtR (M,HjI (N))). A corollary is that Ass (HtI (M,N)? Utj=0 Ass (t?jExtR (M,HjI(N))). In a slightly different direction, we also present some well known results about generalized local cohomology modules.

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 [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.

We show that the first non-finitely generated generalized local cohomology modules of a finite modules M and N over a Noetherian ring R with respect to an ideal of R has only finitely many associated primes. This generalizes the corresponding result which is shown in [1] for standard local cohomology module.

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.

Some properties of graded local cohomology modules

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.

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

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

We study modules which are characteristic submodules in their injective hulls. The author is supported by the Russian Foundation for Basic Research, project 08-01- 00693-a All rings below are assumed to be associative and with nonzero identity ele- ment; all modules are assumed to be unitary. A moduleM is said to be Noetherian if every properly ascending chain of submodules in M is finite. Expressions of the form 'A is a Noetherian ring' mean that AA and AA are Noetherian modules. A moduleM is said to be injective with respect to the moduleX or X-injective if for any submodule X1 in X, every homomorphismX1 → M may be extended to a homomorphismX → M. For a ring A, an A-module M is said to be injective if M is injective with respect to any A-module. A submodule M of the module E is said to be essential if M ∩ E1 6 0 for any nonzero submodule E1 in E. If E is an injective module and M is an essential submodule in E then E is called the injective hull of the module M. The injective hull is unique up to isomorphism. A submoduleM of the module E is called a characteristic submodule if f(M) ⊆ M for every automorphismf of the module E. In the papers (3), (8), and (10), a module M is said to be automorphism- invariant if M is a characteristic submodule of the injective hull of M. It is easy to verify that a module M is an automorphism-invariant if and only if M is a charac- teristic submodule of some injective module; see Remark 6 below. Remark 1. In (3, Theorem 16), it is proved that a module M is an automorphism-invariant module if and only if M is a pseudo-injective module, i.e., for any submodule X in M , every monomorphism X → M may be exten- ded to an endomorphism of the moduleM . For example, see (6), (11), (3) about pseudo-injective modules. Remark 2. It is clear that every injective module is an automorphism-invariant module. The converse is not true, since every simple Abelian group is a finite automorphism-invariant non-injective module over the ring of integers Z.

Let R be a commutative Noetherian ring, I an ideal of R, M and N two R-modules. We characterize the least integer i such that <TEX>$H^i_I(M,N)$</TEX> is not weakly Artinian by using the notion of weakly filter regular sequences. Also, a local-global principle for minimax generalized local cohomology modules is shown and the result generalizes the corresponding result for local cohomology modules.