Local Cohomology Functor Research Articles

Stable categories of spherical modules and torsionfree modules

Auslander and Bridger [Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969] introduced the notions of n n -spherical modules and n n -torsionfree modules. In this paper, we construct an equivalence between the stable category of n n -spherical modules and the category of modules of grade at least n n , and provide its Gorenstein analogue. As an application, we prove that if R R is a Gorenstein local ring of Krull dimension d > 0 d>0 , then there exists a stable equivalence between the category of ( d − 1 ) (d-1) -torsionfree R R -modules and the category of d d -spherical modules relative to the local cohomology functor.

Local cohomology for Gorenstein homologically smooth DG algebras

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of the local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra $${\cal A}$$ , we define a group homomorphism $${\rm{Hdet}}:{\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right) \to {k^ \times }$$ , called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra $${{\cal A}^G}$$ to be Gorenstein, where $${\cal A}$$ is a homologically smooth DG algebra such that $$H\left( {\cal A} \right)$$ is a Noetherian AS-Gorenstein graded algebra and G is a finite subgroup of $${\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right)$$ . Especially, we can apply this result to DG down-up algebras and non-trivial DG free algebras generated in two degree-one elements.

Local cohomology and local homology complexes with respect to a pair of ideals

We introduce local cohomology complexes and local homology complexes with respect to a pair of ideals $(I,J)$ of a commutative noetherian ring $R$, characterize them using Čech complexes and establish "MGM" equivalence between the local cohomology functor $\mathrm{R}\Gamma_{I,J}$ and local homology functor $\mathrm{L}\Lambda_{I,J}$ on $\mathrm{D}(R)$.

Some splitting theorems for local cohomology modules with respect to a pair of ideals

Suppose that [Formula: see text] is a commutative ring with identity, [Formula: see text], [Formula: see text] are ideals of [Formula: see text], and let [Formula: see text] be a finitely generated [Formula: see text]-module. Let [Formula: see text] be the [Formula: see text]th local cohomology functor with respect to [Formula: see text]. In this paper, for fixed integers [Formula: see text] and [Formula: see text], we study the existence of the following isomorphisms of local cohomology modules: (i) [Formula: see text]; (ii) [Formula: see text]; and, (iii) [Formula: see text] for some filter regular element [Formula: see text] on [Formula: see text]. Moreover, we provide some applications of the above isomorphisms.

Local cohomology in Grothendieck categories

Let [Formula: see text] be a locally noetherian Grothendieck category. In this paper, we define and study the section functor on [Formula: see text] with respect to an open subset of [Formula: see text]. Next, we define and study local cohomology theory in [Formula: see text] in terms of the section functors. Finally, we study abstract local cohomology functor on the derived category [Formula: see text].

Structural theorem for gr-injective modules over gr-noetherian G-graded commutative rings and local cohomology functors

It is well known that the decomposition of injective modules over noetherian rings is one of the most aesthetic and important results in commutative algebra. Our aim is to prove similar results for graded noetherian rings. In this paper, we will study the structure theorem for $gr$-injective modules over $gr$-noetherian $G$-graded commutative rings, give a definition of the $gr$-Bass numbers, and study their properties. We will show that every $gr$-injective module has an indecomposable decomposition. Let $R$ be a $gr$-noetherian graded ring and $M$ be a $gr$-finitely generated $R$-module, we will give a formula for expressing the Bass numbers using the functor $Ext$. We will define the section functor $\Gamma_{V}$ with support in a specialization-closed subset $V$ of $Spec^{gr}(R)$ and the abstract local cohomology functor. Finally, we will show that a left exact radical functor $F$ is of the form $\Gamma_V$ for a specialization-closed subset $V$.

Homological Dimensions of Local (Co)homology Over Commutative DG-rings

AbstractLet A be a commutative noetherian ring, let a ⊆ A be an ideal, and let I be an injective A-module. A basic result in the structure theory of injective modules states that the A-module Γa(I) consisting of ɑ-torsion elements is also an injective A-module. Recently, de Jong proved a dual result: If F is a flat A-module, then the ɑ-adic completion of F is also a flat A-module. In this paper we generalize these facts to commutative noetherian DG-rings: let A be a commutative non-positive DG-ring such that H0(A) is a noetherian ring and for each i < 0, the H0(A)-module Hi(A) is finitely generated. Given an ideal ⊆ H0(A), we show that the local cohomology functor R associated with does not increase injective dimension. Dually, the derived -adic completion functor LΛ does not increase flat dimension.

Covariant functors and asymptotic stability

Let R be a commutative Noetherian ring, I,J ideals of R and M a finitely generated R-module. Let F be a covariant R-linear functor from the category of finitely generated R-modules to itself. We first show that if F is coherent, then the sets AssRF(M/InM), AssRF(In−1M/InM) and the values depthJF(M/InM), depthJF(In−1M/InM) become independent of n for large n. Next, we consider several examples in which F is a rather familiar functor, but is not coherent or not even finitely generated in general. In these cases, the sets AssRF(M/InM) still become independent of n for large n. We then show one negative result where F is not finitely generated. Finally, we give a positive result where F belongs to a special class of functors which are not finitely generated in general, an example of which is the zeroth local cohomology functor.

Stable local cohomology

ABSTRACTFor a module having a complete injective resolution, we define a stable version of local cohomology. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this functor behaves like the usual local cohomology functor. Our main result is that when there is only one nonzero local cohomology module, there is a strong connection between that module and the stable local cohomology module; in fact, the latter gives a Gorenstein injective approximation of the former.

DEPTH FOR TRIANGULATED CATEGORIES

Abstract. Recently a construction of local cohomology functors for com-pactly generated triangulated categories admitting small coproducts isintroduced and studied by Benson, Iyengar, Krause, Asadollahi and theircoauthors. Following their idea, we introduce the depth of objects in suchtriangulated categories and get that when (R,m) is a graded-commutativeNoetherian local ring, the depth of every cohomologically bounded andcohomologically finite object is not larger than its dimension. 1. IntroductionDepth is a fundamental numerical invariant of a Noetherian local ring (R,m,k) and a finite R-module M. The depth of a finite module M is classicallyintroduced as the maximal length of an M-regular sequence in mand it can bemeasured by the (non)-vanishing of Ext mR (k,M), see [6]. Later, for non-finitemodules and even complexes of such, Foxby adopted the latter approach todepth, see [8].Local cohomologyfunctors on categoriesof modules are introduced in [5] andit has been shown that the depth and dimension of a module can be expressedin terms of their vanishing and non-vanishing. Furthermore, local cohomologyfunctors of complexes also have been studied by many authors, see [5, 7, 11].Also it is proved in [8] that for any unbounded complex X, the depth can becomputed using either Koszul (co)homology, Ext, or local cohomology. Heregreat attention should be paid to the descriptiondepth

RESULTS OF CERTAIN LOCAL COHOMOLOGY MODULES

Abstract. Let Rbe a commutative Noetherian ring, I and J two idealsof R, and M a finitely generated R-module. We prove thatExt iR (R/I,H tI,J (M))is finitely generated for i = 0,1 where t = inf{i ∈ N 0 : H iI,J (M) isnot finitely generated}. Also, we prove that H iI+J (H tI,J (M)) is Artinianwhen dim(R/I+ J) = 0 and i= 0,1. 1. IntroductionThroughout this paper, we assume that R is a commutative Noetherianring with non-zero identity, I and J two ideals of R, and M a finitely gen-erated R-module. Recently Takahashi, Yoshino and Yoshizawa in [18], intro-duce the module H iI,J (M) as a generalization of the ordinary local cohomologymodule H iI (M) that defined by Grothendieck in [5]. They considered (I,J)-torsion submodule Γ I,J (M) of M which consists of all elements x of M withSupp(Rx) ⊆ W(I,J), where W(I,J) = {p ∈ Spec(R) : I n ⊆ p+ J for aninteger n≥ 1}. Furthermore, they defined the local cohomology functor H iI,J with respect to (I,J) to be the i-th right derived functor of Γ I,J . Notice that ifJ= 0, then H

Generalized Hilbert Functions

Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciupercă. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients 𝔧0,…, 𝔧 d−2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of 𝔧 d−1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the ‘expected’ shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.

RESOLUTIONS AND COHOMOLOGIES OF TORIC SHEAVES: THE AFFINE CASE

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen–Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within this setting, we develop machinery which facilitates the construction of minimal free resolutions for the smooth case as well as resolutions which are acyclic with respect to local cohomology functors for the general case. We give two main applications. First, over the polynomial ring, we determine in explicit combinatorial terms the ℤn-graded Betti numbers and local cohomology of reflexive modules whose associated filtrations form a hyperplane arrangement. Second, for the nonsmooth, simplicial case in dimension d ≥ 3, we construct new examples of indecomposable maximal Cohen–Macaulay modules of rank d – 1.

ON THE BEHAVIOR OF APPLYING DERIVED FUNCTORS OF TORSION FUNCTORS ON LOCAL COHOMOLOGY MODULES

In this note, by using some properties of the local cohomology functors of weakly Laskerian modules, we study the behavior of right and left derived functors of torsion functors. In fact, firstly we gain some isomorphisms in the context of these functors, grade and cohomological dimension. Then we study their supports and their sets of associated prime ideals in special cases.

On the local cohomology and support for triangulated categories

Recently a notion of support and a construction of local cohomology functors for [TR5] compactly generated triangulated categories were introduced and studied by Benson, Iyengar, and Krause. Following their idea, we assign to any object of the category a new subset of Spec(R), again called the (big) support. We study this support and show that it satisfies axioms such as exactness, orthogonality, and separation. Using this support, we study the behavior of the local cohomology functors and show that these triangulated functors respect boundedness. Then we restrict our study to the categories generated by only one compact object. This condition enables us to get some nice results. Our results show that one can get a satisfactory version of the local cohomology theory in the setting of triangulated categories, compatible with the known results for the local cohomology for complexes of modules.

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ HomR(−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x1, …, xn is a regular sequence on M contained in α, then H(x1, …,xnRnD(Han(M))) is a homomorphic image of D(M), where Hbi(−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H(x1, …,xn)Rn(D(Han(M)))) ⋟ D(D(M)).

Tate–Vogel Completions of Half-Exact Functors

We provide a general method to construct the Tate–Vogel homology theory for a general half-exact functor with one variable, aiming at a good generalization of Cohen–Macaulay approximations of modules over commutative Gorenstein rings. For a half exact functor F, using the left and right satellites (S n and S n ), we define F ∨(X)=lim → S n S n F(X) and F ∧(X)=lim ← S n S n F(X), and call F ∨ and F ∧ the Tate–Vogel completions of F. We provide several properties of F ∨ and F ∧, and their relations with the G-dimension and the projective dimension of the functor F. A comparison theorem of Tate–Vogel completions with ordinary Tate–Vogel homologies is proved. If F is a half exact functor over the category of R-modules, where R is a commutative Noetherian local ring inspired by Martsinkovsky's works, we can define the invariants ξ(F) and η(F) of F. If F=Ext R i (M, ), then they coincide with Martsinkovsky's ξ-invariants and Auslander's delta invariants. Our advantage is that we can consider these invariants for any half exact functors. We also compute these invariants for the local cohomology functors.

The Alexander Duality Functors and Local Duality with Monomial Support

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nn-graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual and contains all of the maps in the minimal free resolution of M over every Zn-graded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm-lattices. Using injective resolutions, theorems of Eagon, Reiner, and Terai are generalized to all Nn-graded modules: the projective dimension of M equals the support-regularity of its Alexander dual, and M is Cohen–Macaulay if and only if its Alexander dual has a support-linear free resolution. Alexander duality is applied in the context of the Zn-graded local cohomology functors HiI(−) for squarefree monomial ideals I in the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I=m is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, that Terai's formula for the Hilbert series of HiI(S) is equivalent to Hochster's for Hn−im(S/I).

Local Cohomology for Commutative Banach Algebras

The purpose of this paper is to introduce a local cohomology theory in the unital commutative Banach algebras context and to describe a connection between the local cohomology functor and direct limit of hom functors.

Type of blowing-up over a Buchsbaum ring

Throughout this paper (A, , k) denotes a Noetherian local ring of dimension d and stands for the ith local cohomology functor with respect to . We refer the reader to [16] for any unexplained terminology.