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 G _C -injective dimension. This generalizes a recent result due to Yoshizawa, where the ring is assumed to be complete Cohen-Macaulay.     

Local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

Let ${(R,\mathfrak{m})}$ be a local ring, and let C be a semidualizing R-module. In this paper, we are concerned with the C-injective and G _C -injective dimensions of certain local cohomology modules of R. Firstly, the injective dimension of C and the above quantities are compared. Secondly, as an application of the above comparisons, a characterization of a dualizing module of R is given. Finally, it is shown that if R is Cohen-Macaulay of dimension d such that ${\rm H}_{\mathfrak{m}}^{d}(C)$ is C-injective, then R is Gorenstein. This is an answer to a question which was recently raised.     

Relative Gorenstein injective covers with respect to a semidualizing module

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to C which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every G _C -injective module G, the character module G ⁺ is G _C -flat, then the class \(\mathcal{GI}_{C}(R)\cap\mathcal{A}_C(R)\) is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class \(\mathcal{GI}_{C}(R)\cap\mathcal{A}_C(R)\) is covering.     

Gorenstein injective modules and local cohomology

In this paper we assume that is a Gorenstein Noetherian ring. We show that if is also a local ring with Krull dimension that is less than or equal to 2, then for any nonzero ideal of , is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if is a Gorenstein ring, then for any -module its local cohomology modules can be calculated by means of a resolution of by Gorenstein injective modules. Also we prove that if is -Gorenstein, is a Gorenstein injective and is a nonzero ideal of , then is Gorenstein injective.

     

The injectivity of Frobenius acting on cohomology and local cohomology modules

LetR be a two-dimensional normal graded ring over a field of characteristicp>0. We want to describe the tight closure of (O) in the local cohomology moduleH _R+ ² (R) using the graded module structure ofH _R+ ² (R). For this purpose we explore the condition that the Frobenius mapF: [H _R+ ² (R)]_n→[H _R+ ² (R)]_pninduced on graded pieces ofH _R+ ² (R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisorD onX=Proj (R) such thatR=R (X, D)= ⊕ _n≥0H⁰(X O _X (n D)). Then the above map is identified with the induced Frobenius on the cohomology groups Our interest is the casen<0, and in this case, a generalization of Tango's method for integral divisors enables us to show thatF _n is injective ifp is greater than a certain bound given explicitly byX andnD. This result is useful to studyF-rationality ofR. The notion ofF-rational rings in characteristicp>0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulop reduction of a rational signularity in characteristic 0 isF-rational forp≫0. Our result answers to this question affirmatively and also sheds light to behavior ofF-rationality in smallp.     

Totally reflexive modules with respect to a semidualizing bimodule

Let S and {iaR} be two associative rings, let _S C _R be a semidualizing (S,R)-bimodule. We introduce and investigate properties of the totally reflexive module with respect to _S C _R and we give a characterization of the class of the totally C _R -reflexive modules over any ring R. Moreover, we show that the totally C _R -reflexive module with finite projective dimension is exactly the finitely generated projective right R-module. We then study the relations between the class of totally reflexive modules and the Bass class with respect to a semidualizing bimodule. The paper contains several results which are new in the commutative Noetherian setting.     

Top formal local cohomology module

Periodica Mathematica Hungarica - I give fully detailed proofs of two important theorems—the exact solution of the weak clique game and the compactness theorem—in the theory of...     

Vanishing of stable homology with respect to a semidualizing module

We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces C to be trivial and R to be Gorenstein.     

Modules of finite homological dimension with respect to a semidualizing module

We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of Takahashi and White. This research was conducted while S.S.-W. visited the IPM in Tehran during July 2008. The research of S.Y. was supported in part by a grant from the IPM (No. 87130211).     

Faltings' theorem for the annihilation of local cohomology modules over a Gorenstein ring

In this paper we study the Annihilator Theorem and the Local-global Principle for the annihilation of local cohomology modules over a (not necessarily finite-dimensional) Noetherian Gorenstein ring.

     

Attached primes of the top local cohomology modules with respect to an ideal

Let be a Noetherian local ring and let be an ideal of R. Let M be an R-module of dimension n. In this paper we study the attached primes of the top local cohomology module Received: 12 May 2004     

Local cohomology modules with respect to an ideal containing the irrelevant ideal

Let R=?_n≥0R_n be a homogeneous Noetherian ring, let M be a finitely generated graded R-module and let R₊=?_n>0R_n. Let b?b₀+R₊, where b₀ is an ideal of R₀. In this paper, we first study the finiteness and vanishing of the n-th graded component of the i-th local cohomology module of M with respect to b. Then, among other things, we show that the set becomes ultimately constant, as n→−∞, in the following cases:

(i)

and (R₀,m₀) is a local ring;

(ii)

dim(R₀)≤1 and R₀ is either a finite integral extension of a domain or essentially of finite type over a field;

(iii)

i≤g_b(M), where g_b(M) denotes the cohomological finite length dimension of M with respect to b.

Also, we establish some results about the Artinian property of certain submodules and quotient modules of .     

On coLaskerian module and local cohomology

The notion of generalized Laskerian modules was introduced recently by the authors. In this paper, we continue our investigation of such modules. Moreover, we are going to generalize some results on finiteness of associated primes of local cohomology modules to the category of generalized Laskerian modules.     

Matlis reflexive and generalized local cohomology modules

Let (R,m) be a complete local ring, a an ideal of R and N and L two Matlis reflexive R-modules with Supp(L) ⊆ V(a). We prove that if M is a finitely generated R-module, then Exti _R ⁱ (L, H _a ^j (M,N)) is Matlis reflexive for all i and j in the following cases:

Associated primes and cofiniteness of local cohomology modules

Let be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that is finite R-module. If s is the first integer such that the local cohomology module is non -cofinite, then we show that is finite. In particular, the set of associated primes of is finite. Let be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module is not -cofinite and show that n just depends on the support of M.The research of the first author was supported in part by a grant from IPM (No. 83130114).The second author was supported by a grant from University of Tehran (No. 6103023/1/01).