The local cohomology modules of Matlis reflexive modules are almost cofinite

We show that if and are Matlis reflexive modules over a complete Gorenstein local domain and is an ideal of such that the dimension of is one, then the modules are Matlis reflexive for all and if . It follows that the Bass numbers of are finite. If is not a domain, then the same results hold for .

     

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(−) ≔ Hom _R (−, 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 x ₁, …, x _n is a regular sequence on M contained in α, then H _{(x1, …,xnR} ⁿ D(H _a ⁿ (M))) is a homomorphic image of D(M), where H _b ⁱ (−) 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)R} ⁿ (D(H _a ⁿ (M)))) ⋟ D(D(M)).     

Attached primes and Matlis duals of local cohomology modules

Let J be an ideal of a noetherian local ring R. We show new results on the set of attached primes of a local cohomology module . To prove our results we establish and use new relations between the set of attached primes of a local cohomology module and the set of associated primes of the Matlis dual of the same local cohomology module. Received: 17 March 2006     

On the Matlis duals of local cohomology modules

Let ( ) be a commutative Noetherian local ring with non-zero identity, an ideal of R and M a finitely generated R-module with . Let D(–) := Hom _R (–, E) be the Matlis dual functor, where is the injective hull of the residue field . We show that, for a positive integer n, if there exists a regular sequence and the i-th local cohomology module H ⁱ _a (M) of M with respect to is zero for all i with i > n then The author was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 85130023). Received: 9 August 2006     

Matlis duals of local cohomology modules and their endomorphism rings

Let ${(R, \mathfrak{m})}Let (R, \mathfrakm){(R, \mathfrak{m})} denote a local ring. Let I ì R{I \subset R} be an ideal with c =  grade I. Let D(·) denote the Matlis duality functor. In recent research there is an interest in the structure of the local cohomology module H^c_I : = H^c_I(R){H^c_I := H^c_I(R)}, in particular in the endomorphism ring of D(H^c_I){D(H^c_I)}. Let E _R (k) be the injective hull of the residue field R/\mathfrakm{R/\mathfrak{m}}. By investigating the natural map H^c_I ?D(H^c_I) ? E_R(k){H^c_I \otimes D(H^c_I) \to E_R(k)} we are able to prove that the endomorphism rings of D(H^c_I){D(H^c_I)} and of H^c_I{H^c_I} are naturally isomorphic. This natural homomorphism is related to a quasi-isomorphism of a certain complex. As applications we show results when the endomorphism ring of D(H^c_I){D(H^c_I)} is naturally isomorphic to R generalizing results known under the additional assumption of Hⁱ_I(R) = 0{H^i_I(R) = 0} for i 1 c{i \not= c}.     

On the finiteness properties of Matlis duals of local cohomology modules

Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let D _R (−) := Hom _R (−, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext _R ^j (R/a, D _R (H _a ^t (M))) is finitely generated for all t > n and all j ⩾ 0, then we show that Hom _R (R/a, D _R (H _a ⁿ (M))) is also finitely generated. Specially, the set of prime ideals in Coass _R (H _a ⁿ (M)) which contains a is finite. Next, assume that (R, m) is a complete local ring. We study the finiteness properties of D _R (H _a ^r (R)) where r is the least integer i such that H _a ^r (R) is not Artinian.     

On Artinian generalized local cohomology modules

Let R be a commutative Noetherian ring with non-zero identity and a be a maximal ideal of R. An R-module M is called minimax if there is a finitely generated submodule N of M such that M/N is Artinian. Over a Gorenstein local ring R of finite Krull dimension, we proved that the Socle of H _a ⁿ (R) is a minimax R-module for each n ≥ 0.     

Matlis duals of top Local cohomology modules

In the first section of this paper we present generalizations of known results on the set of associated primes of Matlis duals of local cohomology modules; we prove these generalizations by using a new technique. In section 2 we compute the set of associated primes of the Matlis dual of , where is a -dimensional local ring and an ideal such that and .

     

Finiteness of graded generalized local cohomology modules

We consider two finitely generated graded modules over a homogeneous Noetherian ring $R = \oplus _{n \in \mathbb{N}_0 } R_n$ with a local base ring (R ₀, m₀) and irrelevant ideal R ₊ of R. We study the generalized local cohomology modules H _b ⁱ (M,N) with respect to the ideal b = b₀ + R ₊, where b₀ is an ideal of R ₀. We prove that if dimR ₀/b₀ ≤ 1, then the following cases hold: for all i ≥ 0, the R-module H _b ⁱ (M,N)/a₀ H _b ⁱ (M,N) is Artinian, where $\sqrt {\mathfrak{a}_0 + \mathfrak{b}_0 } = \mathfrak{m}_0$ ; for all i ≥ 0, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is asymptotically stable as n→?∞. Moreover, if H _b ⁱ (M,N) _n is a finitely generated R ₀-module for all n ≤ n ₀ and all j < i, where n ₀ ∈ ? and i ∈ ?₀, then for all n ≤ n ₀, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is finite.     

Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L^′ be R-modules. We investigate the properties of the functors and . For instance, we show the following:

(a)

if L and L^′ are artinian, then is artinian, and is noetherian over the completion ;

(b)

if L is artinian and L^′ is Matlis reflexive, then , , and are Matlis reflexive.

Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.     

Artinianness of local cohomology modules

Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.     

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.

     

On the vanishing and the finiteness of supports of generalized local cohomology modules

Let be a Noetherian local ring, I an ideal of R and M, N two finitely generated R-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that for all j > dim(R), provided M is of finite projective dimension. Next, we study and give characterizations for the least and the last integer r such that Supp is infinite. This work is supported in part by the National Basis Research Programme in Natural Science of Vietnam.