On graded generalized local cohomology

Let be a homogeneous Noetherian ring with local base ring (R₀,m₀) and let M,N be two finitely generated graded R-modules. Let denote the i-th graded generalized local cohomology of N relative to M with support in . We study the vanishing, tameness and asymptotical stability of the homogeneous components of . Received: 22 March 2005; revised: 25 June 2005     

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     

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).     

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 .     

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.     

An avoidance principle with an application to the asymptotic behaviour of graded local cohomology

We present an avoidance principle for certain graded rings. As an application we fill a gap in the proof of a result of Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert–Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring.     

The finiteness dimension of local cohomology modules and its dual notion

Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_a(M), the finiteness dimension of M with respect to a, and, its dual notion q_a(M), the Artinianness dimension of M with respect to a. When (R,m) is local and r?f_a(M) is less than , the m-finiteness dimension of M relative to a, we prove that is not Artinian, and so the filter depth of a on M does not exceed f_a(M). Also, we show that if M has finite dimension and is Artinian for all i>t, where t is a given positive integer, then is Artinian. This immediately implies that if q?q_a(M)>0, then is not finitely generated, and so f_a(M)≤q_a(M).     

The associated primes of local cohomology modules over rings of small dimension

Let R be a commutative Noetherian local ring of dimension d, I an ideal of R, and M a finitely generated R-module. We prove that the set of associated primes of the local cohomology module H ⁱ _I (M) is finite for all i≥ 0 in the following cases: (1) d≤ 3; (2) d= 4 and $R$ is regular on the punctured spectrum; (3) d= 5, R is an unramified regular local ring, and M is torsion-free. In addition, if $d>0$ then H ^{d − 1} _I (M) has finite support for arbitrary R, I, and M. Received: 31 October 2000 / Revised version: 8 January 2001     

On connectedness and indecomposibility of local cohomology modules

Let I denote an ideal of a local Gorenstein ring . Then we show that the local cohomology module , c = height I, is indecomposable if and only if V(I _d ) is connected in codimension one. Here I _d denotes the intersection of the highest dimensional primary components of I. This is a partial extension of a result shown by Hochster and Huneke in the case I the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of is a local Noetherian ring if dim R/I  =  1.     

A characterization of Cohen-Macaulay modules and local cohomology

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated locally quasi-unmixed R-module. In this paper, the main result asserts that N is Cohen-Macaulay if and only if, for any N-proper ideal I of R generated by height_N I elements, the set of asymptotic primes of I with respect to N is equal to the set of presistent primes of I with respect to N. In addition, some applications about local cohomology are included. Received: 3 July 2005     

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     

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.     

A new depth and the annihilation of local cohomology modules

Let J I be two proper ideals of a commutative Noetherian ring and M a finitely generated module. Strong relative depth is defined and characterized. It is proved that this depth is just the maximum integer n such that can be annihilated by some power of J for all i ≤ n. It turns out that the local-global principle for the annihilation of local cohomology modules can be formulated as a natural property of this new depth. Received: 28 July 2006     

Duality and vanishing of generalized local cohomology

In this paper we study local duality and the vanishing and non-vanishing of generalized local cohomology. In particular for a local ring R and a finitely generated R-module N, the category $N^\bot$ of finitely generated R-modules M with $H^{i}_{\frak m}(M, N) = 0$ for all i depth N is studied.Received: 31 October 2002     

Generalized local cohomology and the canonical element conjecture

We study a generalization of the Canonical Element Conjecture. In particular we show that given a nonregular local ring (A,m) and an i>0, there exist finitely generated A-modules M such that the canonical map from to is nonzero. Moreover, we show that even when M has an infinite projective dimension and i>dim(A), studying these maps sheds light on the Canonical Element Conjecture.     

Gorenstein injective, Gorenstein flat modules and the section functor

Let R be a commutative Noetherian ring of Krull dimension d, and let a be an ideal of R. In this paper, we will study the strong cotorsioness and the Gorenstein injectivity of the section functor Γ_a(−) in local cohomology. As applications, we will find new characterizations for Gorenstein and regular local rings. We also study the effect of the section functors Γ_a(−) and the functors on the Auslander and Bass classes.     

Stable cohomology over local rings

For a commutative noetherian ring R with residue field k stable cohomology modules have been defined for each n∈Z, but their meaning has remained elusive. It is proved that the k-rank of any characterizes important properties of R, such as being regular, complete intersection, or Gorenstein. These numerical characterizations are based on results concerning the structure of Z-graded k-algebra carried by stable cohomology. It is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras . Some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings.     

Separating invariants and local cohomology

The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the least possible cardinality of a separating set for a given G-action. Our main result is a lower bound that generalizes the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples.