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     

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     

Bounds on Castelnuovo-Mumford regularity for generalized Cohen-Macaulay graded rings

Supported by a grant from the Centre de Recerca Matematica, Institut d'Estudis Catalans (Spain)     

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.     

Finiteness of graded local cohomology modules

Let R=?_n∈N0R_n be a Noetherian homogeneous ring with local base ring (R₀,m₀) and irrelevant ideal R₊, let M be a finitely generated graded R-module. In this paper we show that is Artinian and is Artinian for each i in the case where R₊ is principal. Moreover, for the case where , we prove that, for each i∈N₀, is Artinian if and only if is Artinian. We also prove that is Artinian, where and c is the cohomological dimension of M with respect to R₊. Finally we present some examples which show that and need not be Artinian.     

A local duality principle in derived categories of commutative Noetherian rings

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors ${\gamma }_W$ with supports in arbitrary subsets W of Spec R. If W is a specialization-closed subset, then ${\gamma }_W$ coincides with the right derived functor ${\mathrm{R}\mathrm{\Gamma }}_W$ of the section functor ${\mathrm{\Gamma }}_W$ with support in W. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for ${\gamma }_W$ with W being an arbitrary subset.     

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     

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.     

On the vanishing of (co)homology over local rings

Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart from each other.     

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.     

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).     

Hochschild cohomology of abelian categories and ringed spaces

This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer-Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf.     

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).     

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.     

The cohomology of the moduli space of controllable linear systems

In this paper we explicitly describe, by generators and relations, the cohomology ring of the manifold _n,m (F) of controllable linear systems having m inputs and state-space dimension n. It is shown that the cohomology ring of _n,m (F) is isomorphic to the invariant cohomology ring of a product of projective spaces. Estimates for the cup length of the cohomology ring are obtained.     

Tight closure of parameter ideals in local rings F-rational on the punctured spectrum

Let $(R,\mathfrak{m},k)$ be an equidimensional excellent local ring of characteristic $p>0$. The aim of this paper is to show that ${?}_R({\mathfrak{q}}^{?}/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$ provided R is an F-injective local ring that is F-rational on the punctured spectrum.