Structure of cohomology modules of linear bundles overG/B

We study the cohomologyG-modules of linear bundles over flag spacesG/B for algebraically closed fields of prime characteristic. Some series of submodules and quotient modules of these modules are described. Translated by S. K. Lando Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 10–17, July, 1997.     

The cohomology group of weak entwining structure

In this paper, we reveal that a weak entwining structure admits a rich cohomology theory. As an application we compute the cohomology of a weak entwining structure associated to a weak coalgebra-Galois extension.     

Varieties and cohomology of infinitely generated modules

Suppose that k is a field of characteristic p and that G is a finite group having p-rank at least three. Given a regular sequence of length 2 in H*(G, k), we show how to construct an infinitely generated module whose support variety is strictly smaller than the small support of its cohomology. Received: 25 January 2008     

Some results on the cofiniteness and annihilators of local cohomology modules

Let (R,𝔪) be a Noetherian local ring. In this paper, we give a new characterization for the universal catenaricity of R and the Cohen–Macaulayness of all its formal fibers, and we study the cofiniteness and annihilators of local cohomology modules.     

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.     

A generalization of the finiteness problem in local cohomology modules

Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H _a ⁱ (N) is a weakly Laskerian R-module for all i < t, then Hom _R (R/a, H _a ^t (M, N)) is weakly Laskerian R-module. Also, we prove that Ext _R ⁱ (R/a, H _a ^t )) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp _R (H _a ⁱ (N)) is a finite set for all i < t, then Ext _R ⁱ (R/a, H _a ^t (N)) is weakly Laskerian R-module for all i = 0, 1.     

Some Weyl Modules for Simple Algebraic Groups of Type B4 and D4

Let G be a simply connected, semisimple algebraic group of type B₄ or D₄ over an algebraically closed field of characteristic p > 0. We determine the characters of certain simple modules for these groups by calculating the composition factors of the Weyl modules.     

Simple modules of the quantum double of the Nichols algebra of unidentified diagonal type 𝔲𝔣𝔬(7)

The finite-dimensional simple modules over the Drinfeld double of the bosonization of the Nichols algebra 𝔲𝔣𝔬(7) are classified.     

On the finiteness properties of extension and torsion functors of local cohomology modules

Let be a commutative Noetherian ring with non-zero identity, and ideals of with , and a finitely generated -module. In this paper, for fixed integers and , we study the finiteness of and in several cases.

     

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.

     

Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules

Let be a Noetherian homogeneous ring with one-dimensional local base ring . Let be an -primary ideal, let be a finitely generated graded -module and let . Let denote the -th local cohomology module of with respect to the irrelevant ideal 0} R_n$"> of . We show that the first Hilbert-Samuel coefficient of the -th graded component of with respect to is antipolynomial of degree in . In addition, we prove that the postulation numbers of the components with respect to have a common upper bound.

     

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

The cohomology of homotopy categories and the general linear group

A homotopy categoryC (of co-H-groups resp.H-groups) represents an element C in the third cohomology ofC. This element determines all Toda brackets and secondary homotopy operations inC. Moreover, in caseC =VS ⁿ consists of all one-point unions ofn-spheres, the bracket is actually a /2-generator which restricts to Igusa's class(1) in casen3; an explicit new cocycle for(1) is obtained by automorphisms of free nil(2)-groups.     

Restricted weak type on maximal linear and multilinear integral maps

It is shown that multilinear operators of the form of restricted weak type are always of weak type whenever the map is a locally integrable -valued function.

     

The grothendieck group of the category of modules of finite projective dimension over certain weakly triangular algebras

In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have finite projective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relations for the Grothendieck group of P ^<∞(Λ) if and only if P ^<∞(Λ) is of finite type. A similar statement is known to hold for the category of all finitely generated modules over an artin algebra, and was proven by C.M.Butler and M. Auslander ( [B] and [A]).     

Essentially normal Hilbert modules and K-homology IV: Quasi-homogenous quotient modules of Hardy module on the polydisks

In this paper,we give a complete characterization for the essential normality of quasi-homogenous quotient modules of the Hardy modules H2 (D2).Also,we show that if d 3,then all the principle homogenous quotient modules of H 2 (Dd) are not essentially normal.