Cotensor products of modules

Let be a coalgebra over a field and its dual algebra. The category of -comodules is equivalent to a category of -modules. We use this to interpret the cotensor product of two comodules in terms of the appropriate Hochschild cohomology of the -bimodule , when is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations.

     

Vertices for characters of -solvable groups

Suppose that is a finite -solvable group. We associate to every irreducible complex character of a canonical pair , where is a -subgroup of and , uniquely determined by up to -conjugacy. This pair behaves as a Green vertex and partitions into ``families" of characters. Using the pair , we give a canonical choice of a certain -radical subgroup of and a character associated to which was predicted by some conjecture of G. R. Robinson.

     

Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group

Applied to a continuous surjection of completely regular Hausdorff spaces and , the Stone-Cech compactification functor yields a surjection . For an -fold covering map , we show that the fibres of , while never containing more than points, may degenerate to sets of cardinality properly dividing . In the special case of the universal bundle of a -group , we show more precisely that every possible type of -orbit occurs among the fibres of . To prove this, we use a weak form of the so-called generalized Sullivan conjecture.

     

Sets of uniqueness for spherically convergent multiple trigonometric series

A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set.

     

Hyponormality of trigonometric Toeplitz operators

In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators with trigonometric polynomial symbols . Our criterion involves the zeros of an analytic polynomial induced by the Fourier coefficients of . Moreover the rank of the selfcommutator of is computed from the number of zeros of in the open unit disk and in counting multiplicity.

     

Summing inclusion maps between symmetric sequence spaces

In 1973/74 Bennett and (independently) Carl proved that for the identity map id: is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a -concave symmetric Banach sequence space the identity map is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in . Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator on with values in a -concave symmetric Banach sequence space is a multiplier from into . Furthermore, we prove an asymptotic formula for the -th approximation number of the identity map , where denotes the linear span of the first standard unit vectors in , and apply it to Lorentz and Orlicz sequence spaces.

     

On the Glauberman and Watanabe correspondences for blocks of finite -solvable groups

If is a finite -solvable group for some prime , a solvable subgroup of the automorphism group of of order prime to such that stabilises a -block of and acts trivially on a defect group of , then there is a Morita equivalence between the block and its Watanabe correspondent of , given by a bimodule with vertex and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe's results).

     

Topological dynamics on moduli spaces II

Let be an orientable genus 0$"> surface with boundary . Let be the mapping class group of fixing . The group acts on the space of -gauge equivalence classes of flat -connections on with fixed holonomy on . We study the topological dynamics of the -action and give conditions for the individual -orbits to be dense in .

     

Uniform and Lipschitz homotopy classes of maps

If is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line into an element of where is the space of uniformly continuous functions from to and is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to is surjective. If is the -dimensional torus, it is bijective, while if is a compact orientable surface of genus 1$">, it is not injective.

In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds to compact smooth manifolds With each such Lipschitz homotopy class we associate an element of where is the dimension of is the space of bounded continuous functions from the positive real axis to and is the set of all such that

     

Extensions for finite Chevalley groups II

Let be a semisimple simply connected algebraic group defined and split over the field with elements, let be the finite Chevalley group consisting of the -rational points of where , and let be the th Frobenius kernel. The purpose of this paper is to relate extensions between modules in and with extensions between modules in . Among the results obtained are the following: for 2$"> and , the -extensions between two simple -modules are isomorphic to the -extensions between two simple -restricted -modules with suitably ``twisted" highest weights. For , we provide a complete characterization of where and is -restricted. Furthermore, for , necessary and sufficient bounds on the size of the highest weight of a -module are given to insure that the restriction map is an isomorphism. Finally, it is shown that the extensions between two simple -restricted -modules coincide in all three categories provided the highest weights are ``close" together.

     

On a class of jointly hyponormal Toeplitz operators

We characterize when a pair of Toeplitz operators is jointly hyponormal under various assumptions--for example, is analytic or is a trigonometric polynomial or is analytic. A typical characterization states that is jointly hyponormal if and only if an algebraic relation of and holds and the single Toeplitz operator is hyponormal, where is a combination of and . More general results for an -tuple of Toeplitz operators are also obtained.

     

A momentum construction for circle-invariant Kähler metrics

Examples of Kähler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kähler metrics of constant scalar curvature by ODE methods. One fruitful idea--the ``Calabi ansatz'--is to begin with an Hermitian line bundle over a Kähler manifold, and to search for Kähler forms in some disk subbundle, where is the logarithm of the norm function and is a function of one variable.

Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kähler metric  arising from the Calabi ansatz. This suggests geometric hypotheses (which we call ``-constancy') to impose upon the base metric  and Hermitian structure  in order that the scalar curvature of  be specified by solving an ODE. We show that -constancy is ``necessary and sufficient for the Calabi ansatz to work' in the following sense. Under the assumption of -constancy, the disk bundle admits a one-parameter family of complete Kähler metrics of constant scalar curvature that restrict to on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of -constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E).

Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kähler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kähler metric on  .

     

Associated primes of graded components of local cohomology modules

The -th local cohomology module of a finitely generated graded module over a standard positively graded commutative Noetherian ring , with respect to the irrelevant ideal , is itself graded; all its graded components are finitely generated modules over , the component of of degree . It is known that the -th component of this local cohomology module is zero for all > 0$">. This paper is concerned with the asymptotic behaviour of as .

The smallest for which such study is interesting is the finiteness dimension of relative to , defined as the least integer for which is not finitely generated. Brodmann and Hellus have shown that is constant for all (that is, in their terminology, is asymptotically stable for ). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that is a homomorphic image of a regular ring): our answer is precisely the set of contractions to of certain relevant primes of whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology.

Brodmann and Hellus raised various questions about such asymptotic behaviour when f$">. They noted that Singh's study of a particular example (in which ) shows that need not be asymptotically stable for . The second main aim of this paper is to determine, for Singh's example, quite precisely for every integer , and, thereby, answer one of the questions raised by Brodmann and Hellus.

     

Automorphisms of finite order on Gorenstein del Pezzo surfaces

In this paper we shall determine all actions of groups of prime order with on Gorenstein del Pezzo (singular) surfaces of Picard number 1. We show that every order- element in ( , being the minimal resolution of ) is lifted from a projective transformation of . We also determine when is finite in terms of , and the number of singular members in . In particular, we show that either for some , or for every prime , there is at least one element of order in (hence is infinite).

     

Partial regularity for the stochastic Navier-Stokes equations

The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary solutions it is proved that at any time the set of singular points is empty. The same result holds true for every martingale solution starting from -a.e. initial condition , where is the law at time zero of a stationary solution. Finally, the previous result is non-trivial when the noise is sufficiently non-degenerate, since for any stationary solution, the measure is supported on the whole space of initial conditions.

     

Trees and branches in Banach spaces

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree of a certain type on a space is presumed to have a branch with some property. It is shown that then can be embedded into a space with an FDD so that all normalized sequences in which are almost a skipped blocking of have that property. As an application of our work we prove that if is a separable reflexive Banach space and for some and every weakly null tree on the sphere of has a branch -equivalent to the unit vector basis of , then for all 0$">, there exists a subspace of having finite codimension which embeds into the sum of finite dimensional spaces.

     

Detection of renewal system factors via the Conley index

Let be an isolating neighborhood for a map . If we can decompose into the disjoint union of compact sets and , then we can relate the dynamics on the maximal invariant set to the shift on two symbols by noting which component of each iterate of a point lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to . In essence, we measure the difference between the Conley index of and the sum of the indices of and .

     

Local subgroups and the stable category

If is a finite group and is an algebraically closed field of characteristic 0$">, then this paper uses the local subgroup structure of to define a category that is equivalent to the stable category of all left -modules modulo projectives. A subcategory of equivalent to the stable category of finitely generated -modules is also identified. The definition of depends largely but not exclusively upon local data; one condition on the objects involves compatibility with respect to conjugations by arbitrary group elements rather than just elements of -local subgroups.

     

Invariant ideals and polynomial forms

Let denote the group algebra of an infinite locally finite group . In recent years, the lattice of ideals of has been extensively studied under the assumption that is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra stable under the action of an appropriate automorphism group of . Specifically, in this paper, we let be a quasi-simple group of Lie type defined over an infinite locally finite field , and we let be a finite-dimensional vector space over a field of the same characteristic . If acts nontrivially on by way of the homomorphism , and if has no proper -stable subgroups, then we show that the augmentation ideal is the unique proper -stable ideal of when . The proof of this result requires, among other things, that we study characteristic division rings , certain multiplicative subgroups of , and the action of on the group algebra , where is the additive group . In particular, properties of the quasi-simple group come into play only in the final section of this paper.