The globally irreducible representations of symmetric groups

Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

A characterization of the hereditary categories derived equivalent to some category of coherent sheaves on a weighted projective line

Let be a connected hereditary abelian category over an algebraically closed field , with finite dimensional homomorphism and extension spaces. There are two main known types of such categories: those derived equivalent to for some finite dimensional hereditary -algebra and those derived equivalent to some category of coherent sheaves on a weighted projective line in the sense of Geigle and Lenzing (1987). The aim of this paper is to give a characterization of the second class in terms of some properties known to hold for these hereditary categories.

     

Singular integrals generated by zonal measures

We study -mapping properties of the rough singular integral operator depending on a finite Borel measure on the unit sphere in . It is shown that the conditions , imply the -boundedness of for all provided that 2$"> and is zonal.

     

Subgroup growth in some pro- groups

For a group let be the number of subgroups of index and let be the number of normal subgroups of index . We show that for 2$">. If and does not divide or if and or , we show that for all sufficiently large . On the other hand if and divides , then is not even bounded as a function of .

     

Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

Bordism groups of special generic mappings

The Pontrjagin-Thom construction expresses a relation between the oriented bordism groups of framed immersions , and the stable homotopy groups of spheres. We apply the Pontrjagin-Thom construction to the oriented bordism groups of mappings n$">, with mildest singularities. Recently, O. Saeki showed that for , the group is isomorphic to the group of smooth structures on the sphere of dimension . Generalizing, we prove that is isomorphic to the -th stable homotopy group , , where is the group of oriented auto-diffeomorphisms of the sphere and is the group of rotations of .

     

Parallel tangent hyperplanes

Let be a smooth strictly convex closed hypersurface in and let be any oriented smooth connected manifold immersed in Suppose that is a continuous function from to Then there is at least one point such that the hyperplane tangent to at is parallel to the hyperplane tangent to the immersed manifold at the point corresponding to If there did not exist at least two such points, would have to be compact and the Hurewicz homomorphism of into would have to be surjective. If in addition our immersion was an embedding, the Euler characteristic of would have to be equal to For any and any immersed we could always get maps for which the number of points satisfying the conditions of our theorem exactly equaled two. An example can be given in which both and are the unit sphere about the origin in and there is only one such point .

     

Fonctions qui operent sur les espaces de Besov

Soient , et trois réels tels que , , et et soit une fonction appartenant à l'espace de Besov . Nous montrons que si est une fonction, de la variable réelle, nulle à l'origine, lipschitzienne et appartenant à l'espace on a alors . La preuve est essentiellement basée sur des résultats d'approximation par des fonctions splines de degré .

     

Eigenvalues of the Laplacian acting on -forms and metric conformal deformations

Let be a compact connected orientable Riemannian manifold of dimension and let be the -th positive eigenvalue of the Laplacian acting on differential forms of degree on . We prove that the metric can be conformally deformed to a metric , having the same volume as , with arbitrarily large for all .

Note that for the other values of , that is and , one can deduce from the literature that, 0$">, the -th eigenvalue is uniformly bounded on any conformal class of metrics of fixed volume on .

For , we show that, for any positive integer , there exists a metric conformal to such that, , , that is, the first eigenforms of are all exact forms.

     

Hochschild cohomology of Frobenius algebras

Let be a field, a finite-dimensional Frobenius -algebra and , the Nakayama automorphism of with respect to a Frobenius homomorphism . Assume that has finite order and that has a primitive -th root of unity . Consider the decomposition of , obtained by defining , and the decomposition of the Hochschild cohomology of , obtained from the decomposition of . In this paper we prove that and that if the decomposition of is strongly -graded, then acts on and .

     

Local cohomology over homogeneous rings with one-dimensional local base ring

Let be a homogeneous Noetherian ring with local base ring and let be a finitely generated graded -module. Let be the -th local cohomology module of with respect to 0}R_n$">. If , the -modules , and are Artinian for all . As a consequence, much can be said on the asymptotic behaviour of the -modules for .

     

On the maximal inequalities for martingales involving two functions

Let and be nonnegative convex functions, and let and be the right continuous derivatives of and respectively. In this paper, we prove the equivalence of the following three conditions: (i) (ii) and (iii) s_0,$">where and are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

On the best possible character of the norm in some a priori estimates for non-divergence form equations in Carnot groups

Let be a group of Heisenberg type with homogeneous dimension . For every we construct a non-divergence form operator and a non-trivial solution to the Dirichlet problem: in , on . This non-uniqueness result shows the impossibility of controlling the maximum of with an norm of when . Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as

where is the dimension of the horizontal layer of the Lie algebra and is the symmetrized horizontal Hessian of .

     

Cancellation of direct sums of countable abelian -groups

Let be abelian groups where is a direct sum of countable -groups. A condition is given on the Ulm-Kaplansky -invariants of and such that .

     

Complemented isometric copies of in dual Banach spaces

Let be a real or complex Banach space and . Then contains a -complemented, isometric copy of if and only if contains a -complemented, isometric copy of if and only if contains a subspace -asymptotic to .

     

Convergence of sequences of sets of associated primes

It is a well-known result of M. Brodmann that if is an ideal of a commutative Noetherian ring , then the set of associated primes of the -th power of is constant for all large . This paper is concerned with the following question: given a prime ideal of which is known to be in for all large integers , can one identify a term of the sequence beyond which will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.

     

Control of radii of convergence and extension of subanalytic functions

Let : denote a real analytic function on an open subset of , and let denote the points where does not admit a local analytic extension. We show that if is semialgebraic (respectively, globally subanalytic), then is semialgebraic (respectively, subanalytic) and extends to a semialgebraic (respectively, subanalytic) neighbourhood of . (In the general subanalytic case, is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series centred at points in the image of an analytic mapping , in terms of the radii of convergence of at points , where denotes the Taylor expansion of at .