A bound on the reduction number of a primary ideal

Let be a local ring of positive dimension and let be an -primary ideal. We denote the reduction number of by , which is the smallest integer such that for some reduction of In this paper we give an upper bound on in terms of numerical invariants which are related with the Hilbert coefficients of when is Cohen-Macaulay. If , it is known that where denotes the multiplicity of If in Corollary 1.5 we prove where is the first Hilbert coefficient of From this bound several results follow. Theorem 1.3 gives an upper bound on in a more general setting.

     

Curvature restrictions on convex, timelike surfaces in Minkowski 3-space

Suppose that and are Minkowski Gauss curvature and Minkowski mean curvature respectively on a timelike surface that is immersed in Minkowski 3-space . Suppose also that and that is complete as a surface in the underlying Euclidean 3-space . It is shown that neither nor can be bounded away from zero on such a surface .

     

On covering translations and homeotopy groups of contractible open -manifolds

This paper gives a new proof of a result of Geoghegan and Mihalik which states that whenever a contractible open -manifold which is not homeomorphic to is a covering space of an -manifold and either or and is irreducible, then the group of covering translations injects into the homeotopy group of .

     

Generalized Matlis duality

Let be a commutative noetherian ring and let be the minimal injective cogenerator of the category of -modules. A module is said to be reflexive with respect to if the natural evaluation map from to is an isomorphism. We give a classification of modules which are reflexive with respect to . A module is reflexive with respect to if and only if has a finitely generated submodule such that is artinian and is a complete semi-local ring.

     

A generalization of Kelley's theorem for -spaces

We prove that if an open map of compacta and has perfect fibers and is a -space, then there exists a -dimensional compact subset of intersecting each fiber of . This is a stronger version of a well-known theorem of Kelley. Applications of this result and related topics are discussed.

     

The number of knot group representations is not a Vassiliev invariant

For a finite group and a knot in the -sphere, let be the number of representations of the knot group into . In answer to a question of D.Altschuler we show that is either constant or not of finite type. Moreover, is constant if and only if is nilpotent. We prove the following, more general boundedness theorem: If a knot invariant is bounded by some function of the braid index, the genus, or the unknotting number, then is either constant or not of finite type.

     

Iwasawa invariants and class numbers of quadratic fields for the prime

Let be a square-free integer with and . Put and . For the cyclotomic -extension of , we denote by the -th layer of over . We prove that the -Sylow subgroup of the ideal class group of is trivial for all integers if and only if the class number of is not divisible by the prime . This enables us to show that there exist infinitely many real quadratic fields in which splits and whose Iwasawa -invariant vanishes.

     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

A criterion for splitting -algebras in tensor products

The goal of the paper is to prove the following theorem: if , are unital -algebras, simple and nuclear, then any -subalgebra of the -tensor product of and , which contains the tensor product of with the scalar multiples of the unit of , splits in the -tensor product of with some -subalgebra of .

     

A note on -bases of rings

Let be a tower of rings of characteristic . Suppose that is a finitely presented -module. We give necessary and sufficient conditions for the existence of -bases of over . Next, let be a polynomial ring where is a perfect field of characteristic , and let be a regular noetherian subring of containing such that . Suppose that is a free -module. Then, applying the above result to a tower of rings, we shall show that a polynomial of minimal degree in is a -basis of over .

     

On Tate-Shafarevich groups of abelian varieties

Let be a finite Galois extension of number fields with Galois group , let be an abelian variety defined over , and let and denote, respectively, the Tate-Shafarevich groups of over and of over . Assuming that these groups are finite, we derive, under certain restrictions on and , a formula for the order of the subgroup of of -invariant elements. As a corollary, we obtain a simple formula relating the orders of , and when is a quadratic extension and is the twist of by the non-trivial character of .

     

Higher order symmetric spaces and the roots of the identity in a Lie group

Let denote the set of all -roots of the identity in a Lie group . We show that is always an embedded submanifold of , having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of -symmetric spaces and we show, more generally, that every -symmetric space of a Lie group is a covering manifold of an embedded submanifold of . We compute also the Hessian of the inclusions of and into , relative to the natural connection on the domain and to the symmetric connection on .

     

Reflection and uniqueness theorems for harmonic functions

Suppose that is harmonic on an open half-ball in such that the origin 0 is the centre of the flat part of the boundary . If has non-negative lower limit at each point of and tends to 0 sufficiently rapidly on the normal to at 0, then has a harmonic continuation by reflection across . Under somewhat stronger hypotheses, the conclusion is that . These results strengthen recent theorems of Baouendi and Rothschild. While the flat boundary set can be replaced by a spherical surface, it cannot in general be replaced by a smooth -dimensional manifold.

     

Completely continuous multilinear operators on spaces

Given a -linear operator from a product of spaces into a Banach space , our main result proves the equivalence between being completely continuous, having an -valued separately continuous extension to the product of the biduals and having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to being weakly compact, and that, for , being weakly compact implies the conditions above but the converse fails.

     

An uncertainty principle for convolution operators on discrete groups

Consider a discrete group and a bounded self-adjoint convolution operator on ; let be the spectrum of . The spectral theorem gives a unitary isomorphism between and a direct sum , where , and is a regular Borel measure supported on . Through this isomorphism corresponds to multiplication by the identity function on each summand. We prove that a nonzero function and its transform cannot be simultaneously concentrated on sets , such that and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

     

Asymptotic behaviour of Castelnuovo-Mumford regularity

Let be a polynomial ring over a field. For a graded -module generated in degree at most , the Castelnuovo-Mumford regularity of each of (i) its symmetric power, (ii) its torsion-free symmetric power and (iii) the integral closure of its torsion-free symmetric power is bounded above by a linear function in with leading coefficient at most . For a graded ideal of , the regularity of is given by a linear function of for all sufficiently large . The leading coefficient of this function is identified.

     

Yang index of the deleted product

For any a -dimensional polyhedron is constructed such that the Yang index of its deleted product equals . This answers a question of Izydorek and Jaworowski (1995). For any a -dimensional closed manifold with involution is constructed such that , but can be mapped into a -dimensional polyhedron without antipodal coincidence.

     

Extreme points of the unit ball of the Fourier-Stieltjes algebra

Let be a locally compact group. Among other things, we proved in this paper that for an IN-group , the extreme points of the unit ball of the Fourier-Stieltjes algebra are not in the Fourier algebra if and only if is non-compact, or equivalently, there is no irreducible representation of which is quasi-equivalent to a subrepresentation of the left regular representation of if and only if is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group , the extreme points of the unit ball of the measure algebra are not in the group algebra if and only if is non-discrete. On the other hand, we also showed that if has the RNP, then there are extreme points of the unit ball of that are in . Since it is well known there are non-compact locally compact group for which has the RNP, there exist non-compact locally compact groups where extreme points of the unit ball of can be in . This shows that the condition be an IN-group cannot be entirely removed.

     

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 .

     

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 .