-identities for the Grassmann algebra: The conjecture of Henke and Regev

Let be an algebraically closed field of characteristic 0, and let be the infinite dimensional Grassmann (or exterior) algebra over . Denote by the vector space of the multilinear polynomials of degree in , ..., in the free associative algebra . The symmetric group acts on the left-hand side on , thus turning it into an -module. This fact, although simple, plays an important role in the theory of PI algebras since one may study the identities satisfied by a given algebra by applying methods from the representation theory of the symmetric group. The -modules and are canonically isomorphic. Letting be the alternating group in , one may study and its isomorphic copy in with the corresponding action of . Henke and Regev described the -codimensions of the Grassmann algebra , and conjectured a finite generating set of the -identities for . Here we answer their conjecture in the affirmative.

     

A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Let and be compact Hausdorff spaces, and , be Banach lattices. Let denote the Banach lattice of all continuous -valued functions on equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism such that is non-vanishing on if and only if is non-vanishing on , then is homeomorphic to , and is Riesz isomorphic to . In this case, can be written as a weighted composition operator: , where is a homeomorphism from onto , and is a Riesz isomorphism from onto for every in . This generalizes some known results obtained recently.

     

On the isolated points of the surjective spectrum of a bounded operator

For a bounded operator acting on a complex Banach space, we show that if is not surjective, then is an isolated point of the surjective spectrum of if and only if , where is the quasinilpotent part of and is the analytic core for . Moreover, we study the operators for which . We show that for each of these operators , there exists a finite set consisting of Riesz points for such that and is connected, and derive some consequences.

     

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree

Let , , be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform of degree . It is proved that if is not in the Maaß space, then there exist infinitely many primes for which the sequence , , has infinitely many sign changes.

     

Strong periodicity of links and the coefficients of the Conway polynomial

Przytycki and Sokolov proved that a three-manifold admits a semi-free action of the finite cyclic group of order with a circle as the set of fixed points if and only if is obtained from the three-sphere by surgery along a strongly -periodic link . Moreover, if the quotient three-manifold is an integral homology sphere, then we may assume that is orbitally separated. This paper studies the behavior of the coefficients of the Conway polynomial of such a link. Namely, we prove that if is a strongly -periodic orbitally separated link and is an odd prime, then the coefficient is congruent to zero modulo for all such that .

     

Global co-stationarity of the ground model from a new countable length sequence

Suppose are models of ZFC with the same ordinals, and that for all regular cardinals in , satisfies . If contains a sequence for some ordinal , then for all cardinals in with regular in and , is stationary in . That is, a new -sequence achieves global co-stationarity of the ground model.

     

The projective -character bounds the order of a -base

All spaces below are Tychonov. We define the projective - character of a space as the supremum of the values where ranges over all (Tychonov) continuous images of . Our main result says that every space has a -base whose order is ; that is, every point in is contained in at most -many members of the -base. Since for compact , this is a significant generalization of a celebrated result of Shapirovskii.

     

Compact weighted composition operators on the Hardy space

A weighted composition operator takes an analytic map on the open unit disk of the complex plane to the analytic map , where is an analytic map of the open unit disk into itself and is an analytic map on the open unit disk. This paper studies how the compactness of depends on the interaction between the two maps and .

     

Groups with a character of large degree

Let be a finite group of order and a simple -module of dimension . For some nonnegative number , we have . If is small, then the character of has unusually large degree. We fix and attempt to classify such groups. For we give a complete classification. For any other fixed we show that there are only finitely many examples.

     

The depth of an ideal with a given Hilbert function

Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .

     

Extreme points, exposed points, differentiability points in CL-spaces

This paper presents a property of geometric and topological nature of Gateaux differentiability points and Fréchet differentiability points of almost CL-spaces. More precisely, if we denote by a maximal convex set of the unit sphere of a CL-space , and by the cone generated by , then all Gateaux differentiability points of are just n-s, and all Fréchet differentiability points of are (where n-s denotes the non-support points set of ).

     

Subgroups generated by small classes in finite groups

Let be the subgroup of generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if is nilpotent, then has nilpotence class at most . Using a slight variation on Mann's methods, we obtain results that do not require us to assume that is nilpotent. We show that if is supersolvable, then is nilpotent with class at most , and in general, the Fitting subgroup of has class at most .

     

Dynamics of the function and the Green-Tao theorem on arithmetic progressions in the primes

Let be the set of all positive integers , where are primes and possibly two, but not all three of them are equal. For any , define a function by where is the largest prime factor of . It is clear that if , then . For any , define , for . An element is semi-periodic if there exists a nonnegative integer and a positive integer such that . We use ind to denote the least such nonnegative integer . Wushi Goldring [Dynamics of the function and primes, J. Number Theory 119(2006), 86-98] proved that any element is semi-periodic. He showed that there exists such that , ind, and conjectured that ind can be arbitrarily large.

In this paper, it is proved that for any we have ind , and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

On the normal bundle of submanifolds of

Let be a submanifold of dimension of the complex projective space . We prove results of the following type.i) If is irregular and , then the normal bundle is indecomposable. ii) If is irregular, and , then is not the direct sum of two vector bundles of rank . iii) If , and is decomposable, then the natural restriction map is an isomorphism (and, in particular, if is embedded Segre in , then is indecomposable). iv) Let and , and assume that is a direct sum of line bundles; if assume furthermore that is simply connected and is not divisible in . Then is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when this fact was proved by M. Schneider in 1990 in a completely different way.

     

Isometries for the Carathéodory metric

Given two open unit balls and in complex Banach spaces, we consider a holomorphic mapping such that and is an isometry. Under some additional hypotheses on the Banach spaces involved, we prove that is a complex closed analytic submanifold of .

     

Polynomials with roots in for all

Let be a monic polynomial in with no rational roots but with roots in for all , or equivalently, with roots mod for all . It is known that cannot be irreducible but can be a product of two or more irreducible polynomials, and that if is a product of irreducible polynomials, then its Galois group must be a union of conjugates of proper subgroups. We prove that for any , every finite solvable group that is a union of conjugates of proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with ) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of .

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Let be an -dimensional compact hypersurface with constant scalar curvature , , in a unit sphere . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral of the mean curvature . In this paper, we first study the eigenvalue of the Jacobi operator of . We derive an optimal upper bound for the first eigenvalue of , and this bound is attained if and only if is a totally umbilical and non-totally geodesic hypersurface or is a Riemannian product , .

     

A note on the injective dimension of local cohomology modules

For a Noetherian ring we call an -module cofinite if there exists an ideal of such that is -cofinite; we show that every cofinite module satisfies . As an application we study the question which local cohomology modules satisfy . There are two situations where the answer is positive. On the other hand, we present two counterexamples, the failure in these two examples coming from different reasons.