Injective modules and linear growth of primary decompositions

The purposes of this paper are to generalize, and to provide a short proof of, I. Swanson's Theorem that each proper ideal in a commutative Noetherian ring has linear growth of primary decompositions, that is, there exists a positive integer such that, for every positive integer , there exists a minimal primary decomposition with for all . The generalization involves a finitely generated -module and several ideals; the short proof is based on the theory of injective -modules.

     

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 Burch's inequality and a reduction system of a filtration

Let be a multiplicative filtration of a local ring such that the Rees algebra is Noetherian. We recall Burch's inequality for and give an upper bound of the a-invariant of the associated graded ring using a reduction system of . Applying those results, we study the symbolic Rees algebra of certain ideals of dimension .

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

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.

     

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 .

     

Hilbert coefficients and the associated graded rings

Let be a -dimensional Cohen-Macaulay local ring with infinite residue field. Let be an -primary ideal of . In this paper, we prove that if for some minimal reduction of , then depth .

     

Castelnuovo-Mumford regularity of simplicial semigroup rings with isolated singularity

Let be the polynomial ring in variables over a field and its graded maximal ideal. Let be homogeneous polynomials of degree generating an -primary ideal, and let be arbitrary homogeneous polynomials of degree . In the present paper it will be proved that the Castelnuovo-Mumford regularity of the standard graded -algebra is at most . By virtue of this result, it follows that the regularity of a simplicial semigroup ring with isolated singularity is at most , where is the multiplicity of and is the codimension of .

     

Hilbert-Samuel functions of modules over Cohen-Macaulay rings

For a finitely generated, non-free module over a CM local ring , it is proved that for the length of is given by a polynomial of degree . The vanishing of is studied, with a view towards answering the question: If there exists a finitely generated -module with such that the projective dimension or the injective dimension of is finite, then is regular? Upper bounds are provided for beyond which the question has an affirmative answer.

     

On polynomial products in nilpotent and solvable Lie groups

We are dealing with Lie groups which are diffeomorphic to , for some . After identifying with , the multiplication on can be seen as a map . We show that if is a polynomial map in one of the two (sets of) variables or , then is solvable. Moreover, if one knows that is polynomial in one of the variables, the group is nilpotent if and only if is polynomial in both its variables.

     

Invariants of semisimple Lie algebras acting on associative algebras

If is a Lie algebra of derivations of an associative algebra , then the subalgebra of invariants is the set In this paper, we study the relationship between the structure of and the structure of , where is a finite dimensional semisimple Lie algebra over a field of characteristic zero acting finitely on , when is semiprime.

     

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.

     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

The measure of holomorphicness of a real submanifold of an almost Hermitian manifold

In this note we define the measure of holomorphicness of a compact real submanifold of an almost Hermitian manifold . The number verifies the following properties: is a complex submanifold iff ; if is odd, then . Explicit examples of surfaces in are obtained, showing that and that , being the Clifford torus.

     

Multi-dimensional versions of a theorem of Fine and Wilf and a formula of Sylvester

Let be vectors in which generate . We show that a body with the vectors as edge vectors is an almost minimal set with the property that every function with periods is constant. For the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma.

Suppose is not a non-trivial linear combination of with non-negative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of over , but infinitely many points on each of its hyperfaces are not. For the result reduces to a formula of Sylvester corresponding to Frobenius' Coin-changing Problem in the case of coins of two denominations.

     

Criteria for irrationality of Euler's constant

By modifying Beukers' proof of Apéry's theorem that is irrational, we derive criteria for irrationality of Euler's constant, . For 0$">, we define a double integral and a positive integer , and prove that with the following are equivalent:

1. The fractional part of is given by for some .

2. The formula holds for all sufficiently large .

3. Euler's constant is a rational number.

A corollary is that if infinitely often, then is irrational. Indeed, if the inequality holds for a given (we present numerical evidence for and is rational, then its denominator does not divide . We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact . A by-product is a rapidly converging asymptotic formula for , used by P. Sebah to compute correct to 18063 decimals.

     

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.

     

Constraints for the normality of monomial subrings and birationality

Let be a field and let be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra and the subring . If the monomials in have the same degree, one of the consequences is a criterion for the -rational map defined by to be birational onto its image.

     

Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

Degenerating families of rank two bundles

We construct families of rank two bundles on , in characteristic two, where for , is a sum of line bundles, and is non-split. We construct families of rank two bundles on , in characteristic , where for , is a sum of line bundles, and is non-split.