On the cofiniteness of local cohomology modules

In this note we show that if is an ideal of a Noetherian ring and is a finitely generated -module, then for any minimax submodule of the -module is finitely generated, whenever the modules are minimax. As a consequence, it follows that the associated primes of are finite. This generalizes the main result of Brodmann and Lashgari (2000).

     

Radical and cyclotomic extensions of the rational numbers

A radical extension of the rational numbers is a field generated by an element having a power in , and a cyclotomic extension is an extension generated by a root of unity. We show that a radical extension that is almost Galois over is almost cyclotomic. More precisely, we prove that if is radical with Galois closure , then contains a cyclotomic field such that the degree is bounded above by an almost linear function of . In particular, if is Galois, it contains a cyclotomic field such that .

     

Artinianess of graded local cohomology modules

Let be a Noetherian homogeneous ring with local base ring and let be a finitely generated graded -module. Let be the largest integer such that is not Artinian. We will prove that are Artinian for all and there exists a polynomial of degree less than such that for all . Let be the first integer such that the local cohomology module is not cofinite. We will show that for all the graded module is Artinian.

     

Basic characters of the unitriangular group (for arbitrary primes)

Let denote the (upper) unitriangular group of degree over the finite field with elements. In this paper we consider the basic (complex) characters of and we prove that every irreducible (complex) character of is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption , where is the characteristic of the field .

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space

For an analytically infinite Riemann surface , the quasiconformal mapping class group always acts faithfully on the ordinary Teichmüller space . However in this paper, an example of is constructed for which acts trivially on its asymptotic Teichmüller space .

     

Stable rank of corner rings

B. Blackadar recently proved that any full corner in a unital C*-algebra has K-theoretic stable rank greater than or equal to the stable rank of . (Here is a projection in , and fullness means that .) This result is extended to arbitrary (unital) rings in the present paper: If is a full idempotent in , then . The proofs rely partly on algebraic analogs of Blackadar's methods and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners . The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if where is a finitely generated projective generator, and can be generated by elements, then .

     

Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Fix integers such that and , and let be the set of all integral, projective and nondegenerate curves of degree in the projective space , such that, for all , does not lie on any integral, projective and nondegenerate variety of dimension and degree . We say that a curve satisfies the flag condition if belongs to . Define where denotes the arithmetic genus of . In the present paper, under the hypothesis , we prove that a curve satisfying the flag condition and of maximal arithmetic genus must lie on a unique flag such as , where, for any , denotes an integral projective subvariety of of degree and dimension , such that its general linear curve section satisfies the flag condition and has maximal arithmetic genus . This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.

     

Decomposable form equations without the finiteness property

Let be a finitely generated (but not necessarily algebraic) extension field of . Let be a form (homogeneous polynomial) in variables with coefficients in , and suppose that is decomposable (i.e., that it factorizes into linear factors over some finite extension of ). We say that has the finiteness property over if for every (here denotes the set of non-zero elements in ) and for every subring of which is finitely generated over , the equation

has only finitely many solutions. This paper proves the following result: Let be a decomposable form in variables with coefficients in , which factorizes into linear factors over . Let denote a maximal set of pairwise linearly independent linear factors of . If has the finiteness property over , then 2(m-1)$">.

     

A simple closure condition for the normal cone intersection formula

In this paper it is shown that if and are two closed convex subsets of a Banach space and , then whenever the convex cone, , is weak* closed, where and are the support function and the normal cone of the set respectively. This closure condition is shown to be weaker than the standard interior-point-like conditions and the bounded linear regularity condition.

     

Fractalized cyclotomic polynomials

For each prime power , we realize the classical cyclotomic polynomial as one of a collection of different polynomials in . We show that the new polynomials are similar to in many ways, including that their discriminants all have the form . We show also that the new polynomials are more complicated than in other ways, including that their complex roots are generally fractal in appearance.

     

Revisiting two theorems of Curto and Fialkow on moment matrices

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence whose moment matrix is positive semidefinite and has finite rank is the sequence of moments of an -atomic nonnegative measure on . We give an alternative proof for this result, using algebraic tools (the Nullstellensatz) in place of the functional analytic tools used in the original proof of Curto and Fialkow. An easy observation is the existence of interpolation polynomials at the atoms of the measure having degree at most if the principal submatrix of (indexed by all monomials of degree ) has full rank . This observation enables us to shortcut the proof of the following result. Consider a basic closed semialgebraic set , where and . If is positive semidefinite and has a flat extension such that all localizing matrices are positive semidefinite, then has an atomic representing measure supported by . We also review an application of this result to the problem of minimizing a polynomial over the set .

     

Free resolutions of parameter ideals for some rings with finite local cohomology

Let be a -dimensional local ring, with maximal ideal , containing a field and let be a system of parameters for . If and the local cohomology module is finitely generated, then there exists an integer such that the modules have the same Betti numbers, for all .

     

On representable linearly compact modules

For a flat module we prove that is a functor from the category of linearly compact modules to itself and is exact. Moreover, is representable when is linearly compact and representable. This gives an affirmative answer to a question of L. Melkersson (1995) for linearly compact modules without the condition of finite Goldie dimension. The set of attached prime ideals of the co-localization of a linearly compact representable module with respect to a multiplicative set in is described.

     

On the topology of the boundary of a basin of attraction

Suppose is a continuous flow on a locally compact metrizable space and is an (asymptotically stable) attractor. Let be the boundary of the basin of attraction of . In the present paper it will be shown how the Conley index of plays an important role in determining the topological nature of and allows one to obtain information about the global dynamics of in .

     

Metric geodesics of isometries in a Hilbert space and the extension problem

We consider the problem of finding short smooth curves of isometries in a Hilbert space . The length of a smooth curve , , is measured by means of , where denotes the usual norm of operators. The initial value problem is solved: for any isometry and each tangent vector at (which is an operator of the form with ) with norm less than or equal to , there exist curves of the form , with initial velocity , which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operator

find all possible extending to all , with . We also consider the problem of finding metric geodesics joining two given isometries and . It is well known that if there exists a continuous path joining and , then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining and .

     

A ``deformation estimate" for the Toeplitz operators on harmonic Bergman spaces

Let denote the open unit ball in for and the Lebesgue volume measure on . For , the (weighted) harmonic Bergman space is the space of all harmonic functions which are in . For , the Toeplitz operator is defined on by , where is the orthogonal projection of onto . In this note, we prove that for radial, .

     

Lipscomb's space

It is known that Lipscomb's space can be imbedded in Hilbert's space . Let be the imbedded version of endowed with the -induced topology. We show how to construct as the attractor of an iterated function system containing an infinite number of affine transformations of . In this way we answer an open question of J.C. Perry.

     

The Banach algebra induced by a double centralizer

Given a Banach algebra , R. Larsen defined, in his book ``An introduction to the theory of multipliers", a Banach algebra by means of a multiplier on , and essentially used it in the case of a commutative semisimple Banach algebra to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra induced by a bounded double centralizer of a Banach algebra . Then, our main concern is devoted to the relationships between , , and the algebras of bounded double centralizers and of and , respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.

     

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 ).