Associated primes and cofiniteness of local cohomology modules

Let be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that is finite R-module. If s is the first integer such that the local cohomology module is non -cofinite, then we show that is finite. In particular, the set of associated primes of is finite. Let be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module is not -cofinite and show that n just depends on the support of M.The research of the first author was supported in part by a grant from IPM (No. 83130114).The second author was supported by a grant from University of Tehran (No. 6103023/1/01).     

Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group

We study the semilinear equationwhere is the Heisenberg Laplacian and is the Heisenberg group. The function f C²(×, ) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S₁²(), which is the Heisenberg analogue of the Sobolev space W^1,2(^N).Mathematics Subject Classification (2000): 22E30, 22E27     

Interior regularity criteria in weak spaces for the Navier-Stokes equations

We study the interior regularity of weak solutions of the incompressible Navier-Stokes equations in ×(0,T), where and 0<T<. The local boundedness of a weak solution u is proved under the assumption that is sufficiently small for some (r,s) with and 3r<. Our result extends the well-known criteria of Serrin (1962), Struwe (1988) and Takahashi (1990) to the weak space-time spaces.Mathematics Subject Classification (2000): 35Q30, 76N10     

The Rees algebra of a positive normal affine semigroup ring

Let R be a positive normal affine semigroup ring of dimension d and let be the maximal homogeneous ideal of R. We show that the integral closure of is equal to for all n ∈ℕ with n ≥ d − 2. From this we derive that the Rees algebra R[t] is normal in case that d ≤ 3. If emb dim(R) = d + 1, we can give a necessary and sufficient condition for R[t] to be normal.     

L p -Estimates for the Navier–Stokes Equations for Steady Compressible Flow

We consider the Navier–Stokes equations for compressible isentropic flow in the steady three-dimensional case. The pressure and the kinetic energy are estimated uniformly in L^q with being the density. This is an improvement of known estimates in the case Mathematics Subject Classification (2000): 35Q30, 76N10     

Eilenberg-Moore spectral sequence calculation of function space cohomology

The purpose of this article is to introduce an Eilenberg-Moore spectral sequence converging to the cohomology algebra of a function space with an adjunction space as its source. Computability of the spectral sequence is shown by determining explicitly the mod p cohomology algebra of the function space of maps from a non-orientable surface S to the classifying space of a simply-connected Lie group G whose homology is p-torsion free. Let M be a closed orientable 3-dimensional manifold. Applying the spectral sequence obtained from a Heegaard splitting of the manifold M, we also prove that is a direct summand of .Mathematics Subject Classification (2000):55T20, 57T35This research was partially supported by a Grant-in-Aid for Scientific Research (C)14540095 from Japan Society for the Promotion of Science.     

On the rank of abelian varieties over function fields

Let be a smooth projective curve defined over a number field k, A/k() an abelian variety and (τ, B) the k()/k-trace of A. We estimate how the rank of A(k())/τB(k) varies when we take a finite geometrically abelian cover defined over k. This work was partially supported by CNPq research grant 304424/2003-0, Pronex 41.96.0830.00 and CNPq Edital Universal 470099/2003-8. I would like to thank Douglas Ulmer for comments on how to treat the case of arbitrary ramification, but the conductor prime to the ramification locus, in the case of elliptic fibrations. I would also like to thank Marc Hindry for comments on the inequality comparing the conductors of A and A'. Finally, I also thank the referee for his comments and criticisms.     

On the commutativity of <Emphasis Type="Italic">C</Emphasis>* -algebras

Let be a C* -algebra. Let f be a non-constant complex-valued continuous function defined on a closed interval I. We shall show that f densely spans As an application, is commutative if f(x)f(y)=f(y)f(x) for all self-adjoint elements x and y in with spectrums contained in I.Mathematics Subject Classification (1991):Primary 46L05     

Invertible modules for commutative -algebras with residue fields

The aim of this note is to understand under which conditions invertible modules over a commutative -algebra in the sense of Elmendorf, Kriz, Mandell & May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative -algebra R has coherent localizations for every maximal ideal , then for every invertible R-module U, U_*=π_*U is an invertible graded R_*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative -algebra has ‘residue fields’ for all maximal ideals if the global dimension of R_* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R₀. We apply these results to finite abelian Galois extensions of Lubin-Tate spectra.     

Diophantine inequalities involving several power sums

Let denote the ring of power sums, i.e. complex functions of the form for some and _iA, where is a multiplicative semigroup. Moreover, let We consider Diophantine inequalities of the form where >1 is a quantity depending on the dominant roots of the power sums appearing as coefficients in F(n,y), and show that all its solutions have y parametrized by some power sums from a finite set. This is a continuation of the work of Corvaja and Zannier [4–6] and of the authors [10, 18] on such problems.Mathematics Subject Classification (2000):11D45,11D61Revised version: 6 May 2004     

Ordinary elliptic curves of high rank over with constant <Emphasis Type="Italic">j</Emphasis>-invariant

We show that under the assumption of Artins Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over with arbitrarily high rank and constant j-invariant. For odd primes p, this result follows from a theorem we prove which states that whenever p is a generator of (/)*/–1 ( an odd prime) there exists a hyperelliptic curve over whose Jacobian is isogenous to a power of one ordinary elliptic curve.Mathematics Subject Classification (2000): Primary: 11G05; Secondary: 11G20, 14H40, 14H52Revised version: 3 February 2004Acknowledgements. The research of Jasper Scholten is funded by the AREHCC project of the European Commission (Fifth Framework Program IST - 2001).     

Tannaka duality on quotient stacks

Let = [X/G] be the quotient stack of a scheme X by an affine group scheme G over a field k. Assume that there is a line bundle on whose underlying line bundle on X is very ample. Let VB() be the category of vector bundles on .We show that is canonically isomorphic to the stack of fiber functors on VB(). This is an analogue of the Tannaka duality for affine groups. Partially supported by CNCSIS contract no. 33079/2004     

q-Skew Derivations and Polynomial Identities

Let R be a prime ring and its left Martindale quotient ring. Assume that a q-skew -derivation of R satisfies the identity relationfor all x R, where the subring of constants of on R. It is proved that if R⁽⁾ satisfies nontrivial polynomial identities, then so does R. This answers affirmatively a problem raised in Bergen and Grzeszczuk [2] by removing the assumption on the algebraicity of .Mathematics Subject Classification (2000): 16W20, 16W25, 16W55Members of Mathematics Division, National Center for Theoretical Sciences at Taipei.Acknowledgement The authors are thankful to the referee for her/his useful suggestions and comments. This research was supported by the National Science Council of Taiwan.     

A class of <Emphasis Type="Italic">p</Emphasis>-adic Galois representations arising from abelian varieties over

Let V be a p-adic representation of the absolute Galois group G of that becomes crystalline over a finite tame extension, and assume p2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module V_p() of an abelian variety defined over . These conditions are stated on the filtered (,G)-module attached to V.Mathematics Subject Classification (2000): 14F30, 11G10, 11F80, 14G20, 14F20     

On the logarithmic plurigenera of complements of plane curves

Let B be a (not necessarily irreducible) plane curve in ². In the present article, we prove that if and only if Moreover, we determine the curve B when and Mathematics Subject Classification (2000): 14R05, 14H50, 14J26     

Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz functions

We describe the possible restrictions of the cotangent bundle to an elliptic curve . We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R₊-primary ideal in the graded section ring .     

Conjugacy classes of affine automorphisms of and linear automorphisms of ℙ n in the Cremona groups

We describe the conjugacy classes of affine automorphisms in the group Aut(n,) (respectively Bir()) of automorphisms (respectively of birational maps) of . From this we deduce also the classification of conjugacy classes of automorphisms of ℙⁿ in the Cremona group Bir().     

On the exact Hausdorff dimension of the set of Liouville numbers

Let denote the set of Liouville numbers. For a dimension function h, we write () for the h-dimensional Hausdorff measure of . In this paper we locate the exact cut-point at which the Hausdorff measure of drops from infinity to zero. Namely, if h is a dimension function that increases faster than any power function near 0, then ()=, and if h is a dimension function that increases slower than some power function near 0, then ()=0. This answers a question asked by R. D. Mauldin.Mathematics Subject Classification (2000): 28A80     

Estimates for stable hypersurfaces of prescribed F-mean curvature

We consider immersed hypersurfaces :Mⁿ→ℝⁿ⁺¹ with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C³-close to the area integrand.     

Inducing W-graphs II

Let be the Hecke algebra associated with a Coxeter group W, and the Hecke algebra associated with W_J, a parabolic subgroup of W. In [5] an algorithm was described for the construction of a W-graph for an induced module where V is an -module derived from a W_J-graph. This note is a continuation of [5], and involves the following results:[] inducing ordered and bipartite W-graphs;[] the relationship between the cell decomposition of a W_J-graph and the cell decomposition of the corresponding induced W-graph;[] a Mackey-type formula for the restriction of an induced W-graph;[] a formula relating the polynomials used in the construction of induced W-graphs to Kazhdan-Lusztig polynomials.The result on cells is a version of a Theorem of M. Geck [4], dealing with cells in W (allowing unequal parameters).Mathematics Subject Classification (2000): Primary 20C