Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.      

Flocks of Infinite Hyperbolic Quadrics

Let K be a field containing a nonsquare and a quadratic extension. Let denote the unique involutory automorphism of F fixing K pointwise. For every field K such that the nonzero squares of K do not form an index 1 or 2 subgroup of , a construction is given which produces large numbers of infinite nearfield and non nearfield flocks of an infinite hyperbolic quadric in PG(3,K).     

On closures of cycle spaces of flag domains

Open orbits D of noncompact real forms G ₀ acting on flag manifolds Z = G/Q of their semisimple complexifications G are considered. Given D and a maximal compact subgroup K ₀ of G ₀, there is a unique complex K ₀–orbit in D which is regarded as a point in the space of q-dimensional cycles in D. The group theoretical cycle space is defined to be the connected component containing C ₀ of the intersection of the G–orbit G(C ₀) with . The main result of the present article is that is closed in . This follows from an analysis of the closure of the universal domain in any G-equivariant compactification of the affine symmetric space G/K, where K is the complexification of K ₀ in G.     

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

Structure of Degenerate Block Algebras

Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map : A A F, we define a Lie algebra = (A, ) over F with basis {e_x | x A/{0}} and Lie product [e_x,e_y] = (x,y)e_x+y. We show that is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der of is a complete Lie algebra. We describe the double extension D( , T) of by T, where T is spanned by the locally finite derivations of , and determine the second cohomology group H²(D( , T),F) using anti-derivations related to the form on D( , T). Finally, we compute the second Leibniz cohomology groups HL²( , F) and HL²(D( , T), F).2000 Mathematics Subject Classification: 17B05, 17B30This work was supported by the NNSF of China (19971044), the Doctoral Programme Foundation of Institution of Higher Education (97005511), and the Foundation of Jiangsu Educational Committee.     

The eigenvalue field is a splitting field

Let K be an algebraically closed field of arbitrary characteristic and F < K a subfield. If is an irreducible semigroup of matrices such that the spectra of all the elements of are contained in F, then is conjugate to a subsemigroup of M _n (F). Research supported in part by the Ministry of Higher Education, Science, and Technology of Slovenia. Received: 6 April 2006     

Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p ∈ S, let be the compositum of all extensions unramified outside S of the form , for . If , let be the intersection of the fixed fields by , for i = 1, . . , n. We provide a wide family of elliptic curves such that the rank of is infinite for all n ≥ 0 and all , subject to the parity conjecture. Similarly, let be a polarized abelian variety, let K be a quadratic number field fixed by , let S be an infinite set of primes of and let be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over . We show that, under certain hypotheses, the -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in . As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.     

Even dimensional higher class groups of orders

Let F be a number field, and let denote the ring of integers in F. Let A be a finite-dimensional central simple F-algebra, and let Λ be an -order in A. In this paper it is shown that the p-torsion of the even dimensional higher class group Cl _2n (Λ) can only occur for primes p, which lie under prime ideals , at which is not maximal, or which divide the dimension of A. Supported by NSFC 10401014, and partially funded by Irish Research Council for Science, Engineering and Technology Basic Research Grant SC/02/265.     

The base components of the dualizing sheaf of a curve on a surface

This note studies the structure of the divisorial fixed part of for a 1-connected curve D on a smooth surface S. It is shown that if the divisorial fixed part F of is non empty then it has arithmetic genus ≤ 0 and each component of F is a smooth rational curve. The structure of curves D, with non empty divisorial fixed part F for , is also described. Received: 16 August 2007     

The frattini subgroup of the absolute Galois group of a local field

LetK be a local field,T the maximal tamely ramified extension ofK, F the fixed field inK _sof the Frattini subgroup ofG(K), andJ the compositum of all minimal Galois extensions ofK containingT. The main result of the paper is thatF=J. IfK is a global field andK _solv is the maximal prosolvable extension ofK, then the Frattini group of % MathType!End!2!1!(K _solv/K) is trivial. Partially supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

K - Theory of the Weil Transfer FunctorDedicated to Daniel Quillen

We consider motivic category of pairs (smooth projective scheme, separable algebra) over a given field. For any separable extension L/Fwe construct the Weil transfer functor . This enables us to compute K-groups of schemes obtained by means of Weil restriction.     

Pólya and Newtonian function fields

Let be a finite function field extension and denote by O _K the integral closure of in K. In this article, we are interested in Pólya fields, that is, fields K, such that the O _K -module Int(O _K ) of integer-valued polynomials over O _K admits a regular basis. We show that the cyclotomic extensions of are Pólya fields, and we characterize some totally imaginary extensions which are Pólya fields. Then, we are interested in Pólya fields K which have a regular basis of the form for some sequences of elements of O _K . For totally imaginary extensions, we show that it is the case if and only if O _K is isomorphic to . This gives a answer to a question raised by Thakur. The author thanks his thesis adviser Jean-Luc Chabert, and Mireille Car for their help, and their valuable advices to do this work. The author thanks also the referee for his valuable remarks.     

Parametrization of tamely ramified maximal tori using bounded subgroups

Let be a reductive group defined over a local complete field F with discrete valuation, and split over some unramified extension of F, and let G be its group of F-points. In this paper, we define a class of abelian “torus-like” subgroups in nonreductive groups, called pseudo-tori, which generalizes the notion of torus, and we establish a correspondence between conjugacy classes of tamely ramified maximal tori of G and association classes of maximal pseudo-tori of the quotients of parahorics of G by their second congruence subgroup, viewed as groups of k-points of algebraic groups defined over the residual field k of F.      

Galois modular representation of associated Jacobians in the tamely ramified cyclic case

Let L/K be an ℓ-cyclic extension with Galois group G of algebraic function fields over an algebraically closed field k of characteristic p ≠  ℓ. In this paper, the -module structure of the ℓ-torsion of the Jacobian associated to L is explicitly determined.     

Algebras whose right nucleus is a central simple algebra

We generalize Amitsur's construction of central simple algebras over a field F which are split by field extensions possessing a derivation with field of constants F to nonassociative algebras: for every central division algebra D over a field F of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is D and whose left and middle nucleus are a field extension K of F splitting D, where F is algebraically closed in K.We then give a short direct proof that every p-algebra of degree m, which has a purely inseparable splitting field K of degree m and exponent one, is a differential extension of K and cyclic. We obtain finite-dimensional division algebras over a field F of characteristic $p>0$ whose right nucleus is a division p-algebra.     

Certain invariants of extension algebras of C({\mathbb{T}}^{2})

In this paper, we compute certain invariants of extension algebras of the torus algebra by , where is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space H. These extension algebras are also constructed up to isomorphism. Received: 5 July 2007, Revised: 14 February 2008     

Cone theorem via Deligne–Mumford stacks

We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne–Mumford stacks. We also obtain a sharper bound on −(K _X + D)-degree of (K _X + D)-negative extremal rays for projective -factorial log terminal threefold pairs (X, D).     

Division by a holomorphic matrix in strictly pseudoconvex Domains

LetF andG be respectively a vector- and a matrix-function in a bounded strictly pseudoconvex domainD, with entries holomorphic inD and continuous in . We prove that ifF can be divided locally byG with holomorphic factors in a neighborhood of a given pointw inD, and the rank ofG is maximal at all points of , then the division ofF byG holds globally, with some factors which are holomorphic inD and continuous in . This method applies also to other function algebras in pseudoconvex domains.     

Positive elimination in valued fields

Let K be an algebraically closed field with a valuation ring or a real closed field with a convex valuation ring . We show that the projection of a basic (see “Introduction”) subset of to K ⁿ is again basic.