The relative separable closure of a valued field in its completion

We consider the completion of a topological field whose topology is defined by the valuations of a dependence class of valuation rings. We characterize the separable closure of the field in its completion through henselization of those valuation rings. Actually, we prove that this relative separable closure is the fixed field of some particular subgroup of continuous automorphisms of the total separable closure. We prove also that this relative separable closure has some properties which are analogous to those of the henselization.This paper is part of the author's doctoral dissertation and was finished during his stay in Konstanz though the CNP_q/GMD program.The results 2.11 and 2.12 were firstly announced in C.R. Acad. Sc. Paris, t.283 (1977).     

On finite intersections of “henselian valued” fields

In this note we study finite intersections of henselian valued fields, i.e. fields carrying either a henselian valuation ring or a henselian absolute value which is (real-) archimedean. To be more precise, we intersect a finite number of henselian valued respectively real closed fields such that the induced valuation rings respectively orderings generate different V-topologies on the intersection, and investigate its algebraic and valuation-theoretic properties.     

Regular closure

Regular closure is an operation performed on submodules of arbitrary modules over a commutative Noetherian ring. The regular closure contains the tight closure when both are defined, but in general, the regular closure is strictly larger. Regular closure is interesting, in part, because it is defined a priori in all characteristics, including mixed characteristic. We show that one can test regular closure in a Noetherian ring by considering only local maps to regular local rings. In certain cases, it is necessary only to consider maps to certain affine algebras. We also prove the equivalence of two variants of regular closure for a class of rings that includes .

     

Going up along absolutely flat morphisms

Flat morphisms from A to B (commutative and unitary rings) such that the multiplication B?_AB→B is flat, have many of the properties of ind-etale morphisms. They don't raise weak dimension. As a consequence they preserve integral closure. In the local case they are the extensions of A that have the same strict henselian extensions as A.     

Some results on the degree of imperfection of complete valued fields

We study the relationship between the degree of imperfection of a filed K of characteristic po and its completion with respect a rank one valuation v. By looking at the topology defined by v in K, a characterization is given of those valued fields (K,v) such that is separable, which in its turn is equivalent to the validity of the so-called fundamental equality, in case v is discrete. Also two applications of Baire category theorem to the theory complete valued fields are given.This work is supported by CNP_q and by contract FINEP/UFC, IF/224–IF/753.     

Genre et genre residuel des corps de fonctions values

Let L be a function field of one variable over a valued field (K,|.|), and (|.|_i), 1is, be distinct absolute values over L extending |.| such that the residue fields ¯Lⁱ are function fields of one variable over the residue field ¯K of (K,|.|). We define the defect of the valued function fields (L,|.|_i)/(K,|.|) and prove an inequality between the genus of L/K and that's of ¯Lⁱ/¯K which takes into account the defect, the ramification index of (L,|.|_i)/(K,|.|) and the constant field of Lⁱ/¯K. Our inequality is better than Mathieu's inequality in discretely valued case.     

The Randomized Integer Convex Hull

Let K R^d be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull I_L(K) = conv (K L), where L is a randomly translated and rotated copy of the integer lattice Z^d. We estimate the expected number of vertices of I_L(K), whose behaviour is similar to the expected number of vertices of the convex hull of Vol K random points in K. In the planar case we also describe the expectation of the missed area Vol (K \ I_L(K)). Surprisingly, for K a polygon, the behaviour in this case is different from the convex hull of random points.     

Sur le défaut de semi-stabilité des courbes elliptiques à réduction additive

Let K be a field of characteristics 0 complete with respect to a discrete valuation v, with a perfect residue field of characteristic p>0. Let be an algebraic closure of K and K_nr its maximal unramified subextension. Let E be an elliptic curve over K with an integral modular invariant. The curve E has potentially good reduction at v, and there exists a smallest extension L of K_nr over which E has good reduction at v. The Galois group, Gal (L/K_nr) is known in the case p≥5. In this paper we give receipts to determine this group in the cases p=2 and p=3.      

Finite separable field extensions with prescribed extensions of valuations. II

Let A¹,...,A^k be pairwise independent valuation rings of K. Prescribing extensions Δ _i ^j . of the value group Γ^j and extensions \(\mathfrak{L}_i^j\) of the residue field \(H^j\) of A^j (i=1,...,r^j) such that \(\sum\limits_{i = 1}^{r^j } {(\Delta _i^j :\Gamma ^j )} \cdot [\mathfrak{L}_i^j :H^j ] = n\) , we provide sufficient conditions for the existence of a separable field extension L of K of degree n with exactly r^j pairwise independent valuation rings B _i ^j lying over A^j, which have Δ _i ^j as value groups and \(\mathfrak{L}_i^j\) as residue fields.     

Separability and the Twisted Frobenius Bimodule

This paper begins with an introduction to -Frobenius structure on a finite-dimensional Hopf subalgebra pair. In Section 2 a study is made of a generalization of Frobenius bimodules and -Frobenius extensions. Also a special type of twisted Frobenius bimodule which gives an endomorphism ring theorem and converse is studied. Section 3 brings together material on separable bimodules, the dual definitions of split and separable extension, and a theorem of Sugano on endomorphism rings of separable bimodules. In Section 4, separable twisted Frobenius bimodules are characterized in terms of data that generalizes a Frobenius homomorphism and a dual base. In the style of duality, two corollaries characterizing split -Frobenius and separable -Frobenius extensions are proven. Sugano"s theorem is extended to -Frobenius extensions and their endomorphism rings. In Section 5, the problem of when separable extensions are Frobenius extensions is discussed. A Hopf algebra example and a matrix example are given of finite rank free separable -Frobenius extensions which are not Frobenius in the ordinary sense.     

Models of Curves and Finite Covers

Let K be a discrete valuation field with ring of integers O _K .Letf : X ! Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over O K of X and of Y. Three such relationships are listed below. Consider a Galois cover f : X ! Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K,thenX achieves semi-stable reduction over some explicit tame extension of K.B/.WhenK is strictly henselian, we determine the minimal extension L=K with the property that X L has semi-stable reduction. Let f : X ! Y be a finite morphism, with g.Y/ > 2. We show that if X has a stable model X over O _K ,thenY has a stable model Y over O _K , and the morphism f extends to a morphism X ! Y. ! Y. Finally, given any finite morphism f : X ! Y, is it possible to choose suitable regular models X and Y of X and Y over O _K such that f extends to a finite morphism X ! Y ?As wasshown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situ-ations, with f a cyclic cover of any order > 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.     

Smooth points of certain operator spaces

Let B(H) denote the Banach space of all bounded linear operators on a separable, infinite dimensional Hilbert space H, and let C be the ideal of compact operators on H. It is shown that the unit ball of the Calkin algebra B(H)/C has no smooth points. We use this to determine the smooth points of the unit ball of B(H).     

A canonical form for self-adjoint pencils in Hilbert space

We characterize those pencils P=A–B of operators on a separable Hilbert space H for which a linear homeomorphism D of H exists satisfying the following: (i) H decomposes into a direct sum F+G, orthogonal in a (perhaps indefinite) inner product induced by A, where F is finite dimensional (ii) D^*PD=P_F(I_G–C) where P_F is a (congruence) canonical form for the general self-adjoint pencil on F, and C is a bounded self-adjoint operator on G. For a given P, explicit constructions are given for C, D, F, G and P_F.     

Topology of Definable Hausdorff Limits

Let A ^n+r be a set definable in an o-minimal expansion S of the real field, let A ^r be its projection, and assume that the non-empty fibers A_a ⁿ are compact for all a A and uniformly bounded, i.e. all fibers are contained in a ball of fixed radius B(0,R). If L is the Hausdorff limit of a sequence of fibers A_ai, we give an upper-bound for the Betti numbers b_k(L) in terms of definable sets explicitly constructed from a fiber A_a. In particular, this allows us to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the relative closure to construct the o-minimal structure S_Pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)₀ in the special case where Y=.     

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula AE, B(x)0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) H₁ according to the formula: L₀y=–y + Q(x)y (–0 is a self-adjoint operator in H₁ under the given assumptions.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 65–72, July, 1969.     

Über kompakte Operatoren in lokalkonvexen Räumen

Let E, F be (B)-spaces and let E have the approximation property. Then the closure of FE in L_b (F,E) is identical with the space K(F,E) of compact maps in L(F,E). The following generalization of this well known statement is proved: Let E, F be complete, locally convex spaces and let E have the approximation property. Then, where denotes a 0-neighborhood base in F and refers to the completion of the tensor product with respect to the topology of bi-equicontinuous convergence.The last part is concerned with the adjoint map of a compact operator.     

Topological K-groups of two-dimensional local fields

A complete two-dimensional local field K of mixed characteristic with finite second residue field is considered. The existence of a completely ramified extension L of K such that L is a standard field is assumed. It is proved that the rank of the quotient U(1)K ₂ ^top K/T_K, where T_K is the closure of the torsion subgroup, is equal to the degree of the constant subfield of K over ℚ_p. I. B. Zhukov constructed a set of generators of this quotient in the case where K is a standard field. In this paper, two natural generalizations of this set are considered, and it is proved that one of them generates the entire group and the other generates its subgroup of finite index. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 206–221.     

<Emphasis Type="Italic">ℓ</Emphasis>-Adic class field theory for regular local rings

In this paper, we prove the ℓ-adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is an ℓ-adic variant of a result of Matsumi (Class field theory for , preprint, 2002).