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.     

About polytopes of valuations on finite distributive lattices

Let L be a finite distributive lattice and V(L) the real vector space of all valuations on L. We verify the conjecture of Geissinger that the extreme points of the convex polytope M(L)={v L : 0 v 1} are precisely the 0–1 valuations.     

Congruence-preserving extensions of finite lattices to sectionally complemented lattices

In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving extension to a finite sectionally complemented lattice.

     

Prolongations to higher jets of Estabrook-Wahlquist coverings for PDE's

Using a definition of the (infinite-dimensional) Lie algebras introduced by Estabrook and Wahlquist (EW) that lives directly on the jet bundle, we establish explicit methods to determine prolongations of these algebras to higher jet spaces, and calculate these prolongations for Burgers' equation and the KdV equation, to arbitrary level on the infinite jet bundle. Each level of prolongation introduces additional generator(s) of the algebra. For Burgers' equation, we first find the Kac-Moody algebra that is the (universal) completion of that algebra on the lowest-order jet space. We then show how each new generator creates a free flow of the lower-order algebra over the jet bundle, without requiring any new commutation relations. For the KdV equation, one new generator creates a flow, but the other introduces into the structure of the prolonged total derivative operators the characteristics for the hierarchy of conserved quantities of the PDE, thereby mixing together these nonlocal symmetries with the local ones in a way that may be valuable. In the limit to the infinite jet bundle, we acquire the EW algebra for the entire KdV hierarchy of PDE's, moving toward a comparison of this approach with the work of the Japanese school.     

Left Kan extensions preserving finite products

We give conditions that ensure the preservation of finite products by left Kan extensions into cocomplete subcanonical sites. The conditions involve a suitable notion of flatness (interpreted in the internal logic of the site) of the functor that is extended and a good behavior of colimits used in the calculation of the left Kan extension. When the recipient category is a Grothendieck topos the good behavior of the colimits is granted and the flatness conditions turn out to be necessary as well as sufficient. We also show that the category of compactly generated Hausdorff spaces and that of small categories are equipped with suitable topologies that account for the well-known phenomenon of preservation of finite products by the geometric and categorical realization, respectively, of simplicial sets.     

Constant-to-one extensions of shifts of finite type

Any transitive shift of finite type has a transitive constant-to-one extension which is not of finite type.

     

Isomorphic trivial extensions of finite dimensional algebras

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. Let T(Λ)=Λ?D(Λ) be the trivial extension of Λ by its minimal injective cogenerator D(Λ). We characterize, in terms of quivers and relations, the algebras Λ^′ such that T(Λ)?T(Λ^′).     

On homogeneous extensions of finite predicate systems

It is shown that any finite n-element predicate system for which any two one-element subsystems are isomorphic is embeddable in a finite (2 ⁿ − 1)-element system having a transitive automorphism group.     

Metrics with finite sets of primitive extensions

In this paper, we give a complete characterization for the class of rational finite metrics with the property that the set () of primitive extensions of is finite. Here, for a metric on a setT, a positive extensionm of to a setV T is calledprimitive if none of the convex combinations of other extensions of toV is less than or equal tom. Our main theorem asserts that the following the properties are equivalent: (i) () is finite; (ii) Up to an integer factor, is a submetric of the path metric d ^H of a graphH with |(d ^H )=1; (iii) A certain bipartite graph associated with contains neither isometrick-cycles withk6 nor induced subgraphsK _3,3 ^– . We then show that () is finite if and only if the dimension of the tight span of is at most two. We also present other results, discuss applications to multicommodity flows, and raise open problems.This research was supported by grant 97-01-00115 from the Russian Foundation of Basic Research and a grant from the Sonderforschungsbereich 343, Bielefeld Universität, Bielefeld, Germany.     

Evaluation codes defined by finite families of plane valuations at infinity

We construct evaluation codes given by weight functions defined over polynomial rings in m ≥ 2 indeterminates. These weight functions are determined by sets of m?1 weight functions over polynomial rings in two indeterminates defined by plane valuations at infinity. Well-suited families in totally ordered commutative groups are an important tool in our procedure.     

Essential finite generation of extensions of valuation rings

Given a generically finite local extension of valuation rings $V\subset W$, the question of whether W is the localization of a finitely generated V-algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when W is essentially of finite type over V in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic 0 using embedded resolution of singularities. In this paper, we prove Knaf's conjecture in full generality.