Related papers: Local Uniformization of Abhyankar Valuations

We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1, then…

We describe integral lifts K(L), indexed by local fields L of degree n = [L:\Q_p], of the extraordinary cohomology theories K(n), and apply the generalized character theory of Hopkins, Kuhn and Ravenel to identify K(L)(BG) \otimes \Q$, for…

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…

Let $k$ be a field with a real valuation $\nu$ and $R$ a $k$-algebra. We show that there exist a $k$-algebra $K$ and a real valuation $\mu$ on $K$ extending $\nu$ such that any real ring valuation of $R$ is induced by $\mu$ via some…

Gr\"obner bases have been generalized by replacing monomial orders with constructions such as valuations and filtrations. We consider suitable valuations on a rational valuation field $K(x,y)$ and analyze their behavior when restricting to…

We work with rational rank 1 valuations centered in regular local rings. Given an algebraic function field $K$ of transcendence degree 3 over $k$, a regular local ring $R$ with $QF(R)=K$ and a $k$-valuation $\nu$ of $K$, we provide an…

K. Kato's conjecture about the cohomological Hasse principle for regular connected schemes $\mathfrak X$ which are flat and proper over the complete discrete valuation rings $\mathcal O_N$ of higher local fields $F_N$ is proven. This…

A well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring are linearly comparable to each other. In the present paper we…

Building on work of Kuhlmann and Lisinski, we study the theory of the Hahn series field $\mathbb{F}_{q}(\!(\mathbb{Q})\!)$, over a finite field $\mathbb{F}_{q}$, equipped with the $t$-adic valuation, in a language of valued fields. We prove…

We provide a new proof of Alesker's Irreducibility Theorem. We first introduce a new localization technique for polynomial valuations on convex bodies, which we use to independently prove that smooth and translation invariant valuations are…

Suppose that (K, $\nu$) is a valued field, f (z) $\in$ K[z] is a unitary and irreducible polynomial and (L, $\omega$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K…

Suppose that $(K,v_0)$ is a valued field, $f(x)\in K[x]$ is a monic and irreducible polynomial and $(L,v)$ is an extension of valued fields, where $L=K[x]/(f(x))$. Let $A$ be a local domain with quotient field $K$ dominated by the valuation…

This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In…

For all simple and finite extension of a valued field, we prove that its defect is the product of the effective degrees of the complete set of key polynomials associated. As a consequence, we obtain a local uniformization theorem for…

We study the algebraic $K$-theory of the ring of continuous functions on a compact Hausdorff space with values in a local division ring, e.g., a local field: We compute its negative $K$-theory and show its $K$-regularity. The complex case…

The main result of this paper is that in order to prove the local uniformization theorem for local rings it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local rings for which we want…

In this note, we show that the algebraic K-theory of generalized archimedean valuation rings occurring in Durov's compactification of the spectrum of a number ring is given by stable homotopy groups of certain classifying spaces. We also…

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed, or algebraically closed field F, we give a sufficient condition for a valued subfield of the field of generalized power…

Let $A$ be an affinoid integral domain over a non-Archimedean field $K$, and let $L$ be its field of fractions. We prove that the normalization of $A$ can be reconstructed from $L$ by taking the intersection of all maximal discrete…

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid…