Obstruction to (1). If the answer to (1) is affirmative, with smooth proper R-model X, then the smooth and proper base change theorems for étale cohomology imply upon choosing a place of Fs over that on F and letting Rsh denote the associated strict henselization inside Fs (so Rsh has residue field ks that is a separable closure of k); it is the “maximal unramified extension” of the henselizati...

Let A be a d-dimensional local ring containing a field. We will prove that the highest Lyubeznik number λd,d(A) (defined in [5]) is equal to the number of connected components of the Hochster-Huneke graph (defined in [2]) associated to B, where B = ̂̂ Ash is the completion of the strict Henselization of the completion of A. This was proven by Lyubeznik in characteristic p > 0. Our statement and p...

We prove that every place of an algebraic function eld of arbitrary characteristic admits a local uniformization in a nite extension of the function eld. We give a valuation theoretical description of these extensions; in certain cases, they can be found in the henselization of the function eld. For places satisfying the Abhyankar equality and for discrete rational places, no extension is neede...

For valued fields K of rank higher than 1, we describe how elements in the henselization K of K can be approximated from within K; our result is a handy generalization of the well-known fact that in rank 1, all of these elements lie in the completion of K. We apply the result to show that if an element z algebraic over K can be approximated from within K in the same way as an element in K, then...

the aim of this paper is to study orders over a valuation ring v with arbitrary rank in acentral simple f-algebra q. the relation between all of the orders is explained with a diagram. it is thenshown that inside bezout order, extremal v-orders are precisely semi-hereditary. in the last section, theeffect of henselization on maximal and semi-hereditary orders is examined.

We classify all possible extensions of a valuation from a ground field K to a rational function field in one or several variables over K. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field e...