Related papers: Extensions of Valuations to the Henselization and …

We study the extension of valuations centered in a local domain to its henseliza-tion. We prove that a valuation $\nu$ centered in a local domain R uniquely determines a minimal prime H($\nu$) of the henselization R h of R and an extension…

In this paper we consider the question of when the associated graded ring along a valuation, ${\rm gr}_{\nu^*}(S)$, is a finite ${\rm gr}_{\nu^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and…

Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$, $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field $K^*$ such that $S$ dominates $R$. Suppose that…

Let (R; m; k) be a local noetherian domain with field of fractions K and R_v a valuation ring, dominating R (not necessarily birationally). Let v|K be the restriction of v to K; by definition, v|K is centered at R. Let \hat{R} denote the…

Let R be a local domain, v a valuation of its quotient field centred in R at its maximal ideal. We investigate the relationship between R^h, the henselisation of R as local ring, and {\~v}, the henselisation of the valuation v, by focussing…

In this paper we consider birational properties of ramification in excellent local rings. We extend earlier results of the author with Olivier Piltant to show that strong local monomialization is true along a valuation dominating a…

Suppose that $R$ is a local domain essentially of finite type over a field of characteristic 0, and $\nu$ a valuation of the quotient field of $R$ which dominates $R$. The rank of such a valuation often increases upon extending the…

Suppose that R\rightarrow S is an extension of local domains and \nu^* is a valuation dominating S. We consider the natural extension of associated graded rings along the valuation gr_{\nu^*}(R)\rightarrow gr_{\nu^*}(S). We give examples…

We study the ring extensions R \subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples…

In this paper we discuss stable forms of extensions of algebraic local rings along a valuation in all dimensions over a field k of characteristic zero, and generalize a formula of Ghezzi, H\`a and Kashcheyeva describing the extension of…

If $R$ is a valuation domain of maximal ideal $P$ with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals $P=L_0\supset L_1\supset...\supset L_m\supseteq 0$ such that…

Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values…

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring $A$ with field of fractions $K$ and for a finite Galois extension $L$ of $K$, the integral closure…

We give a definition, in the ring language, of Z_p inside Q_p and of F_p[[t]] inside F_p((t)), which works uniformly for all $p$ and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula…

We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian…

Let $R$ be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating $R$, there exists a unique sequence $\{R_n\}$ of local quadratic transforms of $R$ along this valuation domain. We consider…

It is shown that every dp-minimal integral domain $R$ is a local ring and for every non-maximal prime ideal $\mathfrak p $ of $R$, the localization $R_{\mathfrak p }$ is a valuation ring and $\mathfrak{p}R_{\mathfrak{p}}=\mathfrak{p}$.…

Let $R$ be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate $R$. We restrict to the class of prime divisors that dominate $R$ and show that if…

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is…

For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper we discuss an extension of $R$ called its perfect closure $R^\infty$. This extension contains all $p^e$-th roots of elements of $R$, and is usually non-Noetherian.…