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\begin{document}
\title{Valuations on equicharacteristic complete noetherian local domains}
\author{Bernard Teissier}
\maketitle

\begin{abstract}Given an equicharacteristic noetherian complete local domain $R$ and a rational valuation $\nu$ on $R$ we show that there exist an algebra $S=\widehat{k[(u_i)_{i\in I}]}$ equipped with a weight, or monomial valuation, and a surjection $\pi\colon S\to R$ such that the valuation $\nu$ is induced by the weight on $S$ in the sense that $\nu (x)$ is the maximum weight of counterinages of $x$ in $S$. Moreover, the kernel of $\pi$ is generated by overweight deformations of binomials corresponding to generators of the relations between generators $(\gamma_i)_{i\in I}$ of the semigroup $\Gamma=\nu (R\setminus\{0\})$. In all this, the index set $I$ is an ordinal $\leq\omega^{{\rm dim}R}$. \end{abstract}



\section{Extended abstract}
 Given an equicharacteristic complete noetherian local domain $R$ with algebraically closed residue field $k$, we study the relation between a zero-dimensional valuation $\nu$ of $R$ centered at the maximal ideal and its associated graded ring ${\rm gr}_\nu R$ with respect to the filtration defined by the valuation. We shall be interested mostly in \textit{rational} valuations, in the sense that the extension $R/m\subset R_\nu/m_\nu$ is trivial, where $R_\nu$ is the valuation ring of $\nu$. Then by general properties of valuations, each nonzero homogeneous component of the $k$-algebra ${\rm gr}_\nu R$ is a $1$-dimensional $k$-vector space. In this case the graded algebra is essentially the semigroup algebra $k[t^\Gamma]$ where $\Gamma=\nu (R\setminus\{0\})$ is the semigroup of values of the valuation. The main idea is that in the event that  ${\rm gr}_\nu R$ is a finitely generated $k$-algebra, some of the birational toric maps which provide embedded pseudo-resolutions for the affine toric variety corresponding to ${\rm gr}_\nu R$ (see \cite{GP-T1}) also provide local uniformizations for $\nu$ on $R$, and when ${\rm gr}_\nu R$ is not finitely generated, then the same should be true for some of the birational toric maps which (pseudo) resolve the affine variety defined by a well chosen finite subset of the set of binomial equations describing the relations between the generators of $\Gamma$. If ${\rm gr}_\nu R$ is a finitely generated $k$-algebra, or equivalently $\Gamma$ is a finitely generated semigroup, the valuation $\nu$ is necessarily Abhyankar (for zero-dimensional valuations this means that the value group is $\Z^r$ with $r={\rm dim}R$). The converse is true up to a birational map on ${\rm Spec}R$ followed by localization at the center of the valuation and completion (see \cite{Te3}). \par
 In general, the semigroup of a valuation on a noetherian local domain is well ordered, so that it has a minimal system of generators $$\Gamma=\langle\gamma_1,\ldots ,\gamma_i,\ldots\rangle = \langle (\gamma_i)_{i\in I}\rangle$$ indexed by an ordinal $I\leq \omega^h$, where $\omega$ is the ordinal of $\N$ and $h$ is the rank, or (archimedian) height, of the valuation. If $\Gamma$ is a numerical semigroup, it is finitely generated by a classical result of Dickson; this is the case when $R$ is one dimensional.\Par
As shown in \cite{Te1}, the fact that $R$ is noetherian and the valuation $\nu$ is rational imply the existence of a presentation of ${\rm gr}_\nu R$ as a quotient of a polynomial ring, possibly in countably many variables, by a binomial ideal:
 $$k[(U_i)_{i\in I}]/(U^{m^\ell}-\lambda_\ell U^{n^\ell})_{\ell\in L}\simeq {\rm gr}_\nu R,\ \ {\rm with}\ \lambda_\ell\in k^*.$$
 Here the variables $U_i$ are in bijection with the generators $\gamma_i$ of $\Gamma$ and the isomorphism sends each $U_i$ to an element $\overline\xi_i$, of degree $\gamma_i$,  of a minimal system of generators of the $k$-algebra ${\rm gr}_\nu R$.\Par
 In order to relate the the embedded pseudo-resolutions for the (generalized) affine toric variety corresponding to ${\rm gr}_\nu R$ with local uniformizations of the valuation $\nu$ on $R$, the main tools are the concept of \textit{overweight deformation} of a prime binomial ideal, and the \textit{valuative Cohen theorem}. \par\medskip\noindent \textbf{Weights and overweight deformations}\Par
 For this summary, a weight on $k[(u_i)_{i\in I}]$ or $k[[(u_i)_{i\in I}]]$ will be a morphism of semigroups $w\colon M(I)\to \Phi_{\geq 0}$, where $M(I)$ denotes the semigroup of monomials in the $u_i$, and $\Phi$ a totally ordered abelian group, which attributes to each variable $u_i$ a weight $w(u_i)=\gamma_i\in\Phi_{\geq 0}$. In our case, $\Phi$ will be the value group of the valuation and the image of the map $w$ is the semigoup $\Gamma\subset\Phi_{\geq 0}$.\par\noindent
 A weight is compatible with a binomial ideal (an ideal generated by binomials) if each generating binomial is homogeneous.\par\noindent
 The weight of a polynomial, or a series,  is the minimum weight of its terms.\par
Given a weight which is compatible with it, an \textit{overweight deformation} of a binomial is an expression
$$F= u^m-\lambda_{mn}u^n+\sum_{w(u^p)>w(u^m)}c_pu^p\in k[[(u_i)_{i\in I}]].$$
If we have any number of binomials and a compatible weight we do the same and consider the deformations
$$F_\ell= u^{m^\ell}-\lambda_\ell u^{n^\ell}+\sum_{w(u^p)>w(u^{m^\ell})}c^{(\ell)}_pu^p\in k[[(u_i)_{i\in I}]].$$
One has to add the condition that \textit{the initial binomials of the $F_\ell$ generate the ideal $F_0$ of initial forms of the elements of the ideal $F$ generated by the $F_\ell$.}\par
Let us consider an overweight deformation $(F_\ell)$ of a prime binomial ideal $F_0$ in the power series ring $k[[u_1,\ldots ,u_N]]$, and the map  
$$\pi\colon k[[u_1,\ldots ,u_N]]\to R=k[[u_1,\ldots ,u_N]]/(F_1,\ldots ,F_s).$$
\begin{proposition} 1) The map which associates to $x\in R$ the \textit{maximum} of the weights of the elements of $\pi^{-1}(x)$ is well defined and is a valuation $\nu$ on $R$.\par\noindent
2) The associated graded ring of $k[[u_1,\ldots ,u_N]]$ with respect to the weight filtration is $k[U_1,\ldots ,U_N]$ and the map $$\Pi\colon k[U_1,\ldots ,U_N]\rightarrow k[U_1,\ldots ,U_N]/F_0={\rm gr}_\nu R$$ is the associated graded map of $\pi$ with respect to the weight and valuation filtrations.\par\noindent
 3) Given $\tilde x\in \pi^{-1}(x)$, we have $w(\tilde x)=\nu(x)$ if and only if ${\rm in}_w(\tilde x)\notin F_0$.
\end{proposition}\noindent
\textbf{The power series ring adapted to $\Gamma$}\Par
Let $(u_i)_{i\in I}$ be variables indexed by the elements of the minimal system of generators $(\gamma_i)_{i\in I}$ of the semigroup $\Gamma$ of the rational valuation $\nu$ on $R$. Give each $u_i$ the weight $w(u_i)=\gamma_i$ and let us consider the set of power series $S=\sum_{e\in E} d_e u^e$ where $(u^e)_{e\in E}$ is any set of monomials in the variables $u_i$ and $d_e\in k$.\par\medskip By a theorem of Campillo-Galindo (see \cite{C-G}), the semigroup $\Gamma$ being well ordered is combinatorially finite, which means that for any $\phi\in\Gamma$ the number of different ways of writing $\phi$ as a sum of elements of $\Gamma$ is finite. This is equivalent to the fact that the set of exponents $e$ such that $w(u^e)=\phi$ is finite: for any given series the map $w\colon E\to \Gamma,\ e\mapsto w(u^e)$ has finite fibers. Each of these fibers is a finite set of monomials in variables indexed by a totally ordered set, and so can be given the lexicographical order and order-embedded into an interval $1\leq i\leq n$ of $\N$. and thus produce an embedding $E\subset (\Gamma\times\N)_{lex}$ which induces a total order on $E$, for which it is well ordered. When $E$ is the set of all monomials, this gives a total monomial order. \par The combinatorial finiteness implies that this set of series $S=\sum_{e\in E} d_e u^e$ is a $k$-algebra, which we denote  by $\widehat{k[(u_i)_{i\in I}]}$. It is endowed with a weight $w(S)$, which is the minimum weight of the terms of $S$, and a topology defined by the weight filtration. It is shown in \cite{Te3} that the algebra is spherically complete with respect to the monomial valuation given by the weight. Since the weights of the elements of a series form a well ordered set and only a finite number of terms of the series have minimum weight, the associated graded ring of $\widehat{k[(u_i)_{i\in I}]}$ with respect to the filtration by weights is the polynomial ring $k[(U_i)_{i\in I}]$.  
\begin{proposition}{{\rm(The valuative Cohen theorem, see [4])}}
Assuming that the local noetherian equicharacteristic domain $R$ is complete, with a rational valuation $\nu$, and fixing a field of representatives $k\subset R$, there exist choices of representatives $\xi_i\in R$ of the $\overline{\xi_i}$ generating the $k$-algebra ${\rm gr}_\nu R$ such that the surjective map of $k$-algebras $k[(U_i)_{i\in I}]\to {\rm gr}_\nu R,\ U_i\mapsto\overline{\xi_i}$, is the associated graded map of a continuous surjective map $$\widehat{k[(u_i)_{i\in I}]}\to R,\ u_i\mapsto\xi_i,$$ of topological $k$-algebras, with respect to the weight and valuation filtrations respectively. The kernel of this map can be generated up to closure by overweight deformations of binomials generating the kernel of $k[(U_i)_{i\in I}]\to {\rm gr}_\nu R$, $U_i\mapsto\overline\xi_i$. \Par If $\nu$ is of rank one or if $\Gamma$ is finitely generated, any choice of representatives $\xi_i$ is permitted.
\end{proposition}
Note that here we have generalized the concept of overweight deformation to the case of countably many binomials.
\begin{corollary}Let $\nu$ be a rational valuation on a complete equicharacteristic noetherian local domain $R$. If the semigroup $\Gamma$ of the valuation on $R$ is finitely generated, the ring $R$ is obtained by an overweight deformation from the quotient of a power series ring by the binomial ideal encoding relations between the generators of $\Gamma$.
\end{corollary}
As a consequence of this, it is a combinatorial problem, relatively easy in the rank one case, to show that if in addition the residue fied $k$ is algebraically closed some of the embedded toric resolutions of ${\rm gr}_\nu R$ give local uniformization for $\nu$ on ${\rm Spec}\ R$, once it is re-embedded in the same ambient space as ${\rm Spec}\ {\rm gr}_\nu R$. Associated with the fact recalled above that the semigroups of values of rational Abhyankar valuations on excellent equicharacteristic local domains  are finitely generated up to $\nu$-modification and completion, this gives a proof of local uniformization for Abhyankar valuations of equicharacteristic excellent local domains with an algebraically closed residue field. \par\medskip
In the general case of a rational valuation $\nu$ of a complete equicharacteristic noetherian local domain the valuative Cohen theorem allows us to produce a sequence of Abhyankar \textit{semivaluations} $\nu_B$ of $R$ of, that is, Abhyankar valuations on $r$-dimensional quotients $R/K_B$ of $R$, where $r$ is the rational rank of $\nu$, which are indexed by finite subsets $B$ ordered by inclusion of the index set $I$ and have the property that for any element $x\in R$, the valuation $\nu(x)$ is equal to $\nu_B(x)$ for large enough $B$. This is an extension of the abyssal phenomenon which was described for the plane in \cite{Te1} and an asymptotic approximation property of rational valuations.\par\medskip\noindent
\textit{All the references below except the first one and the last one are available, with corrections, at} http://webusers.imj-prg.fr/~bernard.teissier/articles-Teissier.html
 
  \begin{thebibliography}{}

\bibitem{C-G} A. Campillo, C. Galindo, On the graded algebra relative to a valuation. \textit{Manuscripta Math.} \textbf{92} (1997), 173-189.
\bibitem{G-T} R. Goldin and B. Teissier, Resolving singularities of plane analytic branches with one toric morphism. In
\textit{Resolution of Singularities, a research textbook in tribute to Oscar Zariski} (ed. by H. Hauser, J. Lipman, F. Oort, and A. Quir\'os). Progress in Math. Vol. 181,
Birkh\"auser, Basel, 2000, 315-340.

\bibitem{GP-T1}
P.~Gonz\'alez P\'erez and B. Teissier, Embedded resolutions of non necessarily normal affine toric varieties.
\textit{Comptes-rendus Acad. Sci. Paris, Ser.1} \textbf{(334)} (2002), 379-382. 

\bibitem{Te1}
B. Teissier, Valuations, deformations, and toric geometry.
In \textit{Valuation Theory and its applications, Vol. II}, Fields Inst. Commun. 33, AMS., Providence, RI., 2003, 361-459.  

\bibitem{Te2} B. Teissier, A viewpoint on local resolution of singularities. Oberwolfach Workshop on Singularities, September 2009. \textit{Oberwolfach Reports}, Vol. 6, No. 3, 2009. European Math. Soc. Publications. 

\bibitem{Te3} B. Teissier, Overweight deformations of affine toric varieties and local uniformization, in "Valuation theory in interaction",  Proceedings of the second international conference on valuation theory, Segovia--El Escorial, 2011. Edited by A. Campillo, F-V. Kuhlmann and B. Teissier. European Math. Soc. Publishing House, Congress Reports Series, Sept. 2014, 474--565.

\bibitem{Tev} J. Tevelev, On a question of B. Teissier. \textit{Collectanea Math.} \textbf{65} (1) (2014), 61-66. (Published on line February 2013. DOI 10.1007/s13348-013-0080-9)


\end{thebibliography}

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