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\begin{document}
\title{Three facts of valuation theory}
\author{Bernard Teissier}
\maketitle
\par\noindent
 Corrected  version of the paper which appeared in the Proceedings of the Japan Academy, Vol.  LXXIV, Ser. A, No. 7,
September 1988. The statement on the $\nu$-adic completion depended on  two corollaries in [Z-S] which, as I learned later from
reading [Ve], are false. The correct statement is below.
\vskip.6truecm

\begin{section}{Statement of results} Let $k$ be a field and $(R,m)$ a local integral $k$-algebra with field of
fractions $K$. We study $k$-valuations $\nu$ of $K$ with a center in $R$, that is, such that
their valuation ring $(R_\nu, m_\nu)$ contains $R$ and $m_\nu\cap k=(0)$. We denote by
$\Phi$ the totally ordered group of the valuation and set $\Gamma=\nu
(R\setminus\{0\})\subset
\Phi_+\cup\{0\}$. The valuation determines on
$R$ filtrations defined by the ideals
$$\P_\phi (R)=\{x\in R/\nu (x)\geq \phi\}\ \ \ \hbox{\rm or}\ \ \P^+_\phi (R)=\{x\in R/\nu
(x)> \phi\},$$ and the associated graded ring introduced by Spivakovsky ([S], see also [L-T],
[T1]):
$$\hbox{\rm gr}_\nu R=\bigoplus_{\phi\in \Gamma }\P_\phi/\P^+_\phi ,$$ which is a
$\Gamma$ (or $\Phi_+$)-graded $(R/m_\nu\cap R)$-algebra. We assume throughout that
$\Phi$ has finite rational rank $\hbox{\rm r}(\Phi)$ (and therefore is countable) and finite height
(or rank)
$\hbox{\rm h}(\Phi)$.  The three facts, extracted from [T2], are the
following:\par\medskip\noindent 1) {\bf A connexion between valuation theory and toric
geometry}\begin{proposition} For any valuation
$\nu_0$ with a center in $R$ and such that $m_{\nu_0}\cap R=m$ and the residue field extension
$k_R\rightarrow k_{\nu_0}=R_{\nu_0}/m_{\nu_0}$ induced by the inclusion $R\subset
R_{\nu_0}$ is trivial, the algebra $\hbox{\rm gr}_{\nu_0} R$ is isomorphic to a quotient of a
polynomial ring
$k_R[(U_i)_{i\in I}]$ with coefficients in
$k_R$ and possibly countably many indeterminates by a binomial ideal, i.e, an
ideal with (possibly countably many) generators of the form $U^m-\lambda_{mn}U^n$ where
$U^m=U_1^{m_1}\ldots U_s^{m_s}$ and $\lambda_{mn}\in k_R^*$. It means that
it is a deformation of a (non normal) toric variety (see [E-S]), possibly of infinite embedding
dimension, which is nothing but $\hbox{\rm Spec}k_R[t^\Gamma]$, where
$k_R[t^\Gamma]$ is the semigroup algebra of $\Gamma$, obtained by replacing all
$\lambda_{mn}$ by 1.\end{proposition}
2) {\bf Structure of valuation semigroup algebras and regularity of $\hbox{\rm gr}_\nu
R_\nu$}\begin{proposition} The graded $k_\nu$-algebra
$\hbox{\rm gr}_\nu R_\nu$ is a filtering direct limit of termic maps (i.e mapping a variable to a
term, of the form constant times a monomial) between polynomial subalgebras in $\hbox{\rm
r}(\Phi)$ variables. The semigroup algebra $k_\nu[t^{\Phi_+}]$ is the direct limit of the
corresponding system of toric (or monomial) maps, obtained by replacing all the constants by
1.\end{proposition} 3) {\bf N\oe therianity of certain $\nu$-adic completions}\begin{proposition}
Assume that $R$ is an analytically irreducible n\oe therian local ring. If $\nu_1$ denotes the
valuation of height one with which $\nu$ is composed, and ${\bf p}=m_{\nu_1}\cap R$ the center of
$\nu_1$ on $R$, then the completion $\hat R^\nu$ of the ring $R$ with respect
to the topology defined by the $(\P_\phi)_{\phi\in \Phi_+}$-filtration, is isomorphic as
topological ring to a quotient of the symbolic ${\bf p}$-adic completion $\hat R^{({\bf p)}}$ of $R$; if $R$
is regular, it is n\oe therian. \end{proposition}
 In particular, if $R$ is excellent and regular, the ring $\hat R^\nu$ is excellent since $\hat R^{(\bf p)}$ is
excellent in this case by ([Ro], [Ve]).\end{section}
\begin{section}{Ideas of proofs}
1) Since $R_\nu$ is a valuation ring, the $\Phi_+$-graded
$k_\nu$-algebra $\hbox{\rm gr}_\nu R_\nu$ has the property that each of its homogeneous
components is a 1-dimensional vector space over $k_\nu$. If the residual
extension is trivial, the same is true over $k_R$, and since
the $k_R$-algebra $\hbox{\rm gr}_\nu R$ is a graded subalgebra of $\hbox{\rm
gr}_\nu R_\nu$, each of its homogeneous components is a $k_R$-vector space of
dimension $\leq 1$. By an observation of Korkina ([Ko], see also [E-S]), this implies the result:
taking a (possibly countable) system of homogeneous generators of the algebra gives a graded
surjection $k_R[(U_i)_{i\in I}]\rightarrow \hbox{\rm gr}_\nu R$ once
$U_i$ is given the degree of its image. The kernel is generated by homogeneous polynomials,
but any two terms of such a polynomial have non zero $k_R$-proportional
images, which shows that the kernel is generated by binomials.
\par\medskip\noindent 2) Let $\nu$ be a valuation of height one, i.e with archimedian value
group $\Phi\subset {\bf R}$ (see [Z-S], Vol. II). Assume first that $\Phi$ is generated by
$m$ rationally independent positive real numbers $\tau_1,\ldots ,\tau_m$. We use the Perron
algorithm as expounded in ([Z], B. I, p. 861), but with a somewhat different interpretation. The
algorithm consists in writing
$$\tau_1=\tau_m^{(1)} ,\tau_2=\tau_1^{(1)}+a_2^{(0)}\tau_m^{(1)},\ldots
,\tau_m=\tau_{m-1}^{(1)}+a_m^{(0)}\tau_m^{(1)},$$ where $$a_j^{(0)}=[\tau_j/\tau_1],\ \ \
j=2,\ldots ,m,$$ and repeating this operation after replacing $(\tau_1,\ldots ,\tau_m)$ by
$(\tau_1^{(1)},\ldots ,\tau_m^{(1)})$, and so on. After
$h$ steps, one has written
$$\tau_i=A_i^{(h)}\tau_1^{(h)}+\cdots +A_i^{(h+m-1)}\tau_m^{(h)}$$ or,  if we
denote by $w$ the (weight) vector $(\tau_1,\ldots ,\tau_m)\in {\bf R}^m$ and by $A^{(h)}$ the vector 
$(A_1^{(h)},\ldots ,A_m^{(h)})$,
$$w=\tau_1^{(h)}A^{(h)}+\tau_2^{(h)}A^{(h+1)}+\cdots +\tau_m^{(h)}A^{(h+m-1)}$$
where the $\tau^{(h)}_j$ are positive, the coefficients $A^{(j)}_i$ are non negative integers, and
the matrix of the vectors
$$A^{(h)},A^{(h+1)},\ldots ,A^{(h+m-1)}$$ has determinant $(-1)^{h(m-1)}$. Moreover, the
directions in ${\bf P}^{m-1}({\bf R})$ of the vectors $A^{(h)}$  tend to the direction of $w$. So we
have a sequence of vectors $A^{(h)}$ with positive integral coordinates (becoming longer and
longer) whose directions in ${\bf P}^{m-1}({\bf R})$ spiral to the direction of $w$ and such that
any consecutive $m$ of them as above form a basis of the integral lattice such that $w$ is
contained in the convex cone $\sigma^{(h)}=<A^{(h)},A^{(h+1)},\ldots ,A^{(h+m-1)}>$
which they generate. The convex dual
$\check\sigma^{(h)}$ of $\sigma^{(h)}$ (see [E], V, 2, p. 149) is contained in the half
space $\sum_{j=1}^ma_j\tau_j\geq 0$, the integral points of which form the
semigroup $\Phi_+$. For any commutative ring $A$, the $A$-algebra of the semigroup $\check
\sigma^{(h)}\cap \Z^m$ is a polynomial algebra $A[x_1^{(h)}, \ldots,x_m^{(h)}]$ (\it loc.cit \rm , VI,2)
contained in
$A[t^{\Phi_+}]$, and since by assumption there are no integral points on the hyperplane 
$\sum_{j=1}^ma_i\tau_i= 0$ except the origin, the semigroup $\Phi_+$ is the union of
the $\check \sigma^{(h)}\cap \Z^m$ as $h\rightarrow \infty$. This proves that $A[t^{\Phi_+}]$ is the
union,
 or direct limit, of these polynomial subalgebras.\par If we now consider a group 
with one more generator $\tau_{m+1}>0$ which is rationally dependent
on $\tau_1,\ldots ,\tau_m$, Zariski shows in ([Z], B. I, p. 862) that the new weight vector 
$w=(\tau_1,\ldots ,\tau_m ,\tau_{m+1})\in {\bf R}^{m+1}$ is contained in a rational simplicial 
cone $\sigma \subset {\bf R}^{m+1}$ generated by $m$ integral vectors of the first quadrant 
forming part of a basis of the integral lattice. Indeed $w$ is contained in a unique rational
hyperplane. The dual cone $\check \sigma\subset\check {\bf R}^{m+1}$  is the product of an
$m$ dimensional strictly convex cone by a 1-dimensional vector space (see [E], V, 2) generated
by an integral vector which is the dual of the rational hyperplane containing $w$. By refining as
above by the Perron algorithm for $w$ inside the linear span
$<\sigma >$ of $\sigma$, starting with the barycentric coordinates of $w$ in $\sigma$,
 we find a sequence of regular simplicial cones $\sigma^{(h)}\subset \sigma $ whose duals $\check
\sigma^{(h)}\subset \Phi_+$ correspond ([E], VI, Th. 2.12) to algebras of the form $A[x_1^{(h)},
\ldots ,x_m^{(h)}, x_{m+1}^{\pm 1}]\subset A[t^{\Phi_+}]$. Note that the map $f: \Z^{m+1}\rightarrow
{\bf R}$ defined by
$(a_1,\ldots ,a_{m+1})\mapsto \sum_{i=1}^{m+1}a_i\tau_i$ is no longer injective; 
the primitive vector corresponding to the variable $x_{m+1}$ is in the kernel.  The $\check
\sigma^{(h)}\cap \Z^{m+1}$ fill up $f^{-1}(\Phi_+\cup\{ 0\})$ as $h\rightarrow \infty$ since the only 
rational points of the hyperplane $\sum_{j=1}^{m+1}a_j\tau_j=0$ are on the dual of the hyperplane containing $w$,
which is contained in all the $\check\sigma^{(h)}$. So the direct limit of the images of the maps
$A[t^f]\colon A[x_1^{(h)},
\ldots ,x_m^{(h)}, x_{m+1}^{\pm 1}]\rightarrow A[t^{\Phi_+}]$ is $A[t^{\Phi_+}]$. But these images are
isomorphic to $ A[x_1^{(h)},
\ldots ,x_m^{(h)}, x_{m+1}^{\pm 1}]/(x_{m+1}-1)$ so that they are again polynomial rings
 $A[x_1^{(h)}, \ldots,x_m^{(h)}]$. If we have more generators rationally
dependent on $\tau_1,\ldots ,\tau_m$, we can repeat the argument after taking as new
generators the coordinates of the weight vector with respect to the $m$ primitive vectors of
$\sigma$. A group $\Phi\subset{\bf R}$ of finite rational rank is a direct limit of such
extensions.\par\medskip Let now $\Phi$ be a totally ordered group of finite height
$h>1$. We have a surjective ordered map
$\lambda\colon \Phi\rightarrow \Phi_1$ where $\Phi_1$ is of height $h-1$, and the kernel
$\Psi$ of $\lambda$ is of height $1$. By induction on the height we may assume that
$\Phi_{1+}$ is the union of sub-semigroups isomorphic to $\N^{\hbox{\rm r}(\Phi_1)}$, and we
know from the lemma above that the same is true for $\Psi_+$. \par
Let us denote the semigroups that fill $\Phi_{1+}$ by $F_i$, and let $\tilde F_i\subset \Phi_+$ be
the subsemigroup generated by elements $e_1, \ldots ,e_{r_i}$ which lift to $\Phi_+$ the
generators of $F_i$. Similarly let us denote by $G_j\subset \Psi_+$ free semigroups which fill
$\Psi_+$, generated say by
$f_1,\ldots ,f_{s_j}$. Note that for $\phi\in \Phi_+,\ \psi\in \Psi,\ \phi +\psi\in \Phi_+\cup \{ 0\}$, 
and consider for $r_is_j$-tuples $n=(n_{s,t},1\leq s\leq r_i,\ 1\leq t\leq s_j)$ of non negative
integers, the free semigroups 
$\tilde F_i(n)\subset \Phi_+\cup \{ 0\}$ generated by $e_1-\sum_tn_{1t}f_t,\ldots
,e_{r_i}-\sum_t n_{r_it}f_t$. The $\tilde F_i(n)\oplus G_j$ are of rank equal to the rational rank of
$\Phi$ and fill up
$\Phi_+$. This proves the result.
\par\noindent
\begin{corollary}\label{goodeq} The semigroup algebra $k_\nu [t^{\Phi_+}]$ is a quotient of a
polynomial ring over $k_\nu$ in countably
 many indeterminates $k_\nu [(U_j)_{j\in J}]$ by a binomial ideal with generators of the form 
$$(U_j- U^{m(j)})_{j\in J'}$$\noindent
for some $J'\subset J$, with $\vert m(j)\vert\geq 2$.\end{corollary}\noindent
The proof above shows that $k_\nu [t^{\Phi_+}]$ is a union of polynomial algebras with
monomial maps. Each member of a system of homogeneous generators of the $k_\nu $-algebra
$k_\nu [t^{\Phi_+}]$ must appear in one of the polynomial subalgebras, and we may
assume that it appears as one of the generators of that algebra. The only relations between
these generators are those corresponding to the toric inclusions $k_\nu [x_1,\ldots , x_m]\to
k_\nu [y_1,\ldots ,y_m]$, and they are of the announced type.
\begin{corollary}. Given the ring $R_\nu$ of a valuation of finite rational rank, the graded algebra 
$\hbox{\rm gr}_\nu R_\nu$ is a quotient of a polynomial ring $k_\nu[(U_j)_{j\in J}]$ in countably
 many indeterminates over the residue field $k_\nu$ by a binomial ideal of the form 
$$(U_j-\lambda_j U^{m(j)})_{j\in J'}\ ,\ \lambda_j\in k_\nu^*$$\noindent
for some $J'\subset J$, with $\vert m(j)\vert\geq 2$.\end{corollary}
it suffices to observe that setting all constants equal to 1 in the binomial relations given by fact
1) for $\hbox{\rm gr}_\nu R_\nu$ must give $k_\nu[t^{\Phi_+}]$.
\par\medskip\noindent
3) We note that since $\Phi$ is countable, the $\nu$-adic topology has a countable basis of
neighborhoods for every point. The quotients
$R/\P_\phi$ form a projective system since the semigroup $\Phi_+$ is totally ordered. One can
then define the $\nu$-adic completion of $R$ as:
$$\hat R^\nu=\displaystyle\limproj_{\phi\in \Phi_+} R/\P_\phi .$$
There is a natural map $R\rightarrow \hat R^\nu$, which is injective since the filtration is separated. The valuation $\nu$
has a canonical extension $\hat \nu$ to a valuation of $\hat R^\nu$ with values in
$\Phi$. \par\noindent Let us assume that $R$ is a n\oe therian local ring and that $\nu$ is a
valuation of height one. Set
${\bf p}=m_\nu\cap R$; since $R$ is n\oe therian, we have $\nu ({\bf p})>0$, say $\nu ({\bf
p})=\phi_0$, and since $\nu$ is of height one, its group
$\Phi$ is archimedian, so that for any $\phi\in \Phi$ there exists an integer $N(\phi )$ such that
$N(\phi )\phi_0\geq \phi$, that is ${\bf p}^{N(\phi )}\subset \P_\phi$. Note that since by definition of the symbolic powers ${\bf
p}^{(n)}={\bf p}^nR_{\bf p}\cap R$ of ${\bf p}$ we have $${\bf p}^{(n)}=\{x\in R/\exists s\notin {\bf p}:sx\in {\bf p}^n\},$$ we have
the inclusion ${\bf p}^{(N(\phi ))}\subset \P_\phi$. Then we have a morphism of completions
$\hat R^{({\bf p})}\rightarrow \hat R^\nu$, where $\hat R^{({\bf p})}$ is the completion of $R$ for the
 topology defined by the symbolic powers. Since
$\hat R^\nu$ is an integral domain, the kernel of this map is a prime ideal
$H\subset\hat R^{({\bf p})}$.  By the definition of completions, this kernel is
$\bigcap_{\phi\in\Phi_+}\P_\phi\hat R^{({\bf p})}$. Let us show that the natural injection
$$\hat R^{({\bf p})}/H\to
\hat R^\nu$$\noindent is an isomorphism. The injection
$$\hat R^{({\bf p})}/H\subset \hat R^\nu $$ implies that the valuation $\nu$ extends to
$\hat R^{({\bf p})}/H$ as a valuation $\hat \nu$ of height one.\par\noindent
Recall that when  ${\bf p}$ is the maximal ideal, the symbolic powers coincide with the ordinary ones. The valuation $\nu$ is still
$\geq 0$ on the localization
$R_{\bf p}$, and so we have a morphism of completions 
$\hat R^{{\bf p}R_{\bf p}}_{\bf p}/H_{{\bf p}}\subset \hat R_{\bf p}^\nu$. Let us replace
$R$ by $R_{\bf p}$ for a moment, so that ${\bf p}$ is the maximal ideal $m$. By [Z-S], Vol. 2,
Appendix 3, Lemma 3 p. 343, the distinct valuation ideals (i.e our
$\P_\phi (\hat R^m/H)$ written without repetition) form a simple infinite descending chain of
ideals which are primary for the maximal ideal $\hat m$ of
$\hat R^{m}/H$ and have intersection zero. We denote them by $\hat\P_j$. Now we can apply
Chevalley's Theorem ([Z-S], Vol. 2, Chap. VIII, \S 5, Th. 13, p. 270), which asserts that there
exists an integer-valued function
$s(n)$ tending to infinity with $n$ and such that for each valuation ideal $\hat\P_j$,
we have $\hat\P_j\subset
\hat m^{s(j)}$. This, added to the fact that the $\hat\P_j$ are primary for
$\hat m$, proves that in the ring
$\hat R^{m}/H$ the $\hat\nu$-adic topology coincides with the
$\hat m$-adic topology, so that it is complete for both, and therefore has to be equal to
$\hat R^\nu$. \par
When the center of $\nu$ is not necessarily $m$, this shows that on $R$ we have the
inclusions $\P_j\subset {\bf p}^{(s(j))}$. This gives the result in the general case.\par
It now suffices to note that whenever a valuation $\nu$ is composed with a valuation $\nu_1$,
the topologies which they define are equivalent. Finally, if $R$ is regular, by [Ve] the symbolic and usual topologies coincide.
\end{section}
\vskip.3truecm
\centerline{Bibliography}\par\medskip
\par\noindent
[E] G. Ewald, \it Combinatorial Convexity and Algebraic Geometry, \rm Graduate Texts in Math.,
No. 168, Springer 1996.\par\noindent
[E-S] D. Eisenbud and B. Sturmfels, Binomial ideals, \it Duke Math. J. \rm {\bf 84}, I, 1996,
1-45.\par\noindent 
[Ko] E. Korkina, classification of $A$-graded algebras with 3
generators, \it Indag. Math., New Series, \rm {\bf 3}, 1992, 27-40. \par\noindent 
[L-T] M. Lejeune-Jalabert et B. Teissier, Cloture
int\'egrale des id\'eaux et
\'equisingularit\'e. S\'eminaire Lejeune-Teissier  1974, Publ. Institut Fourier,
Grenoble. \par\noindent
[Ro] C. Rotthaus, Kompletierung semilokaler quasiausgezeichneter ringe, \it
Nagoya Math. J.
\rm {\bf 76}, 1980, 173-180.\par\noindent
[S] M. Spivakovsky, Valuations in function fields of surfaces, \it Amer. J. of Math. \rm {bf 112}
(1990), 107-156.\par\noindent
[T1] B. Teissier, ''Appendice'', in  ''Le probl\`eme des modules
pour les branches planes'' by O. Zariski. Centre de Math\'ematiques de l'Ecole polytechnique,
1975, republished by Hermann, Paris 1986.\par\noindent
[T2] B. Teissier, \it Valuations, deformations, and toric geometry,
\rm in preparation.
\par\noindent
[Ve] J-K. Verma, {\it On the symbolic topology of an ideal}, J. of Algebra, {\bf 112} (1988), 416-429.
\par\noindent
[Z] O. Zariski, Local uniformization on algebraic varieties, \it Annals
of Math. \rm Vol. 41, No. 4, October 1940, 852-886.\par\noindent
[Z-S] O. Zariski and P. Samuel, Commutative Algebra, Van Nostrand 1960.\par\noindent
\par\vskip.2truecm
Bernard Teissier\par
D.M.I, E.N.S, URA No. 762 du CNRS.\par
45 Rue d'Ulm, F75005 Paris\hfill email: teissier@dmi.ens.fr\par
\par\medskip\centerline{\copyright Bernard Teissier 1999}
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