Fields Institute Communications Volume 00, 0000

Valuations, deformations, and toric geometry

Bernard Teissier

Institut de Mathématiques, UMR 7586 du CNRS, 175 Rue du Chevaleret, 75013 Paris, France

To David Rees, on the occasion of his eightieth Birthday, and Pierre Samuel, on the occasion of his seventy-seventh Birthday

Summary

Given a nœtherian local integral domain R and a valuation ν of its field of fractions which is non negative on R, i.e., an inclusion R ⊂ R_{ν} of R in a valuation ring, I study a geometric specialization of R to the graded ring gr_{ν}R determined by the valuation. If R_{ν} dominates R and the residue field extension k = R/m → R_{ν}/m_{ν} is trivial, this graded ring corresponds to an essentially toric variety, of Krull dimension ≤ dimR but possibly of infinite embedding dimension; it is of the form:

gr_{ν}R = k[(U_{i})_{i∈I}]/(U^{m} − λ_{mn}U^{n})_{(m,n)∈E} , λ_{mn} ∈ k^{∗},

where I and E are countable sets and U^{m} = U_{i_{1}}^{m_{i_{1}}} … U_{i_{r}}^{m_{ir}}. In order to apply this fact to a characteristic-blind proof of local uniformization by deformation of a partial resolution of singularities of Specgr_{ν}R, in the case where R is equicharacteristic and excellent, with k algebraically closed, I explore the following strategy:

1) Extend the valuation ν to a valuation ν̂ of a suitable nœtherian scalewise ν-adic completion R̂^{(ν)} of R such that the natural map

gr_{ν}R → grν̂R̂^{(ν)} is scalewise birational.

2) Obtain a presentation, that is a surjective continuous morphism of k-algebras

{ \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)},

where the left hand term is a suitable scalewise completion of the polynomial ring, and the kernel is generated up to closure by elements whose initial forms for the term order t deduced from ν̂ are the binomial equations defining grν̂R̂^{(ν)}, that is, which are of the form:

w^{m} −λ_{mn}w^{n} +  ∑_{t(s)>t(m)=t(n)} c_{s}^{(mn)}w^{s}  with (m, n) ∈ Ê, λ_{mn} ∈ k^{∗}, c_{s}^{(mn)} ∈ k.

0000 American Mathematical Society

1991 Mathematics Subject Classification. 14B07,14E15, 14M25, 32S45.

3) Show that all but finitely many of these equations serve only to express the images of all w_{j}’s in the nœtherian ring R̂^{(ν)} in terms of finitely many of them. Then show that a toric map in the coordinates (w_{j})_{j∈J} which resolves the singularities of the finitely many binomial equations of grν̂R̂^{(ν)} involving these finitely many variables will uniformize ν̂ on R̂^{(ν)}, and that such resolving toric maps exist.

4) Use the excellence of R and the birationality of the map gr_{ν}R → grν̂R^{(ν)} to lift this to a uniformization of ν on R.

Contents

1. Introduction 4 2. Valuative sorites 10 2.1. The valuation algebra 10 2.2. The valuation μ_{A} 11 2.3. The specialization to the graded algebra 12 2.4. The valuation ν_{A} 16 3. Valuation algebras and composition of valuations 18 3.1. The composition of valuations 18 3.2. The valuation ν̃_{A} 23 3.3. The valuation algebra of the residual valuation 24 3.4. Comparison of graded algebras for composed valuations 24 3.5. Comparison of residual valuations 26 3.6. Specialization of valuations and rational valuations 27 3.7. Monomial valuations 27 4. The toric structure of the graded algebras of valuation rings

and nœtherian rings 30 4.1. The graded algebra of a rational valuation is essentially toric 30 4.2. More on the structure of gr_{ν}R 32 4.3. gr_{ν}R_{ν} as a limit of polynomial algebras 33 4.4. A selection of examples 38 5. Completion problems 45 5.1. The ν-adic completion of R 46 5.2. Scalewise completions of R 50 5.3. The ν_{A}-adic completion of A_{ν}(R) 62 5.4. The specialization of R̂^{(ν)} to ĝr_{ν}^{(ν)}R 66 5.5. The abyssal phenomenon 70 5.6. An example: complex plane branches 77 6. Toric modifications 78 6.1. Valuative motivation: an example 79 6.2. Embedded toric resolution of orbit closures and binomial ideals in

finite dimensions 79

6.3. An example: branches revisited 86 6.4. A question on surfaces 88 6.5. Another example, in dimension three 88 7. Local uniformization (speculative sketch) 89 7.1. The approximation process 90 7.2. Toric modifications and transforms of R̂^{(ν)} 92 References 99

1 Introduction

Lord Calversham: Do you always really understand what you say, sir? Lord Goring (after some hesitation): Yes, father, if I listen attentively.

Oscar Wilde, An ideal husband, third Act.

This text is an attempt to clarify the problems which one meets when trying to prove local uniformization of a valuation at a singular point of an algebraic or analytic variety along the lines of what is done for plane branches in [G-T], by re-embedding the singularity in such a way that the valuation can be uniformized (in an embedded manner) by a single toric modification of the ambient space. It will help the reader to be aware of the basics of valuation theory contained in [V1], of the theory of toric varieties found in [Cox] and of the results of [T1] and [G-T].

Originally intended for the proceedings of “Mountains and singularities, Working Week on Resolution of singularities,” and in agreement with the spirit that Herwig Hauser wished to give to that Working Week, this text is somewhat programmatic, even speculative, in nature since there are several statements which may not be in their final form, and some proofs are only sketched. The corresponding statements are between asterisks. I am grateful that the Editors of this volume have kindly agreed to publish it in the same spirit, with some asterisks left.

There is therefore no claim to give here a complete detailed proof of local uniformization for an excellent equicharacteristic local ring with an algebraically closed residue field, but only to sketch a new and possibly useful way of looking at the problem of local uniformization and, perhaps more importantly, at a simultaneous uniformization process for all valuations “close enough” to a given one.

Given an integral domain R with fraction field K, a valuation of R is a valuation ν: K^{∗} → Φ of K, with values in a totally ordered group Φ, which is non-negative on R. It is also the datum of a valuation ring R_{ν} of K containing R. To such a datum (R, ν) are associated the semigroup Γ = ν(R \ {0}) ⊂ Φ_{+} ∪ {0} and a filtration by ideals P_{ϕ}(R) = {x ∈ R/ν(x) ≥ ϕ}, the “successor” of P_{ϕ}(R) being P_{ϕ}^{+}(R) = {x ∈ R/ν(x) > ϕ}, and therefore a graded algebra

gr_{ν}R = ⨁_{ϕ∈Γ} \frac{P_{ϕ}(R)}{P_{ϕ}^{+}(R)}.

The problem of local uniformization is, given the local ring R of an algebraic variety or more generally an excellent local ring, and assuming that it is an integral domain, to find for each valuation ν of R a regular local R-algebra R′, essentially of finite type over R and contained in R_{ν}. We wish R′ to be obtained by localizing an affine chart of a proper algebraic map Z → SpecR which is described as precisely as possible, for example as a composition of blowing ups with non singular centers or as a proper and birational toric map with respect to some system of generators of the maximal ideal of R.

The basic idea here is to view our local ring R as a deformation of the graded ring gr_{ν}R of R with respect to the filtration associated to the valuation, and to obtain the uniformization of the valuation ν as a deformation of an initial partial toric uniformization of the valuation ν_{gr} induced by ν on gr_{ν}R. In the case where the residue field of R is algebraically closed, this initial partial uniformization of ν_{gr} should contain all the combinatorics associated to the uniformization of ν.

The motivating example is the case of complex plane branches studied as deformations of monomial curves in [T1] and [G-T]. The analytic algebra of a plane branch has a unique valuation, since its normalization is the valuation ring C{t}. It has a semigroup Γ and a graded ring as above, and we chose to abandon the simplicity of embedding dimension two in exchange for the simplicity of dealing only with the monomial curve C^{Γ} with algebra C[t^{Γ}] in affine (g + 1)-space , where g + 1 is the number of generators of the semigroup of the branch (in characteristic zero, g is the number of its Puiseux exponents), because the resolution of singularities of the monomial curve is a purely combinatorial problem. We proved that after re-embedding a plane branch C in C^{g+1} in the way given by the elements of its algebra whose valuations generate the semigroup it specializes to the monomial curve C^{Γ} with the same semigroup, which corresponds to the graded algebra gr_{ν}R of the ring R of C with respect to its unique valuation.

The equations of the corresponding family of curves in C^{g+1} are of a special form which makes it possible to prove, using the implicit function theorem, that a toric embedded resolution of C^{Γ} ⊂ C^{g+1} is also an embedded resolution for C ⊂ C^{g+1}. In higher dimensions, the first serious difficulty is that the graded ring gr_{ν}R associated to the filtration of R determined by the valuation ν is not nœtherian in general; its structure depends on the semigroup of the values which the valuation takes on the ring, which is not always finitely generated.

The good news is that for rational valuations, that is for valuations of a local ring R such that the inclusion R ⊂ R_{ν} in the ring R_{ν} of the valuation induces a trivial residue field extension^{1}, the space Specgr_{ν}R is, up to a very simple deformation, a toric variety, of finite Krull dimension but possibly of infinite embedding dimension (see Proposition 4.2); it is this toric variety which generalizes the monomial curve of [T1] and [G-T]. The uniformization of the valuation ν_{gr} induced by ν on gr_{ν}R amounts to resolving the singularities of this toric variety, which is a combinatorial problem (see §6) with a simple solution in the finitely generated case (see subsection 6.2) and no solution in the usual sense otherwise, but fortunately we need only to partially resolve in that case; see below. Moreover, it is blind to the characteristic. The combinatorial structure of the binomial equations defining the toric variety reflects the complexity of the valuation in R, including its rational rank and its rank (or height). The fact that however complicated this structure the variety defined by the binomial equations involving finitely many variables can be resolved by a toric map, is crucial.

In order to relate the uniformizations of ν and ν_{gr}, the main tool is the valuation algebra (see 2.1) associated to a valuation ν of a domain R; its spectrum is the total space of a (faithfully flat) specialization of SpecR to Specgr_{ν}R (see Proposition 2.3).

However, just like in the plane branch case, where we had to re-embed our branch in C^{g+1} before we could write the specialization, I need to write explicit equations for this deformation, in suitable variables.

This leads to the second serious difficulty, that of finding suitable completions of R, inside which one can lift generators of the graded k-algebra gr_{ν}R to coordinates, as Cohen’s Theorem does for the usual graded algebra gr_{m}R and the usual madic completion. Here the key result is that Laurent monomials \overline{η}_{j} = \overline{ξ}^{d(j)} in the generators (\overline{ξ}_{i})_{i∈I} of the graded k-algebra gr_{ν}R generate the graded algebra of a suitable completion R̂^{(ν)} of R with respect to an extension ν̂ of the valuation ν, and can be lifted to elements (η_{j})_{j∈J} of R̂^{(ν)}, which are in some sense “coordinates”; the surjection

1this meaning is different from that used by Abhyankar, e.g., in [A3], where it means “of rational rank one”. Abhyankar writes “residually rational” for what I call “rational”.

k[(W_{j})_{j∈J}] ⟶ grν̂R̂^{(ν)}, W_{j} ↦ \overline{η}_{j}

expressing the fact that the \overline{η}_{j} generate the k-algebra grν̂R̂^{(ν)} lifts to a surjection

{ \atop k[(w_{j})_{j∈J}]} ⟶ R̂^{(ν)}, w_{j} ↦ η_{j}.

Here the first candidate for R̂^{(ν)} is the completion R̂^{ν} of R with respect to the topology defined by the rational valuation ν. However this is not enough because the quotients of this ν-adic completion are not complete for the quotient topology in general and therefore ν-adic completeness does not suffice to ensure convergence of series with terms of increasing valuation (see subsection 5.2). This difficulty is circumvented by extending the valuation ν to a valuation ν̂ of a suitable quotient R̂^{(ν)} of the m-adic completion R̂^{m} of R; it is complete for the ν̂-adic topology and its quotients are also complete, while the natural morphism gr_{ν}R → grν̂R̂^{(ν)} is birational in a scalewise sense if the valuation ν is rational. The ring { \atop k[(w_{j})_{j∈J}]} is a scalewise completion of the polynomial ring (see 5.4).

The next important fact is that one can describe by equations the specialization of R̂^{(ν)} to grν̂R̂^{(ν)}. One can choose generators (up to closure) of the kernel of the surjection { \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)} such that their initial forms in a suitable sense are the binomials which generate the kernel in k[(W_{j})_{j∈J}] of the surjective homomorphism of graded k-algebras k[(W_{j})_{j∈J}] → grν̂R̂^{(ν)} sending W_{j} to the j-th generator. These generators of the kernel whose initial forms generate the kernel of the associated graded map may be interpreted as a generalized Gröbner, or standard, basis for this kernel with respect to the monomial order on the -countably manyvariables w_{j} defined by deciding that w^{s} < w^{t} if the valuation of the image of w^{s} in R̂^{(ν)} is less than the valuation of the image of w^{t}. From the viewpoint taken here, however, the important thing is that they can be resolved by toric maps.

This presentation of the scalewise complete nœtherian local ring R̂^{(ν)} as a quotient of a generalized power series ring may be seen as a valuative analogue of Cohen’s structure theorem, and manifests an important difference with Spivakovsky’s and Zariski’s approaches. I do not start from a set of coordinates and equations for the singularity and try to improve them, then follow them through a sequence of blowing-ups, but I produce directly from the ring of the singularity, given a valuation, a “natural” embedding in a (possibly infinite-dimensional) space equipped with a natural class of coordinates (w_{j})_{j∈J}, liftings of (Laurent monomials in) the generators of the graded algebra gr_{ν}R, and a “natural” set of equations, which are as simple as one might wish (deformations of binomials) and in particular are “nondegenerate” in the natural coordinates (see Propositions 5.43 and 5.44). A substantial part of the work is then to show that they truly are coordinates and equations in a useful sense with respect to the given valuation.

This viewpoint is illustrated in subsection 4.4 by a geometric interpretation of MacLane’s ”Key polynomials” associated to a valuation of k(x)[y] extending the x-adic valuation of k(x) and having value group Q as equations of plane approximations of a transcendental plane curve which is a deformation of an essentially monomial curve of infinite embedding dimension.

The specialization of the space defined by all the equations to the space defined by their binomial initial forms, inside the space with coordinates (w_{j})_{j∈J}, is the explicit form of the geometric specialization of the ring R̂^{(ν)} to its associated graded ring with respect to the valuation ν̂.

Finally the nœtherian and henselian properties of the completion R̂^{(ν)} are used again to ensure that the lifting to R̂^{(ν)} of finitely many steps of the (possibly infinite) combinatorial process of resolution of grν̂R̂^{(ν)} suffices to uniformize the extension ν̂ of ν to R̂^{(ν)}.

Here the main point is that all but finitely many of the equations defining SpecR̂^{(ν)} in the possibly infinite-dimensional affine space with coordinates (w_{j})_{j∈J} serve only to express in R̂^{(ν)} the coordinates (w_{j})_{j∉F} in terms of finitely many coordinates (w_{j})_{j∈F} which suffice to generate its maximal ideal and by their valuations, to rationally generate the value group, and have some other properties. If there were also infinitely many equations not serving this purpose, they would indefinitely add singularities, and the method could not work. This finiteness implies that it suffices that a toric modification partially resolves grν̂R̂^{(ν)}, in a precise sense, to uniformize ν̂. This uniformization occurs in spite of the fact that grν̂R̂^{(ν)} may be of a smaller Krull dimension than R̂^{(ν)}, because of the “abyssal” phenomenon generated by the form of the equations defining SpecR̂^{(ν)}; see example 4.16 in subsection 4.4, subsections 5.5, 6.5 and 7.2, and Proposition 5.53.

In general, for a rational valuation ν the Krull dimension of gr_{ν}R is equal, by a theorem of Piltant (see 3.1) to the rational rank r(ν) of ν. By Abhyankar’s inequality r(ν) ≤ dimR. The cases of strict inequality may seem to contradict the fact that there is a specialization of R to gr_{ν}R, which is even faithfully flat. But the semicontinuity of fiber dimensions is proved only for morphisms which are essentiallyof finite type. The abyssal phenomenon provides explicit examples of failure of this semicontinuity in the general case.

It is also necessary to make precise what initial partial resolution of Specgr_{ν}R means, that is, to approximate in a useful way the (possibly infinite-dimensional) toric map which resolves our toric variety by finite-dimensional ones. This is possible thanks to the fact that for any valuation ring R_{ν} with residue field k_{ν} and finite rational rank r(ν), the graded algebra gr_{ν}R_{ν} is the direct limit of a system of essentially toric injective maps between polynomial k_{ν}-subalgebras in r(ν) variables, and this system can be described rather explicitly; for archimedian valuations, it is essentially an avatar of Perron’s algorithm (see section 4).

This result may also be seen as a graded version of (non-embedded) local uniformization, since local uniformization in the rational case over a field k means that a valuation ring over k with residue field k is a direct limit of regular local k-subalgebra with the same field of fractions which are essentially of finite type. Perron’s algorithm also appears in Zariski’s proof, in an apparently different role.

Then there remains to show that this process can be extended to a toric modification of R provided that it is excellent. This last step uses the scalewise birationality of the map gr_{ν}R → grν̂R̂^{(ν)} and an observation, also used by Spivakovsky, which implies in the language used here that if R is excellent, SpecR̂^{(ν)} is not contained in the singular locus of SpecR̂^{m} (see Proposition 7.6).

Another difference with Zariski’s approach is that I must go directly to the “worst” valuations from the viewpoint of the value group, the rational ones, to take advantage of the toric nature of the corresponding graded algebra, while Zariski and Spivakovsky both begin by the case of height one (archimedian) valuations (see [Z1], top of page 858). That the center of gravity of the proof, as Zariski puts it, is now the uniformization of rational valuations has the great advantage that it makes sense to assume that the residue field is algebraically closed, whereas in Zariski’s approach, in order to deal with valuations of height > 1, one localizes, and so one cannot make such an assumption. We may therefore, in this first phase, confine our attention to the case of an algebraically closed residue field. The rational valuations are also the only ones which appear naturally in the complex-analytic case.

The cases of valuations which are not rational are essentially deformations of the rational case, as I show using the simple behavior of the valuation algebra under composition of valuations. The reason is that, if we assume that the residue field of R is algebraically closed, for any valuation ν_{0} there are rational valuations ν composed with ν_{0}, of which ν_{0} may be deemed to be a deformation.

The specialization to the graded ring changes the field of fractions, which neither Zariski’s proof in [Z1] nor Spivakovsky in [S2] did, but it brings us to a situation where two of the principal complications have disappeared: they are the behaviour of transformed equations under translation after a birational map (see [Z1], pp. 884-885), and the proof that one reaches non singularity after finitely many steps, using the fact that some partial derivative is not zero (see [Z1], pp. 886-887). Both are replaced by the results of section 6, and the implicit function theorem then allows us, via the abyssal phenomenon, to pass from the graded ring to a suitable completion of the original ring. From there one finally gets back to the ring itself using excellence.

I believe that this strategy clarifies the nature of the problem by separating as much as possible combinatorial problems from choices of coordinates and equations. Once the toric setup is understood, finding centers of blowing-up in a given embedding of the singularity should be considerably easier.

This work was initially partly motivated by the desire to understand from the viewpoint of [T1] the inspiring attempt of Spivakovsky in [S2] to prove local uniformization and resolution of singularities; because of this change in approach, substantial differences soon appeared, which were outlined in [G-T] for the case of plane branches. The viewpoint developed here is therefore different from that of [S2], although I am indebted to its author for some specific statements, as the text evidences. Spivakovsky generalizes to valuations of height one the construction which for plane branches describes resolution as a composition of plane toric maps, each of these being in turn a composition of blowing ups of points in the plane (see sections 7 and 8 of [G-T]). The graded algebra gr_{ν}R also plays an important, although quite different, role (see the introduction and §7 of [G-T], as well as [GP]). Many difficulties in that approach are due to the fact whose avatar for curves is that after each plane toric map one must choose again an origin and local coordinates in which the next toric map will be described. I allow from the start a change of embedding, even to infinite dimensions, such that in this new embedding a single toric map in natural coordinates will uniformize, and I do not seek, in the first phase, to construct a sequence of blowing-ups with non singular centers. The existence of such a sequence will follow from a theorem of De Concini-Procesi ([DC-P]) on the domination of toric birational maps by sequences of blowing ups.

In the redaction, some facts must be applied to valuation rings R_{ν}, which are not nœtherian in general, and so parts are written without that assumption. Basically, I try to do here what is needed for local uniformization of an arbitrary valuation on a local equicharacteristic excellent integral domain with an algebraically closed residue field. The case where the residue field is not algebraically closed requires other techniques in this approach. In the first sections I assume the existence of a base field, since it is only after scalewise ν-adic completion that one may use the existence of a field of representatives.

The point of view of this paper also avoids the use of the ramification and extension theory of valuations in the study of the local uniformization problem, replacing it by the relations between the ring and its associated graded ring. The ramification is in general much worse for the graded ring, but is of no import since it is a toric variety and as such can be resolved by toric maps impervious to the ramification. There have been remarkable advances in the ramification theory (see [C-P], [K1], [K2]) and the extension theory (see [V2]), some of which indicate that the graded rings may play a useful role there too, especially when the graded ring extension associated to a separable extension of valued rings is inseparable.

It was during Heisuke Hironaka’s wonderful Summer meeting at Ohnuma-Hokkaido in 1993 that this train of thought started. I wish to thank Antonio Campillo, Dale Cutkosky, David Eisenbud, Pedro González Pérez, Heisuke Hironaka, Franz-Viktor Kuhlmann, Monique Lejeune-Jalabert, Olivier Piltant, Mark Spivakovsky, Bernd Sturmfels and Michel Vaquié for stimulating comments, some important corrections, and suggestions. I am especially grateful to Mark Spivakovsky, whose work rekindled the ever living embers of my interest for the role of the monomial curve and the Riemann-Zariski manifold, and to Franz-Viktor Kuhlmann, Michel Vaquié and the referees who made very useful comments on a very preliminary version and later on a more elaborated one. I am also grateful to Nicolas Bourbaki for his incomparably careful and lucid exposition of completion problems.

During the preparation of this work, I enjoyed the hospitality of Brandeis University, the Fields Institute in Toronto, the University of Valladolid, Tokyo Metropolitan University, the MSRI at Berkeley and the University of Saskatchewan at Saskatoon.

Finally, I wish to thank warmly Herwig Hauser for conceiving the idea of the Tirol meeting and for his determination in realizing it, and Murray Marshall, Franz-Viktor and Salma Kühlmann for doing the same for the Saskatoon Conference and Workshop.

2 Valuative sorites

This section^{2} introduces the specialization of a ring R to the graded ring associated to a valuation. It generalizes to filtrations indexed by a totally ordered group a construction found, for filtrations indexed by Z, in the work of Gerstenhaber [G1], [G2], Rees [R], and in [T1]. One difference is that I do not use the valuation algebra as an algebraic construction describing the interaction of the filtration with the multiplication law, but rather as a way to encode a geometric object.

2Its title is that of an embryo version of this paper. The following was added at the request of a referee: the origin of the word sorite is an abbreviation (found e.g. in Galien) of the Greek which refers to a syllogistic type of reasoning based on the accumulation of premises; means “I accumulate”. It is used by some mathematicians to designate such an accumulation, adding a slightly depreciatory nuance. According to the Concise Oxford Dictionary, it also evokes “a form of sophism leading by gradual steps from truth to absurdity”.

2.1 The valuation algebra. Recall that a valuation of a commutative field K is a map ν: K^{∗} → Φ to a totally ordered commutative group Φ (the value group) such that ν(xy) = ν(x) + ν(y), ν(x + y) ≥ inf(ν(x), ν(y)), with equality if ν(x) ≠ ν(y). One may extend the map to K by giving to the element 0 ∈ K a valuation ∞ greater than any element of Φ. The set {x ∈ K | ν(x) ≥ 0} is the ring R_{ν} of the valuation; it is local with maximal ideal m_{ν} = {x ∈ K | ν(x) > 0}. If the first condition is relaxed to ν(xy) ≥ ν(x) + ν(y), one has a loose valuation; the constructions explained below in this section which use only the fact that the ideals P_{ϕ} form a filtration of the ring R are valid if one replaces “valuation” by “loose valuation”, the only difference being that the graded algebras are not integral domains in that case. Loose valuations are useful because under certain conditions a valuation of a ring R induces a loose valuation on a quotient ring. The trivial valuation has value group {0}.

Let R be an integral domain with field of fractions K and let ν be a valuation of K such that its valuation ring R_{ν} contains R. Except when there is explicit mention of the contrary, all valuations are assumed to be non trivial. Let us denote by Φ the totally ordered value group of the valuation ν and by k_{ν} its residue field R_{ν}/m_{ν}. Denote by Φ_{+} (resp. Φ_{−}) the semigroup of positive (resp. negative) elements of Φ and set Γ = ν(R \ {0}) ⊂ Φ_{+} ∪ {0}; it is the semigroup of (R, ν); since Γ generates the group Φ, it is cofinal in the ordered set Φ_{+}. For ϕ ∈ Φ, set

\begin{aligned}P_{ϕ}(R) & = & \{x ∈ R | ν(x) ≥ ϕ\} \\ P_{ϕ}^{+}(R) & = & \{x ∈ R | ν(x) > ϕ, \} \end{aligned}

where we agree that 0 ∈ P_{ϕ} for all ϕ, since its value is larger than any ϕ, so that by the properties of valuations the P_{ϕ} are ideals of R. Note that the intersection ⋂_{ϕ∈Φ_{+}} P_{ϕ} = (0) and that if ϕ is in the negative part Φ_{−} of Φ, then P_{ϕ}(R) = P_{ϕ}^{+}(R) = R. Consider the graded algebra à la Gerstenhaber-Rees (see the papers quoted above), which we may call the valuation algebra of (R, ν):

A_{ν}(R) = ⨁_{ϕ∈Φ} P_{ϕ}(R)v^{−ϕ} ⊂ R[v^{Φ}]

where R[v^{Φ}] is the group algebra of Φ with coefficients in R, the element x_{ϕ}[ϕ] being written x_{ϕ}v^{ϕ}.

When there is no ambiguity we shall denote this algebra simply by A; it is a domain, not nœtherian in general even if R is, as we shall see below. Note that when ν is the trivial valuation, with Φ = {0}, we have A_{ν}(R) = R.

2.2 The valuation μ_{A}. We now define a valuation μ_{A} on A with value group Φ by

μ_{A}(∑ x_{ϕ}v^{−ϕ}) = min(ν(x_{ϕ}) − ϕ).

By construction of A this valuation is non-negative on A. There are several rings connected with A_{ν}(R): its quotient field is the quotient field K(v^{Φ}) of K[v^{Φ}], and it is not difficult to describe the valuation ring R_{μ_{A}} of μ_{A}; it is the set of quotients (∑ x_{ϕ}v^{−ϕ})(∑ y_{ψ}v^{−ψ})^{−1} such that min(ν(x_{ϕ}) − ϕ) ≥ min(ν(y_{ψ}) − ψ).

We set P_{ϕ}(R_{ν}) = {x ∈ R_{ν} | ν(x) ≥ ϕ} as above, and similarly for P_{ϕ}^{+}(R_{ν}) and P_{ϕ}(K), and remark that P_{ϕ}(K), P_{ϕ}^{+}(K) are R_{ν}-submodules of K, and that for ϕ ∈ Φ_{+} we have P_{ϕ}(K) = P_{ϕ}(R_{ν}). With this notation, we have:

R_{μ_{A}} ∩ K[v^{Φ}] = A_{ν}(K) = ⨁_{ϕ∈Φ_{−}} P_{ϕ}(K)v^{−ϕ} ⨁ R_{ν} ⨁ ⨁_{ϕ∈Φ_{+}} P_{ϕ}(R_{ν})v^{−ϕ}

and

R_{μ_{A}} ∩ R_{ν}[v^{Φ}] = A_{ν}(R_{ν}) = ⨁_{ϕ∈Φ} P_{ϕ}(R_{ν})v^{−ϕ} ⊂ R_{ν}[v^{Φ}].

The maximal homogeneous ideal of A_{ν}(R_{ν}) containing m_{ν} is:

M_{ν} = ⨁_{ϕ∈Φ_{−}} R_{ν}v^{−ϕ} ⨁ m_{ν} ⨁ ⨁_{ϕ∈Φ_{+}} P_{ϕ}(R_{ν})v^{−ϕ},

so that the corresponding residue field is the residue field k_{ν} of the valuation, while the ideal of elements of positive value is

Q̃_{0}^{+} = ⨁_{ϕ∈Φ_{−}} R_{ν}v^{−ϕ} ⨁ m_{ν} ⨁ ⨁_{ϕ∈Φ_{+}} P_{ϕ}^{+}(R_{ν})v^{−ϕ}.

The maximal homogeneous ideal of A_{ν}(K) containing m_{ν} is

⨁_{ϕ∈Φ} P_{ϕ}^{+}(K)v^{−ϕ}

which induces Q̃_{0}^{+} on A_{ν}(R_{ν}). In any case, the ideal of elements of positive μ_{A}-value

of A_{ν}(R) is

Q_{0}^{+} = ⨁_{ϕ∈Φ_{−}} Rv^{−ϕ} ⨁(m_{ν} ∩ R) ⨁ ⨁_{ϕ∈Φ_{+}} P_{ϕ}^{+}(R)v^{−ϕ},

and we remark that it is exactly the ideal of A_{ν}(R) generated by the family of elements (v^{Φ_{+}}). Indeed, it is generated by elements x_{ϕ}v^{−ϕ} with ν(x_{ϕ}) > ϕ; but then,

x_{ϕ}v^{−ϕ} = v^{ν(x_{ϕ})−ϕ}x_{ϕ}v^{−ν(x_{ϕ})} ∈ (v^{Φ_{+}})A_{ν}(R).

The other inclusion is clear.

For any commutative ring A, let us denote by A[v^{Φ_{+}}] the algebra of the semigroup Φ_{+} ∪ {0} with coefficients in A.

2.3 The specialization to the graded algebra. The graded algebra associated with the valuation ν was introduced in ([L-T], [T1]) for the very special case of a plane branch (see [G-T]), and in [S1], [S2] in full generality. It is

gr_{ν}R = ⨁_{ϕ∈Γ} P_{ϕ}(R)/P_{ϕ}^{+}(R).

For ϕ ∉ Γ, P_{ϕ}(R) = P_{ϕ}^{+}(R); it is sometimes convenient to view it as graded by Φ_{+} ∪ {0}, with a zero homogeneous component for ϕ ∉ Γ. For each non zero element x ∈ R, there is a unique ϕ ∈ Γ such that x ∈ P_{ϕ} \ P_{ϕ}^{+}; the image of x in the quotient (gr_{ν}R)_{ϕ} = P_{ϕ}(R)/P_{ϕ}^{+}(R) is the initial form in_{ν}(x) of x. When there is no ambiguity, I will often denote it by \overline{x}. The graded ring gr_{ν}R inherits naturally from (R, ν) a valuation ν_{gr}, defined by

ν_{gr}(∑ \overline{x_{ϕ}}) = min(ν(x_{ϕ})),

where x_{ϕ} ∈ R is a representative of \overline{x_{ϕ}}, and the valuation does not depend on its choice.

Remarks 2.1 1) Since R is an integral domain and ν is a valuation, the ring gr_{ν}R is an integral domain.

2) If the value group Φ has finite rational rank, it is countable, since it is torsion free and hence the map Φ → Q ⊗_{Z} Φ = Q^{r} is injective. Therefore the R/(m_{ν} ∩ R)- algebra gr_{ν}R has a countable system of generators, and so does the k_{ν}-algebra gr_{ν}R_{ν}.

3) Each P_{ϕ}(R)/P_{ϕ}^{+}(R) is a torsion free R/(m_{ν} ∩ R)-module, which is of finite type if the ring R is nœtherian.

4) If R is a local ring with maximal ideal m, by definition of domination, the valuation ring R_{ν} dominates R if m_{ν} ∩ R = m, and we will sometimes be in that situation. Then gr_{ν}R is a k_{R}-algebra, where k_{R} is the residue field of R. If we want to emphasize that R and R_{ν} have the same field of fractions, we say that R_{ν} birationally dominates R.

Let us denote by (v^{Φ_{+}})A_{ν}(R) the ideal generated in A_{ν}(R) by the elements of (v^{Φ_{+}}).

Proposition 2.2 a) The natural map

A_{ν}(R) ⟶ gr_{ν}R

defined by

x_{ϕ}v^{−ϕ} ↦ x_{ϕ} mod.P_{ϕ}^{+}(R)

induces an isomorphism of graded rings

A_{ν}(R)/Q_{0}^{+} = A_{ν}(R)/(v^{Φ_{+}})A_{ν}(R) ⟶^{≃} gr_{ν}R.

b) The natural inclusion R[v^{Φ_{+}}] ⟶ A_{ν}(R) obtained by considering only the part of negative degree (i.e., ϕ ∈ Φ_{−}) of A induces, after taking rings of fractions by the multiplicative subset (v^{Φ_{+}}), an isomorphism

(v^{Φ_{+}})^{−1}R[v^{Φ_{+}}] ⟶^{≃} (v^{Φ_{+}})^{−1}A_{ν}(R);

the quotient A_{ν}(R)/R[v^{Φ_{+}}] is a torsion Z[v^{Φ_{+}}]-module.

c) Given a homomorphism ϕ ↦ e(ϕ) from Φ to the multiplicative group of units of R, it induces a surjection A_{ν}(R) → R defined by ∑ x_{ϕ}v^{−ϕ} ↦ ∑ x_{ϕ}e(−ϕ). The kernel of this surjection is the ideal ((v^{ϕ} − e(ϕ))_{ϕ∈Φ_{+}})A_{ν}(R). If R contains a field k, this is true in particular for homomorphisms Φ → k^{∗}.

Proof the map in a) is clearly surjective, and its kernel is homogeneous and generated by elements x_{ϕ}v^{−ϕ} with ν(x_{ϕ}) > ϕ; but we saw in 2.2 that it is (v^{Φ_{+}})A_{ν}(R) = Q_{0}^{+}. To prove b), consider an element v^{−ψ} ∑_{ϕ∈Φ_{−}} x_{ϕ}v^{−ϕ} of the ring of fractions (v^{Φ_{+}})^{−1}R[v^{Φ_{+}}], and rewrite it as v^{−ψ} ∑_{ϕ∈Φ_{−}} v^{−ν(x_{ϕ})}x_{ϕ}v^{−ϕ+ν(x_{ϕ})}. Since −ϕ ≥ 0 and ν(x_{ϕ}) ≥ 0, this is clearly an element of (v^{Φ_{+}})^{−1}A_{ν}(R), and every element can be written in this way; if ∑ x_{ϕ}v^{−ϕ} ∈ A_{ν}(R), let ϕ_{+} be the largest degree appearing in this sum. If it is negative, the element is in R[v^{Φ_{+}}]. Otherwise, rewrite the sum as v^{−ϕ_{+}} ∑ x_{ϕ}v^{−(ϕ−ϕ_{+})} ∈ (v^{Φ_{+}})^{−1}R[v^{Φ_{+}}]. This proves both assertions of b). Assertion c) follows from the remarks that any x ∈ R of valuation ϕ is the image of e(ϕ)xv^{−ϕ} and that if ∑ x_{ϕ}e(−ϕ) = 0, it is also the image of ∑ x_{ϕ}(v^{−ϕ} − e(ϕ). □

To interpret this geometrically, let us assume in this section that R contains^{3} a field k ; we have a composed map of algebras k[v^{Φ_{+}}] → R[v^{Φ_{+}}] → A_{ν}(R). Then:

Proposition 2.3 The k[v^{Φ_{+}}]-algebra A_{ν}(R) is faithfully flat.

Proof The criterion for faithful flatness of [B3], Chap. I. §3, No. 7, Proposition 13 is that every solution (y_{k} ∈ A_{ν}(R), 1 ≤ k ≤ n) of a system

(∗)  ∑_{k=1}^{n} c_{ki}y_{k} = d_{i} (1 ≤ i ≤ m)

of linear equations with coefficients and right-hand sides in k[v^{Φ_{+}}] can be written in the form

y_{k} = x_{k} + ∑_{j=1}^{p} z_{jk}e_{j} (1 ≤ k ≤ n)

where (x_{k}) is a solution of (∗) in k[v^{Φ_{+}}], the e_{j} belong to A_{ν}(R) and the z_{jk} for each fixed j are solutions in k[v^{Φ_{+}}] of the homogeneous system associated to (∗). Writing y_{k} = ∑ y_{kψ}v^{−ψ}, we notice that the elements y_{kψ} with ψ > 0 are such that ν(y_{kψ}) > 0 and therefore if we write

y_{k} = ∑_{ψ≤0} y_{kψ}v^{−ψ} + ∑_{ψ>0} y_{kψ}v^{−ψ} = y_{k}^{≤0} + y_{k}^{>0}

we must have, since the d_{i} are in k[v^{Φ_{+}}] and so have all their coefficients of valuation zero,

a) ∑_{k} c_{ki}y_{k}^{≤0} = d_{i} , b) ∑_{k} c_{ki}y_{k}^{>0} = 0 (1 ≤ i ≤ m).

We may view the system a) as a system of linear equations with coefficients and right-hand side in k[v^{Φ_{+}}], and for which we have a solution y_{k}^{≤0} ∈ R[v^{Φ_{+}}]. By [B3], Chap. 1, §3, No.3, Prop. 5, the k[v^{Φ_{+}}]-module R[v^{Φ_{+}}] is faithfully flat since it is obtained from the faithfully flat map k → R by the base extension k → k[v^{Φ_{+}}]. By the criterion quoted above, we may write

y_{k}^{≤0} = x_{k}^{≤0} + ∑_{j=1}^{p} z_{jk}^{≤0}e_{j}^{≤0} (1 ≤ k ≤ n)

with the same conditions as above except that A_{ν}(R) is replaced by R[v^{Φ_{+}}].

Let us now consider the second system; here we have to deal with restrictions on the valuations of the elements; a solution of this system defines a finite dimensional k-vector subspace V of R, generated by the coefficients y_{kψ}^{>0} of the solutions y_{k}^{>0} = ∑_{ψ} y_{kψ}^{>0}v^{−ψ} of system b). Let us choose a basis (ẽ_{q}^{>0})_{1≤q≤r} of this vector space which is compatible with the finite filtration of V by the vector subspaces P_{ϕ} ∩ V. We can now write

y_{kψ}^{>0} = ∑_{q=1}^{r} w_{kqψ}^{>0}ẽ_{q}^{>0} with w_{kqψ}^{>0} = 0 if ν(ẽ_{q}^{>0}) < ψ,

3In this paper, whenever I introduce a base field k, it will be tacitly assumed that the valuations studied are trivial on k.

and the coefficients w_{kqψ}^{>0} satisfy the system of linear equations

∑ c_{ki}(∑_{ψ} w_{kqψ}^{>0}v^{−ψ}) = 0 ∀q.

This means that we may write

y_{k}^{>0} = ∑_{q}(∑_{ψ} w_{kqψ}^{>0}v^{ν(ẽ_{q}^{>0})−ψ})ẽ_{q}^{>0}v^{−ν(e_{q}^{>0})},

and this is indeed a combination of elements of A_{ν}(R) with coefficients in k[v^{Φ_{+}}] satisfying b). Let us denote by e_{q}^{>0} the product ẽ_{q}^{>0}v^{−ν(ẽ_{q}^{>0})} and by z_{kq}^{>0} the sum

∑_{ψ} w_{kqψ}^{>0}v^{ν(ẽ_{q}^{>0})−ψ} ∈ k[v^{Φ_{+}}]. Adding up the decompositions:

y_{k} = y_{k}^{≤0} + y_{k}^{>0} = x_{k}^{≤0} + ∑_{j=1}^{p} z_{jk}^{≤0}e_{j}^{≤0} + ∑_{q} z_{kq}^{>0}e_{q}^{>0},

we see that we have satisfied the criterion for faithful flatness. □

Remark 2.4 Let (ξ_{i})_{i∈I} be elements of R such that their images (initial forms) in gr_{ν}R generate it as a R/(m_{ν} ∩R)-algebra. Let us denote by S the k-subalgebra of R generated by the elements (ξ_{i})_{i∈I} and let B_{ν}(S) the sub k[v^{Φ_{+}}]-algebra of A_{ν}(S) generated by the elements ξ_{i}v^{−ν(ξ_{i})}. The proof we have just seen can be adapted to show that the k[v^{Φ_{+}}]-algebra B_{ν}(S) is faithfully flat.

The inclusion k[v^{Φ_{+}}] → A defines a map of schemes

SpecA ⟶ Speck[v^{Φ_{+}}].

Part a) of Proposition 2.2 is equivalent to the fact that the special fiber of this map is Specgr_{ν}R and parts b) and c) are ways of stating that its generic and general fibers are isomorphic to SpecR. Since this map is faithfully flat by Proposition 2.3, we can say that we have made R appear as a deformation of its associated graded ring, in a family parametrized by Speck[v^{Φ_{+}}].

Remark 2.5 Without assuming that R contains a field of representatives, we have an isomorphism

A_{ν}(R)/mA_{ν}(R) ≃ k[v^{Φ_{+}}]

corresponding to an inclusion Speck[v^{Φ_{+}}] ⊂ SpecA_{ν}(R). The localization of SpecA_{ν}(R) at the general point corresponding to localization by (v^{Φ_{+}}) induces a general point of this subscheme and produces the “general fiber ” seen above, which is essentially SpecR, while the subscheme of SpecA_{ν}(R) defined by the inverse image in A_{ν}(R) of the ideal (v^{Φ_{+}})k[v^{Φ_{+}}] is Specgr_{ν}R.

The center of the valuation μ_{A} on SpecA_{ν}(R) is the subscheme Specgr_{ν}R; as we saw, this subscheme is integral since ν is a valuation and not a loose valuation.

Let us now compute the associated graded ring of A = A_{ν}(R) with respect to the valuation μ_{A}. As usual I implicitely add {0} to the semigroups. Set P_{ϕ} = P_{ϕ}(R) and

Q_{δ} = {a ∈ A | μ_{A}(a) ≥ δ};

it is a graded ideal generated by the {x_{ϕ}v^{−ϕ} | ν(x_{ϕ}) − ϕ ≥ δ}, and similarly for Q_{δ}^{+}. So we have

Q_{δ} = A ∩ ⨁_{ϕ∈Φ} P_{ϕ+δ}v^{−ϕ},

Q_{δ}^{+} = A ∩ ⨁_{ϕ∈Φ} P_{ϕ+δ}^{+}v^{−ϕ}.

So Q_{δ} = A for δ ≤ 0 and we can write:

gr_{μ_{A}}A = ⨁_{δ∈Φ_{+}∪{0}}(Q_{δ}/Q_{δ}^{+}) = ⨁_{δ∈Φ_{+}}( ⨁_{{ϕ|ϕ+δ∈Φ_{+}∪{0}}} P_{ϕ+δ}/P_{ϕ+δ}^{+}).

Thus gr_{μ_{A}}A has a Φ_{+} ⊕ Φ_{+}-graduation by (δ, ϕ + δ) if we give degree ϕ + δ to the elements of P_{ϕ+δ}/P_{ϕ+δ}^{+} as suggested by the graduation of A.

Let us denote by \overline{R} the quotient R/(m_{ν} ∩ R), and remark that gr_{ν}R is an \overline{R}-algebra.

We have a map of \overline{R}-algebras

gr_{μ_{A}}A → (gr_{ν}R) ⊗\overline{R} \overline{R}[v^{Φ_{+}}]

induced by the map Q_{δ} → (gr_{ν}R) ⊗\overline{R} \overline{R}[v^{Φ_{+}}] defined by:

∑_{ν(x_{ϕ})−ϕ≥δ} x_{ϕ}v^{−ϕ} ↦ ∑_{ν(x_{ϕ})−ϕ=δ} \overline{x_{ϕ}} ⊗ v^{δ},

where \overline{x_{ϕ}} is the initial form in P_{ϕ+δ}/P_{ϕ+δ}^{+} of x_{ϕ}. This map is homogeneous with respect to the Φ_{+} ⊕ Φ_{+} graduation of gr_{μ_{A}}A described above if we give (gr_{ν}R) ⊗\overline{R} \overline{R}[v^{Φ_{+}}] the tensor product graduation where elements of degree δ ⊕ ψ ∈ Φ_{+} ⊕ Φ_{+} are tensor products with v^{δ} of elements of degree ψ of gr_{ν}R.

Proposition 2.6 With the gradings just described, this map is a (Φ_{+} ⊕ Φ_{+})- graded isomorphism of \overline{R}-algebras

gr_{μ_{A}}A →^{≃} (gr_{ν}R) ⊗\overline{R} \overline{R}[v^{Φ_{+}}].

Proof The map is clearly surjective, and its injectivity is easily verified degree by degree. Note that upon taking ϕ + δ = 0, this map induces the identity of \overline{R}[v^{Φ_{+}}]. Note also that the valuation μ_{A} takes its positive values in Φ_{+}, and not in Γ. □

Remark 2.7 The meaning of this Proposition is that the deformation of graded \overline{R}-algebras corresponding as in Proposition 2.2 to the valuation algebra associated to A and to μ_{A} is “trivial” in the sense that the total space Specgr_{μ_{A}}A is the product by the base space Spec\overline{R}[v^{Φ_{+}}] of the special fiber Specgr_{ν}R.

2.4 The valuation ν_{A}. There is another natural valuation on the ring A_{ν}(R): it is the valuation ν_{A} defined by

ν_{A}(∑ x_{ϕ}v^{−ϕ}) = min(ν(x_{ϕ})).

Its center is the subscheme of SpecA_{ν}(R) defined by the ideal of A_{ν}(R) generated by (m_{ν} ∩ R, ⨁_{ϕ∈Φ_{+}} P_{ϕ}v^{−ϕ}). This subscheme is isomorphic to Spec\overline{R}[v^{Φ_{+}}]. In the special case where R is a local k-algebra dominated by R_{ν} and the extension k → R/m is trivial, this means that the center of ν_{A} is the image of a section

σ: Speck[v^{Φ_{+}}] → SpecA_{ν}(R).

Let us examine the valuation algebra of A_{ν}(R) with respect to ν_{A}. Set

S_{δ} = {∑ x_{ϕ}v^{−ϕ} ∈ A_{ν}(R) | min(ν(x_{ϕ})) ≥ δ} = ⨁_{ϕ∈Φ} P_{max(δ,ϕ)}(R)v^{−ϕ},

S_{δ}^{+} = {∑ x_{ϕ}v^{−ϕ} ∈ A_{ν}(R) | min(ν(x_{ϕ})) > δ} = ⨁_{ϕ∈Φ}(P_{δ}^{+}(R) ∩ P_{ϕ}(R))v^{−ϕ};

we have the equalities

gr_{ν_{A}}A_{ν}(R) = ⨁_{δ∈Φ} S_{δ}/S_{δ}^{+} = ⨁_{ϕ∈Φ}(⨁_{δ≥ϕ}(P_{δ}(R)/P_{δ}^{+}(R))).

Let us define the “degree algebra” of a graded algebra H = ⨁_{δ∈Φ_{+}∪{0}} H_{δ} in a way similar to the definition of the valuation algebra, replacing valuation by degree, and setting H_{δ} = 0 for δ ∈ Φ_{−}:

A_{deg}(H) = ⨁_{ϕ∈Φ}(⨁_{δ≥ϕ} H_{δ})v^{−ϕ} ⊂ H[v^{Φ}].

This algebra is “trivial” in view of the:

Lemma 2.8 The map

A_{deg}(H) → H ⊗_{H_{0}} H_{0}[v^{Φ_{+}}]

determined by

∑_{ϕ}(∑_{δ≥ϕ} x_{δ})v^{−ϕ} ↦ ∑_{ϕ,δ∈Φ, δ≥ϕ} x_{δ} ⊗_{H_{0}} v^{δ−ϕ}

is a graded isomorphism of H_{0}-algebras when one gives the right hand side the twisted Φ ⊕ Φ_{+} graduation for which a homogeneous element of degree δ ⊕ ψ is of the form x_{δ} ⊗ v^{δ−ψ} where x_{δ} is of degree δ.

Proof Indeed, the inverse map is given by x_{δ} ⊗ v^{ψ} ↦ x_{δ}v^{−(δ−ψ)}. □

In order to show that the valuation algebra gr_{ν_{A}}A_{ν}(R) is “trivial”, as in the case of the valuation μ_{A}, it only remains to check the:

Lemma 2.9 If we view gr_{ν}(R) as a graded algebra indexed by Φ_{+} ∪ {0} with a 0 component for ϕ ∉ Γ, we have the equality:

gr_{ν_{A}}A_{ν}(R) = Adeg(gr_{ν}(R)).

Proof It follows directly from the definitions. □

Finally, we have proved

Proposition 2.10 With the twisted grading on (gr_{ν}R) ⊗\overline{R} \overline{R}[v^{Φ_{+}}] described above, the map of Lemma 2.8 is a graded isomorphism of graded \overline{R}-algebras

gr_{ν_{A}}A_{ν}(R) →^{≃} (gr_{ν}R) ⊗\overline{R} \overline{R}[v^{Φ_{+}}],

determined by the fact that the ν_{A}-initial form of ξv^{−ν(ξ)} is mapped to \overline{ξ} ⊗ 1, where \overline{ξ} is the ν-initial form of ξ. Note that

A_{ν}(R)/S_{0} = \overline{R}[v^{Φ_{+}}] = \overline{R} ⊗\overline{R} \overline{R}[v^{Φ_{+}}].

Remarks 2.11 1) The isomorphisms of the two graded algebras of A_{ν}(R) with respect to μ_{A} and ν_{A} with (gr_{ν}R) ⊗\overline{R} \overline{R}[v^{Φ_{+}}] have quite different geometric significations, as one can see by considering the analogous constructions for the Rees algebra R(I) = ⨁_{n∈Z} I^{n}v^{−n} ⊂ R[v, v^{−1}] of an ideal I of a ring R, where I^{n} = R for n ≤ 0. The valuation μ corresponds to the v-adic filtration; it is the (loose) valuation associated to this filtration, while the valuation ν corresponds to the (loose) valuation induced by the I-adic filtration of R[v, v^{−1}]. The triviality of the gr_{μ_{A}}A_{ν}(R) generalizes the fact that since the special fiber Specgr_{I}R in this case is a divisor, its normal cone in SpecR(I) is Spec(gr_{I}R)[v]. The triviality for ν_{A} generalizes the fact, more specific to the Rees algebra, that the graded ring of R(I) with respect to the I-adic filtration is trivial in the sense that it is isomorphic to gr_{I}R ⊗_{R/I} R/I[v]. This reflects the fact that in the specialization of SpecR to Specgr_{I}R provided by SpecR(I), the graded ring is constant, or more precisely that the induced specialization on the graded ring is trivial.

2) We shall see below in subsection 3.2 that there is a valuation ν̃_{A} on A_{ν}(R) which is composed with the natural valuation ν_{gr} on the graded ring gr_{ν}R. Moreover, this valuation ν̃_{A} induces in a natural way the valuation ν on R. This will gives a precise meaning to the intuition that ν_{gr} is the specialization of ν to gr_{ν}R.

3) The graded ring gr_{ν_{gr}}gr_{ν}R is isomorphic to gr_{ν}R.

An example: Let R be a discrete valuation ring, and u a generator of its maximal ideal. The value group of the valuation of R is Z and we have P_{n}(R) = u^{n}R for n ≥ 0, P_{n}(R) = R for n ≤ 0, so that

A_{ν}(R) = R[v, uv^{−1}] ⊂ R[v, v^{−1}].

In this case, gr_{ν}R = (R/uR)[U], where U is the initial form of u. I leave it as an exercise to determine the maps A_{ν}(R) → R and A_{ν}(R) → gr_{ν}R in this case (hint: map uv^{−1} to U). This applies in particular when R = C{u}, as we shall see in subsection 5.6, or to k[[u]] where k is any field. It also applies to the nonequicharacteristic case, for example to R = Ẑ_{p}, the ring of p-adic integers, and more generally when R = W (k) is the ring of Witt vectors associated to a perfect field k of characteristic p, with u = p (see [B3], Chap. IX, §§1,2, [Ei], §7). We shall see below in subsection 5.3 that the Spectrum of the p-adic completion of A_{ν}(W (k)) contains both Speck[[t]] and SpecW (k). This is the very first step in the extension of the method presented here to the non-equicharacteristic case.

3 Valuation algebras and composition of valuations

In this section I examine the behavior of valuation algebras when one changes a valuation for an other one with which it is composed, or for the corresponding residual valuation. This section also contains two results which are very useful in the sequel: Zariski’s description of valuation ideals with respect to two consecutive valuations, and Piltant’s theorem on the Krull dimension of gr_{ν}R.

3.1 The composition of valuations. In this paragraph, we study the relation of composition of valuations; if ν is composed with ν_{1}, both non-negative on our local ring R, it means that we have inclusions of local rings

R ⊂ R_{ν} ⊂ R_{ν_{1}},

and of ideals

m_{ν_{1}} ⊂ m_{ν} ⊂ R_{ν},

m_{ν_{1}} ∩ R ⊆ m_{ν} ∩ R ⊆ m.

I assume known the theory of composition of valuations, referring to ([V1], section 4, [Z-S], Vol 2, Ch. VI, [A1], [B2]), and fix notations.

Let us recall that the rank r(Φ) of the Q-vector space Q ⊗_{Z} Φ is called the rational rank of Φ; if Φ is the group of a valuation ν, it is also denoted by r(ν). Recall also that in the case where R_{ν} dominates R, the transcendence degree t_{k_{R}}(ν) of the extension (k_{ν} : k_{R}) is called the residual transcendence degree, and that the height h(ν) (often also called the rank) of the valuation ν is the number (in general the ordinal type) of elements of a maximal chain

R_{ν} ⊂ R_{ν_{1}} ⊂ ⋯ ⊂ R_{ν_{h−1}}

of valuation rings of K containing R_{ν}; it is also the Krull dimension of the ring R_{ν}. Each inclusion R_{ν} ⊂ R_{ν_{i}} corresponds to a monotone (non-decreasing) map of totally ordered groups λ_{i}: Φ → Φ_{i} such that ν_{i} = λ_{i} ∘ ν. A monotone nondecreasing map means here that if ϕ ≤ ψ, then λ(ϕ) ≤ λ(ψ). The kernel Ψ_{i} of such a map is an isolated (or convex) subgroup of Φ, so that h(ν) is also the ordinal type of the chain of isolated subgoups of Φ; it depends only on Φ and we will speak of the height of Φ.

A totally ordered group of height one is archimedian and can be embedded as an ordered subgroup of R. Following [A1], we will say that a group of height one is discrete if it is well ordered (i.e., isomorphic to Z, see [A1]), and we will say that a group Φ of finite height is discrete if the quotient of two successive elements of the sequence

(0) ⊂ Ψ_{h−1} ⊂ ⋯ ⊂ Ψ_{1} ⊂ Φ

of the isolated subgroups of Φ is discrete.

The additivity of the rational rank in exact sequences implies the inequality

h(ν) ≤ r(ν)

and for a valuation ring R_{ν} birationally dominating a nœtherian local ring R, we have Abhyankar’s inequality (see [V1], section 9, Th. 9.2)

r(ν) + t_{k_{R}}(ν) ≤ dimR.

It is important to note that given a valuation of height ν of a ring R, the prime ideals of R_{ν} form a chain

m_{h(ν)−1} ⊂ m_{h(ν)−2} ⊂ ⋯ ⊂ m_{ν_{1}} ⊂ m_{ν}

of length h(ν) but their intersections with R are not necessarily distinct, and the length of the chain of distinct prime ideals among the

m_{h(ν)−1} ∩ R ⊆ ⋯ ⊆ m_{ν_{1}} ∩ R ⊆ m_{ν} ∩ R,

which may be called the height of ν in R and denoted by h_{R}(ν), is at most h(ν). Abhyankar’s inequality has a nice interpretation in terms of the graded algebra:

Proposition 3.1 (Piltant, [P]) If R is a nœtherian local domain with uncountable residue field k_{R}, birationally dominated by the valuation ring R_{ν}, the Krull dimension of gr_{ν}R is equal to r(ν) + t_{k_{R}}(ν). If t_{k_{R}}(ν) = 0, the assumption on the cardinality is unnecessary.

In the cases where k_{R} is uncountable, Abhyankar’s inequality then becomes

dimgr_{ν}R ≤ dimR.

Note that if we have R ⊂ R_{ν}, the local ring R_{ν} dominates R_{m_{ν}∩R}. Spivakovsky had remarked that the transcendence degree over R/m of gr_{ν}R is equal to r(ν) + t_{k_{R}}(ν) (see [S3], Remark 3.7).

Remark. I will use Piltant’s result only in the case where t_{k_{R}}(ν) = 0. For the convenience of the reader, I sketch Piltant’s proof of his result in this case. In the beginning, I suppose only that m_{ν} ∩ R = m, i.e. that the valuation ring R_{ν} dominates R. Let S be the multiplicative set of homogeneous elements of positive degree in gr_{ν}R, and let [F]_{0} denote the part of degree zero of an homogenous ring of fractions F of a graded algebra by a multiplicative set of homogeneous elements.

Lemma 3.2 The map

[S^{−1}gr_{ν}R]_{0} → k_{ν}, \begin{matrix}\overline{x} & ϕ \\ \overline{y} & ϕ \end{matrix} ↦ \frac{x_{ϕ}}{y_{ϕ}} mod.m_{ν}

is well defined and is an isomorphism.

Proof This follows from a direct verification. □

Let now (u_{1}, …, u_{r}) be elements of R whose valuations (δ_{1}, …, δ_{r}) rationally generate the group Φ, where r is the rational rank r(ν) of Φ; this means that (δ_{1}, …, δ_{r}) form a basis of the Q-vector space Φ ⊗_{Z} Q. Let Φ^{(u)} = Zδ_{1} ⊕ ⋯ ⊕ Zδ_{r} ⊆ Φ be the subgroup generated by the δ_{i}; set Φ_{+}^{(u)} = Φ^{(u)} ∩ Φ_{+} and define the subalgebra

gr_{ν}^{(u)}R = ⨁_{ϕ∈Φ_{+}^{(u)}} P_{ϕ}(R)/P_{ϕ}^{+}(R).

Let us denote by S^{(u)} the intersection S ∩ gr_{ν}^{(u)}R and by k^{(u)} the induced field [(S^{(u)})^{−1}gr_{ν}^{(u)}R]_{0}.

Lemma 3.3 The extensions gr_{ν}^{(u)}R → gr_{ν}R and k^{(u)} → k_{ν} are integral.

Proof By [B3], Chap.V, No.1, Cor. to Prop.4, it suffices to check that any homogeneous element \overline{x}_{ϕ} ∈ gr_{ν}R has a power which is gr_{ν}^{(u)}R. Since Φ^{(u)} rationally generates Φ, there is an integer n such that nϕ ∈ Φ^{(u)}, so that \overline{x}^{n} ∈ gr_{ν}^{(u)}R. □

Given an element ϕ ∈ Φ^{(u)}, it can be written uniquely as

ϕ = a_{1}δ_{1} + ⋯ + a_{r}δ_{r} with a_{i} ∈ Z.

Lemma 3.4 With the notation just introduced, the map

ι_{u}: gr_{ν}^{(u)}R → k^{(u)}[t^{Φ_{+}^{(u)}}], \overline{x}_{ϕ} ↦ \begin{matrix}\overline{x} & ϕ \\ \overline{u} & {a_{1} \atop 1} & … & \overline{u} & {a_{r} \atop r} \end{matrix} t^{ϕ}

induces an isomorphism

(S^{(u)})^{−1}gr_{ν}^{(u)}R ⟶^{≃} k^{(u)}[x_{1}^{±1}, …, x_{r}^{±1}].

If k_{ν} is algebraic over k_{R}, the map ι_{u} is integral.

Proof For the first part, since Φ^{(u)} is a free abelian group of rank r, it suffices to show that (S^{(u)})^{−1}gr_{ν}^{(u)}R → k^{(u)}[t^{Φ^{(u)}}] is an isomorphism. Remarking that ι_{u}(S^{(u)}) ⊆ {λt^{γ}, γ ∈ Φ_{+}^{(u)}, λ ∈ k^{(u)}^{∗}}, we see that (S^{(u)})^{−1}gr_{ν}^{(u)}R ⊂ k^{(u)}[t^{Φ^{(u)}}]. The surjectivity follows from the fact that Φ_{+}^{(u)} rationally generates Φ^{(u)} = Z^{r}. If the extension k_{R} → k_{ν} is algebraic, so is k_{R} → k^{(u)}. To prove that ι_{u} is integral, it suffices to show that any element λt^{γ} with λ ∈ k^{(u)}, γ ∈ Φ^{(u)}, is integral. This follows from the fact that λ is algebraic over k_{R}. □

From these lemmas we deduce —————————————————————— ————————————————————————

Proposition 3.5 (Spivakovsky, Piltant) The transcendence degree over k_{R} of gr_{ν}R is equal to r(ν) + t_{k_{R}}(ν).

Proof Because the extension gr_{ν}^{(u)}R → gr_{ν}R is integral by Lemma 3.3, this transcendence degree is equal to that of gr_{ν}^{(u)}R, hence to that of (S^{(u)})^{−1}gr_{ν}^{(u)}R, which is that of k^{(u)}[t^{Φ^{(u)}}], and also of k_{ν}[t^{Φ^{(u)}}] since the extension k^{(u)} → k_{ν} is integral by Lemma 3.3. □

Now Piltant uses the following corollary to Cohen’s dimension inequality:

Proposition 3.6 Let B be an integral domain containing a nœtherian ring A. Then

dimB ≤ dimA + t_{A}(B),

where t_{A}(B) is the transcendence degree of B over A.

Proof The proof is like the classical one (see [Ei], Chap. 13) except that we do not suppose that B is finitely generated over A. But if (0) = p_{0} ⊂ p_{1} ⊂ p_{2} ⊂ ⋯ ⊂ p_{d} is a chain of distinct prime ideals of B, choosing for each i, 1 ≤ i ≤ d an element y_{i} ∈ p_{i} \ p_{i−1}, we see that the p_{i} ∩ A[y_{1}, …, y_{d}] form a chain of distinct prime ideals, so that dimA[y_{1}, …, y_{d}] ≥ d, and we are reduced to the case of a finitely generated algebra. □

Applying this result with A = k_{R} and B = gr_{ν}R, remarking that we have used the nœtherianity of R only to ensure that r(ν) + t_{k_{R}}(ν) < ∞ and using Proposition 3.5 gives the first part of the: ———————————————————————— ——————————————————————

Proposition 3.7 (Piltant, [P]) Whenever the valuation ring R_{ν} birationally dominates the local ring R and r(ν) + t_{k_{R}}(ν) < ∞, the following inequality holds:

dimgr_{ν}R ≤ r(ν) + t_{k_{R}}(ν).

If t_{k_{R}}(ν) = 0,

dimgr_{ν}R = r(ν).

Proof To prove the second part, we use the lemmas above: the map

(S^{(u)})^{−1}gr_{ν}^{(u)}R → k^{(u)}[x_{1}^{±1}, …, x_{r}^{±1}]

is an isomorphism, so these two rings have the same dimension r(ν). Since dimension cannot increase by localization, this gives dimgr_{ν}^{(u)}R ≥ r(ν) and finally, because the extension gr_{ν}^{(u)}R → gr_{ν}R is integral, using ([B3], Chap. VIII, No.3. Th.1), dimgr_{ν}R ≥ r(ν). But we have the opposite inequality by the first part of the proposition. □

Remark 3.8 Piltant’s result implies that if r(ν) < ∞ we have:

dimgr_{ν}R_{ν} = r(ν)

and if t_{k_{R}}(ν) = 0,

dimgr_{ν}R = t_{k_{R}}(gr_{ν}R),

the transcendence degree of gr_{ν}R over k_{R}. I will use this last equality in several examples. The first one should be contrasted with the fact that dimR_{ν} = h(ν) ≤ r(ν).

Question. Does the proof in the case where t_{k_{R}}(ν) = 0 also show that gr_{ν}R is catenary? The proof of the fact that equality in the inequality of Proposition 3.7 continues to hold without the assumption that t_{k_{R}}(ν) = 0 provided that k_{R} is uncountable is more delicate. Piltant has given in [P] an example of a valuation ν on the ring R = \overline{Q}[s, t, x, y]_{(s,t,x,y)} with value group Z_{lex}^{2} and residue field \overline{Q}(\frac{s}{t}) such that gr_{ν}R has Krull dimension 3 = r(ν) + t_{k_{R}}(ν) but is not catenary; it possesses a saturated chain of prime ideals of length two. It would be interesting to decide whether the completion ĝr_{ν}^{(ν)}R with respect to ν_{gr} which we shall meet in subsection 5.3 is catenary. Strict inequality in Abhyankar’s inequality may happen, but this does not contradict the semi-continuity theorem for the dimensions of the fibers of the map SpecA_{ν}(R) → Speck[v^{Φ_{+}}] since semicontinuity is proved only for morphisms which are essentially of finite type. Under the assumptions of Piltant’s theorem, for A_{ν}(R) to be a k[v^{Φ_{+}}]-algebra of finite type it is necessary that equality holds in Abhyankar’s inequality.

Since R is assumed to be nœtherian we will in this text remain in the situation where r(Φ), t_{k_{R}}(ν), etc.. are finite.

Recall (see [B3], Chap. VI, §10, No. 2) that if Φ is finitely generated and if r(ν) = h(ν), then Φ is order isomorphic to Z^{h(ν)} with the lexicographic order.

I will often make use of the following fact:

Proposition 3.9 (Krull, [Kr], Zariski; see ([Z-S], Appendix 3, [S2], remark 2.7, [V1], Prop. 9.1.) If R is nœtherian, the value semigroup Γ = ν(R\{0}) ⊂ Φ_{+} ∪{0} is well ordered and its ordinal type (see [B5]) is ω^{h_{R}(ν)}, where h_{R}(ν) is the height in R of the valuation ν.

Proof Zariski’s proof is in terms of valuation ideals of R, which are the distinct P_{ϕ}(R) and therefore in monotonous bijection with the elements of Γ by the map γ ↦ P_{γ} for the reverse inclusion ordering on ideals; a decreasing sequence of elements of Γ gives an increasing sequence of ideals P_{ϕ}(R), which has to be stationary since R is nœtherian. The second part of the statement follows from [Z-S], Appendix 3, Proposition 2. □

Corollary 1 For a valuation ν of a nœtherian ring R, the value semigroup admits a unique minimal system of generators

Γ = ⟨γ_{1}, γ_{2}, …, γ_{i}, …⟩

satisfying

γ_{i} < γ_{i+1}.

It is indexed by ordinals < ω^{h_{R}(ν)}. If the height of ν in R is one, in particular if the height of ν is one, given γ ∈ Γ, there are finitely many generators γ_{i} ≤ γ.

Proof Define γ_{1} to be the smallest non zero element of Γ, and then inductively γ_{i+1} to be the smallest non zero element of Γ which is not in the semigroup

Γ_{i} = ⟨γ_{1}, γ_{2}, …, γ_{i}⟩

generated by the previous ones. If after finitely many steps, Γ_{i} = Γ, our semigroup is finitely generated; otherwise, the elements obtained after iterating the successor construction countably many times generate a semigroup Γ_{<ω} ⊆ Γ. If it is not equal to Γ, let γ_{ω} be the smallest non zero element of Γ not contained in Γ_{<ω}, set Γ_{ω} = ⟨Γ_{<ω}, γ_{ω}⟩ and continue in this manner. This is clearly a minimal system of generators of Γ. If for some i we had γ_{i+1} < γ_{i} it would contradict the definition of γ_{i} since γ_{i+1} ∉ ⟨γ_{1}, …, γ_{i−1}⟩. The second assertion also follows [Z-S], Appendix 3, Lemma 4. □

Viewing a valuation as a map ν: K^{∗} → Φ as in subsection 2.1, we see that a surjective monotone non-decreasing map of ordered groups λ: Φ → Φ_{1} corresponds to a composite valuation ν_{1} = λ ∘ ν and another valuation ring R_{ν_{1}} ⊃ R_{ν}. However the classical terminology, referring to a geometric notion of composition, is that ν is composed with ν_{1} and we will respect it. The map λ extends to a map

λ̃: k[v^{Φ_{+}}] → k[w^{Φ_{1+}}].

We have a map A_{ν}(R) → A_{ν_{1}}(R) and in fact a commutative diagram

λ̃_{A}(R): A_{ν}(R) ⟶ A_{ν_{1}}(R)
↑  ↑
λ̃:  k[v^{Φ_{+}}] ⟶ k[w^{Φ_{1+}}]

where the top arrow is induced by the inclusions P_{ϕ}(R) ⊂ P_{λ(ϕ)}(R), i.e., x_{ϕ}v^{−ϕ} ↦ x_{ϕ}w^{−λ(ϕ)}, and the bottom arrow is λ̃. The map λ̃ is surjective since λ: Φ → Φ_{1} is order preserving, and its kernel is the ideal generated by the (v^{ψ} − 1)_{ψ∈Ψ_{+}} where Ψ is the kernel of the map Φ → Φ_{1}. Let us show that λ̃_{A}(R) is also surjective: given xw^{−μ}, where μ = λ(ϕ) and ν_{1}(x) ≥ μ, either ν(x) ≥ ϕ and xw^{−μ} is the image of xv^{−ϕ}, or we have not chosen ϕ well, but since λ is order preserving, if ν(x) < ϕ, we have ν_{1}(x) = λ(ν(x)) ≤ λ(ϕ) = μ so that λ(ν(x)) = μ and ϕ − ν(x) ∈ Ψ. Finally, we see that xw^{−μ} is the image of xv^{−ν(x)} and we have shown the surjectivity. The kernel is the ideal of A_{ν} generated by (v^{ψ} − 1)_{ψ∈Ψ_{+}}.

Finally, we have shown that the formation of k[v^{Φ_{+}}] → A_{ν}(R) commutes with the composition of valuations, as follows;

Proposition 3.10 The natural morphism

\begin{matrix}A_{ν}(R) ⊗_{k[v^{Φ_{+}} ]} k[w^{Φ_{1+}}] & → A_{ν_{1}}(R) \\ x_{ϕ}v^{−ϕ} ⊗ w^{ϕ_{1}} & ↦ x_{ϕ}w^{ϕ_{1}−λ(ϕ)} \end{matrix}

is an isomorphism of graded k[v^{Φ_{+}}]-algebras.

Remark 3.11 The fact that

A_{ν_{1}}(R) = A_{ν}(R)/((v^{ψ} − 1)_{ψ∈Ψ_{+}})

is true without assuming that R contains a field.

Corollary 2 The rings gr_{ν}R and gr_{ν_{1}}R are both specializations of the same ring over the polynomial ring k[(t_{ψ})_{ψ∈Ψ_{+}}].

Proof The ring gr_{ν_{1}}R is isomorphic to the quotient ring of A_{ν}(R) by ((v^{ψ} − 1)_{ψ∈Ψ_{+}}, (v^{ϕ})_{ϕ∈Φ_{+}\Ψ_{+}})), and gr_{ν}R = A_{ν}(R)/((v^{ϕ})_{ϕ∈Φ_{+}}). These rings are specializations of

A_{ν}(R) ⊗_{k} k[(t_{ψ})_{ψ∈Ψ_{+}}]/((v^{ψ} − t_{ψ})_{ψ∈Ψ_{+}}, (v^{ϕ})_{ϕ∈Φ_{+}\Ψ_{+}})).
□

Remarks 3.12 1) If we have a sequence of valuation rings as above, with R ⊂ R_{ν},

R_{ν} ⊂ R_{ν_{1}} ⊂ ⋯ ⊂ R_{ν_{h−1}},

where h is the height of the valuation ν, all the graded algebras gr_{ν_{s}}R are specializations of the same algebra of the form

A_{ν}(R) ⊗_{k} k[t_{ω}]/((v^{ω} − t_{ω})_{ω∈Ω_{1}}, (v^{ω})_{ω∈Ω_{2}}),

and we can in a number of cases use a system of generators of gr_{ν}R to produce a system of “formal” generators for each gr_{ν_{s}}R (see below).

2) Denoting by λ: Φ → Φ_{1} the morphism of value groups, notice that the graph (ϕ, λ(ϕ)) ⊂ Φ × Φ_{1} is isomorphic to Φ and that we have an isomorphism of graded algebras:

A_{ν}(R) →^{≃} ⨁_{ϕ_{1}∈Φ_{1}}( ⨁_{λ(ϕ)=ϕ_{1}} P_{ϕ}(R)w^{−ϕ})w_{1}^{−ϕ_{1}}.

3.2 The valuation ν̃_{A}. Let us denote by ν̃_{A} the valuation on A_{ν}(R) with values in the group Φ ⊕ Φ ordered lexicographically, and defined by

ν̃_{A}(∑ x_{ϕ}v^{−ϕ}) = (μ_{A}(∑ x_{ϕ}v^{−ϕ}), ν_{A}(∑ x_{ϕ}v^{−ϕ})).

According to what we saw in §1, the center of this valuation is defined by the ideal ((v^{Φ_{+}})A_{ν}(R), ⊕_{ϕ∈Φ_{+}}P_{ϕ}v^{−ϕ}) of A_{ν}(R). Geometrically, this center is Spec\overline{R}.

This valuation is composed with μ_{A} in the manner corresponding to the first projection Φ ⊕ Φ → Φ, and the residual valuation of ν̃ on the center gr_{ν}R of μ_{A} is ν_{gr}.

This valuation is of interest because it is a “specialization” of the valuation ν on R to the valuation ν_{gr}. Indeed, the valuation ν_{A} extends to a valuation of (v^{Φ_{+}})^{−1}A_{ν}(R) = (v^{Φ_{+}})^{−1}R[v^{Φ_{+}}] (Proposition 2.2, b)) which is the algebra of the “generic fiber” of the faithfully flat map defined in §2.3, and as we saw it induces the natural valuation on the special fiber. Heuristically, we may think of the inclusion A_{ν}(R) → A_{ν}(R_{ν}) as a uniformization of the valuation ν̃_{A} (see below, §4), forgetting for a moment the fact that it is not an extension of finite type. The center of ν̃_{A} in A_{ν}(R_{ν}) is the origin, and, still heuristically, A_{ν}(R_{ν}) is “regular”. Then the corresponding map of schemes is a simultaneous uniformization for ν and ν_{gr} since the induced map gr_{ν}R → gr_{ν}R_{ν} is also (in the same heuristic sense) a uniformization of ν_{gr}.

3.3 The valuation algebra of the residual valuation. Let us now come back to the inclusion R_{ν} ⊂ R_{ν_{1}} associated to an monotone non-decreasing surjection λ: Φ → Φ_{1} of ordered groups, with kernel Ψ; it is a classical fact of valuation theory (see [A1], pp. 56-57 and [V1]) that m_{ν_{1}} ⊂ R_{ν} and the image \overline{R}_{ν} = R_{ν}/m_{ν_{1}} ⊂ R_{ν_{1}}/m_{ν_{1}} is a valuation ring of the field R_{ν_{1}}/m_{ν_{1}}, corresponding to a valuation \overline{ν} with value group Ψ and center m_{ν} ∩ R/m_{ν_{1}} ∩ R in R/m_{ν_{1}} ∩ R.

If we endeavor to find the valuation algebra of \overline{R}_{1} = R/(m_{ν_{1}} ∩ R) with respect to \overline{ν}, we see that if we denote by A_{Ψ}(R) the subalgebra ⨁_{ϕ∈Ψ} P_{ϕ}(R)v^{−ϕ} of A_{ν}(R), then A\overline{ν}(\overline{R}_{1}) is nothing else than the image of A_{Ψ}(R) in \overline{R}_{1}[v^{Φ}] by the canonical map R[v^{Φ}] → \overline{R}_{1}[v^{Φ}]. It is important to note that, setting Γ = ν(R \ \{0\}), the semigroup Γ ∩ Ψ may be zero. This happens if and only if m_{ν_{1}} ∩ R = m_{ν} ∩ R; in that case the valuation \overline{ν} is trivial on \overline{R}_{1}.

A typical example is the valuation on R = k[u_{1}, u_{2}]_{(u_{1},u_{2})} with values in Z^{2} ordered lexicographically determined by setting ν(u_{1}) = (1, 0), ν(u_{2}) = (1, 1) and deciding that the valuation of a polynomial is the smallest valuation of the monomials which it contains; it is composed with the (u_{1})-adic valuation, Γ is the semigroup generated by (1, 0), (1, 1) and Ψ is the subgroup 0 ⨁ Z. Note that the two valuations have the same center in R and that for a given ϕ ∈ Φ_{+} there are only finitely many ideals P_{ϕ′} between P_{λ(ϕ)} and P_{λ(ϕ)}^{+}.

3.4 Comparison of graded algebras for composed valuations. The result of Proposition 3.10 suggests that we seek a closer relation between gr_{ν}R and gr_{ν_{1}}R when ν is composed with ν_{1}. First, let us remark that with the usual notations, we have for ϕ ∈ Φ_{+} ∪ {0} the following inclusions^{4}

P_{λ(ϕ)}^{+} ⊂ P_{ϕ}^{+} ⊂ P_{ϕ} ⊂ P_{λ(ϕ)},

so that we get a filtration of each (R/m_{ν_{1}} ∩ R)-module

(gr_{ν_{1}}R)_{ϕ_{1}} = \frac{P_{ϕ_{1}}}{P_{ϕ_{1}}^{+}}

by the submodules (P_{ϕ}/P_{λ(ϕ)}^{+})_{λ(ϕ)=ϕ_{1}}. In the case where ϕ_{1} = 0, it is nothing else than the filtration of \overline{R}_{1} = R/(m_{ν_{1}} ∩ R) by the images of the (P_{ϕ})_{ϕ∈Ψ} where Ψ = Kerλ, i.e., the filtration corresponding to the ideals (\overline{P}_{ψ})_{ψ∈Ψ} associated to the residual valuation \overline{ν} on \overline{R}_{1}. Let us denote by \overline{P}(ϕ_{1}) this filtration, for each ϕ_{1} ∈ Φ_{1+} ∪ {0}. We have immediately

Lemma 3.13 The natural map

P_{ϕ} → \frac{P_{ϕ}}{P_{λ(ϕ)}^{+}}

induces an isomorphism

gr_{ν}R →^{≃} ⨁_{ϕ_{1}∈Φ_{1+}∪\{0\}} gr_{\overline{P}(ϕ_{1})}(gr_{ν_{1}}R)_{ϕ_{1}}.

It is graded in the sense that an element of degree ϕ is mapped to an element of bidegree (ϕ, λ(ϕ)). Moreover, for x ∈ R, we have

in_{ν}x = in_{\overline{P}(ν_{1}(x))}(in_{ν_{1}}x).

4The fact that λ(ϕ) ∈ Φ_{1} is deemed sufficient, here and in the sequel, to indicate that the first and last ideals correspond to the valuation ν_{1}.

In particular, taking λ(ϕ) = 0, we find that it induces a graded isomorphism

⨁_{ψ∈Ψ_{+}∪\{0\}}(gr_{ν}R)_{ψ} →^{≃} gr\overline{ν} \overline{R}_{1},

which can also be deduced from 3.3. Note that when R is nœtherian, and so Γ is well ordered by Proposition 3.9, the topology on R/(m_{ν_{1}} ∩ R) corresponding to the \overline{ν}-adic filtration is the same as the image by the natural surjection R → R/(m_{ν_{1}} ∩R) of the ν-adic topology on R. This follows from the inclusions P_{ψ}(R) ⊃ m_{ν_{1}} ∩ R for ψ ∈ Ψ. Thus, the image of P_{ψ}(R) in R/(m_{ν_{1}} ∩ R) is the valuation ideal \overline{P}_{ψ}(\overline{R}) for \overline{ν}. Note that it may be the trivial topology; more generally, if ν and ν_{1} have the same center in R, the filtrations \overline{P}(ϕ_{1}) are finite ([Z-S], Vol. II, Appendix 3, Corollary). I will make considerable use of the following:

Proposition 3.14 (Zariski) With the same notations, assume ν is a valuation on the nœtherian local domain R and that the valuation ν_{1} has height one less than ν, so that \overline{ν} is of height one. Given ϕ_{1} ∈ Φ_{1,+}, we have, whenever λ(ϕ) = ϕ_{1} (see Lemma 3.13)

P_{ϕ_{1}}^{+} ⊆ P_{ϕ} ⊆ P_{ϕ_{1}}.

There are only finitely many ideals P_{ϕ′} between P_{ϕ_{1}} and P_{ϕ}. The ideals P_{ϕ} between P_{ϕ_{1}} and P_{ϕ_{1}}^{+} are finite in number if the centers of ν and ν_{1} are equal, and form a simple infinite sequence if the centers of ν and ν_{1} are different; in this last case, denoting by ϕ(ϕ_{1}) the least element of Γ ∩ λ^{−1}(ϕ_{1}), the R/(m_{ν_{1}} ∩ R)-modules (P_{ϕ(ϕ_{1})+ψ}/P_{ϕ_{1}}^{+})_{ψ∈Γ∩Ψ_{+}} are cofinal in the sequence of the (P_{ϕ}/P_{ϕ_{1}}^{+})_{λ(ϕ)=ϕ_{1}}, and for each ϕ ∈ Γ ∩ λ^{−1}(ϕ_{1}) there is an integer q such that (m_{ν} ∩ R)^{q}.P_{ϕ_{1}} ⊂ P_{ϕ}.

Proof Let Γ ⊂ Φ_{+} ∪ {0} be the semigroup of R. Given ϕ_{1} ∈ Γ_{1} = λ(Γ), for any ϕ ∈ Γ such that λ(ϕ) = ϕ_{1} we can write ϕ = ϕ(ϕ_{1}) + ψ with ψ ∈ Ψ_{+} ∪ {0}, and the result follows from the fact that if \overline{ν} is not trivial, Γ ∩ Ψ_{+} is cofinal in Ψ_{+} (see subsection 2.1), and the fact that \overline{ν} has center (m_{ν} ∩ R)/(m_{ν_{1}} ∩ R) and is archimedian. See [Z-S], Appendix 3, lemma 4 and its corollary. □

Remark 3.15 In fact, whenever we have a specialization R ⊂ R_{ν} ⊂ R_{ν_{1}} of valuations of R, the valuation ν induces, as described above, a filtration on the ring gr_{ν_{1}}R, to which we can associate as in subsection 2.1 a “filtration algebra” which determines a specialization of gr_{ν_{1}}R to gr_{ν}R. This specialization is the same as that coming from the embedding of SpecA_{ν_{1}}(R) in SpecA_{ν}(R).

Example 3.16 Let R be a local integral domain, and f ∈ R \ {0} be such that R/(f) is again integral and ⋂_{n≥0} f^{n}R = \{0\}. Let us choose a valuation \overline{ν} of R/(f) with group Ψ. Since each element x of R \ {0} can be written in a unique way x = f^{n}y with y ∉ (f), we may define a valuation ν on R with group Z ⊕ Ψ ordered lexicographically by setting

ν(f^{n}y) = (n, \overline{ν}(\overline{y})) where \overline{y} = y mod.(f).

The map f^{n}y ↦ n ∈ N defines a valuation ν_{1} on R with value group Z and with which ν is composed. With the notations introduced above, we have \overline{R} = R/(f), and

gr_{ν_{1}}R = \overline{R}[ξ] with ξ = in_{ν_{1}}(f),

and from the considerations above we see that because it is a polynomial ring in one variable, we have:

gr_{ν}R = (gr\overline{ν} \overline{R})[ξ].

We shall see the general form of this result in subsection 4.2.

3.5 Comparison of residual valuations. In this subsection I recall some direct consequences of the basic facts on the composition of valuations found in ([A1], pp. 56-57, [V1], section 4, [Z-S]). Keeping the notations of the beginning of this section, let us consider a sequence

R_{ν} ⊂ R_{ν_{1}} ⊂ R_{ν_{2}}

of valuation rings; it corresponds to a sequence

Φ → Φ_{1} → Φ_{2}

of value groups; let Ψ_{2} ⊂ Ψ_{1} ⊂ Φ be the kernels of the maps Φ → Φ_{1} and Φ → Φ_{2} respectively. To this situation is associated an inclusion

R_{ν}/(m_{ν_{2}} ∩ R_{ν}) ⊂ R_{ν_{1}}/(m_{ν_{2}} ∩ R_{ν_{1}})

of valuation rings of the residue field k_{ν_{2}} of R_{ν_{2}}. Note that in fact m_{ν_{2}} ∩ R_{ν} = m_{ν_{2}} since m_{ν_{2}} ⊂ R_{ν} by the general properties of composed valuations (see [V1]), and similarly for R_{ν_{1}}; recall also that R_{ν_{i}} is the localization of R_{ν} at the prime m_{ν_{i}}. The value group of the first valuation ring is Ψ_{1}, and the value group of the second is Ψ_{1}/Ψ_{2}. If R_{ν_{1}} and R_{ν_{2}} are consecutive valuation rings, in the sense that there is no other valuation ring between them, the group Ψ_{1}/Ψ_{2} is of height one.

Assuming that this is the case, given a subring R ⊂ R_{ν}, we see that ν_{1} induces a valuation of height at most one, with valuation ring R_{ν_{1}}/(m_{ν_{2}} ∩ R_{ν_{1}}), on the quotient R/(m_{ν_{2}} ∩ R); the center of this valuation is the ideal (m_{ν_{1}} ∩ R)/(m_{ν_{2}} ∩ R), which is also the kernel of the natural morphism

R/(m_{ν_{2}} ∩ R) → R/(m_{ν_{1}} ∩ R).

More precisely, it may happen that m_{ν_{1}} ∩ R = m_{ν_{2}} ∩ R, and then the residual valuation is trivial on R/(m_{ν_{2}} ∩ R).

3.6 Specialization of valuations and rational valuations. Let k be a field, and (R, m) a local k-algebra which is an integral domain. Let R_{ν_{0}} be a valuation ring in the fraction field of R, containing R, with residue field k_{ν_{0}}, and assume that the transcendence degree over k of k_{ν_{0}} is greater than zero. Then, if R/(m_{ν_{0}} ∩R) ≠ k_{ν_{0}}, there exists a k-valuation ring V of k_{ν_{0}} containing R/(m_{ν_{0}} ∩R) and different from k_{ν_{0}}. If we denote by π: R_{ν_{0}} → k_{ν_{0}} the canonical projection, we see that π^{−1}(V) is a valuation ring R_{ν_{1}} containing R and contained in R_{ν_{0}}. The residue field of R_{ν_{1}} is equal to the residue field of V, and therefore its transcendence degree over k is less than that of k_{ν_{0}}. We may iterate this construction as long as the residue field of R_{ν_{1}} is transcendental over k and R/(m_{ν_{1}} ∩ R) ≠ k_{ν_{1}}. When we stop, we have built a sequence

R ⊂ R_{ν_{t}} ⊂ R_{ν_{t−1}} ⊂ ⋯ ⊂ R_{ν_{0}}

such that t ≤ t_{k}(ν_{0}) and either R_{ν_{t}} dominates R and the residual extension is trivial, or the residue field of R_{ν_{t}} is algebraic over k. In this case, the ring R/(m_{ν_{t}} ∩ R) contains k and is contained in k_{ν_{t}}, so it is a field. This means that m_{ν_{t}} ∩R = m, so R_{ν_{t}} dominates R. So at worse we end up the specialization process with a valuation ring dominating R and an algebraic residual extension.

This is a special case of the construction found in [Z-S], Vol. 2, Chap VI, §16. In the case where R_{ν} dominates R and the residual extension is trivial, one

says that the valuation ν on R is rational. So we have:

Proposition 3.17 If the residue field of R is algebraically closed, we can specialize any valuation ν_{0} of R to a rational valuation.

Remarks 3.18 1) Assuming that R is excellent, and so in particular a Nagata ring, we may (see [V1], Proposition 10.1), at each step in the preceding construction, choose a discrete divisorial valuation; then the transcendence degree of the residual valuation drops exactly by one at each step, while the height of the value group increases exactly by one since the value group of V is the kernel of the map Φ_{ν_{1}} → Φ_{ν_{0}}. Finally we have:

h(ν_{t}) = h(ν_{0}) + t_{k_{R}}(ν_{0}),

and since h(ν_{t}) ≤ r(ν_{t}), in order to prove Abhyankar’s inequality in this case it suffices to check that r(ν) ≤ dimR when R_{ν} dominates R, but this is readily done, at least in the algebraic or formal case, if we present the fraction field K of R as an algebraic extension of a field of rational functions of transcendence degree dimR over k_{R}, since extending valuations to algebraic extensions does not affect the rational rank. This suggests a proof of Abhyankar’s inequality, not very different from that described in [V1].

3.7 Monomial valuations. This section begins to explain in what sense the liftings (ξ_{i})_{i∈I} of generators (\overline{ξ}_{i})_{i∈I} of the graded R/(m_{ν} ∩ R)-algebra gr_{ν}R form a system of coordinates “adapted” to the valuation. The idea is that, at least in the case of trivial residual extension, valuations which are monomial in a certain system of coordinates can be uniformized by a single toric modification of the ambiant space in which this system of coordinates embeds our singularity.

Given a valuation ν of a local or graded ring R, we generalize and modify the definition given in [S2] for a regular local ring as follows

Definition 3.19 1) The valuation ν of the local (or graded) integral domain R is algebraically monomial with respect to a system of generators (ξ_{i})_{i∈I} of the center m_{ν} ∩ R of ν in R if (ξ_{i})_{i∈I} is a generating sequence for ν in the sense (see [S2]) that any element x ∈ R can be written as a finite sum

x = ∑_{ν(ξ^{γ})≥ν(x)} c_{γ}ξ^{γ}

with γ ∈ I^{b}, for some b ∈ N, c_{γ} ∈ R.

2) If the valuation ν is rational and R is complete with respect to the ν-adic topology and has a field of representatives, we say that ν is analytically monomial with respect to (ξ_{i})_{i∈I} if the same holds with “finite sum” replaced by “ν-adically convergent series” and “c_{γ} ∈ R” by “c_{γ} ∈ k”. In particular, in both cases, ν(x) = min_{{γ/c_{γ}≠0}}(ν(ξ^{γ})).

Remarks 3.20 1) To say that ν is algebraically monomial with respect to the ξ_{i} is equivalent, when R is nœtherian, to asking that for each ϕ ∈ Φ_{+} the ideal P_{ϕ}(R) is generated by monomials in the (ξ_{i})_{i∈I}, and implies that the initial forms (\overline{ξ}_{i})_{i∈I} in gr_{ν}R of the elements (ξ_{i})_{i∈I} generate it as R/(m_{ν} ∩ R)-algebra. Analytic monomiality implies that they generate the k-algebra gr_{ν}R. In particular, if the valuation ν is algebraically (resp. analytically) monomial with respect to a finite set of generators of m_{ν} ∩R (resp. the maximal ideal m), the R/(m_{ν} ∩R)-algebra (resp. k-algebra) gr_{ν}R is finitely generated. Since the expansion x = ∑_{ν(ξ^{γ})≥ν(x)} c_{γ}ξ^{γ} is not unique in general, one must stress the fact that the definition requires the existence of one expansion with the stated properties.

2) If R is a graded k-algebra generated by the elements ξ_{i} and we assume that they all have positive valuation and that each homogeneous component of R is a finite-dimensional vector space, the valuation ν is monomial with respect to the ξ_{i}. Indeed, it suffices to check the definition for homogeneous elements. Every homogeneous element \overline{x}_{ϕ} can be written as a finite sum of monomials of the same degree \overline{x}_{ϕ} = ∑_{γ} c_{γ}ξ^{γ} with c_{γ} ∈ R/p and ν_{gr}(c_{γ}ξ^{γ}) = ϕ, as required.

3) If R is a one-dimensional excellent henselian equicharacteristic local domain, and ν its unique non-trivial valuation, there are, as we shall see (subsection 6.3 and Proposition 14), finite systems of generators of the maximal ideal of R with respect to which ν is analytically monomial; in this case, the valuation is also algebraically monomial with respect to any such system of generators.

The argument in the case where R is complete is as follows: Let k be the residue field of R and let (ξ_{1}, …, ξ_{g+1}) be elements of R whose images (\overline{ξ}_{i}) constitute a system of generators of the graded k-algebra gr_{ν}R (see subsection 6.3 below). Let us fix a field of representatives k ⊂ R. Then, by the usual method of successive approximations, using the fact that the ν-adic and m-adic topologies coincide in this case, we have a surjection of k-algebras

k[[u_{1}, …, u_{g+1}]] → R

mapping each u_{i} to ξ_{i} and inducing, by passing to the graded rings with respect to ν and the corresponding monomial order on k[[u_{1}, …, u_{g+1}]], the surjective map

k[U_{1}, …, U_{g+1}] → gr_{ν}R

mapping U_{i} to \overline{ξ}_{i}. Then each element x of R can be written

(∗∗) x = ∑_{α/ν(ξ^{α}≥ν(x)} c_{α}ξ^{α} with c_{α} ∈ k,

which shows that the valuation ν is analyticaly monomial with respect to (ξ_{1}, …, ξ_{g+1}). For each index 2 ≤ i ≤ g +1, in the one-dimensional subring k[[ξ_{1}, ξ_{i}]] ⊂ R with the valuation induced by ν, the minimality of the valuation of ξ_{1} implies that ξ_{i} is integral over the ideal ξ_{1}k[[ξ_{1}, ξ_{i}]], so that we have a relation ξ_{i}^{s_{i}} +b_{1}^{(i)}(ξ_{1}, ξ_{i})ξ_{i}^{s_{i}−1}+⋯+ b_{s_{i}}^{(i)}(ξ_{1}, ξ_{i}) = 0 with b_{ℓ}^{(i)}(ξ_{1}, ξ_{i}) ∈ (ξ_{1}^{ℓ})k[[ξ_{1}, ξ_{i}]]. Using the Weierstrass preparation theorem, we see that we have relations

ξ_{i}^{s_{i}} + a_{1}^{(i)}ξ_{i}^{s_{i}−1} + ⋯ + a_{s_{i}}^{(i)} = 0 with a_{ℓ}^{(i)} ∈ (ξ_{1})^{ℓ}k[[ξ_{1}]].

Using this relation to eliminate from the representation (∗∗) all terms of order ≥ s_{i} in ξ_{i} for i ≥ 2, we see that we can now write

x = ∑ ; finite a_{α}(ξ_{1})ξ_{2}^{α_{2}} … ξ_{g+1}^{α_{g+1}},

where a_{α}(ξ_{1}) ∈ k[[ξ_{1}]] and the sum involves only terms such that

ν(a_{α}) + ∑_{i=2}^{g+1} α_{i}ν(u_{i}) ≥ ν(x).

Writing a_{α}(ξ_{1}) = ξ_{1}^{α_{1}}v_{α}(ξ_{1}) with v_{α}(ξ_{1}) a unit in k[[ξ_{1}]], this shows that the (ξ_{i})_{1≤i≤g+1} form a system of generators for the maximal ideal of R with respect to which ν is algebraically monomial. The proof in the henselian case is similar. For a different proof in the case of plane branches, see [PP].

We shall see in Proposition 5.43 that if ν is a rational valuation, after extending it to a suitable completion R̂^{(ν)} one always has a (possibly infinite) system of generators of the maximal ideal of R̂^{(ν)} with respect to which this extension of ν is analytically monomial. One may then, by analogy with the curve case, choose a system of parameters (ξ_{1}, …, ξ_{s}) with s = dimR̂^{(ν)} such that the integral closure of the ideal (ξ_{1}, …, ξ_{s})R̂^{(ν)} is the maximal ideal, and examine under which conditions the valuation is not only analytically monomial but also algebraically monomial with respect to this system of generators, or becomes so after a birational extension.

Example 3.21 1) Let (S, m) be a regular local ring and ν be a valuation of S having the property of the above definition with respect to a system of regular parameters (ξ_{0}, …, ξ_{n}) of S. If we denote by R_{ϕ} the ideal of R generated by the monomials in (ξ_{0}, …, ξ_{n}) of valuation ≥ ϕ, we have x ∈ R_{ν(x)} and therefore ν is monomial in the sense of [S2]. In this case gr_{ν}R is a polynomial ring in n + 1 variables, and this is clearly the only case where gr_{ν}R is regular, that is, a polynomial ring.

2) If R is the algebra of a plane branch C, and ν its unique valuation, then ν is monomial with respect to two well chosen generators of the maximal ideal of R if and only if the semigroup ν(R \ {0}) of C has two generators. In characteristic zero, this means that C has only one characteristic pair (see [G-T], [T1]).

Proposition 3.22 Let R be an integral domain and ν a valuation on R, i.e., R ⊂ R_{ν}. Set p = m_{ν} ∩ R. The valuation ν_{gr} on gr_{ν}R is monomial with respect to any system of homogeneous generators (ξ_{i})_{i∈I} of the graded R/p-algebra gr_{ν}R.

Proof by the definition of ν_{gr}, this follows from one of the remarks made above. □

4 The toric structure of the graded algebras of valuation rings and nœtherian rings

In this section I show that the graded ring gr_{ν}R is, when ν is a rational valuation of finite rational rank for the local integral domain R, a quotient of a polynomial ring in countably many indeterminates over the residue field by a binomial ideal, and that gr_{ν}R_{ν} is the direct limit of a nested sequence of polynomial subalgebras in r(ν) variables over the residue field k_{ν} of the valuation. This means in particular that it is “regular”, and hence can be deemed to be a resolution of singularities of the algebra gr_{ν}R, if we forget for a moment that gr_{ν}R_{ν} is not an extension of finite type of gr_{ν}R, and remember only that they have the same field of fractions. It means also that the binomial equations defining gr_{ν}R_{ν} are of a very special form.

It turns out that the result for gr_{ν}R_{ν} is equivalent to its counterpart for k[t^{Φ_{+}}] where k is any field.

The result for k[t^{Φ_{+}}] in turn follows from the stronger statement that the semigroup Φ_{+} is a direct limit of a system of linear maps between finitely generated free monoids, which will give us toric maps between the corresponding polynomial algebras.

We shall see that this is true, even with finitely generated free submonoids, whenever Φ is of finite height.

4.1 The graded algebra of a rational valuation is essentially toric. Here and in the sequel, unless otherwise specified, the notation V^{m} stands for a monomial V_{j_{1}}^{m_{j_{1}}} … V_{j_{s}}^{m_{js}}, with m_{j_{i}} ∈ N ∪ {0}.

Proposition 4.1 Assume that a valuation ring R_{ν} birationally dominates a local ring R and that the residual extension k_{R} = R/m → R_{ν}/m_{ν} = k_{ν} is trivial. Then each homogeneous component of the graded k_{R}-algebra gr_{ν}R is a vector space of dimension ≤ 1 over k_{R}.

Proof Indeed, by definition of the ideals P_{ϕ}(R), we have the graded inclusion

gr_{ν}R ⊂ gr_{ν}R_{ν}

of k_{R}-algebras and the result is true for the second algebra since the initial forms of two elements of R_{ν} with the same valuation differ by a factor which is the initial form of a unit, i.e., an element of k_{ν}^{∗}; each homogeneous component of gr_{ν}R_{ν} is a k_{ν}-vector space of dimension one. □

From Proposition 4.1 above follows by a direct generalization of a result of Korkina ([Ko], see also [E-S]) the

Proposition 4.2 In the situation of Proposition 4.1, assuming that r(ν) is finite, the graded k_{R}-algebra gr_{ν}R is the quotient of a polynomial ring in countably many indeterminates k_{R}[(U_{j})_{j∈J}] by a binomial prime ideal.

Proof By assumption, Φ is of finite rational rank, and therefore countable. Let (\overline{ξ}_{j})_{j∈J} be countable system of nonzero homogeneous generators for the k_{R}-algebra gr_{ν}R (note that the number of these generators may be finite); consider the kernel K of the surjective homomorphism

k_{R}[(U_{j})_{j∈J}] → gr_{ν}R

determined by U_{j} ↦ \overline{ξ}_{j}. It is a homogeneous ideal for the Φ_{+}-grading giving to each U_{j} the degree of \overline{ξ}_{j}. Let P = ∑ c_{m}U^{m} be a homogeneous generator of K, say of degree ϕ. All the monomials occurring in P have their images in the one dimensional k-vector space (gr_{ν}R)_{ϕ} and the sum of these images is zero. By induction on the number of monomials of P, we see that P is a sum of binomial terms c_{m,m′}(U^{m} − λ_{m,m′}U^{m′}) with c_{m,m′}, λ_{m,m′} ∈ k. The ideal K is generated by binomials of the form U^{m} −λ_{m,m′}U^{m′} and there is one such for each (m, m′) such that U^{m} and U^{m′} have the same degree. By construction, all the λ_{m,m′} are ≠ 0. Since gr_{ν}R is integral, the ideal K is prime. □

Corollary 3 For any valuation ring R_{ν} with countable value group, the graded algebra gr_{ν}R_{ν} is a quotient of a polynomial algebra k[(U_{i})_{i∈I}] in countably many variables over k_{ν} = R_{ν}/m_{ν} by a prime ideal generated by binomials U^{m} − λ_{mn}U^{n} with λ_{mn} ∈ k^{∗}.

Remark 4.3 This corollary is used in the proof of Proposition 4.11, but does not tell us anything precise about the form of the binomial equations.

Assume now that R is nœtherian.

Let ν(R \ {0}) = Γ ⊂ Φ_{+} ∪ {0} be the semigroup of R. Since R is nœtherian, the semigroup Γ is well ordered (Proposition 3.9), and we have a countable system of generators, the valuations of the generators \overline{ξ}_{i} of gr_{ν}R, i.e., γ_{i} = νgr(\overline{ξ}_{i}):

Γ = ⟨γ_{1}, γ_{2}, …, γ_{j}, …⟩ with γ_{j} ∈ Φ_{+} , j an ordinal < ω^{h_{R}(ν)}, γ_{j} < γ_{j+1}.

(Corollary to Proposition 3.9). In the plane branch case, the generators are denoted by \overline{β}_{i} to underline their kinship with the Puiseux exponents β_{i}, according to a tradition initiated by Zariski; we followed that tradition in [G-T] and in the example treated above. Anyway, to this situation is associated the map of groups

Z^{N} → Φ determined by ∑ λ_{j}e_{j} ↦ ∑ λ_{j}γ_{j},

where e_{j} is the j-th basis vector, such that the image of (Z_{≥0})^{N} is Γ. It defines a map of k_{ν}-algebras

k_{ν}[(U_{j})_{j∈J}] → k_{ν}[t^{Φ_{+}}] determined by U_{j} ↦ t^{γ_{j}}.

The image of this map is the semigroup algebra k_{ν}[t^{Γ}] ⊂ k_{ν}[t^{Φ_{+}}], and its kernel is the binomial ideal obtained from that of of Proposition 4.2 by setting all λ_{m,m′} equal to 1.

Remark 4.4 We note that since the γ_{i} are assumed to be a minimal system of generators, none of the binomial equations is of the form

U_{i} − λ_{i}U^{m(i)} = 0.

Let (R, ν) be a valued nœtherian ring; assume that the center of ν in R is the maximal ideal m; set k = R/m and let

k[(U_{i})_{i∈I}] → gr_{ν}R determined by U_{i} ↦ \overline{ξ}_{i}

be a presentation of the k-algebra gr_{ν}R corresponding to a minimal system of generators of the semigroup Γ of ν. Let (ξ_{i}) ∈ R be a system of representatives of the \overline{ξ}_{i}. Let R_{ν} be the valuation ring of ν and consider all the images λ_{mn} in R_{ν}/m_{ν} of the ratios of valuation zero ξ^{m−n} of monomials in the ξ_{i}. For each λ_{mn} which is algebraic over the field k, consider the minimal polynomial P_{mn} ∈ k[T] of λ_{mn}. Setting T = \frac{U^{m}}{U^{n}}, after multiplication by a suitable power of U^{n}, P_{mn} may be considered as the image in k[(U_{i})_{i∈I}] of an irreducible polynomial. The polynomials P_{mn} belong to the kernel I of the surjective map k[(U_{i})_{i∈I}] → gr_{ν}R. After extension of scalars to R_{ν}/m_{ν}, the ideal IR_{ν}/m_{ν}[(U_{i})_{i∈I}] is contained, in view of Proposition 4.2 in the ideal of R_{ν}/m_{ν}[(U_{i})_{i∈I}] generated by all the binomials U^{m} − λ_{mn}U^{n}, with λ_{mn} ∈ R_{ν}/m_{ν} and ν(ξ^{m−n}) = 0. By the faithful flatness of the extension k[(U_{i})_{i∈I}] → R_{ν}/m_{ν}[(U_{i})_{i∈I}], the ideal I is contained in the intersection of that ideal with k[(U_{i})_{i∈I}], which is the ideal generated by all the polynomials P_{mn}(U) for λ_{mn} algebraic over k; the ideal I is equal to the ideal generated by the P_{mn}(U). For an arbitrary valuation, say with center p, we may apply this to the extension of ν to the localization R_{p} with residue field κ(p) = R_{p}/pR_{p}:

Corollary 4 Let (R, ν) be a valued nœtherian ring; let p be the center of ν in R and extend ν to the localization R_{p}. The kernel I of the map of κ(p)-algebras κ(p)[(U_{i})_{i∈I}] → gr_{ν}R_{p} determined by U_{i} ↦ \overline{ξ}_{i} is generated by the polynomials P_{mn} obtained by clearing denominators in the minimal polynomials over κ(p) of the images in R_{ν}/m_{ν} of the ratios ξ^{m−n} of valuation 0, viewed as polynomials in \frac{U^{m}}{U^{n}}.

Proposition 4.5 Let Λ denote the multiplicative subset of k_{ν}[(Λ_{m,m′})_{(m,m′)}] generated by the Λ_{m,m′}. The natural map of k_{ν}-algebras

Λ^{−1}k_{ν}[(Λ_{m,m′})_{(m,m′)}] → Λ^{−1}k_{ν}[(Λ_{m,m′})_{(m,m′)}][(U_{j})_{j∈J}]/(U^{m} − Λ_{m,m′}U^{m′})_{(m,m′)} is faithfully flat and corresponds to a flat family of schemes having one fiber equal to Speck_{ν}[t^{Γ}] and another to Specgr_{ν}R.

Indeed, if the second algebra is graded by giving to U_{j} the degree γ_{j}, then by definition of the binomial relations, each of its homogeneous components becomes a free module of rank one over Λ^{−1}k_{ν}[(Λ_{m,m′})_{(m,m′)}].

*This family parametrized by the Λ_{m,n} has simultaneous resolution in any reasonable sense; it is locally analytically trivial.* We shall see below that it has simultaneous embedded resolution by a toric map; it follows from the results of subsection 6.2.

4.2 More on the structure of gr_{ν}R. Let Φ be a group of height h associated to a valuation of a local integral domain R centered at the maximal ideal m of R; set k = R/m. Let

(0) = Ψ_{h} ⊂ Ψ_{h−1} ⊂ ⋯ ⊂ Ψ_{1} ⊂ Ψ_{0} = Φ

be the sequence of isolated subgroups of Φ (including the trivial ones). Let Γ be the semigroup of ν on R (note that it may be equal to Φ_{+}, if R = R_{ν}) and set Γ_{i} = Γ∩Ψ_{i}. Considering the valuation ν_{1} of height h−1 with center p_{1} with which ν is composed, we have seen in subsection 3.4 that we could identify gr\overline{ν} \overline{R}_{1}, where \overline{R}_{1} = R/p_{1}, with the subalgebra ⨁_{ψ∈Ψ_{h−1+}∪\{0\}}(gr_{ν}R)_{ψ} of gr_{ν}R. If we distinguish between the generators (U_{i})_{i∈I_{1}}, where I_{1} ⊂ I, of the k-algebra gr_{ν}R whose degree lies in Ψ_{h−1} and those whose degree lies in Φ \ Ψ_{h−1}, we have:

Proposition 4.6 If R ⊂ R_{ν} ⊂ R_{ν_{1}}, where R_{ν} dominates the local ring R without residual extension and ν_{1} is of height one less than ν, the natural map

gr\overline{ν} \overline{R}_{1}[(U_{i})_{i∈I\I_{1}}] → gr_{ν}R

is surjective and its kernel is generated by the images in gr\overline{ν} \overline{R}_{1}[(U_{i})_{i∈I\I_{1}}] of those binomials U^{n} − λ_{mn}U^{n} which appear in Proposition 4.2 and which involve at least one variable U_{i} with i ∈ I \ I_{1}.

Proof The subalgebra gr\overline{ν} \overline{R}_{1} contains all the generators of the k-algebra gr_{ν}R whose degree is in I_{1}; we need to add only the generators U_{i} whose degree lies in I \ I_{1}, and the relations between them are those indicated. □

Corollary 5 If R_{ν} dominates R without residual extension, setting I_{t} = {i ∈ I/degU_{i} ∈ Ψ_{h−t} \ Ψ_{h−t+1}}, we can write

\begin{matrix}gr_{ν}R = \\ (⋯ (k[(U_{i})_{i∈I_{1}}]/(U^{n} − λ_{mn}U^{n}))[(U_{i})_{i∈I_{2}}]/(U^{n} − λ_{mn}U^{n})) ⋯)[(U_{i})_{i∈I_{h}}]/(U^{n} − λ_{mn}U^{n}), \end{matrix}

where at each step t the binomials which are written involve only variables U_{i} with degrees in Ψ_{h−t} and must involve one which has degree in I_{t}. Each monomial of such binomials must then involve variables U_{i} with i ∈ I_{t}.

Proof The first part is only the iteration of the preceding Proposition. The second part follows from the fact that there can be no linear expression of an element of I_{t} in terms of those of Ψ_{h−t+1}. □

4.3 gr_{ν}R_{ν} as a limit of polynomial algebras. In this subsection, I show that gr_{ν}R_{ν} is the union of a nested sequence of polynomial subalgebras in r(Φ) variables over k_{ν}, generated by homogeneous elements, the inclusions sending variables to terms, i.e., constants times monomials. Technically this is essentially Perron’s algorithm. This result, however, may be considered as a graded version of local uniformization; each finitely generated k_{ν}-subalgebra of gr_{ν}R_{ν} is contained in a polynomial subalgebra in r(Φ) variables. However this inclusion is birational only in special cases, for example if our subalgebra contains a finitely generated gr_{ν}R with R ⊂ R_{ν} birational and residually rational.

I first prove the corresponding result for semigroup algebras k[t^{Φ_{+}}], and I begin with an example: the case when Φ = Z^{d} with the lexicographic order. Indeed, consider for each (d − 1)-uple (r_{1}, r_{2}, …, r_{d−1}) of positive integers the map

e_{r_{1},r_{2},...,r_{d−1}} : N^{d} → Z^{d}

defined by

e_{r_{1},r_{2},...,r_{d−1}}(i_{1}, …, i_{d}) = (i_{1}, i_{2} − r_{1}i_{1}, …, i_{d} − r_{d−1}i_{d−1});

its image is in the semigroup Z_{+}^{d} of positive elements, and it is injective. It defines an injective map of semigroup algebras

e_{r_{1},r_{2},...,r_{d−1}} : k[z_{1}, …, z_{d}] → k[t^{Z_{+}^{d}}],

and clearly this last algebra is the union of these subalgebras as r_{1}, r_{2}, …, r_{d−1} vary. Note that whenever (r_{1}, r_{2}, …, r_{d−1}) ≤ (s_{1}, s_{2}, …, s_{d−1}) in the sense that r_{j} ≤ s_{j} , 1 ≤ j ≤ d − 1, the map e_{s_{1},s_{2},...,s_{d−1}} factors through a map of algebras e_{s_{1}−r_{1},s_{2}−r_{2},...,s_{d−1}−r_{d−1}} : k[z_{1}, …, z_{d}] → k[w_{1}, …, w_{d}] which is toric, i.e., given by monomials, and that k[t^{Z_{+}^{d}}] is the direct limit of this system of toric maps.

When Φ is a subgroup of R, consisting of numbers of the form ∑_{i=1}^{m} a_{i}τ_{i} with given τ_{i} > 0 and a_{i} ∈ Z, it is also true that k[t^{Φ_{+}}] is a direct limit of polynomial subalgebras over k. This is again because the half space in R^{m} defined by ∑_{i=1}^{m} a_{i}τ_{i} ≥ 0 can be approximated by regular rational simplicial cones (generated by integral vectors which form a basis of the integral lattice).

More precisely, let ν be a valuation of height one, i.e., with archimedian value group Φ ⊂ R (see [A1], p.46, [Z-S], Vol. II). Assume first that Φ is generated by m rationally independent real numbers τ_{1}, …, τ_{m}, which we may assume to be positive. I use the Perron algorithm as expounded in ([Z1], B. I, p. 861), but with a somewhat different interpretation. The algorithm consists in writing

τ_{1} = τ_{m}^{(1)}, τ_{2} = τ_{1}^{(1)} + a_{2}^{(0)}τ_{m}^{(1)}, …, τ_{m} = τ_{m−1}^{(1)} + a_{m}^{(0)}τ_{m}^{(1)},

where

a_{j}^{(0)} = [τ_{j}/τ_{1}], j = 2, …, m,

and repeating this operation after replacing (τ_{1}, …, τ_{m}) by (τ_{1}^{(1)}, …, τ_{m}^{(1)}), and so on. After h steps, one has written

τ_{i} = A_{i}^{(h)}τ_{1}^{(h)} + ⋯ + A_{i}^{(h+m−1)}τ_{m}^{(h)}

or, if we denote by w the (weight) vector (τ_{1}, …, τ_{m}) ∈ R^{m} and by A^{(h)} the vector (A_{1}^{(h)}, …, A_{m}^{(h)}),

w = τ_{1}^{(h)}A^{(h)} + τ_{2}^{(h)}A^{(h+1)} + ⋯ + τ_{m}^{(h)}A^{(h+m−1)}

where the τ_{j}^{(h)} are positive, the coefficients A_{i}^{(j)} are non negative integers, and the matrix of the vectors

A^{(h)}, A^{(h+1)}, …, A^{(h+m−1)}

has determinant (−1)^{h(m−1)}. Moreover, as h grows the directions in P^{m−1}(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 whose directions in P^{m−1}(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 σ^{(h)} = ⟨A^{(h)}, A^{(h+1)}, …, A^{(h+m−1)}⟩ which they generate. The convex dual σ̌^{(h)} of σ^{(h)} (see [Cox], §2, [E], V, 2, p. 149) is contained in the half space ∑_{i=1}^{m} a_{i}τ_{i} ≥ 0, the integral points of which form the semigroup Φ_{+}. The algebra of the semigroup σ̌^{(h)} ∩ Z^{m} is a polynomial algebra k[x_{1}^{(h)}, …, x_{m}^{(h)}] (loc.cit., VI,2) contained in k[t^{Φ_{+}}], and since by assumption there are no integral points on the hyperplane ∑_{i=1}^{m} a_{i}τ_{i} = 0 except the origin, the semigroup Φ_{+} is the union of the σ̌^{(h)} ∩ Z^{m} as h → ∞. This proves that k[t^{Φ_{+}}] is the union, or direct limit, of these polynomial subalgebras.

Note that by construction we have for each h ≥ 1 the equality

A^{(h)} − A^{(h+m)} + a_{2}^{(h)}A^{(h+1)} + ⋯ + a_{m}^{(h)}A^{(h+m−1)} = 0,

which shows, since the a_{j}^{(h)} = [τ_{j}^{(h)}/τ_{1}^{(h)}] are non-negative, that we have

A^{(h+m)} ∈ σ^{(h)} = ⟨A^{(h)}, …, A^{(h+m−1)}⟩

and therefore σ^{(h+1)} ⊂ σ^{(h)}, that is σ̌^{(h)} ⊂ σ̌^{(h+1)} and

k[x_{1}^{(h)}, …, x_{m}^{(h)}] ⊂ k[x_{1}^{(h+1)}, …, x_{m}^{(h+1)}],

so that our direct system is in fact a nested sequence of polynomial subalgebras. The morphisms between these polynomial algebras correspond by duality to the expressions of the A^{(h+m)} as linear combinations with non negative integral coefficients of (A^{(h)}, …, A^{(h+m−1)}) and are therefore monomial as announced.

If we now consider a group with one more generator τ_{m+1} > 0 which is rationally dependent on τ_{1}, …, τ_{m}, Zariski shows in ([Z1], B. I, p. 862) that the new weight vector w = (τ_{1}, …, τ_{m}, τ_{m+1}) ∈ R^{m+1} is contained in a rational simplicial cone σ ⊂ R^{m+1} generated by m integral vectors v_{1}, …, v_{m} 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 σ̌ ⊂ Ř^{m+1} is the product of an m dimensional strictly convex cone generated by vectors e_{1}, …, e_{m} by a 1-dimensional vector space (see [E], V, 2), generated by a primitive integral vector e_{m+1}, which is the dual of the rational hyperplane containing w. The vectors e_{1}, …, e_{m+1} are a basis of the integral lattice, and correspond to the variables x_{1}, …, x_{m+1} generating a polynomial ring. Note that the map f : Z^{m+1} → R defined by (a_{1}, …, a_{m+1}) ↦ ∑_{i=1}^{m+1} a_{i}τ_{i} is no longer injective; the primitive vector e_{m+1} corresponding to the variable x_{m+1} is in the kernel. Let us set Φ̃_{+} = f^{−1}(Φ_{+}∪{0}). By refining as above by the Perron algorithm for w inside the linear span < σ > of σ, starting with the coordinates of w in σ, we find a sequence of regular simplicial cones σ^{(h)} ⊂ σ whose duals σ̌^{(h)} ⊂ Φ̃_{+} correspond ([E], VI, Th. 2.12) to algebras of the form k[x_{1}^{(h)}, …, x_{m}^{(h)}, x_{m+1}^{±1}] ⊂ k[tΦ̃_{+}]. The free semigroups σ̌^{(h)} ∩ Z^{m+1} fill up Φ̃_{+} as h → ∞ since the only rational points of the hyperplane ∑_{i=1}^{m+1} a_{i}τ_{i} = 0 are on the dual of the hyperplane containing w, which is contained in all the σ̌^{(h)}. So the direct limit of the images of the maps k[t^{f}]: k[x_{1}^{(h)}, …, x_{m}^{(h)}, x_{m+1}^{±1}] → k[t^{Φ_{+}}] is k[t^{Φ_{+}}]. But these images are isomorphic to k[x_{1}^{(h)}, …, x_{m}^{(h)}, x_{m+1}^{±1}]/(x_{m+1} − 1) so that they are again polynomial rings k[x_{1}^{(h)}, …, x_{m}^{(h)}]. If we have more generators rationally dependent on τ_{1}, …, τ_{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 σ. In both examples, the fact that k is a field plays no role, so we have proved:

Lemma 4.7 Let Φ be a totally ordered finitely generated group of height one (i.e., archimedian), or Z^{d} with the lexicographic order. For any commutative ring A the semigroup algebra A[t^{Φ_{+}}] of Φ_{+} with coefficients in A is the direct limit of a direct system of graded subalgebras which are polynomial algebras A[x_{1}, …, x_{m}] over A with m = r(Φ).

In addition, the maps between these algebras are toric maps, i.e., each variable of one is sent to a monomial in the variables of the other, and there is a cofinal subsystem which is a chain of nested subalgebras.

Remarks 4.8 1) In both cases, the smooth subalgebras are produced by an algorithm.

2) The result, especially in view of its algorithmic nature, is much more useful than its consequence that SpecA[t^{Φ_{+}}] is a pro-object in the category of smooth affine toric schemes over SpecA with toric maps.

3) The proof also shows that the semigroup algebra of the semigroup

Φ̃_{+} = {(a_{1}, …, a_{m}, a_{m+1}, …, a_{m+r}) ∈ Z^{m+r} | ∑_{i=1}^{m+r} a_{i}τ_{i} ≥ 0},

where τ_{1}, …, τ_{m} are rationally independent and the others are rationally dependent upon them, is a toric direct limit of toric subalgebras of the form

A[x_{1}, …, x_{m}, x_{m+1}^{±1}, …, x_{m+r}^{±1}].

4) The result for subgroups of R holds without the finiteness assumption provided the rational rank is finite, since any abelian group is a direct limit of its subgroups of finite type. If we allow m to vary, even that last assumption is unnecessary.

Let now Φ be a totally ordered group of finite height h > 1. We have a surjective monotone non-decreasing map λ: Φ → Φ_{1} where Φ_{1} is of height h − 1, and the kernel Ψ of λ is of height 1. By induction on the height we may assume that Φ_{1+} is the union of sub-semigroups isomorphic to N^{m}, and we know from the lemma above that the same is true for Ψ_{+}.

Let us denote the semigroups that fill Φ_{1+} by F_{i}, and let F̃_{i} ⊂ Φ_{+} be the subsemigroup generated by elements e_{1}, …, e_{r_{i}} which lift to Φ_{+} \ Ψ the generators of F_{i}. Similarly let us denote by G_{j} ⊂ Ψ_{+} free semigroups which fill Ψ_{+}, generated say by f_{1}, …, f_{s_{j}}. Note that for ϕ ∈ Φ_{+} \ Ψ, ψ ∈ Ψ, ϕ + ψ ∈ Φ_{+}, and consider for r_{i}s_{j}-tuples n = (n_{s,t}, 1 ≤ s ≤ r_{i}, 1 ≤ t ≤ s_{j}) of non negative integers, the free semigroups F̃_{i}(n) ⊂ Φ_{+} ∪ {0} generated by e_{1} − ∑_{t} n_{1t}f_{t}, …, e_{r_{i}} − ∑_{t} n_{r_{i}t}f_{t}. Let us check that the direct sums of free semigroups F̃_{i}(n) ⊕ G_{j} fill up Φ_{+}; the proof generalizes that given in the case of the lexicographic Z^{d}. Given ϕ ∈ Φ_{+}, there exists an index i and a ϕ_{1} ∈ F̃_{i} such that ϕ − ϕ_{1} ∈ Ψ. If we have ϕ − ϕ_{1} ∈ Ψ_{+}, there exists an index j such that ϕ − ϕ_{1} ∈ G_{j} so that indeed ϕ ∈ F̃_{i} ⊕ G_{j}. This happens in particular if ϕ ∈ Ψ. If ϕ − ϕ_{1} ∈ Ψ_{−}, there exists an index j such that we can write:

ϕ = ∑_{s=1}^{r_{i}} k_{s}e_{s} − ∑_{t=1}^{s_{j}} ℓ_{t}f_{t} with non negative integers k_{s}, ℓ_{t} and f_{t} ∈ G_{j}.

Since ϕ ∉ Ψ, at least one of the k_{s} is not zero, say k_{1}. Choosing positive integers ℓ̃_{t} such that k_{1}ℓ̃_{t} > ℓ_{t}, we may rewrite ϕ as follows

ϕ = k_{1}(e_{1} − ∑_{t=1}^{s_{j}} ℓ̃_{t}f_{t}) + ∑_{s=2}^{r_{i}} k_{s}e_{s} + ∑_{t=1}^{s_{j}}(k_{1}ℓ̃_{t} − ℓ_{t})f_{t}.

This shows that ϕ is indeed in F̃_{i}(n) ⊕ G_{j} with n = ℓ̃.

Now if we assume, as we may by induction, that the F_{i} and G_{j} are nested sequences, and choose n(i) given by (n_{s,t} = i, 1 ≤ s ≤ r_{i}, 1 ≤ t ≤ s_{j}), we see that the corresponding groups F̃_{i}(n(i)) ⊕ G_{i} form a nested sequence which is cofinal in the direct system. So we have:

Proposition 4.9 For any totally ordered group Φ of finite height, and any commutative ring A, the semigroup algebra A[t^{Φ_{+}}] is a direct limit of a direct system of graded polynomial subalgebras over A with monomial maps, and there are cofinal nested subsystems of such polynomial subalgebras. If Φ is of finite rational rank r(Φ), all the polynomial subalgebras may be chosen isomorphic to A[x_{1}, …, x_{r(Φ)}].

Proof There remains only to prove the last sentence. This is done by induction on the height; we have shown above that the result is true for valuations of height one. Assume that the result is true for Ψ and Φ_{1}, and so the rank of the free monoid F̃_{i}(n) ⊕ G_{j} is the sum of the rational ranks of Ψ and Φ_{1}. But since the rational rank is additive in an exact sequence because Q is a flat Z-module, this is the rational rank of Φ. □

Corollary 6 If the rational rank of Φ is finite, the semigroup algebra A[t^{Φ_{+}}], endowed with its natural grading, is a quotient of a polynomial ring over A in countably many indeterminates A[(V_{j})_{j∈J}], graded by Φ_{+}, by a homogeneous binomial ideal of the form

(V_{j} − V^{m(j)})_{j∈J′},

where J′ is a subset of J and |m(j)| ≥ 2 for j ∈ J′.

Proof We may choose as a system of homogeneous generators of the A-algebra A[t^{Φ_{+}}], the union of the generators of the polynomial subalgebras of which A[t^{Φ_{+}}] is the direct limit. The only relations between these generators are those corresponding to the toric inclusions A[x_{1}, …, x_{r}] → A[y_{1}, …, y_{r}], and they are of the announced type after we discard the trivial relations x_{i} = y_{j} by removing some generators. □

Remarks 4.10 1) There is in general no minimal system of generators for A[t^{Φ_{+}}] and we can of course discard an arbitrary number of those generators which appear as V_{j} in a relation V_{j} − V^{m(j)} = 0. For example consider Φ = Z_{lex}^{2}.

2) We can apply this to the subalgebra k[v^{Φ_{+}}] of A_{ν}(R), and view the v^{ϕ} as coordinates for Speck[v^{Φ_{+}}] subjected to the binomial relations described in this Corollary.

Proposition 4.11 Given the ring R_{ν} of a valuation of finite rational rank r(ν):

a) the graded k_{ν}-algebra gr_{ν}R_{ν} is a quotient of a polynomial ring k_{ν}[(V_{j})_{j∈J}] in countably many indeterminates over the residue field k_{ν}, graded by Φ_{+}, by a homogeneous binomial ideal of the form

(V_{j} − λ_{j}V^{m(j)})_{j∈J′} , λ_{j} ∈ k_{ν}^{∗}

where J′ is a subset of J and |m(j)| ≥ 2 for j ∈ J′.

b) The graded k_{ν}-algebra gr_{ν}R_{ν} is the union of a nested sequence of graded polynomial subalgebras k_{ν}[x_{1}^{(h)}, …, x_{r(ν)}^{(h)}], where the inclusions are given by maps sending

each variable x_{i}^{(h)} to a constant times a monomial in the x_{j}^{(h+1)}, 1 ≤ j ≤ r(ν).

Proof See Proposition 4.9 and the previous Corollary. We have seen in Proposition 4.2 that gr_{ν}R_{ν} is isomorphic to a quotient of a polynomial algebra k[(V_{i})_{i∈I}] by a binomial ideal whose generators are of the form V^{m} − λ_{mn}V^{n}. Setting all the constants λ_{mn} equal to one gives the semigroup algebra k[t^{Φ_{+}}], hence assertion a). Assertion b) follows from this and the correspondence between the direct system of polynomial subalgebras and the binomial equations exhibited in the proof of the Corollary to Proposition 4.9. □

Corollary 7 Given any finite set of homogeneous elements \overline{x}_{1}, \overline{x}_{2}, ⋯, \overline{x}_{s} in gr_{ν}R_{ν} and a cofinal nested system of polynomial subalgebras as above, there is an algebra in our nested system such that not only do we have \overline{x}_{i} ∈ A[x_{1}^{(h)}, …, x_{r(Φ)}^{(h)}] for 1 ≤ i ≤ s, but the element of least degree divides all the others in this subalgebra.

Remark 4.12 It is important to note that the direct system built in this way is rather precisely determined by Φ; we shall see on the next examples that at least in dimension two the structure of the binomial equations, or equivalently that of the direct system of polynomial subalgebras converging to gr_{ν}R_{ν}, reflects the structure of the system of points in birational modifications of SpecR corresponding to the valuation ν, as well as the height and rational rank of ν.

Problem. It is an interesting problem to make these correspondences precise, especially in dimensions ≥ 3. A very useful reference for dimension two is [A2].

Corollary 8 The dimension of gr_{ν}R_{ν} is r(ν).

Proof By construction of the subalgebras k_{ν}[x_{1}^{(h)}, …, x_{r(ν)}^{(h)}], the valuations of their generators rationally generate the group Φ. This means that the injection k_{ν}[x_{1}^{(h)}, …, x_{r(ν)}^{(h)}] → gr_{ν}R_{ν} is an integral extension, and the result follows from [B3], Chap. VIII, §2, No. 3, Theorem 1. □

Remark 4.13 We have seen another proof in the remark following Proposition 3.7.

4.4 A selection of examples.

Example 4.14 1) I give a slightly different proof of the fact, which we saw at the beginning of this section, that the algebra of the semigroup Z_{+}^{d} for the lexicographic order is a toric union of polynomial algebras; the relation between the two proofs is the idea for the induction on the height shown above. Let Φ be Z^{d} with the lexicographic order; the semigroup Φ_{+} is generated by the following elements e_{j}^{(1)} = (1, −j, 0, …, 0), e_{j}^{(2)} = (0, 1, −j, …, 0), …, e_{j}^{(d−1)} = (0, 0, …, 1, −j), e_{0}^{(d)} = (0, 0, …, 0, 1) for j = 0, 1, 2, . . .. To check this, we proceed by induction on d; the case d = 1 is obvious, so let d ≥ 2 and ϕ = (a_{1}, …, a_{d}) ∈ Z_{+}^{d}. If we have a_{1} = 0 we are reduced to Z^{d−1}, so we may assume a_{1} > 0. If a_{2} ≥ 0, we may write ϕ = a_{1}e_{0}^{(1)} + (0, a_{2}, …, a_{d}) and we are again reduced to the case of Z^{d−1}. If a_{2} < 0, let ℓ be the smallest positive integer such that ℓa_{1} + a_{2} ≥ 0; we can write ϕ = a_{1}e_{ℓ}^{(1)} + (0, ℓa_{1} + a_{2}, a_{3}, …, a_{d}) and again we may conclude by induction. One can check, again by induction, that the relations are generated by e_{j}^{(i)} = e_{j+1}^{(i)} + e_{0}^{(i+1)} for 1 ≤ i ≤ d − 1, j ≥ 0. The semigroup algebra A[t^{Z_{+}^{d}}] is isomorphic to

A[(V_{j_{1}}^{(1)})_{j_{1}∈N_{0}}, …, (V_{j_{d−1}}^{(d−1)})_{j_{d−1}∈N_{0}}, V_{0}^{(d)}]/((V_{j}^{(i)} − V_{j+1}^{(i)}V_{0}^{(i+1)})_{j∈N_{0},1≤i≤d−1}),

where N_{0} = N ∪ {0}. Note that from these equations we can read the fact that the valuation is of height d; they show that for each (i, j), 1 ≤ i ≤ d − 1, j ≥ 1 and each integer n we have V_{j}^{(i)} = V_{j+n}^{(i)}(V_{0}^{(i+1)})^{n}, so that the value in Φ_{+} of V_{j}^{(i)} is greater than n times the value of V_{0}^{(i+1)} for all n ≥ 1. This implies the existence of d − 1 non trivial convex subgroups of Φ; since we know that h(Φ) ≤ r(Φ), we have h(Φ) = d. In particular, for d = 2, we see that

A[t^{Z_{+}^{2}}] = A[W, V_{0}, V_{1}, …, V_{j}, …]/((V_{j} − W V_{j+1})_{j∈N∪{0}}).

Upon taking for A a field k, we see that it corresponds to the sequence of infinitely near points in the plane A^{2}(k) obtained by endlessly repeating the following process: blow up the origin of an ordered set of coordinates (u_{1}, u_{2}), choose the chart of the blow up where the ideal (u_{1}, u_{2}) is generated by u_{1}, and in this chart, with coordinates (u_{1}, u_{2}/u_{1}), blow up the origin. We see that the direct limit of the monomial maps k[u_{1}, u_{2}] → k[u_{1}, u_{2}/u_{1}] obtained by repeating this operation is k[t^{Z_{+}^{2}}] presented as above, with V_{j} = u_{2}u_{1}^{−j}, W = u_{1}.

By the correspondence between sequences of points in blowing ups and valuations ([V1]), this sequence of points corresponds to a valuation ν of the local ring R = k[u_{1}, u_{2}]_{(u_{1},u_{2})}, the monomial valuation in the coordinates (u_{1}, u_{2}) with value group (Z^{2}, lex.) determined by ν(u_{1}) = (1, 0), ν(u_{2}) = (0, 1). The graded algebra gr_{ν}R_{ν} is equal to k[t^{Z_{+}^{2}}]. The valuation ν is monomial with respect to the coordinates (u_{1}, u_{2}), so that we have gr_{ν}R = k[U_{1}, U_{2}], where U_{1}, U_{2} are the initial forms of u_{1}, u_{2}. Things are quite similar for d > 2.

Example 4.15 Given a positive real number τ, which I assume to be irrational, let us denote by

τ = [s_{1}, s_{2}, …, s_{i}, s_{i+1}, …] = s_{1} + \frac{1}{s_{2} + ⋯}

its continued fraction expansion. Let p_{i}/q_{i} = [s_{1}, …, s_{i}] be the i-th approximant. Using the inductive relations

p_{i+1} = p_{i−1} + s_{i+1}p_{i}, q_{i+1} = q_{i−1} + s_{i+1}q_{i} with p_{0} = 1, q_{0} = 0, p_{1} = s_{1}, q_{1} = 1,

the reader will see that Perron’s algorithm applied to (1, τ) produces the vectors A^{(h)} = (p_{h}, q_{h}) ∈ R_{+}^{2} for h ≥ 1. If Φ denotes the subgroup of R generated by (1, τ) endowed with the induced order, the algorithm gives the following presentation, for any commutative ring A:

A[t^{Φ_{+}}] = A[(V_{i})_{i∈N}]/((V_{i} − V_{i+1}^{s_{i}}V_{i+2})_{i∈N}).

It is equivalent to say that it presents A[t^{Φ_{+}}] as the direct limit of the maps

A[V_{i}, V_{i+1}] → A[V_{i+1}, V_{i+2}] given by V_{i} ↦ V_{i+1}^{s_{i}}V_{i+2}, V_{i+1} ↦ V_{i+1}.

The case where τ ∈ Q_{+} is left as an exercise. We shall see below in examples 4.16 and 4.17 the interpretation of these maps in terms of blowing ups, the corresponding valuation of k[u_{1}, u_{2}]_{(u_{1},u_{2})} and the graded algebra of its valuation ring.

Let ν be the monomial valuation of R = k[u_{1}, u_{2}]_{(u_{1},u_{2})} determined by ν(u_{1}) = 1, ν(u_{2}) = τ. The group of the valuation is Φ = Z + Zτ with the order induced by that of R, so that gr_{ν}R_{ν} is presented as above, with A = k:

gr_{ν}R_{ν} = k[(V_{i})_{i∈N}]/((V_{i} − V_{i+1}^{s_{i}}V_{i+2})_{i∈N}).

The semigroup of the values of ν on R is Γ = {a + bτ/a, b ∈ Z_{+} ∪ {0}} and the graded algebra gr_{ν}R is the polynomial algebra k[U_{1}, U_{2}], where U_{1} is of degree one and U_{2} of degree τ.

Example 4.16 Recall the examples given in [Z-S]. Vol.2, Chap. VI, §15, p.102: let k be a field, {s_{1}, s_{2}, …} a sequence of positive integers such that the products s_{1}s_{2} ⋯ s_{i} tend to infinity with i, and {c_{1}, c_{2}, …} a sequence of elements of k^{∗}. We define an infinite sequence of elements v_{i} in k(u_{1}, u_{2}) by

v_{1} = u_{1}, v_{2} = u_{2}, …, v_{i+2} = \frac{v_{i} − c_{i}v_{i+1}^{s_{i}}}{v_{i+1}^{s_{i}}} , ….

Setting R_{i} = k[v_{i−1}, v_{i}], m_{i} = (v_{i−1}, v_{i})R_{i}, Zariski shows that the ring

R_{ν} = ⋃_{i=2}^{∞}(R_{i})_{m_{i}}

is the valuation ring of a valuation ν of the field k(u_{1}, u_{2}) with value group

Φ = \{\frac{n}{s_{1} ⋯ s_{k}} , n ∈ Z, k = 1, 2, …\},

that is, all rational numbers whose denominator is a product of s_{i}’s.

One checks that the elements ((v_{i})_{i≥1}) generate the maximal ideal of R_{ν}, and their initial forms (V_{i}) generate the k-algebra gr_{ν}R_{ν}. As ν(v_{i+2}) > 0, we have ν(v_{i} − c_{i}v_{i+1}^{s_{i}}) > ν(v_{i+1}^{s_{i}}), which gives the relations V_{i} − c_{i}V_{i+1}^{s_{i}} = 0 for i ≥ 1; they generate all relations between the (V_{i}), so that the graded algebra of R_{ν} is given by

gr_{ν}R_{ν} = k[V_{1}, V_{2}, …, V_{i}, …]/((V_{i} − c_{i}V_{i+1}^{s_{i}})_{i≥1}).

Since the rational rank of Φ is one, we expect the k-algebra gr_{ν}R_{ν} to be the direct limit of polynomial algebras in one variable, and indeed this shows that gr_{ν}R_{ν} is the direct limit of the system of maps k[V_{i}] → k[V_{i+1}] given by V_{i} ↦ c_{i}V_{i+1}^{s_{i}}. This corresponds, replacing k by a commutative ring A and taking c_{i} = 1 for all is, to a presentation of the semigroup algebra of the semigroup Φ_{+} with coefficients in A as

A[t^{Φ_{+}}] = A[(V_{i})_{i∈N}]/((V_{i} − V_{i+1}^{s_{i}})_{i≥1}).

This corresponds also to the fact that Φ_{+} is the direct limit of its subsemigroups generated by \frac{1}{s_{1}....s_{i}}, the inclusion between two consecutive semigroups being given by multiplication by s_{i+1}.

Note that this valuation is not monomial for R = k[u_{1}, u_{2}]_{(u_{1},u_{2})} with respect to the coordinates (u_{1}, u_{2}).

In fact it is an interesting exercise to begin to compute gr_{ν}R as a subalgebra of gr_{ν}R_{ν} in this case. Recall that R = k[u_{1}, u_{2}]_{(u_{1},u_{2})}, and assume for simplicity that all s_{i} are > 1 for i ≥ 2. From the equations defining the v_{i}, we see that if we set ν(u_{1}) = 1, then ν(u_{2}) = \frac{1}{s_{1}} and u_{2}^{s_{1}} − c_{1}^{−1}u_{1} = −c_{1}^{−1}u_{2}^{s_{1}}v_{3} ∈ R, I set u_{3} = −c_{1}^{−1}u_{2}^{s_{1}}v_{3}, so we have the following equation defining u_{3}

u_{2}^{s_{1}} − c_{1}^{−1}u_{1} = u_{3},

and we see similarly that u_{3}^{s_{2}} − c_{2}^{−1}(−c_{1})^{s_{2}}u_{2}^{s_{1}s_{2}+1} = −c_{2}^{−1}u_{3}^{s_{2}}v_{4} ∈ R. I set u_{4} = −c_{2}^{−1}u_{3}^{s_{2}}v_{4} and so we have the equation

u_{3}^{s_{2}} − c_{2}^{−1}(−c_{1})^{s_{2}}u_{2}^{s_{1}s_{2}+1} = u_{4}

which we may use to define u_{4}. If we apply the same method to find u_{5}, we first find the equation

u_{4}^{s_{3}} − (−1)^{s_{3}(s_{2}−1)}c_{1}^{s_{2}s_{3}}c_{2}^{−s_{3}}c_{3}^{−1}u_{3}^{s_{2}s_{3}}v_{3} = (−1)^{s_{3}(s_{2}−1)}c_{1}^{s_{2}s_{3}}c_{2}^{−s_{3}}c_{3}^{−1}u_{3}^{s_{2}s_{3}}v_{4}^{s_{3}}v_{5}.

But the monomial u_{3}^{s_{2}s_{3}}v_{3} is not in R; its valuation, however, is in the semigroup Γ of R; it is the valuation of u_{2}^{s_{3}(s_{1}s_{2}+1)−s_{1}}u_{3}. Working out the computations gives an equation

u_{4}^{s_{3}} − d_{4}u_{2}^{s_{3}(s_{1}s_{2}+1)−s_{1}}u_{3} = ∗u_{4}^{s_{3}}v_{5} + ∗u_{4}v_{3} ∏_{ω∈μ_{s_{3}} \{1}}(u_{3}^{s_{2}} − ∗ωu_{2}^{s_{1}s_{2}+1}),

where d_{4} and the asterisks are monomials in c_{1}, c_{2}, c_{3}. We choose the right-hand term as u_{5}, note that its valuation is that of u_{4}^{s_{3}}v_{5}, and continue in this way. The minimal system of generators of the semigroup Γ corresponding to the values of ν on R is

Γ = ⟨\frac{1}{s_{1}} , 1 + \frac{1}{s_{1}s_{2}} , s_{2} + \frac{1}{s_{1}} + \frac{1}{s_{1}s_{2}s_{3}} , …, γ_{i}, …⟩,

where γ_{i} = ν(u_{i+1}) and

γ_{i+1} = s_{i}γ_{i} + \frac{1}{s_{1}s_{2} ⋯ s_{i+1}} .

From this definition follows that for all i ≥ 2

s_{i}γ_{i} ∈ ⟨γ_{1}, …, γ_{i−1}⟩.

We have seen the result for i = 2. Proceed now by induction on i, assuming the result true until γ_{i−1} and writing

s_{i}γ_{i} = (s_{i}s_{i−1} + 1)γ_{i−1} − γ_{i−1} + \frac{1}{s_{1} … s_{i−1}} = (s_{i}s_{i−1} + 1)γ_{i−1} + s_{i−2}γ_{i−2},

the last equality being provided by the induction assumption. Since we know by our induction hypothesis that s_{i}s_{i−1}γ_{i−1} ∈ ⟨γ_{1}, …, γ_{i−2}⟩, we see that in fact s_{i}γ_{i} − γ_{i−1} ∈ ⟨γ_{1}, …, γ_{i−2}⟩ for i ≥ 3. So we have relations, which we may write

s_{i}γ_{i} = γ_{i−1} + ∑_{k=1}^{i−2} ℓ_{k}^{(i+1)}γ_{k} with ℓ_{k}^{(i+1)} < s_{k} for all k ≥ 2.

The ℓ are indexed by i + 1 which corresponds to the variable U_{i} instead of i in order to give naturality to the equations given below. These relations give us the equations for the monomial curve associated to Γ in the infinite-dimensional space with coordinates (U_{j})_{j≥2} (note that we have eliminated U_{1}), and therefore also for its avatar Specgr_{ν}R; the latter are

U_{3}^{s_{2}} − d_{3}U_{2}^{s_{1}s_{2}+1} = 0, and U_{i+1}^{s_{i}} − d_{i+1}U_{i} ∏_{k=1}^{i−2} U_{k+1}^{ℓ_{k}^{(i+1)}} = 0 for i ≥ 3,

where the d_{i} are, up to sign, Laurent monomials in the c_{k}’s. Going back to our ring R, note that in the first few relations defining the u_{i} inductively another expression appeared naturally for the inclusion s_{3}γ_{3} ∈ ⟨γ_{1}, γ_{2}⟩. In any case the important fact is the following : the equations defining inductively the u_{j} are of the form

u_{j}^{s_{j−1}} − d_{j}u_{1}^{ℓ_{1}^{(j)}} … u_{j−1}^{ℓ_{j−1}^{(j)}} = u_{j+1}

with d_{j} ∈ k^{∗}, the d_{j}’s being Laurent monomials in the c_{k}’s, and the term on the right of the equation has greater valuation than the monomials appearing in the left hand side.

When we compute gr_{ν}R from these equations, we need to take as coordinates U_{2}, U_{3}, … but not U_{1}, because of the equation U_{2}^{s_{1}} − c_{1}^{−1}U_{1} = 0. All the righthand sides disappear because they have greater valuation, and the equations give algebraic relations between the initial forms U_{j} of u_{j} and those of u_{2}, …, u_{j−1}, which decreases the transcendence degree over k, and therefore the Krull dimension, of gr_{ν}R to one in agreement with Proposition 3.7 (see also the remark which follows it). In the ring R however each equation is captured, via its right-hand side u_{j+1}, in an abyssal^{5} sequence of relations, and never gets a chance to decrease the dimension of R by one. If we stopped after a finite number of steps, setting

u_{j}^{s_{j−1}} − d_{j}u_{1}^{ℓ_{1}^{(j)}} … u_{j−1}^{ℓ_{j−1}^{(j)}} = P (u_{1}, …, u_{j}),

where P is any polynomial (for example 0) which is not a zero divisor modulo the

ideal generated by u_{j}^{s_{j−1}} −d_{j}u_{1}^{ℓ_{1}^{(j)}} … u_{j−1}^{ℓ_{j−1}^{(j)}} and the previous equations, the dimension would decrease to one. This example displays for us a basic phenomenon: the decrease of dimension of the special fiber in the specialization of R to gr_{ν}R in the case where Abhyankar’s inequality is strict. In both rings, all variables of index ≥ 4 are algebraically dependent upon u_{2} and u_{3}, but in gr_{ν}R these two variables are dependent, while in R they are independent and all others are polynomials in them, which shows “from the equations” that R is regular of dimension two. In fact, we have given a presentation

k[u_{2}, u_{3}] = k[u_{2}, u_{3}, …, u_{i}, …]/(u_{3}^{s_{2}} − c_{1}^{−s_{2}}c_{2}^{−1}u_{2}^{s_{1}s_{2}+1} − u_{4}, u_{4}^{s_{3}} − …) of our polynomial ring which may seem complicated, but is in fact an explicit description of the valuation, allowing us to compute the valuation of every element, as follows: Given a polynomial P (u_{2}, u_{3}), replace every occurrence of u_{3}^{s_{2}} by c_{1}^{−s_{2}}c_{2}^{−1}u_{2}^{s_{1}s_{2}+1} +

5From the Greek meaning without bottom; in spite of a discrepancy in meaning I have preferred this to abysmal, which does not exist in French.

u_{4}, then every occurrence of u_{4}^{s_{3}} by d_{4}u_{2}^{ℓ_{2}^{(4)}}u_{3}^{ℓ_{3}^{(4)}} + u_{5}, still replacing occurences of u_{3}^{s_{2}} as above, and so on. This stops after finitely many steps because the product s_{2}s_{3} ⋯ s_{i} eventually exceed the degree of the polynomial P. We obtain finally a polynomial P̃(u_{2}, u_{3}, …, u_{k}) where no variable u_{i} appears with an exponent ≥ s_{i−1} for i ≥ 3, and now the valuation of P is the smallest valuation of the monomials of P̃ simply because relations between their initial forms in the graded ring have become impossible. We shall see below in subsection 5.5 that since ν is rational of height one, a similar result is always true at least after a suitable completion of the ring R, that is

R̂^{(ν)} = { \atop k[u_{2}, u_{3}, …, u_{i}, …]}/\overline{(F, (u_{j}^{n_{j}}u^{n(j)} − λ_{nm}u^{m(j)} + c_{j+1}u_{j+1} + ⋯)_{j≥j_{0}})},

where the second term is a quotient of a suitable completion of the polynomial ring, F is a finitely generated ideal, the c_{j+1} are in k^{∗} and the bar means a topological closure. If h(ν) > 1, a similar result is true, where the variables u_{i} are indexed by ordinals. Another important phenomenon seen here is that because of the minimality the initial form U_{1} of u_{1} does not appear among the generators of gr_{ν}R, so that if we wish to find generators of the maximal ideal of R whose initial forms are among a minimal system of generators of gr_{ν}R, we have to write R = k[u_{2}, u_{3}]_{(u_{2},u_{3})}. The valuation of u_{3} is maximal among those of elements which, together with u_{2}, generate the center of the valuation. Heuristically, the process leads us to coordinates which have, with respect to the given valuation, a property analogous to Hironaka’s maximal contact (see [H1] and below).

We can describe the valuation in terms of blowing-ups as follows: starting with A^{2}(k) with ordered coordinates (v_{1}, v_{2}), we perform the following sequence of steps. Step 1: reverse the order of coordinates. Step 2: blow-up the origin and localize in the chart where v_{2}, now the first coordinate, generates the ideal (v_{1}, v_{2}), taking as coordinates (v_{2}, v_{1}v_{2}^{−1}). Step 3: repeat this second step s_{1} − 1 times. Step 4: substract c_{1} from the second coordinate so that the point with coordinate c_{1} on the axis of the second coordinate (the exceptional divisor just created) is the origin of coordinates, and localize at this point. Repeat from step 1, replacing (s_{1}, c_{1}) by (s_{2}, c_{2}), and so on. The valuation ring R_{ν} is the direct limit of the direct system of local rings corresponding to the sequence of infinitely near points obtained in this way. Note that the strict transform of u_{3} is a coordinate at the point obtained after step 4; and so on; it is one of the attributes of maximal contact. According to ([Z-S], Vol. 2, p. 104), any subgroup Φ of the additive group Q of rational numbers can be obtained in this manner. This applies in particular to Φ = Q, obtained by taking s_{i} = i for all i ≥ 1.

Another construction of valuations whose value groups are given subgroups of Q is given by Zariski in [Z3], p.648, which leads to the distinction between those prime numbers which divide only a finite number of the s_{i}’s and those which divide an infinite number of them. If we worked over a field whose characteristic belongs to the second group and tried to effectively build inductively elements like the v_{i} by solving equations of the type c_{i}v_{i+1}^{s_{i}} − v_{i} + ⋯ = 0, we would run into substantial difficulties. However, this is not at all what I do here; I only use the direct system as a means to approximate gr_{ν}R_{ν} by polynomial subalgebras. Note anyway that in these examples, gr_{ν}R is not regular.

Example 4.17 One may ask what happens if, in the sequence of steps just described, one makes no translation; it amounts to setting all the c_{i} equal to zero. Since the equations describing the formation of the v_{i} are

v_{i} = v_{i+1}^{s_{i}}(c_{i} + v_{i+2}),

we see that the graded algebra of our new valuation ν_{0} is

gr_{ν_{0}}R_{ν_{0}} = k[V_{1}, V_{2}, …, V_{i}, …]/((V_{i} − V_{i+1}^{s_{i}}V_{i+2})_{i∈N}).

It follows from what we saw in example 4.15 that ν_{0} is the monomial valuation of height one and rational rank two defined by ν_{0}(u) = 1, ν_{0}(v) = τ, where τ is the number with continued fraction expansion

τ = [s_{1}, s_{2}, …, s_{i}, s_{i+1}, …].

Its value group is

Φ_{τ} = Z + Zτ ⊂ R

with the order induced by that of R. We have gr_{ν_{0}}R = k[U_{1}, U_{2}], with U_{1} of degree 1 and U_{2} of degree τ; it is regular. In particular, the valuation with value group Q constructed above becomes, upon setting all the c_{i} equal to zero, the monomial valuation corresponding to ν_{0}(u) = 1, ν_{0}(v) = ρ, where

ρ = [1, 2, 3, …, i, i + 1, …].

Problem. It would be interesting if, given any rational valuation ν of a nœtherian local integral domain R, one could specialize it (in the sense used just above, which differs from the usual one) to a valuation of R of rational rank equal to the dimension of R. Heuristically this means we can always “suppress the translations” continuously. As we shall see in Section 5 it is reasonable to try first for a valuation ν_{0} of rational rank equal to the dimension of the scalewise ν-adic completion of R. Can one use the results of [A2] to verify this statement in dimension two?

Example 4.18 In [Z3], Zariski gives yet another construction, also found in [McL-S], of valuations on k[u_{1}, u_{2}]_{(u_{1},u_{2})} with a value group Φ ⊂ Q given in advance. Let k[[u^{Q_{+}}]] be the ring of power series with exponents forming a well ordered subset of Q_{+}, and with the same s_{i} as above, given a sequence of integers m_{i} and a sequence of elements e_{i} ∈ k^{∗}, consider the series

w(u) = e_{1}u^{\frac{m_{1}}{s_{1}}} + e_{2}u^{\frac{m_{2}}{s_{1}s_{2}}} + ⋯ + e_{j}u^{\frac{m_{j}}{s_{1}...s_{j}}} + · · · .

The map u_{1} ↦ u, u_{2} ↦ w(u) induces an injection

k[u_{1}, u_{2}]_{(u_{1},u_{2})} ⊂ k[[u^{Q_{+}}]]

and the u-adic valuation induces on k(u_{1}, u_{2}) a valuation having as group of values the group of rational numbers whose denominator is a product of s_{i}’s. The correspondence between this construction and the previous one is as follows.

In our set of equations u_{j}^{s_{j−1}} − d_{j}u_{1}^{ℓ_{1}^{(j)}} … u_{j−1}^{ℓ_{j−1}^{(j)}} = u_{j+1} for k[[u_{1}, u_{2}]], stop at j = ℓ and set u_{ℓ+1} = 0, then using the previous equations, eliminate the variables u_{i}, i > 2 to finally get an equation Q_{ℓ}(u_{1}, u_{2}) = 0 in u_{2} and u_{1} which has as solution a Puiseux series u_{2} = w_{ℓ}(u_{1}) whose exponents have denominators s_{1} … s_{ℓ−1}. As we let ℓ go to infinity, the series w_{ℓ}(u_{1}) converge to a series of the form w(u_{1}) in k[[u^{Q_{+}}]]. This is a generalization of the procedure given in [T1] to compute the equation of a branch having a given semigroup.

What I have just done is to describe the transcendental branch defined by u_{1} = u, u_{2} = w(u) as a plane deformation of the avatar corresponding to the constants d_{j} of the monomial curve associated to the semigroup Γ described above, which lives in an infinite-dimensional space. One can see, with a little work, that the Q_{ℓ}(u_{1}, u_{2}), viewed as polynomials in u_{2}, are “key polynomials” in the sense of [McL 2] and [Ka] for the valuation ν viewed as extending to k(u_{1})(u_{2}) the (u_{1})-adic valuation of k(u_{1}) (the reader may also consult [F-J] and [V2] ). This provides a geometric interpretation for these polynomials. Note that, as ℓ goes to infinity, the parametric representation u_{2} = w_{ℓ}(u_{1}) of our approximating plane branches tend to a transcendental series, while their equations Q_{ℓ} tend to zero in the (u_{1}, u_{2})-adic topology; this last fact explains why the algebraic description, which is blind to transcendental series, gives us the whole (formal germ of the) affine plane as a limit.

Example 4.19 A power series in one variable with rational exponents having unbounded denominators is not necessarily very transcendental in positive characteristic: Let k be a perfect field of finite characteristic p; set K = ⋃_{n≥1} k(u^{\frac{1}{p^{n}}}), the perfect closure of k(u). There is a unique extension to K of the u-adic valuation of k(u), and its valuation ring is not nœtherian. Consider the series

v = ∑_{i=1}^{∞} u^{1− \frac{1}{p^{i}}} ∈ k[[u^{Q_{+}}]];

it is a solution of the polynomial equation

v^{p} − u^{p−1}(1 + v) = 0.

This equation is an Artin-Schreier equation: it is obtained from the standard Artin-Schreier y^{p}−y = \frac{1}{u} by replacing y by \frac{v}{u}. If we set L = K(v), it is shown in [K3] that the extension L/K has degree p and defect p. More precisely, the unique extension ν to K of the u-adic valuation of k(u) has a unique extension ν′ to L, with the same group of values, so that the ramification index e = [Φ′ : Φ] is equal to one, and no residual extension so that the inertia degree f = [κ(ν′) : κ(ν)] is also equal to one. The extension is of degree p so that the Ostrowski ramification formula (see [K3], [Roq]), which is [L : K] = def, where d is the defect, gives d = p. This defect complicates the parametrization but does not make it more difficult to create a non-singular model. We remark that our curve is a deformation of the monomial curve v^{p} − u^{p−1} = 0, and apply to this monomial curve the toric resolution process of [G-T] and subsection 6 below: it gives us a chart u = y_{1}^{p}y_{2}, v = y_{1}^{p−1}y_{2}. Our equation then becomes y_{1}^{p(p−1)}y_{2}^{p−1}(y_{2} − 1 − y_{1}^{p−1}y_{2}), so that the strict transform y_{2} − 1 − y_{1}^{p−1}y_{2} = 0 is non singular. It can be parametrized in a neighborhood of the exceptional divisor y_{1}y_{2} = 0 by y_{2} = _{\frac{1}{1−y_{1}^{p−1}}} , so that we have the following parametrization of our curve in a neighborhood of the origin:

u = \frac{y_{1}^{p}}{1 − y_{1}^{p−1}} ; v = \frac{y_{1}^{p−1}}{1 − y_{1}^{p−1}} .

The fact that the extension K ⊂ K(v) has defect seems to be related to the fact that while the extension of fields k(u) → k(u)[v]/(v^{p} −u^{p−1}(1+v)) is separable, the extension of graded rings associated to the u-adic valuation of k[u] and its extension to k[u, v]/(v^{p} − u^{p−1}(1 + v)), which is

k[U] → k[U, V]/(V^{p} − U^{p−1})

is purely inseparable of degree p.

5 Completion problems

In this section I study the ν-adic completion of a nœtherian local ring R and show that it is nœtherian in certain cases. Although completions of valued fields are a classical topic, the algebraic structure of the ν-adic completions of nœtherian rings does not seem to have been studied. Perhaps the reason is that the nœtherian hypothesis is not natural from the viewpoint of the theory of valued fields, while completions other than those with respect to an ideal are not commonplace in algebraic geometry. This is a meeting place for the number-theoretic/henselian and the algebro-geometric traditions of valuation theory.

The general fact is that the completion of R for the ν-adic topology is a quotient of the completion of R for the topology of the symbolic powers of the center of the height one valuation with which ν is composed. If R is analytically irreducible this topology is finer than the m-adic topology.

Being complete for the ν-adic topology, however, does not suffice to imply the convergence of the sequences which occur when one tries to lift elements from the graded ring to R. In order to overcome this difficulty I consider, in the case of a rational valuation, a quotient R̂^{(ν)} of the m-adic completion R̂^{m}, which is complete for a valuation ν̂ extending ν and such that the inclusion R → R̂^{(ν)} induces a scalewise birational morphism gr_{ν}R → grν̂R̂^{(ν)}. This ring R̂^{(ν)} is not only complete for the ν-adic topology but also scalewise complete, which ensures the convergence of the sequences mentioned above. It is also henselian, and in the equicharacteristic case admits a field of representatives.

I also show how to specialize the ring R̂^{(ν)} to a suitable (scalewise) ν̂_{gr}-adic completion of grν̂R̂^{(ν)} by a suitable completion of the valuation algebra, and the existence of a coordinate system and equations for the ring of this specialization, which is a fundamental fact in this approach.

5.1 The ν-adic completion of R. In general, even if the local ring R has a field of representatives, it is not possible to lift generators of the graded algebra gr_{ν}R to generators of R; we need a hypothesis of completeness (see [B3], Chap. III). The arguments using the fact that a deformation preserves smoothness and transversality also use some form of the implicit function theorem, or Hensel’s Lemma, the validity of which is another attribute of complete rings. This leads us to the study of the completion of a ring R with respect to the (P_{ϕ})_{ϕ∈Φ_{+}}-filtration. We note that the quotients R/P_{ϕ} form a countable projective system since the semigroup Φ_{+} is countable and totally ordered. One can then define the ν-adic completion of R as

R̂^{ν} = \underleftarrow{lim}_{ϕ∈Φ_{+}} R/P_{ϕ}.

There is a natural map R → R̂^{ν}, which is injective since the filtration is separated. The ring R̂^{ν} is an integral domain and the valuation ν has a canonical extension ν̂ to a valuation of R̂^{ν} with values in Φ. To see this it is enough to apply directly the explicit form of the definition of the ν-adic completion as a subset of the product of the quotients R/P_{ϕ}. We note that R̂^{ν} is the closure of R in the completion K̂^{ν} of its field of fractions with respect to the topology T_{ν} defined by ν (see [B3], Chap VI, §5, No. 3). In particular, we have:

Rν̂ = R̂_{ν}^{ν}.

One sees also immediately that if we set

P̂_{ϕ} = {x ∈ R̂^{ν} | ν̂(x) ≥ ϕ}, P̂_{ϕ}^{+} = {x ∈ R̂^{ν} | ν̂(x) > ϕ},

they are respectively the closure of P_{ϕ}, P_{ϕ}^{+} in R̂^{ν} ([B3], Chap. VI, §5 No. 2) and the natural inclusion P_{ϕ}/P_{ϕ}^{+} ⊂ P̂_{ϕ}/P̂_{ϕ}^{+} is an equality, so that the inclusion R ⊂ R̂^{ν} induces an equality of graded rings

gr_{ν}R = grν̂R̂^{ν}.

We note that the semigroup of values ν̂(R̂^{ν} \ {0}) ⊂ Φ_{+} ∪ {0} is the same as that of ν on R, and so is well ordered when R is nœtherian (Proposition 3.9).

Let us assume that R is a nœtherian local ring with maximal ideal m and ν is a valuation of K of height one such that its valuation ring R_{ν} contains R. Set p = m_{ν} ∩ R; since R is nœtherian, we have ν(p) > 0, say ν(p) = ϕ_{0}, and since ν is of height one, its group Φ is archimedian, so that for any ϕ ∈ Φ there exists an integer N(ϕ) such that N(ϕ)ϕ_{0} ≥ ϕ, that is p^{N(ϕ)} ⊂ P_{ϕ}. Then we have a projective system of surjective maps

R/p^{N(ϕ)} → R/P_{ϕ},

which induces a continuous morphism of completions R̂^{p} → R̂^{ν}, where R̂^{p} is the completion of R for the p-adic topology. Since R̂^{ν} is an integral domain, the kernel of this map is a prime ideal of R̂^{p}. By the definition of completions, this kernel is ⋂_{ϕ∈Φ_{+}} P_{ϕ}R̂^{p}, so that in the case where ν is of height one and R_{ν} dominates R, i.e., p = m, it coincides with the ideal called the implicit ideal of R̂ by Spivakovsky ([S2]). Recall that the symbolic power p^{(n)} = p^{n}R_{p} ∩ R is the set of elements x ∈ R such that there exists s ∉ p such that sx ∈ p^{n}. Because our ring R is a nœtherian domain, the symbolic powers define a separated topology on R (see [Z-S], Chap. IV, Th. 23), for which there is a completion R̂^{(p)}. Note that, since when s ∉ p we have ν(sx) = ν(x), with the notations just introduced we have p^{(N(ϕ))} ⊂ P_{ϕ}, so that we have natural continuous maps

R̂^{p} → R̂^{(p)} → R̂^{ν}.

Let us denote by H the kernel of the map R̂^{(p)} → R̂^{ν} and remark that H ∩R = (0). We have the following

Proposition 5.1 For a valuation ν of height one on a nœtherian local ring R, with the notations introduced above, the natural continuous injection ^{6}

R̂^{(p)}/H → R̂^{ν}

is an isomorphism of topological rings inducing the identity on R. If the center of ν in R is the maximal ideal m, we may drop the assumption that R is analytically irreducible, and moreover R̂^{m} → R̂^{(m)} is an isomorphism, so that R̂^{ν} is a quotient of R̂^{m} and is nœtherian. If R is a regular local ring, the map R̂^{p} → R̂^{(p)} is an isomorphism for all prime ideals, R̂^{ν} is a quotient of R̂^{p} and therefore is nœtherian.

Proof The injection

R̂^{(p)}/H ⊂ R̂^{ν}

implies that the valuation ν extends to R̂^{(p)}/H as a valuation ν̂ of height one. The valuation ν is still ≥ 0 on the localization R_{p}, and so we have an injection R̂_{p}^{pR_{p}}/H_{p} ⊂ R̂_{p}^{ν} corresponding to R_{p}. Let us replace R by R_{p} for a moment, so that 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_{ϕ}(R̂^{m}/H) written without repetition) form a simple infinite descending chain of ideals, which are primary for the maximal ideal m̂ of R̂^{m}/H by the archimedian property, and have intersection zero. I denote them by P̂_{j}. Now we can apply Chevalley’s Theorem ([Z-S], Vol. 2, Chap. VIII, §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 P̂_{j}, we have P̂_{j} ⊂ m̂^{s(j)}. This, added to the fact that the P̂_{j} are primary for m̂, proves that in the ring R̂^{m}/H the ν̂-adic topology coincides with the m̂-adic topology, so that it is complete for both, and therefore has to be equal to R̂^{ν}.

Now coming back to the case where the center of ν is not necessarily m, we can remark that by definition of the symbolic powers the inclusion R → R_{p} extends to an inclusion

R̂^{(p)} → R̂_{p}^{pR_{p}}.

In fact R̂^{(p)} is the closure of R in R̂_{p}^{pR_{p}}. The p^{(n)}-adic topology of the first ring is induced by the p^{n}-adic topology of the second since p^{(n)}R̂^{(p)} = p^{n}R̂_{p}^{pR_{p}} ∩ R̂^{(p)} as one can check with Cauchy sequences. The valuation ν extends to R_{p}, and so by the special case where the center is the maximal ideal, the ν-adic completion of R_{p} is R̂_{p}^{pR_{p}}/H_{p} where H_{p} = ⋂_{ϕ∈Φ_{+}} P_{ϕ}(R_{p})R̂_{p}^{pR_{p}}. Setting H = H_{p} ∩ R̂^{(p)}, we have an injection

R̂^{(p)}/H → R̂_{p}^{pR_{p}}/H_{p}.

The quotient topology on R̂^{(p)}/H is the p^{(n)}-adic topology and is induced by the quotient topology of R̂_{p}^{pR_{p}}; this last one coincides with the ν̂-adic topology by the special case, so that in the end the ν̂-adic topology and the p^{(n)}-adic topology coincide on R̂^{(p)}/H. From this follows that this quotient is equal to R̂^{ν}. Since P_{ϕ}(R) is finitely generated, its closure in R̂^{(p)}/H is P_{ϕ}(R)R̂^{(p)}/H and we have H = ⋂_{ϕ∈Φ_{+}} P_{ϕ}(R)R̂^{(p)}. This gives the first result in the general case. We have already seen that m^{(n)} = m^{n} and finally, in a regular ring the adic and symbolic topologies coincide for all ideals ([Ve], Th. 3.5). □

6This corrects an error in the statement of Proposition 1.3 of [T2]

Remarks 5.2 1) This shows that the ring R̂^{m}/H to which Spivakovsky ([S2]) extends the valuation ν in the special case of a valuation of height one such that R_{ν} dominates R is nothing but R̂^{ν}.

2) In particular if R is a regular local ring, since its completion R̂^{p} for the p-adic topology is a nœtherian ring ([B3], Chap. III, §3, No.4, Prop.8), for every valuation of height one ν of the field of fractions of R whose valuation ring contains R, the completion R̂^{ν} of R for the ν-adic valuation is nœtherian; as a topological ring it is in fact a Zariski ring with respect to the center of the valuation ν̂.

Proposition 5.3 Let R be an analytically irreducible nœtherian local ring contained in a valuation ring R_{ν} of its field of fractions. The completion R̂^{ν} is isomorphic to the completion R̂^{ν_{1}} of R with respect to the height one valuation ν_{1} with which the valuation ν is composed, and is a quotient of the symbolic p-adic completion of R, where p is the center of ν_{1} in R.

Proof We saw above that is true for valuations of height one. Because of Abhyankar’s inequality, the valuation ν is of finite height. Let us assume that h(ν) > 1. By assumption we have a surjective monotone map λ: Φ → Φ_{1} with Φ_{1} of height one and Kerλ of height h − 1, a maximal isolated subgroup of Φ.

Denote by ν_{1} the valuation λ ∘ ν. Then we have the inclusion P_{ϕ} ⊂ P_{λ(ϕ)}. Whenever ϕ_{1} > λ(ϕ), we have P_{ϕ_{1}} ⊂ P_{ϕ} ⊂ P_{λ(ϕ)}, which shows that the ν-adic topology on R coincides with the ν_{1}-adic topology, hence the equality of the completions. We are thus reduced to the case of height one treated above, and we see that the completion R̂^{ν} is a quotient of the completion of R with respect to the symbolic (m_{ν_{1}} ∩ R)-adic filtration. □

Remarks 5.4 1) In a nœtherian domain R with a non negative valuation ν, the ideal of elements of R which are topologically nilpotent with respect to the ν-adic topology is the center in R of the height one valuation with which ν is composed.

2) The equality of the topologies determined on a ring by two valuations, one of which is composed with the other, is well known; in the nœtherian case, a more precise result can be found in [Z-S], Vol. 2, Lemma 4 of Appendix 3, p. 344.

Corollary 9 Given a nœtherian local integral domain R and a valuation ν of its field of fractions, non negative on R, if ν is composed with a valuation ν_{1}, the natural map of completions

R̂^{ν} ⟶ R̂^{ν_{1}}

induced by the inclusions P_{ϕ} ⊂ P_{λ(ϕ)} is an isomorphism of topological rings. In particular, the valuation ν_{1} extends to a valuation ν̂_{1} on R̂^{ν}.

Corollary 10 If R is a regular excellent local ring, for any valuation ν of its field of fractions which is non negative on R, the ring R̂^{ν} is excellent.

Proof By Proposition 5.3, and the second part of Proposition 5.1, the ν-adic completion R̂^{ν} is a quotient of the completion R̂^{p} of the local ring R with respect to an ideal, so it is excellent as a quotient of an excellent ring (see [Ro1], [Ro2]). □

When trying to understand how to complete some non-nœtherian rings, essentially semigroup algebras, one encounters questions such as this:

Question. (Exercise; Compare with [Ha], [Ka]) Let A be a commutative ring and Γ the value semigroup of a valuation of height one on a nœtherian ring R. When is it true that the completion of the semigroup algebra A[t^{Γ}] with respect to the topology deduced from the canonical valuation ν(∑ a_{ϕ}t^{ϕ}) = min{ϕ|a_{ϕ} ≠ 0} is the ring of power series A[[t^{Γ}]] in the sense of [McL-S]? Remark that A[[t^{Γ}]], which a priori is only a group, is a ring, since in Γ there are only finitely many ways to write an element as a sum of other elements (because Γ is well ordered by Proposition 3.9; see [Ka], [B2], [C-G], Theorem 1), so that we can multiply two series.

Example 5.5 1) Take the usual example of a rank one valuation not satisfying Abhyankar’s equality (see above and [Z-S], Chap VI, §15, Example 2, [S1], [S2]): let k be a field, and take R = k[u_{1}, u_{2}]_{(u_{1},u_{2})} so that R̂ = k[[u_{1}, u_{2}]]. Let ∑_{i≥1} a_{i}u_{1}^{i} ∈ k[[u_{1}]] be transcendental over k(u_{1}) and set f(u_{1}, u_{2}) = u_{2} − ∑_{i≥1} a_{i}u_{1}^{i} ∈ R̂. Define a valuation ν on R by means of the injective map R → k[[t]] determined by u_{1} ↦ t , u_{2} ↦ ∑ a_{i}t^{i}. The residual transcendence degree of ν is 0 and its rational rank is one. The completion R̂^{ν} is isomorphic to the quotient k[[u_{1}, u_{2}]]/(f); it is one-dimensional. This shows that the injective map R → R̂^{ν} is not flat in general. The group of the valuation is Z and for each n ∈ N ∪ {0} we have P_{n}(R) = (t^{n})k[[t]] ∩ R. We see that for each n ≥ 1 we have u_{2} − ∑_{1}^{n−1} a_{i}u_{1}^{i} ∈ P_{n}(R), which shows that there is no inclusion P_{n}(R) ⊂ m^{s(n)} with s(n) > 1. Note that we could have taken any irreducible element f ∈ (u_{1}, u_{2})k[[u_{1}, u_{2}]] not having a root algebraic over k(u_{1}) and the composed map k[u_{1}, u_{2}]_{(u_{1},u_{2})} → k[[u_{1}, u_{2}]]/(f) → k[[t]] where the last map is the normalization.

2) On the same ring R, consider the valuation ν with value group Z^{2} ordered lexicographically, which attributes value (1, 0) to u_{1}, value (0, 1) to u_{2}, and such that the valuation of a polynomial is the infimum of the valuations of its monomials. There is a natural monotone surjective map λ = pr_{1} : Z^{2} → Z, hence a valuation ν_{1}, by the u_{1}-adic order. The ideal P_{(i,j)} is the product ideal u_{1}^{i}(u_{1}, u_{2}^{j}) of R, while the ideal P_{i} for ν_{1} is (u_{1}^{i}). Since (u_{1}^{i+1}) ⊂ u_{1}^{i}(u_{1}, u_{2}^{j}) ⊂ (u_{1}^{i}), these two filtrations define the same topology on R, so that in this case,

R̂^{ν} = R̂^{ν_{1}}.

Remark that the residual ideal (P_{(i,j)} : P_{i}) = (u_{1}, u_{2}^{j}) is a simple ν-ideal in R in the sense of Zariski’s theory (see [Z-S], Vol.2, Appendix 3). Notice however that, if we look at the primary decomposition as in loc.cit., we have P_{(i,j)} = (u_{1}^{i+1}, u_{2}^{j}) ∩ (u_{1}^{i}). In this case, the completions of k[u_{1}, u_{2}]_{(u_{1},u_{2})} with respect to ν and ν_{1} are isomorphic to

{ \atop k[u_{1}, u_{2}]_{(u_{1},u_{2})}}^{ν} = \underleftarrow{lim}_{i≥0}k[u_{1}, u_{2}]_{(u_{1},u_{2})}/(u_{1})^{i} = k[u_{2}]_{(u_{2})}[[u_{1}]];

the ideal p is (u_{1})k[u_{1}, u_{2}]_{(u_{1},u_{2})} and the ideal H is zero. Remark that the the quotient R̂^{ν}/pR̂^{ν} = k[u_{2}]_{(u_{2})} is not complete for the \overline{ν}-adic (in this case the (u_{2})-adic) topology.

3) (a non-discrete case) Given a subgroup Φ ⊂ R ordered by the order of R, consider, following H. Hahn ([Ha]), the ring k[[t^{Φ_{+}}]] of formal power series with coefficients in k and exponents in Φ_{+}, such that the set of their exponents is a well ordered subset of Φ_{+} (see also [Ka], [B2]). This ring is naturally equipped with the t-adic valuation, with values in Φ_{+}. Choose algebraically independent series without constant term u_{i}(t) ∈ k[[t^{Φ_{+}}]], 1 ≤ i ≤ r (see [McL-S]) and let R be the localization of the sub k-algebra of k[[t^{Φ_{+}}]] generated by u_{1}(t), …, u_{r}(t) at the maximal ideal generated by these elements. Since R is the image of an injective morphism ι: k[u_{1}, …, u_{r}]_{(u_{1},...,u_{r})} → k[[t^{Φ_{+}}]] determined by u_{i} ↦ u_{i}(t), the t-adic valuation of k[[t^{Φ_{+}}]] induces a rational valuation of height one on k[u_{1}, …, u_{r}]_{(u_{1},...,u_{r})}. The morphism ι extends, by substitution in formal power series, to a map k[[u_{1}, …, u_{r}]] → k[[t^{Φ_{+}}]], with a certain kernel H, which describes the formal relations between the u_{i}(t). By Proposition 5.1, the completion R̂^{ν} is the image of this map.

5.2 Scalewise completions of R. The gist of this subsection is the construction of ν-adic completions of R such that their quotients by the centers of the valuations with which ν̂ is composed are complete for the image of the ν̂-adic topology. The natural choice is a quotient of the m-adic completion of R. The connexion with the previous subsection is provided by the following:

Lemma 5.6 (Zariski, [Z5], p.29) If the nœtherian local ring R is analytically irreducible, for any prime ideal p high symbolic powers of p are contained in high powers of the maximal ideal m. □

From this and the description given above of the ν-adic completion follows the:

Proposition 5.7 If the nœtherian local integral domain R is complete for the m-adic topology, for any non negative valuation ν it is complete for the ν-adic topology as well.

Proof Let (x_{n})_{n∈N} be a Cauchy sequence in R for the symbolic p-adic topology, where p is the center of the valuation of height one with which ν is composed. By the preceding lemma 5.6, it is also a Cauchy sequence for the m-adic topology, and so converges, since R is complete, to some x ∈ R. Let us show that it converges to x for the symbolic p-adic topology as well. Say that we have x−x_{n} ∈ m^{s(n)}, with s(n) tending to infinity with n. Since (x_{n}) is a Cauchy sequence for the symbolic p-adic topology, for j ≥ n we have x_{j} − x_{n} ∈ p^{(t(n))}, with t(n) tending to infinity with n. Since x − x_{j} ∈ m^{s(j)}, for all j ≥ n we have x − x_{n} ∈ p^{(t(n))} + m^{s(j)}. But p^{(t(n))} is closed for the m-adic topology since R is nœtherian, so letting j increase gives x − x_{n} ∈ p^{(t(n))}, which shows that (x_{n}) converges to x in the symbolic p-adic topology. Therefore we have R̂^{(p)} = R and the result follows from Proposition 5.1. □

Since the property of being m-adically complete is preserved under taking quotients, it is reasonable to try to extend the valuation ν to an m-adically complete ring; it will then be complete for ν, and is quotients as well. The case of rational valuations of height one suggests to try to extend ν to a quotient of R̂^{m}.

Definition 5.8 Let R be a nœtherian local domain. Given a family (ξ_{i})_{i∈I} of elements of the maximal ideal m of R, we say that a birational map R → R′ to a local ring R′ is a birational toric extension of R associated to the family if there exists a finite subset F ⊂ I and monomials (ξ^{b^{j}})_{j∈F} with b^{j} ∈ Z^{F} such that R′ is a localization of R[(ξ^{b^{j}})_{j∈F}] at a maximal ideal.

Remarks 5.9 1) In this text the elements ξ_{i} will be representatives in R of a system of generators \overline{ξ}_{i} of the k-algebra gr_{ν}R for a rational valuation ν of R. We will consider only toric extensions of the following form: there exists a regular rational cone σ = ⟨a^{1}, …, a^{N}⟩ in R_{+}^{N} with N = #F, such that if we denote by M the matrix with columns (a^{1}, …, a^{N}), the matrix with columns (b^{1}, …, b^{N}) is M^{−1}.

2) Each vector b^{j} can be written as a difference b^{j} = b_{+}^{j} − b_{−}^{j} of vectors with non negative coordinates. The equality ν(ξ^{b^{j}}) = 0 means that there is a constant

c_{j} ∈ k^{∗} such that \overline{ξ}^{b_{+}^{j}} − c_{j}\overline{ξ}^{b_{−}^{j}} = 0 in gr_{ν}R. If we have presented, according to Corollary 3 the graded algebra gr_{ν}R as k[(U_{i})_{i∈I}]/((U^{m} − λ_{mn}U^{n})_{(m,n)∈L}), this shows that the binomial U^{b_{+}^{j}} −c_{j}U^{b_{−}^{j}} must belong to the binomial ideal, and c_{j} is a monomial in the λ_{mn}. Heuristically, this means that some of the binomial relations in gr_{ν}R correspond to translations of the center of the valuation ν with respect to the origin of the new coordinates in toric extensions subordinate to ν, and the constants λ_{mn} correspond to the coordinates of these translations.

If R is endowed with a rational valuation ν, it will be assumed implicitely that ν(ξ^{b^{j}}) ≥ 0 for j ∈ F; the maximal ideal with respect to which we localize R[ξ^{b^{1}}, …, ξ^{b^{N}}] will then be the center of ν in this ring. In this manner we obtain a birational toric extension

R ⊂ R′ ⊂ R_{ν}.

In this case we say that the toric extension is subordinate to ν. Let us now consider a valuation ν_{1} of R, with which ν is composed; denote by Φ_{1} the group of ν_{1}, by p_{1} the center of ν_{1} in R and by \overline{ν} the valuation on R/p_{1} which is the image ofν.

Lemma 5.10 Let ν be a valuation of R and R → R′ be a birational toric extension subordinate to ν associated to a finite family (ξ_{j})_{j∈F} of elements such that ν_{1}(ξ_{j}) = 0 for j ∈ F. For each ϕ_{1} ∈ Φ_{1} ∪ {0}, each element of the the kernel and cokernel of the natural map of R/p_{1}-modules

P_{ϕ_{1}}(R)/P_{ϕ_{1}}^{+}(R) ⊗_{R/p_{1}} R′/p_{1}R′ → P_{ϕ_{1}}(R′)/P_{ϕ_{1}}^{+}(R′)

is annihilated by a monomial in the (ξ_{j})_{j∈F}. In particular they are torsion modules, and the kernel is the torsion submodule of P_{ϕ_{1}}(R)/P_{ϕ_{1}}^{+}(R) ⊗_{R/p_{1}} R′/p_{1}R′.

Proof Let x = ∑_{α} c_{i}ξ^{α_{i+}−α_{i−}} be a representative of an element \overline{x} of P_{ϕ_{1}}(R′)/P_{ϕ_{1}}^{+}(R′). Set α_{+} = ∑_{i} α_{i+}, α_{−} = ∑_{i} α_{i−}. The element ξ^{α_{−}}x is in P_{ϕ_{1}}(R), so that ξ^{α_{+}}x = ξ^{α_{+}−α_{−}}.ξ^{α_{−}}x is in P_{ϕ_{1}}(R)R′. This shows that the cokernel of our map is a torsion module. The proof for the kernel is analogous if we notice that P_{ϕ_{1}}(R′)/P_{ϕ_{1}}^{+}(R′) is a torsion-free R/p_{1}-module. This follows from the fact that the R′/P_{0}^{+}(R′)-module P_{ϕ_{1}}(R′)/P_{ϕ_{1}}^{+}(R′) is torsion-free (see subsection 2.3) and the map R/p_{1} → R′/P_{0}^{+}(R′) is injective since by construction P_{0}^{+}(R′) ∩ R = P_{0}^{+}(R) = p_{1}. The last sentence also follows from this remark. □

Remark 5.11 For ϕ_{1} = 0, our map becomes R′/p_{1}R′ → R′/P_{0}^{+}(R′). This map is clearly surjective; its kernel consists of the torsion elements of the R/p_{1}Rmodule R′/p_{1}R′. The ideals p_{1}R′ and P_{0}^{+}(R′) correspond respectively to the total transform and the strict transform under our toric extension of the center of ν_{1} in SpecR.

Definition 5.12 Assertion TLU(d) (Toric local uniformization in dimension ≤ d) is the following: For every local equicharacteristic excellent integral domain R of dimension ≤ d, given a rational valuation ν on R, and elements (ξ_{i})_{i∈I} in R whose initial forms generate the graded k-algebra gr_{ν}R, there is a birational toric (in the ξ_{i}) extension R ⊂ R′ ⊂ R_{ν} such that:

a) The local ring R′ is regular, with a system of coordinates containing elements whose images in gr_{ν}R′ are part of a minimal system of generators.

b) The kernel of the scalewise completion map R̂′^{m′} → R̂′^{(ν)} (see Proposition 5.16 below and the remarks which follow) is generated by part of a regular system of generators of the maximal ideal of the regular local ring R̂′^{m′}; in particular the complete local ring R̂′^{(ν)} is regular.

Definition 5.13 Assertion TP(d) (Toric principalization in dimension ≤ d) is the following: For every local equicharacteristic excellent integral domain R of dimension ≤ d, given a rational valuation ν and an ideal I of R there is a birational toric extension R′ ⊂ R_{ν} such that IR′ is of the form (ξ′)^{δ}u′, where u′ is a unit of R′, the ξ′_{j} are elements of the maximal ideal m′ and their images in gr_{ν}R′ are part of a system of generators of this k-algebra.

Definition 5.14 Let k be a ring and Φ be a totally ordered group of height h and

(0) ⊂ Ψ_{h−1} ⊂ ⋯ ⊂ Ψ_{1} ⊂ Ψ_{0} = Φ

the ordered sequence of its convex subgroups. A graded map G → G′ of Φ-graded k-algebras is said to be scalewise graded if for each Ψ_{k} the image of the subalgebra G_{k} ⊂ G generated by homogeneous elements whose degree is in Ψ_{k} is contained in the subalgebra G′_{k} generated by homogeneous elements of G′ whose degree is in Ψ_{k}.

Lemma 5.15 If k is a field, if the map G → G′ is injective, if all homogeneous components of G and G′ are k-vector spaces of dimension ≤ 1 and if the semigroups Γ = {ϕ ∈ Φ/G_{ϕ} ≠ (0) and Γ′ = {ϕ ∈ Φ/G′_{ϕ} ≠ (0) generate Φ as a group, then a scalewise graded inclusion is birational.

Proof It suffices to show that any non zero homogeneous element of G′ is the quotient of the images of two homogeneous elements of G. Such an element is of the form cy_{ϕ} where y_{ϕ} is a generator of the k-vector space G′_{ϕ} and c ∈ k^{∗}. Since ϕ is the difference of two elements of Γ, we may write y_{ϕ} = d_{ϕ}x_{ϕ_{1}}x_{ϕ_{2}}^{−1} where x_{ϕ_{i}} is a generator of G_{ϕ_{i}} and d_{ϕ} ∈ k. The result follows. □

A rather special case (valuations of height one and center m) of the first part of the following Proposition was proved en passant by Zariski (see [Z6], from the bottom of p. 63) and has been proved again by Spivakovsky ([S2]) with a different perspective, in both cases without the assumption that the residue field is algebraically closed; see the remark after Proposition 5.1. Spivakovsky has also communicated to me that he had proved an extension of his result to arbitrary finite heights.

Proposition 5.16 *Let R be a local equicharacteristic excellent integral domain with maximal ideal m and an algebraically closed residue field, and let ν a rational valuation on R with value froup Φ. a) There exists a prime ideal H of the m-adic completion R̂^{m} of R such that ν extends to a valuation ν̂ of R̂^{m}/H in such a way that:

b) the associated graded map

gr_{ν}R → grν̂R̂^{m}/H

is a scalewise birational extension of Φ-graded R/m-algebras, uniquely determined by (R, ν). If the valuation ν is of height one, the map gr_{ν}R → grν̂R̂^{m}/H is an isomorphism.*

Remarks 5.17 1) The proof depends on toric local uniformization, an argument for which is only sketched at the end of the paper.

2) The fact that ν̂ extends ν implies that H ∩ R = (0).

3) The assumption that the residue field is algebraically closed is due to the use of local uniformization in lower dimensions in the proof. It should eventually disappear when the method is generalized to excellent equicharacteristic local rings.

4) Although the ideal H is unique in the case considered by Zariski and Spivakovsky, it is not uniquely determined in general; it is only the graded map associated to the map R → R̂^{m}/H which is uniquely determined. Once H is fixed, the extension of ν is unique. A quotient R̂^{m}/H as in the Proposition will nevertheless be denoted by R̂^{(ν)}.

4) The fact that the extension of the graded rings is scalewise birational implies that any toric modification we make in R̂^{(ν)} with respect to representatives (η_{j})_{j∈J} of the generators of the graded algebra grν̂R̂^{(ν)} can be viewed as coming from a toric modification in R with respect to the chosen system of representatives (ξ_{i})_{i∈I}.

Proof (Sketch) The method is to work by induction on the dimension of R. Let us therefore assume first that R is a one-dimensional nœtherian local domain with a rational valuation ν, which is necessarily of height one. Let us set H = ⋂_{ϕ∈Φ_{+}} P_{ϕ}(R)R̂^{m}. Then we have H ∩ R = {0} and if we set P̂_{ϕ} = P_{ϕ}R̂^{m}/H, we see that P̂_{ϕ} ∩ R = P_{ϕ} since by the faithful flatness of the extension R → R̂^{m} we have P_{ϕ}R̂^{m} ∩ R = P_{ϕ}, and that the natural inclusion

gr_{ν}R ⊂ ⨁_{ϕ∈Φ_{+}} \frac{P̂_{ϕ}}{P̂_{ϕ}^{+}}

is an equality. This shows that R̂^{m}/H is an integral domain, since its associated graded ring for the filtration by the P̂_{ϕ} is an integral graded algebra, and that the P̂_{ϕ} are the valuation ideals of a valuation ν̂ on R̂^{m}/H which extends ν through the inclusion R ⊂ R̂^{m}/H. Since H ∩ R = (0), the ideal H is a minimal prime ideal of R̂^{m} uniquely determined by the valuation ν and the quotient R̂^{m}/H is a one-dimensional local domain; we may call it the formal branch of the curve SpecR corresponding to ν. The proof in the dimension one case is partly inspired by a proof in [S2], Theorem 4.3.

Assume now that the result is true for excellent local domains of dimension < d = dimR. Let us choose a set of elements (ξ_{i})_{i∈I} of R whose images in gr_{ν}R generate it as a k-algebra. It may happen that the (ξ_{i})_{i∈I} do not generate the maximal ideal of R. We choose a minimal set of elements y_{1}, …, y_{a} such that the (ξ_{i})_{i∈I}, y_{1}, …, y_{a} generate the maximal ideal. Our toric modifications will always be in terms of the (ξ_{i})_{i∈I}. Let ν_{h−1} denote the valuation of height one with which ν is composed, and p_{h−1} denote its center in R. The valuation ν projects as a rational valuation \overline{ν} of height h(ν) − 1 and value group Ψ_{1} of the ring \overline{R} = R/p_{h−1}, and we assume the result to be true for \overline{ν}.

The proof now uses TLU(d-1) and TP(d-1). By using TLU(d-1) we may make a toric birational extension \overline{R} → \overline{R}′ of \overline{R} in the coordinates which are the images of the \{ξ_{i})|ν(ξ_{i}) ∈ Ψ_{1}\} such that the conclusion of TLU is satisfied for \overline{R}′. Such a toric modification lifts in a natural way to a birational toric modification R → R′. By applying TP to the relevant Fitting ideals as \overline{R}-modules of any finite number of the quotients P_{ϕ_{h−1}}/P_{ϕ_{h−1}}^{+} we may assume that in addition their

transforms modulo torsion, in the sense of Lemma 5.10, are free \overline{R}′-submodules of P_{ϕ_{h−1}}(R′)/P_{ϕ_{h−1}}^{+}(R′) with a cokernel annihilated by some monomials in the images

in \overline{R}′ of the \{ξ_{i})|ν(ξ_{i}) ∈ Ψ_{1}\}.

In order to extend the valuation ν on R to a quotient R̂^{m}/H in the way required by the proposition it suffices to do the same for R′; the ideal H ∈ R̂^{m} will then be

the kernel of the map R̂^{m} → R̂′^{m′}/H′. From now on, I assume therefore that \overline{R} is regular, that \overline{H} ⊂ ˆ \overline{R}^{m} is generated by part of a system of coordinates, and that P_{ϕ_{h−1}}/P_{ϕ_{h−1}}^{+} contains a free submodule such that the quotient is annihilated by monomials in the \{ξ_{i})|ν(ξ_{i}) ∈ Ψ_{1}\}. Moreover, the ideal \overline{H} is generated by elements \overline{b}_{1}, …, \overline{b}_{s} of the form \overline{y}_{k} −a_{k}(\overline{η}) where the \overline{η} are representatives in ˆ \overline{R}^{m} of generators

of the graded algebra grˆ\overline{ν} ˆ\overline{R}^{(\overline{ν})}.

Denote by p̂ the kernel of the natural map R̂^{m} → ˆ \overline{R}^{(\overline{ν})}. Let \overline{b}_{1}, …, \overline{b}_{t} be the generators of \overline{H} which are not contained in p_{h−1}R̂m ; p̂ ∩ R̂^{m}/p_{h−1}R̂^{m}. We can lift them to elements b_{1}, …, b_{t} of R̂^{m}, of the form y_{k} − a_{k}(η). Now set

H = (b_{1}, …, b_{t})R̂^{m} + ⋂_{ϕ_{h−1}∈Φ_{h−1+}}(P_{ϕ_{h−1}} R̂m ; p̂ ∩ R̂^{m}).

The idea now is to extend the valuation ν_{h−1} to a valuation ν̂_{h−1} of the ring (R̂^{m}/H)_{p̂/H} which induces a valuation with center p̂/H in the ring R̂^{m}/H. The valuation ν̂ which we seek will be the composition of ˆ \overline{ν} and ν̂_{h−1}. For each ϕ_{h−1} ∈ Φ_{h−1+} let us set

P̂_{ϕ_{h−1}} = (P_{ϕ_{h−1}} R̂m ; p̂ ∩ R̂^{m} + (b_{1}, …, b_{t})R̂^{m})/H.

Let us first prove that the graded algebra ⨁_{ϕ_{h−1}∈Φ_{h−1}} P̂_{ϕ_{h−1}}/P̂_{ϕ_{h−1}+} is a ˆ \overline{R}^{(\overline{ν})}-

algebra, which is a birational extension of gr_{ν_{h−1}}R⊗\overline{R} ˆ\overline{R}^{(\overline{ν})} and is an integral domain. We see that p̂ = (b_{1}, …, b_{t})R̂^{m} + p_{h−1}R̂m ; p̂ ∩ R̂^{m}, and it follows from this that in

R̂^{m}/H we have p̂P̂_{ϕ_{h−1}} ⊂ P̂_{ϕ_{h−1}+}, so that our algebra is indeed a ˆ \overline{R}^{(\overline{ν})}-algebra. By construction, it is contained in gr_{ν_{h−1}}R ⊗\overline{R} κ(p̂_{h−1}), where κ(p̂_{h−1}) is the field of

fractions of ˆ \overline{R}^{(\overline{ν})}, so that it is a birational extension of gr_{ν_{h−1}}R ⊗\overline{R} ˆ \overline{R}^{(\overline{ν})}.

Now we have a candidate graded algebra of R̂^{m}/H with respect to the ν̂_{h−1} valuation. We are going to prove that it is an integral domain by building a filtration on it such that the associated graded ring is a domain. This filtration will turn out to be the filtration associated to a valuation ν̂ extending ν to R̂^{m}/H, and its associated graded ring will be a scalewise birational extension of gr_{ν}R. First, one must make a distinction concerning birational extensions of rings such as R̂^{m}/H; this distinction is essentially due to the fact that if R → R′ ⊂ R_{ν} is a birational extension, the extension R/p_{h−1} → R′/p′_{h−1} is not in general birational:

Given a birational toric extension ˆ \overline{R}^{(\overline{ν})} → ( ˆ\overline{R}^{(\overline{ν})})′ of ˆ \overline{R}^{(\overline{ν})}, we can lift it to a birational toric extension of R̂^{m}/H since the first ring is the quotient of the second by the ideal (\overline{H}R̂^{m} + p_{h−1}R̂^{m})/H. Let us call such an extension R̂^{m}/H → (R̂^{m}/H)′

a basic birational extension of R̂^{m}/H. There is a natural map (R̂^{m}/H)′ → ( ˆ\overline{R}^{(\overline{ν})})′. We use the fact that after a toric modification \overline{R} → \overline{R}′, for all further toric exten-

sions \overline{R}′ → \overline{R}′′ the ideal \overline{H}′′ is the strict transform of \overline{H}′ under the map ˆ \overline{R}′ → ˆ \overline{R}′′. This is part of the embedded resolution ”package” in lower dimension, see Proposition 7.7. There is another type of birational extension: Given ϕ_{h−1}, let us choose elements (e_{1}, …, e_{r}) of P_{ϕ_{h−1}}(R) such that ν(e_{1}) < ν(e_{2}) < … < ν(e_{r}) and their images modulo P_{ϕ_{h−1}}^{+} form a system of generators (\overline{e}_{1}, …, \overline{e}_{r}) of the R/p_{h−1}-module P_{ϕ_{h−1}}/P_{ϕ_{h−1}}^{+} . Let κ: R → R′ be the morphism of local rings obtained by blowing up the ideal P_{ϕ_{h−1}}(R) in SpecR and localizing at the point picked by the valuation ν. We have a map (not necessarily injective)

R̂^{m} → R̂′^{m′}.

The idea now is that to an element x ∈ P̂_{ϕ_{h−1}} may be attributed a ν̂-valuation as follows: There is an element s ∉ p̂ such that we may write sx = ∑_{i=1}^{s} \overline{c}_{i}e_{i} with \overline{c}_{i} ∈

R̂^{m}/H; in the ring R̂′^{m′}, we may write e_{i} = d_{i}e_{1} (2 ≤ i ≤ s), and therefore sx = (c_{1} + ∑_{i=2}^{s} c_{i}c_{i})e_{1}.

By our induction hypothesis, the valuation \overline{ν}′ on R′/p′_{h−1} induced by the valu-

ation ν on R′ extends to a valuation ˆ \overline{ν}′ with value group Ψ_{1} on { \atop R′/p′_{h−1}}^{(\overline{ν}′)}, which

is a quotient of R̂′^{m′}.

Now we may view c = c_{1} + ∑_{i=2}^{s} λ_{i}c_{i} as an element of R̂′^{m′}; its image in

{ \atop R′/p′_{h−1}}^{(\overline{ν}′)} is not zero by the construction of the ideals \overline{H}′ and we can define the ν̂-valuation of sx as the sum of ν(e_{1}) and the ˆ \overline{ν}′-valuation of the image of c

in { \atop R′/p′_{h−1}}^{(\overline{ν}′)}. This is well defined since H′R̂′^{m′} ∩ R̂^{m} and H are equal modulo p_{h−1}R̂^{m}. Then define the ν̂ valuation of x as ν̂(sx) − ν̂(s). If we replace R′ by an R′′ such that p_{h−1}R′′ = e_{1}R′′, the ˆ \overline{ν}′ value of c and its ˆ \overline{ν}” value are equal. For ϕ ∈ Φ_{+} ∪{0} having image ϕ_{h−1} in Φ_{h−1}, define the ideal P̂_{ϕ} ⊂ P̂_{ϕ_{h−1}} ⊂ R̂^{m}/H to be the set of those elements x ∈ P̂_{ϕ_{h−1}} such that in the local ring of some (not necessarily basic) birational extension (R̂^{m}/H)′ of R̂^{m}/H we have sx = ce_{1}, with

s ∉ p̂′/H′ and ϕ ≤ ˆ \overline{ν}(\overline{c}) + ν(e_{1}) − ˆ \overline{ν}(\overline{s}), where \overline{c} is the image of c in ( ˆ\overline{R}^{(\overline{ν})})′. By what we have just seen, the P̂_{ϕ}/P̂_{ϕ_{h−1}} form an exhaustive filtration of the quotient P̂_{ϕ_{h−1}}/P̂_{ϕ_{h−1}}^{+}. The quotient P̂_{ϕ}/P̂_{ϕ}^{+} is either (0) or k. Given x ∈ P̂_{ϕ_{h−1}} and y ∈ P̂_{ψ_{h−1}}, let f_{1}, g_{1} be generators of minimal valuation of P_{ψ_{h−1}} and P_{ϕ_{h−1}+ψ_{h−1}} respectively; after blowing up the product of ideals P_{ϕ_{h−1}}P_{ψ_{h−1}}P_{ϕ_{h−1}+ψ_{h−1}} and localizing at the point picked by ν, we may write e_{1}f_{1} = qg_{1}, together with x = ce_{1}, y = df_{1}. Using the definitions above, we see that ν̂(xy) = ˆ \overline{ν}(\overline{cdq})+ν(g_{1}), which means ν̂(xy) = ν̂(x)+ν̂(y). This shows that the graded algebra ⨁_{ϕ∈Φ_{+}∪{0}} P̂_{ϕ}/P̂_{ϕ}^{+} is a graded domain; it is therefore the graded algebra grν̂R̂^{m}/H with respect to the valuation ν̂. In particular, the ideal gr_{ν_{h−1}}R ⊗\overline{R} \overline{H} of gr_{ν_{h−1}}R ⊗\overline{R} ˆ \overline{R}^{m} is prime.

Now we have to show that the valuation ν̂ induces ν on R. Let us first prove that for all ϕ_{h−1} ∈ Φ_{h−1} we have that the inverse image in R of P̂_{ϕ_{h−1}} ∩ R is equal to P_{ϕ_{h−1}}. The inverse image clearly contains P_{ϕ_{h−1}}. If we did not have the opposite inclusion, there would be an element x ∈ R such that ν_{h−1}(x) = μ_{h−1} < ϕ_{h−1} and an element s ∈ R̂^{m} \ p̂ such that sx ∈ (b_{1}, …, b_{t})R̂^{m} + P_{ϕ_{h−1}}^{+} R̂^{m}. The image of sx in

P_{μ_{h−1}} R̂^{m}/P_{μ_{h−1}}^{+} R̂^{m} = P_{μ_{h−1}}/P_{μ_{h−1}}^{+} ⊗\overline{R} R̂^{m}/p_{h−1}R̂^{m}

has to come from an element y of (b_{1}, …, b_{t})R̂^{m}. Let us write y = ∑_{i=1}^{t} b_{i}c_{i} with c_{i} ∈ R̂^{m}. If the least P_{ϕ}R̂^{m} to which all c_{i} belong is μ_{h−1}, the image of y in P_{μ_{h−1}}/P_{μ_{h−1}}^{+} ⊗\overline{R} R̂^{m}/p_{h−1}R̂^{m} is in P_{μ_{h−1}}/P_{μ_{h−1}}^{+} ⊗\overline{R} \overline{H}. Now \overline{s} ∉ \overline{H} since s ∉ p̂ and so we must have that the initial form of x is in P_{μ_{h−1}}/P_{μ_{h−1}}^{+} ⊗\overline{R} \overline{H}. Since \overline{H} ∩ (R/p_{h−1}) = (0), and the image of y is also that of x ∈ R, this is impossible, and so an element such as x cannot exist. If the smallest ϕ such that c_{i} ∈ P_{ϕ} for all i is κ < μ_{h−1}, there must be a relation

∑_{j∈J⊂\{1,...,t\}} \overline{b}_{j}\overline{c}_{j} = 0,

where \overline{c}_{j} is the image of c_{j} modulo P_{κ}^{+}R̂^{m}. After possibly making a new toric modification we may assume, since ν_{h−1} is of height one, that all P_{κ}/P_{κ}^{+} for κ ≤ ϕ_{h−1} contain a free submodule M_{κ} such that the quotient by this submodule is annihilated by monomials in the {ξ_{i}|ν(ξ_{i}) ∈ Ψ_{1}}. Multiplying x by a suitable monomial ξ^{e}, and since ξ^{e}x ∈ R, we may assume that the \overline{c}_{j} are in M_{κ}, in view of the fact that the R̂^{m}/p_{h−1}R̂^{m}-module P_{κ}R̂^{m}/P_{κ}^{+}R̂^{m} is torsion-free. Since \overline{H} is generated by a regular sequence, the relation between the \overline{c}_{j} implies that for all j ∈ J we can write \overline{c}_{j} = ∑_{k} \overline{e}_{jk}\overline{b}_{k} with \overline{e}_{jk} = −\overline{e}_{kj}. Taking representatives e_{jk} ∈ R̂^{m} such that e_{jk} = −e_{kj}, we can finally write

ξ^{e}x = ∑_{i∉J} ξ^{e}c_{i}b_{i} + ∑_{j∈J}(ξ^{e}c_{j} − ∑_{k} e_{jk}b_{k})b_{j},

and the minimum of the ν_{h−1}-valuations has increased; in a finite number of such steps, we reach the situation where the minimum valuation of the coefficients is μ_{h−1} and a contradiction as above. This also shows that H ∩ R = ⋂_{ϕ_{h−1}∈Φ_{h−1+}} P_{ϕ_{h−1}} = (0). Finally, the valuation ν̂ on R̂^{m}/H is indeed the composition of the extension ν̂_{h−1} of ν_{h−1} to R̂^{m}/H and ˆ \overline{ν}

on ˆ \overline{R}^{(\overline{ν})}.

Since we have by the induction hypothesis that the graded algebra gr ˆ\overline{ν}′( ˆ\overline{R}′^{(\overline{ν})})′ is a scalewise birational extension of gr\overline{ν}(\overline{R}′), the group of values of ˆ \overline{ν}′ is Ψ_{1}, and therefore the group of values of ν̂ is Φ. The inclusion gr_{ν}R ⊂ ⨁_{ϕ∈Φ_{+}∪{0}} P̂_{ϕ}/P̂_{ϕ}^{+} is scalewise graded. From this and Lemma 5.15 follows that the inclusion

gr_{ν}R ⊂ grν̂R̂^{m}/H

is scalewise birational.

From the fact that the graded algebra grν̂_{h−1} R̂^{m}/H is independent from the choice of represntatives and that inside each of its homogeneous components the definition of P̂_{ϕ}/P̂_{ϕ_{h−1}} is itself by construction independent of the choice of the representatives ξ_{i}, we see that the inclusion gr_{ν}R ⊂ ⨁_{ϕ∈Φ_{+}∪{0}} P̂_{ϕ}/P̂_{ϕ_{h−1}} is independent of the choice of the representatives

If the valuation ν is of height one, we set H = ⋂_{ϕ∈Φ_{+}} P_{ϕ}R̂^{m} and P̂_{ϕ} = P_{ϕ}R̂^{m}/H. Since for all ϕ the ideal P̂_{ϕ} contains H and mP_{ϕ} ⊂ P_{ϕ}^{+}, one sees that P̂_{ϕ}/P̂_{ϕ}^{+} = P_{ϕ}/P_{ϕ}^{+}, and the fact that in this case the graded algebra does not change. □

Remarks 5.18 1) Unlike the graded k-algebra ⨁_{ϕ∈Φ_{+}∪{0}} P̂_{ϕ}/P̂_{ϕ}^{+}, the ideal H is not uniquely determined by (R, ν). Let R = k[x_{1}, x_{2}, x_{3}]_{(x_{1},x_{2},x_{3})}, let w(x_{1}) = ∑_{k=1}^{∞} c_{k}x_{1}^{k} be a series which is transcendental over k(x_{1}), consider the ring \overline{R} = k[x_{1}, x_{2}]_{(x_{1},x_{2})}, its injection ι: \overline{R} ⊂ k[[x_{1}]] defined by x_{1} ↦ x_{1}, x_{2} ↦ w(x_{1}) and the valuation \overline{ν} of \overline{R} defined by the restriction via the injection ι of the x_{1}-adic valuation. Let ν be the rational valuation of R composed of \overline{ν} and the x_{3}-adic valuation. As we have seen in example 5.5 of subsection 5.1, the ideal \overline{H} is the ideal of k[[x_{1}, x_{2}]] generated by x_{2} − w(x_{1}). For given series without constant terms y(x_{3}), z(x_{3}) we may choose as representatives of x_{1}, x_{2} the elements x_{1} + y(x_{3}), x_{2} + z(x_{3}). The corresponding ideal H_{y,z} of k[[x_{1}, x_{2}, x_{3}]] is (x_{2} + z(x_{3}) − w(x_{1} + y(x_{3})))k[[x_{1}, x_{2}, x_{3}]]. For each pair y, z ∈ k[[x_{3}]] of series without constant term the valuation ν̂ on k[[x_{1}, x_{2}, x_{3}]]/H_{y,z} which is composed of the valuation ˆ \overline{ν} (which is the x_{1}-adic valuation on k[[x_{1}]]), and the x̃_{3}-adic valuation, where x̃_{3} is the image of x_{3} modulo H_{y,z}, induces the valuation ν on R.

In spite of this non uniqueness, I shall denote by R̂^{(ν)} a quotient R̂^{m}/H as in the Proposition and sometimes even speak of “the” scalewise completion of (R, ν). In a sense they are all ”equisingular” with respect to ν̂, just as general sections by non singular spaces transverse to an equisingularity stratum through a singular point are all equisingular.

2) A completion which does not induce an isomorphism on the associated graded rings may be somewhat surprising; let us take example 3.16 and assume that R̂^{m} is an integral domain but fR̂^{m} is not a prime ideal; say that we have fR̂^{m} = ĝĥR̂^{m} with ĝR̂^{m} and ĥR̂^{m} prime ideals. By Proposition 7.6 below, an ideal \overline{H} of R̂^{m}/fR̂^{m} as in the Proposition contains a unique minimal prime of R̂^{m}/fR̂^{m}. Assume that we are in the simple situation where \overline{H} = ĝ(R̂^{m}/fR̂^{m}). The valuation ν extends to a valuation on R̂^{m} which is composed of the ĝ-adic valuation, of height one, of R̂^{m}, and the valuation ˆ \overline{ν} on R̂^{m}/ĝR̂^{m} which extends the valuation \overline{ν} on R/(f). Remark that this is a case where \overline{H} ≠ (0) but H = (0), but of course p_{h−1}R̂^{m} is not prime, and is distinct from p_{h−1}R̂m ; p̂ ∩ R̂^{m}. Now by example 3.16, the graded ring grν̂R̂^{m} is equal to grˆ\overline{ν}(R^{m}/ĝR̂^{m})[Ĝ], where Ĝ is the initial form of ĝ with respect to ν̂.

The inclusion

gr_{ν}R = gr\overline{ν}(R/fR)[F] ⊂ grˆ\overline{ν}(R^{m}/ĝR̂^{m})[Ĝ]

is described by saying that gr\overline{ν}(R/fR) → grˆ\overline{ν}(R^{m}/ĝR̂^{m}) is a scalewise birational map by induction on the dimension, and F ↦ \overline{ĥ}Ĝ, where \overline{ĥ} is the image of ĥ in R̂^{m}/ĝR̂^{m}. This last step introduces a scalewise birational character in the graded rings extension if it was not already present for R/fR.

Corollary 11 Abhyankar’s inequality can be made more precise in the case of a rational valuation ν of an excellent local ring R; with the notations of section 3, we have:

r(ν) ≤ dimR̂^{(ν)} ≤ dimR.

In particular, if r(ν) = dimR the ideal H must be a minimal prime of R̂^{m} and if in addition R is analytically irreducible, we have R̂^{(ν)} = R̂^{m}. In all cases, in view of Proposition 7.6, the ideal H contains a unique minimal prime of R̂^{m}.

Remarks 5.19 1) If ν is a valuation of height one whose ring birationally dominates R, we have R̂^{(ν)} = R̂^{ν}. One may ask, if we only assume that ν is of height one, what is the relationship of the ring R̂^{(ν)} with the m-adic Hausdorff completion of R̂^{ν}.

2) The Proposition can be used to study extensions of a rational valuation ν of an excellent local ring R to the completion R̂^{m}; by composition with ν̂, every valuation of the regular local ring R̂_{H}^{m} centered at is maximal ideal gives rise to a valuation of R̂^{m} extending ν. As we saw in Corollary 11, if r(ν) = dimR, the ideal H must be a minimal prime of R̂^{m} and if R is analytically irreducible, ν extends to R̂^{m} with the same group. This extension is then unique since P̂_{ϕ} = P_{ϕ}R̂^{m}.

If r(ν) = dimR − 1, and we assume that H is not a minimal prime of R̂^{m}, then the localization R̂_{H}^{m} is one-dimensional because of the Proposition, and it is a regular local ring by Proposition 7.6 below. It is a valuation ring with group Z. Denote by μ its valuation. The valuation ν̃ on R̂^{m} with values in Z ⊕ Φ ordered lexicographically, defined by ν̃(x) = (μ(x), ν̂(x mod.H)) is an extension of ν to R̂^{m}.

In order to check the uniqueness of the extension, assuming that R is analytically irreducible, remark that ⋂_{ϕ∈Φ_{+}} P_{ϕ}R̂^{m} ⊆ H by construction of H. The first ideal is prime because the graded ring associated to the filtration of R̂^{m} by the P_{ϕ}R̂^{m} is equal to gr_{ν}R, which is a domain. Since R is analytically irreducible, and in view of the dimension condition, we have ⋂_{ϕ∈Φ_{+}} P_{ϕ}R̂^{m} = H. Then, given an extension ν̃ ′ of ν to R̂^{m} with group Φ̃, the ideal H appears as the set of elements z ∈ R̂^{m} such that ν̃ ′(z) > ϕ ∀ ϕ ∈ Φ ⊂ Φ̃, and Φ ⊂ Φ̃ is a convex subgroup of the group Φ̃. Then ν̃ ′ is composed of a valuation on the localization R̂_{H}^{m}, which is regular of dimension one, and a valuation on R̂^{m}/H, which is uniquely determined since its valuation ideals must be the P_{ϕ}R̂^{m}/H, and coincides with the extension ν̂ of the Proposition. This shows that if we assume R analytically irreducible, in the case where r(ν) = dimR − 1 there is a unique extension of ν to R̂^{m}, described as above. These two results on extensions of valuations to R̂^{m} in the case where it is a domain are due to Heinzer and Sally ([H-S]) under an assumption weaker than excellence.

3) After replacing R by R̂^{(ν)} and ν by ν̂, we may assume that the local ring R is complete for the m-adic topology, with a valuation ν, so that it is henselian by Hensel’s lemma, admits (in the equicharacteristic case) a field of representatives by Cohen’s theorem and is complete for the ν-adic topology.

Definition 5.20 Given R and a rational valuation as above, we call a map R → R̂^{(ν)} such as we have built in Proposition 5.16 a scalewise ν-adic completion of (R, ν).

Definition 5.21 We say that the valuation ν is normally flat in degree ϕ if either P_{ϕ}/P_{ϕ}^{+} is zero, or it is a flat \overline{R}-module, where \overline{R} = R/p with p the center of ν in R.

Question: Given a rational valuation ν on R, and a valuation μ with semigroup Δ, with which ν is composed, is it true that for any δ ∈ Δ there exits a birational toric extension R → R′ compatible with ν, determined by elements of μ-valuation zero, and such that μ is normally flat in degree δ on R′?

I now show that with this definition of scalewise completion one has a Cohentype theorem.

Lemma 5.22 Let R be an excellent equicharacteristic local integral domain, and ν a rational valuation on R. Assume that the valuation ν is of height one. Lift a system of generators (\overline{ξ}_{i})_{i∈I} of the k-algebra grν̂R̂^{(ν)} to elements ξ_{i} ∈ R and choose a field of representatives k ⊂ R̂^{(ν)}. The k-algebra k[(ξ_{i})_{i∈I}] is dense in R̂^{(ν)} for the ν̂-adic topology.

Proof We retain the notations of Proposition 5.16, remarking that our assumption implies that ν̂ is trivial on k and remembering that in this case grν̂R̂^{(ν)} = gr_{ν}R. If we pick x ∈ R̂^{(ν)}, its ν̂-adic initial form is in gr_{ν}R, and therefore inν̂x = P_{1}(\overline{ξ}_{i}) , with P_{1} ∈ k[(X_{i})_{i∈I}]. The polynomial P_{1} is in fact a term cX^{a} with c ∈ k^{∗} and X^{a} a monomial, but I will not use this here. Abbreviating (ξ_{i})_{i∈I} to (ξ), we set x^{(0)} = x, x^{(1)} = x − P_{1}(ξ), and iterate this process. We have for all j ≥ 1 the inequality ν̂(x^{(j+1)}) > ν̂(x^{(j)}) and thus ν̂(x^{(j+1)} − x^{(j)}) = ν̂(x^{(j)}). Now remark that since R̂^{(ν)} is nœtherian by Proposition 5.16, the valuation ν̂(m̂) is > 0, and since ν̂ is of height one, for any ϕ ∈ Φ_{+} there is an integer N(ϕ) such that m̂^{N(ϕ)} ⊂ P̂_{ϕ}. Then there are only finitely many distinct P̂_{ϕ′} containing P̂_{ϕ}, so that in a finite number of steps we have ν̂(x^{(j+1)}) > ϕ, which proves that the sequence x^{(j)} converges to x, and the Lemma. □

Remark 5.23 Without the height one assumption, it could happen that the valuations of the differences x−∑_{j=1}^{r} P_{j}(ξ) increase at each step of the construction but remain bounded in the semigroup Φ_{+}.

I assume until the next Corollary that the local ring R is complete for its m-adic topology; see remark 2) at the end of subsection 5.1.

Example 5.24 Let k be a field, let R be a complete nœtherian local k-algebra with residue field k, and let f ∈ R generate a prime ideal. We go back to example 3.16 of subsection 3.4 and its notations. Let x ∈ R and let us write it in the form x = af^{ℓ} with a ∉ (f). We have ν_{1}(x) = ℓ and its ν-initial form is (in\overline{ν} \overline{a})ξ^{ℓ}. Let us fix a field of representatives k ⊂ R; in general, the element a will not be a polynomial with coefficients in k in the liftings (ξ_{i}) to R of the generators of gr\overline{ν} \overline{R}, so that we will have, denoting by ã a lifting in R of in\overline{ν} \overline{a} which is a polynomial in the (ξ_{i}), and choosing f as representative of ξ, a lifting ĩn_{ν}x of in_{ν}x which is polynomial in the ξ_{i} and f, and such that

x − ĩn_{ν}x = (a − ã)f^{ℓ}

and the ν_{1}-value will not have changed.

However, if we can approximate \overline{a} in \overline{R} by a sequence of polynomials P_{s}((\overline{ξ}_{i})_{i∈I_{1}}) in k[(\overline{ξ}_{i})_{i∈I_{1}}], we will have after replacing the \overline{ξ}_{i} by their representatives ξ_{i} in R lim_{s→∞}(a−P_{s}((ξ_{i})_{i∈I_{1}})) ∈ fR so that we find as limit of a sequence of polynomials P_{s}((\overline{ξ}_{i})_{i∈I_{1}})f^{ℓ}, with the notations of the first remark of this subsection, an element z^{(1)} ∈ R congruent to af^{ℓ} modulo the ideal f^{ℓ+1}R. We may write z^{(1)} = a_{ℓ}f^{ℓ}. Then we have to repeat the process with x_{1} = x − z^{(1)} and the next term. At each stage we build by the convergence of a series an element z^{(k)} ∈ R and x is the limit of the series ∑_{k} z^{(k)}. In fact, the element x then appears as the sum of a series ∑_{i=ℓ}^{∞} a_{ℓ}f^{ℓ}. I call this situation “scalewise density”^{7} for the polynomial ring k[(ξ_{i})_{i∈I}; f] in R.

Now for the general case:

Let R be a complete integral nœtherian local ring with a field of representatives k, and ν a k-valuation of its field of fractions, positive on R and whose center is the maximal ideal m of R. Let (ξ_{i})_{i∈I} be a set of elements of R whose images in gr_{ν}R generate it as a k-algebra. Consider the set of isolated subgroups of the value group Φ of ν

(0) = Ψ_{h} ⊂ Ψ_{h−1} ⊂ ⋯ ⊂ Ψ_{t} ⊂ ⋯ ⊂ Ψ_{1} ⊂ Ψ_{0} = Φ,

and the corresponding valuations ν_{t} with value group Φ/Ψ_{t} indexed by their height. Let

p_{r} ⊂ p_{r−1} ⊂ ⋯ ⊂ p_{1} ⊂ m

be the sequence of the distinct centers of the valuations ν_{t} in R. Let us partition the set I as follows: I = ⋃_{k=1}^{h} I_{t} where

I_{t} = {i ∈ I | ν(ξ_{i}) ∈ Ψ_{h−t} \ Ψ_{h−t+1}},

remarking that some of the sets I_{t} may be empty. Let us also define the following sequence of subrings of R: R̂_{1} is the closure of k[(ξ_{i})_{i∈I_{1}}] in R for the topology defined by the filtration by the ideals (P_{ψ})_{ψ∈Ψ_{h−1}}. We can continue and define inductively R̂_{t} to the closure of R̂_{t−1}[(ξ_{i})_{i∈I_{t}}] in R for the topology defined by the filtration by the ideals (P_{ψ})_{ψ∈Ψ_{h−t}}.

Proposition 5.25 For each t, 1 ≤ t ≤ h, the ring R̂_{t} is local and its maximal ideal is generated by the (ξ_{i})_{i∈⋃_{ℓ=1}^{t} I_{ℓ}}.

Proof The ring R̂_{1} is local because if y = ∑ a_{α}ξ^{α} is not in the closure of the ideal (ξ_{i})_{i∈I_{1}}, we can write y = a_{0}(1 + ∑_{α>0} a_{0}^{−1}a_{α}ξ^{α}) with a_{0} ∈ k^{∗}, and to show that y^{−1} is in R̂_{1} we have to show that the series

7F.-V. Kuhlmann has suggested to me that this idea could be related to Ostrowski’s pseudo-Cauchy sequences (see [Roq], [Ka]) and Ribenboim’s notion of “complet par étages” ([Ri]) in the theory of completion of valued fields. The problem is similar, but I did not compare the two definitions; there is a major difference in that I work with nœtherian rings.

1 − ∑_{α>0} a_{0}^{−1}a_{α}ξ^{α} + (∑_{α>0} a_{0}^{−1}a_{α}ξ^{α})^{2} + ⋯ + (∑_{α>0} a_{0}^{−1}a_{α}ξ^{α})^{n} + ⋯

converges, which is clear because Ψ_{h−1} is a group of height one. The passage from R̂_{t−1} to R̂_{t} is similar, if we notice the following

Lemma 5.26 If Ψ is an isolated subgroup of Φ such that Φ/Ψ is of height one, the semigroup Φ_{+} \ Ψ_{+} is archimedian.

Proof It follows directly from the fact that Φ/Ψ is archimedian. □

So, if ν(ξ) ∈ I_{t}, then as n ∈ N tends to infinity, nν(ξ) eventually exceeds any element of Ψ_{h−t}. □

Definition 5.27 We say that k[(ξ_{i})_{i∈I}] is scalewise dense in R for the ν-adic topology if R̂_{h} is equal to R.

Proposition 5.28 Let R_{ν} be the ring of a rational valuation of a complete equicharacteristic nœtherian local integral domain R. If we fix a field of representatives k of R and choose representatives (ξ_{i})_{i∈I} in R of a system of generators (\overline{ξ}_{i})_{i∈I} of the k-algebra gr_{ν}R, where k = R/m = R/(m_{ν} ∩ R), the algebra k[(ξ_{i})_{i∈I}] is scalewise dense in R for the ν-adic topology.

Proof It is useful to keep in mind Corollary 5, which gives us the structure of the initial forms which appear here. We have to prove that, given x ∈ R, there is a sequence (r_{j})_{j∈N} of elements of R̂_{h−1}[(ξ_{i})_{i∈I_{h}}] converging to x in the ν̃_{h}-adic topology. Define the sequence of x^{(j)} and P_{j}(ξ_{i}) as in the proof of Lemma 5.22, and let us build from it a sequence (r_{j})_{j∈N} of elements of R̂_{h−1}[(ξ_{i})_{i∈I_{h}}] such that the valuations ν(x − r_{j}) exceed, for large j, any ϕ ∈ Φ_{+} given in advance. Let s be the least integer such that for infinitely many values of the integer j we have ν_{s}(x^{(j+1)}) > ν_{s}(x^{(j)}). If s = 1, since the valuation ν_{1} is of height one, for any ϕ_{1} ∈ Φ_{1+}, after finitely many steps we have ν_{1}(x^{(j)}) > ϕ_{1}, hence for any ϕ ∈ Φ_{+}, after finitely many steps we have ν(x^{(j)}) > ϕ, so we may take r_{j} = x^{(j)} and get ν(x − ∑_{r=0}^{j−1} P_{r}(ξ_{i})) > ϕ and this shows that x ∈ R̂_{h}. If s > 1, by definition, there exists a j_{0} such that ν_{s−1}(x^{(j+1)}) = ν_{s−1}(x^{(j)}) for j ≥ j_{0}. By substracting from x a polynomial in the ξ_{i}, we may assume that j_{0} = 1. Note that the center of ν_{s−1} is necessarily distinct from the center of ν_{s}, since if the two centers are equal, there are only finitely many distinct ν_{s}-ideals between two consecutive ν_{s−1}-ideals of R, according to [Z-S], Vol. II, Appendix 3, Corollary p.345. We consider the initial forms in_{ν_{s−1}}x^{(j)} ∈ gr_{ν_{s−1}}R; for j ≥ j_{0}, they all have the same degree, say ϕ_{s−1}. Since the residual valuation \overline{ν}_{s} on R/(m_{ν_{s−1}} ∩ R) is of height one, with group Ψ_{s−1}/Ψ_{s}, and each homogeneous component of gr_{ν_{s−1}}R is a R/(m_{ν_{s−1}} ∩R)-module of finite type, and therefore complete for the m/(m_{ν_{s−1}} ∩R)- topology, so that it is also complete for the \overline{ν}_{s} topology by Proposition 5.7, the sequence of the initial forms in_{ν_{s−1}}(x − x^{(i)}) converges in (gr_{ν_{s−1}}R)_{ν_{s−1}(x)} to a

unique limit \overline{x}_{1}^{(1)}, which we may lift to an element x_{1}^{(1)} ∈ R̂_{h}. This element has the property that for all j ≥ 1, we have ν_{s−1}(x_{1}^{(1)}) = ν_{s−1}(x^{(j)}) and ν_{s−1}(x − x_{1}^{(1)}) > ν_{s−1}(x − x^{(j)}). Replacing now x by x − x_{1}^{(1)}, and so on, we build a sequence of elements x_{1}^{(q)} ∈ R_{h} such that ν_{s−1}(x − x_{1}^{(q)}) increases at each step. We may now repeat the whole process, replacing the sequence x^{(j)} by x_{1}^{(q)}; s is replaced by s − 1 and ultimately we are reduced to the case where s = 1, which is already settled. This proves the Proposition. □

I stop assuming that the local ring R is complete, but I assume until the end of this section that its residue field is algebraically closed.

Corollary 12 Let ν be a rational valuation on the local equicharacteristic excellent integral domain R; let k be the residue field of R and let (ξ_{i})_{i∈I} be representatives in R of a system of generators (\overline{ξ}_{i})_{i∈I} of the graded k-algebra gr_{ν}R. Let k ⊂ R̂^{(ν)} be a field of representatives in the scalewise completion of R built in Proposition 5.16 and let (η_{j})_{j∈J} be representatives in R̂^{(ν)} of a system of generators (\overline{η}_{j})_{j∈J} of the graded k-algebra grν̂R̂^{(ν)}, which are Laurent monomials in the (\overline{ξ}_{i})_{i∈I} (see 5.16). The subalgebra k[(η_{j})_{j∈J}] is scalewise dense in R̂^{(ν)}. The local subrings R̂_{t} ⊂ R̂^{(ν)} of Proposition 5.25 are complete and nœtherian, and the valuation ν restricted to R̂_{t} is analytically monomial with respect to the (η_{j})_{j∈⋃_{ℓ=1}^{t} I_{ℓ}}.

Proof The first part follows from the Proposition in view of the fact that R̂^{(ν)} is complete for its m̂-adic topology. To prove that the rings R_{t} are nœtherian the last statement, use descending induction on k: the ring R̂_{h} = R̂^{(ν)} is nœtherian, and each R̂_{t−1} is isomorphic to the quotient of R̂_{t} by the closure of the ideal generated by (ξ_{i})_{i∈I_{t}}. It also reproves that the R̂_{t} are local.

To prove the last two assertions, we can check that m_{ν_{h−1}} ∩ R̂_{1} = (0) so that an element of R/(m_{ν_{h−1}} ∩ R), which is in the closure of k[(\overline{ξ}_{i})_{i∈I_{1}}], lifts uniquely to an element of this closure in R, and we have seen in Proposition 3.22 that the valuation \overline{ν}_{1} is monomial with respect to the (\overline{η}_{j})_{j∈J_{1}}. One passes from R_{t−1} to R_{t} in the same way. □

5.3 The ν_{A}-adic completion of A_{ν}(R). The main result of this subsection is, given a system (\overline{ξ}_{i})_{i∈I} of generators of the k-algebra gr_{ν}R, giving rise to a surjective map k[(U_{i})_{i∈I}] → gr_{ν}R, a field of representatives k ⊂ R̂^{(ν)} and representatives (ξ_{i})_{i∈I} in R of the \overline{ξ}_{i}, the extension of the map k[(u_{i})_{i∈I}] → R̂^{(ν)} mapping u_{i} to ξ_{i} to a continuous surjective map

{ \atop k[(u_{i})_{i∈I}]} → R̂^{(ν)}

from the scalewise completion of the polynomial ring to the scalewise ν-adic completion of R. When { \atop k[(u_{i})_{i∈I}]} is endowed with the term order obtained by giving to u_{i} the valuation of its image ξ_{i} ∈ R̂^{(ν)}, the associated graded map is the map

k[(U_{i})_{i∈I}] → gr_{ν}R

mapping U_{i} to \overline{ξ}_{i}. This is a ν-adic analogue, called here T C(∗), of Cohen’s structure theorem ([B3], Chap. IX, §3, No.3, Th. 2, a)). Thus, T C(∗) is the generalization of the embedding of a plane complex branch in C^{g+1}, while the similar result for the valuation algebra corresponds to the construction of the family degenerating the re-embedded branch to the monomial curve.

For brevity I construct the map directly in the case of the valuation algebra (see Proposition 5.43), which is only slightly more complicated than the ν-adic Cohen theorem.

Proposition 5.29 In the situation of 5.1, the completion { \atop A_{ν}(R)}^{ν_{A}} is the closure of Aν̂(R̂^{ν}) in the algebra of restricted power series R̂^{ν}{v^{Φ}} defined by

R̂^{ν}\{v^{Φ}\} = \underleftarrow{lim}_{ϕ∈Φ_{+}}(R/P_{ϕ})[v^{Φ}].

Proof This follows from the definitions, the fact that S_{δ} = P_{δ}R[v^{Φ}] ∩ A_{ν}(R) and [B4], Chap II, §3, No. 9, Cor.1. □

Note that the closure in R̂^{ν} of P_{ϕ} is P̂_{ϕ} and that, with the notations of subsection 2.4, we have:

{ \atop A_{ν}(R)}^{ν_{A}} = \underleftarrow{lim}_{δ∈Φ_{+}} A_{ν}(R)/S_{δ} = \underleftarrow{lim}_{δ∈Φ_{+}} ⨁_{ϕ∈Φ} (P_{ϕ}(R)/P_{ϕ}(R) ∩ P_{δ}(R))v^{−ϕ}.

So we may view { \atop A_{ν}(R)}^{ν_{A}} as the subring of the R̂^{ν}-module R̂^{ν}[[v^{Φ}]] of formal power series ∑ x_{ϕ}v^{−ϕ}, consisting of power series ∑ x_{ϕ}v^{−ϕ} such that ν̂(x_{ϕ}) ≥ ϕ and which are restricted, i.e., such that for any ψ ∈ Φ_{+}, all but a finite number of the coefficients x_{ϕ} are in P_{ψ}. This last condition implies that when we formally multiply two series

( ∑ x_{ϕ}v^{−ϕ})( ∑ x_{ψ}v^{−ψ}) = ∑_{η∈Φ} ( ∑_{ϕ+ψ=η} x_{ϕ}y_{ψ})v^{−η},

the coefficient of v^{−η} is a convergent series in R̂^{ν}.

Lemma 5.30 The closure

\overline{(v^{Φ_{+}}){ \atop A_{ν}(R)}^{ν_{A}}} in { \atop A_{ν}(R)}^{ν_{A}} of the ideal (v^{Φ_{+}}){ \atop A_{ν}(R)}^{ν_{A}}

is the set of series

∑ x_{ϕ}v^{−ϕ} ∈ { \atop A_{ν}(R)}^{ν_{A}} such that ν̂(x_{ϕ}) > ϕ.

Proof The proof follows immediately from 5.29. □

Definition 5.31 With the notations of subsection 2.3, let us define on the model of ([B4], exerc. 27 for §2) the algebra

{ \atop A_{ν}(R)}^{ν_{A}}\{(v^{Φ_{+}})^{−1}\} = \underleftarrow{lim}_{δ∈Φ_{+}}(v^{Φ_{+}})^{−1}A_{ν}(R)/S_{δ} = { \atop Aν̂(R̂^{ν})}ν̂Â\{(v^{Φ_{+}})^{−1}\}.

This ring is also the Hausdorff completion of (v^{Φ_{+}})^{−1}A_{ν}(R) for the topology having as a fundamental system of neighborhoods the (v^{Φ_{+}})^{−1}P_{δ}. I follow Bourbaki’s convention according to which writing A{S^{−1}} entails the completion of A, i.e., A{S^{−1}} = Â{S′^{−1}} where S′ is the image of S in Â. The inclusion A_{ν}(R) ⟶ A_{ν}(R){(v^{Φ_{+}})^{−1}} is universal with respect to continuous maps A_{ν}(R) → B for linearly topologized Hausdorff and complete rings B such that the images in B of the elements of the multiplicative set (v^{Φ_{+}}) are invertible. I also consider the Hausdorff completion ĝr_{ν}R of the graded algebra gr_{ν}R with respect to the valuation ν_{gr}.

According to the general properties of completions of linearly topologized modules ([Ma], Prop. in 23.I, p.167), the completion of { \atop A_{ν}(R)}^{ν_{A}}/(v^{Φ_{+}}){ \atop A_{ν}(R)}^{ν_{A}} for the

quotient topology is { \atop A_{ν}(R)}^{ν_{A}}/\overline{(v^{Φ_{+}}){ \atop A_{ν}(R)}^{ν_{A}}}; it is also the completion, for the quotient topology, of A_{ν}(R)/(v^{Φ_{+}})A_{ν}(R). I can now state

Proposition 5.32 a) The natural map

A_{ν}(R) ⟶ gr_{ν}R

defined by

x_{ϕ}v^{−ϕ} ↦ x_{ϕ} mod.P_{ϕ}^{+}(R)

induces an isomorphism of filtered rings

{ \atop A_{ν}(R)}^{ν_{A}}/\overline{(v^{Φ_{+}}){ \atop A_{ν}(R)}^{ν_{A}}} ⟶^{≃} ĝr_{ν}R.

b) The natural inclusion R[v^{Φ_{+}}] ⟶ A obtained by considering only the part of negative degree ( i.e., ϕ ∈ Φ_{−}) of A induces an isomorphism

R[v^{Φ_{+}}]\{(v^{Φ_{+}})^{−1}\} = R̂^{ν}\{v^{Φ_{+}}\}\{(v^{Φ_{+}})^{−1}\} ⟶^{≃} { \atop A_{ν}(R)}^{ν_{A}}\{(v^{Φ_{+}})^{−1}\}.

Here, as above, R̂^{ν}{v^{Φ_{+}}} denotes the ring of restricted power series in v^{ϕ}, ϕ ∈ Φ_{+} and coefficients in R̂^{ν}.

c) Given a homomorphism ϕ ↦ c(ϕ) from Φ to the multiplicative group of units of R, it induces a surjection { \atop A_{ν}(R)}^{ν_{A}} → R̂^{ν} defined by ∑ x_{ϕ}v^{−ϕ} ↦ ∑ x_{ϕ}c(−ϕ). The kernel of this surjection is the closure of the ideal

((v^{ϕ} − c(ϕ))_{ϕ∈Φ_{+}}){ \atop A_{ν}(R)}^{ν_{A}}.

Proof Part a) follows from 5.29 and the general facts on completion, and Part b) follows from 2.2 and the remark following definition 5.31. Part c) follows from 2.2 which contains the fact that the corresponding result for A_{ν}(R) is true. □

Corollary 13 If ν is composed with a valuation ν_{1}, according to a surjective monotone non-decreasing group homomorphism λ: Φ → Φ_{1}, setting Ψ = Kerλ, we have:

{ \atop A_{ν_{1}}(R)}^{ν_{1,A}} = { \atop A_{ν}(R)}^{ν_{A}}/\overline{((v^{ψ} − 1)_{ψ∈Ψ_{+}})},

where the bar denotes the closure.

Proof This follows from Proposition 5.29 and the fact that in view of 9 the continuous map R̂^{ν}{v^{Φ}} → R̂^{ν_{1}}{v^{Φ_{1}}} is surjective and its kernel is the ideal \overline{((v^{ψ} − 1)_{ψ∈Ψ_{+}})}. □

In fact this corollary is the generalization to valuation algebras of the first corollary of Proposition 5.3. In spite of the fact that none of this applies when ν_{1} is the trivial valuation, we have:

{ \atop A_{ν}(R)}^{ν_{A}}/\overline{((v^{ϕ} − 1)_{ϕ∈Φ_{+}})} = R̂^{ν},

because the natural map A_{ν}(R) ⟶ R with kernel (v^{ϕ} − 1)_{ϕ∈Φ_{+}} is continuous not only when R has the trivial topology given by the trivial valuation, but also when R has the ν-adic topology. Assuming now, as we did in subsection 2.3, that R is a k-algebra for some field k, we have:

Proposition 5.33 *The k[v^{Φ_{+}}]-algebra { \atop A_{ν}(R)}^{ν_{A}} is faithfully flat.*

Proof This is proved exactly like Proposition 2.3 except that we have to accept infinite sums ∑ y_{kψ}v^{−ψ} and an infinite-dimensional vector space V in R̂^{ν}, but everything is convergent in { \atop A_{ν}(R)}^{ν_{A}} because we deal with restricted power series.

□

Remark that since ν is trivial on k, we have k[v^{Φ_{+}}] = k{v^{Φ_{+}}}.

In order to control the specialization we need in fact to consider the scalewise completion of both A_{ν}(R) and gr_{ν}R, which correspond to the scalewise completion R̂^{(ν)} of R. I only give the definitions and state the properties we need. Let us denote by

Φ →^{λ_{1}} Φ_{1} →^{λ_{2}} … →^{λ_{h−1}} Φ_{h−1}

the sequence of groups of the valuations with which ν is composed. Notice that the subset

Λ = {(ϕ, ϕ_{1}, …, ϕ_{h−1}) ∈ Φ × Φ_{1} × ⋯ × Φ_{h−1}/ϕ_{i} = λ_{i}(ϕ_{i−1})for 1 ≤ i ≤ h − 1}

is isomorphic to Φ. Iterating the last remark before subsection 3.2, we can write:

A_{ν}(R) = ⨁_{ϕ_{h−1}∈Φ_{h−1}}( ⨁_{λ_{h−1}(ϕ_{h−2})=ϕ_{h−1}}(… P_{ϕ}(R)w^{−ϕ} …)w_{h−2}^{−ϕ_{h−2}})w_{h−1}^{−ϕ_{h−1}}.

Definition 5.34 The scalewise completion of A_{ν}(R) consists in completing first with respect to the ν_{h−1,A}-adic topology, then completing the space of coefficients of each w^{−ϕ_{h−1}} with respect to the ν_{h−2,A}-adic topology, and so on. Note that each behaves like the filtration associated to a valuation of height one. I will denote the result by

{ \atop A_{ν}(R)}^{(ν_{A})}.

I define similarly the scalewise completion of gr_{ν}R after iterating the construction of Lemma 3.13 to obtain an isomorphism

gr_{ν}R →^{≃} ⨁_{(ϕ,ϕ_{1},...,ϕ_{h−1})∈Λ}(gr\overline{P}gr_{\overline{P}_{1}(ϕ_{1})} … (gr_{ν_{h−1}}R)_{ϕ_{h−1}})_{ϕ},

where gr_{\overline{P}_{i}} is the graded ring corresponding to the filtration \overline{ν}_{i} on each homogeneous component of gr_{ν_{i+1}} induced by ν_{i}. Again each behaves like in the case of height one.

Definition 5.35 The scalewise completion of gr_{ν}R is obtained by successively completing with respect to the filtrations ν_{h−1,gr}, \overline{ν}_{h−2,gr}, ..., \overline{ν}gr. It will be denoted by

ĝr_{ν}^{(ν)}R.

Proposition 5.36 *The k[v^{Φ_{+}}]-algebra { \atop Aν̂(R̂^{(ν)})}^{(ν_{A})} is faithfully flat; the special fiber of the map

Spec{ \atop Aν̂(R̂^{(ν)})}^{(ν_{A})} → Speck[v^{Φ_{+}}]

is Specĝr_{ν}^{(ν)}R and its “general” fiber, over Spec(v^{Φ_{+}})^{−1}k[v^{Φ_{+}}], is isomorphic to Spec(v^{Φ_{+}})^{−1}R̂^{(ν)}[v^{Φ_{+}}].* In any case, we shall see below why it is for the family

Spec{ \atop Aν̂(R̂^{(ν)})}^{(ν_{A})} → Speck[v^{Φ_{+}}]

that we may indeed hope to have simultaneous embedded uniformization; the k[v^{Φ_{+}}]-

algebra { \atop Aν̂(R̂^{(ν)})}^{(ν_{A})} is a quotient of the scalewise completion of k[v^{Φ_{+}}][(w_{j})_{j∈J}], which corresponds to a simultaneous embedding of its fibers. This will put us in a position to apply the implicit function theorem to extend a partial uniformization of ν_{gr} on gr_{ν}R to a uniformization of ν̂ on R̂^{(ν)}.

5.4 The specialization of R̂^{(ν)} to ĝr_{ν}^{(ν)}R. By Proposition 2.10 we have a graded isomorphism

gr_{ν_{A}}A_{ν}(R) →^{≃} (gr_{ν}R) ⊗\overline{R} \overline{R}[v^{Φ_{+}}].

If we take a system of generators \overline{ξ}_{i} of the \overline{R}-algebra gr_{ν}R, we obtain generators

\overline{υ}_{i} = \overline{ξ}_{i} ⊗1 of the \overline{R}[v^{Φ_{+}}]-algebra gr_{ν_{A}}A_{ν}(R) = grν̂_{A} { \atop A_{ν}(R)}^{(ν_{A})} and we can lift them

to homogeneous elements υ_{i} ∈ A_{ν}(R) ⊂ { \atop A_{ν}(R)}^{(ν_{A})}. Note that according to Proposition 2.10, each element υ_{i} is of the form ξ_{i}v^{−ν(ξ_{i})} where ξ_{i} ∈ R has initial form \overline{ξ}_{i}. Let us begin by assuming that R is local and has a field of representatives k. Then we may consider the morphism of k[v^{Φ_{+}}]-algebras

k[v^{Φ_{+}}][(u_{i})_{i∈I}] → A_{ν}(R)

determined by u_{i} ↦ υ_{i} = ξ_{i}v^{−ν(ξ_{i})}. We can consider the scalewise completion of k[v^{Φ_{+}}][(u_{i})_{i∈I}] defined as follows: define, with the notations introduced in Subsection 4.2, the sequence of rings

Ŝ_{1} = k[v^{Φ_{+}}][(u_{i})_{i∈I_{1}}]^{ν̃_{1}}, …, Ŝ_{t} = Ŝ_{t−1}[(u_{i})_{i∈I_{t}}]^{ν̃_{t}}, …, Ŝ_{h} = Ŝ_{h−1}[(u_{i})_{i∈I_{h}}]^{ν̃_{h}}

where ν̃_{t} is the monomial valuation on Ŝ_{t−1}[(u_{i})_{i∈I_{t}}] whose values lie in Ψ_{h−t}/Ψ_{h−t+1} defined by ν̃_{t}(∑ a_{α}u^{α}) = min(ν_{t}(ξ^{α})), and where the hat means the completion.

Definition 5.37 We can inductively define a valuation ν̃ on Ŝ_{h} as follows: it is trivial on k[v^{Φ_{+}}] and, assuming that it has been defined on Ŝ_{t−1}, for an element ∑ a_{α}u^{α} ∈ Ŝ_{t} we set

ν̃(∑ a_{α}u^{α}) = min_{α}{ν̃(a_{α}) + ν(ξ^{α})} ∈ Ψ_{h−t},

which is well defined because the semigroup Γ = ν(R \ {0}) is well ordered. It extends the monomial valuation on the polynomial ring with the (u_{i})_{i∈I} as system of generators and ν̃(u_{i}) = ν(ξ_{i}) = γ_{i}, the i-th generator of the semigroup Γ.

Note that ν̃ is not a rational valuation in general.

We can apply the same procedure to k[(u_{i})_{i∈I}] instead of k[v^{Φ_{+}}][(u_{i})_{i∈I}]:

Definition 5.38 The ring { \atop k[(u_{i})_{i∈I}]} obtained from the polynomial ring by the process just described is called the scalewise completion of the polynomial ring k[(u_{i})_{i∈I}] .

Proposition 5.39 The scalewise completion { \atop k[(u_{i})_{i∈I}]} is a local ring and its maximal ideal is the closure of the ideal generated by the (u_{i})_{i∈I}. The valuation ν̃ is rational in this case and analytically monomial with respect to (u_{i})_{i∈I}.

Proof The proof of the first two statements is the same as that of Proposition 5.25, and the last one follows from the very definition of ν̃. The graded algebra with respect to ν̃ is given by

grν̃{ \atop k[(u_{i})_{i∈I}]} = k[(U_{i})_{i∈I}].

□

Remarks 5.40 1) If the k-algebra gr_{ν}R is finitely generated, say by (\overline{ξ}_{i})_{i∈F}, the ring { \atop k[(u_{i})_{i∈I}]} is the usual power series ring k[[(u_{i})_{i∈F}]], and ν̃ is the usual monomial valuation determined by ν̃(u_{i}) = ν(ξ_{i}) = γ_{i}, the i-th generator of the semigroup Γ.

2) In the general case, but with Φ of height one say, { \atop k[(u_{i})_{i∈I}]} contains elements such as u_{1} + u_{2} + ⋯ + u_{i} + · · ·.

Proposition 5.41 The ring Ŝ_{h} is complete for the topology defined by ν̃.

Proof Again this is proved by induction on t. For t = 1 is follows from the definitions. Assume the proposition true up to t − 1 and let ∑_{α} a_{α}^{(i)}u^{α} ∈ Ŝ_{t} be a Cauchy sequence. The minimum over α of the ν̃ valuations of the differences (a_{α}^{(i)} − a_{α}^{(j)})u^{α} must become arbitrarily large in Ψ_{h−t} as min(i, j) grows. For each α the a_{α}^{(i)} must form a Cauchy sequence for the ν̃ topology, which converges by the induction hypothesis. □

Definition 5.42 We will denote by { \atop k[v^{Φ_{+}}][(u_{i})_{i∈I}]} the ring Ŝ_{h} endowed with the topology determined by the ν̃-adic filtration, and call it the scalewise completion of the ring k[v^{Φ_{+}}][(u_{i})_{i∈I}].

Assume that the local ring R is complete and let k ⊂ R be a field of representatives. The map k[(u_{i})_{i∈I}] → R defined by u_{i} ↦ ξ_{i}v^{−ν(ξ_{i})} induces a map

k[v^{Φ_{+}}][(u_{i})_{i∈I}] → A_{ν}(R); v^{−ϕ} ↦ v^{−ϕ}, u_{i} ↦ ξ_{i}v^{−ν(ξ_{i})},

which in turn induces a surjective map of k[v^{Φ_{+}}]-algebras

{ \atop k[v^{Φ_{+}}][(u_{i})_{i∈I}]} ⟶ { \atop A_{ν}(R)}^{(ν_{A})},

where the first hat designates the scalewise completion defined above. The surjectivity follows from Proposition 5.28 or rather from repeating its proof in this situation where k is replaced by k[v^{Φ_{+}}]. This map induces, modulo the closures of the ideals generated by (v^{Φ_{+}}), the surjective map

{ \atop k[(U_{i})_{i∈I}]} → ĝr_{ν}^{(ν)}R

which extends to the completions the original map U_{i} ↦ \overline{ξ}_{i}. We can summarize this as follows

Proposition 5.43 * Let R be a complete equicharacteristic nœtherian local integral domain with residue field k, and ν a rational valuation of R with value group Φ. Let k ⊂ R be a field of representatives for R and (ξ_{i})_{i∈I} be representatives in R of a system of homogeneous generators (\overline{ξ}_{i})_{i∈I} of the k-algebra gr_{ν}R. Then the map of k[v^{Φ_{+}}]-algebras k[v^{Φ_{+}}][(u_{i})_{i∈I}] → A_{ν}(R) determined by the injection k[v^{Φ_{+}}] ⊂ A_{ν}(R) deduced from k ⊂ R and the applications u_{i} ↦ ξ_{i}v^{−ν(ξ_{i})} extends to a continuous surjective map of topological k[v^{Φ_{+}}]-algebras

{ \atop k[v^{Φ_{+}}][(u_{i})_{i∈I}]} → { \atop A_{ν}(R)}^{(ν_{A})},

which induces, modulo the closures in each ring of the ideal (v^{Φ_{+}}), the map

{ \atop k[(U_{i})_{i∈I}]} → ĝr_{ν}^{(ν)}R.∗

(I write capital U’s for the initial forms of the u’s). The kernel of this last map is the closure in { \atop k[(U_{i})_{i∈I}]} of the binomial ideal (U^{m} − λ_{mn}U^{n})k[(U_{i})_{i∈I}] defining gr_{ν}R; it is the ideal consisting of possibly infinite sums

∑_{m,n} A_{m,n}(U)(U^{m} − λ_{mn}U^{n}) ∈ { \atop k[(U_{i})_{i∈I}]}, with A_{m,n}(U) ∈ { \atop k[(U_{i})_{i∈I}]}. Using the

fact that A_{ν}(R) is a flat k[v^{Φ_{+}}]-algebra, one can show

Proposition 5.44 * a) In the situation of the preceding Proposition, there exist elements of the form

F_{mn} = u^{m} − λ_{mn}u^{n} + ∑_{s} c_{s}^{(mn)}(v^{ϕ})u^{s} ∈ { \atop k[v^{Φ_{+}}][(u_{i})_{i∈I}]},

with, for all s appearing in F_{mn}, ν̃(u^{s}) > ν̃(u^{n}) = ν̃(u^{m}) and c_{s}^{(mn)} a term (=constant times a monomial) in the v^{ϕ}, and such that the closure in { \atop k[v^{Φ_{+}}][(u_{i})_{i∈I}]} of the ideal generated by the F_{mn} is the kernel of the surjection

{ \atop k[v^{Φ_{+}}][(u_{i})_{i∈I}]} → { \atop A_{ν}(R)}^{(ν_{A})}.

b) The F_{mn} may be chosen so that, with the notations of Proposition 5.25, if U^{m} −

λ_{mn}U^{n} ∈ k[(U_{i})_{i∈Ψ_{h−t}}], then F_{mn} ∈ { \atop k[v^{Φ_{+}}][(u_{i})_{i∈Ψ_{h−t}}]}*.

Proof By the remark following Proposition 2.3, and with its notations, after replacing R by R̂^{(ν)}, the k[v^{Φ_{+}}]-algebra B(S), which is the image of the map of k[v^{Φ_{+}}]-algebras

k[v^{Φ_{+}}][(u_{i})_{i∈I}] → A_{ν}(R), u_{i} ↦ ξ_{i}v^{−ν(ξ_{i})}

is faithfully flat. It follows that, denoting by J the kernel of this map, we can lift the generators U^{m} −λ_{mn}U^{n} of the kernel of the map k[(U_{i})_{i∈I}] → gr_{ν}R to elements F_{mn} ∈ k[v^{Φ_{+}}][(u_{i})_{i∈I}] generating an ideal J′ ⊆ J such that J ⊆ J′ + (v^{Φ_{+}}).J. The equality (v^{Φ_{+}}) ∩ J = (v^{Φ_{+}}).J holds by flatness since it is equivalent to

Tor_{1}^{k[v^{Φ_{+}} ][(u_{i})_{i∈I}]}(k[v^{Φ_{+}}][(u_{i})_{i∈I}]/(v^{Φ_{+}}), k[v^{Φ_{+}}][(u_{i})_{i∈I}]/J) = 0.

It means that we can write in A_{ν}(R) generators of J which begin with

u^{m} − λ_{mn}u^{n} + ∑_{s} c_{s}(v^{ϕ})u^{s} + ∑_{i} v^{ϕ_{i}}A_{i}(u^{m^{i}} − λ_{m^{i}n^{i}}u^{n^{i}} + ⋯).

Now we observe that since this sum must give zero in { \atop A_{ν}(R)}^{(ν_{A})}, the w-order of ∑_{i} v^{ϕ_{i}}A_{i}(u^{m^{i}} −λ_{m^{i}n^{i}}u^{n^{i}} +⋯) must be greater than that of u^{m}. Following through the process of scalewise completion, one sees that the closure in { \atop k[v^{Φ_{+}}][(u_{i})_{i∈I}]} of the ideal J′ is the kernel of the surjection

{ \atop k[v^{Φ_{+}}][(u_{i})_{i∈I}]} → { \atop A_{ν}(R)}^{(ν_{A})}.

To prove part b), one needs only apply this argument to each R̂_{t}. □

Before the next corollary, let us record the

Definition 5.45 Assertion TC(d) (Toric coordinatization in dimension ≤ d) is the following: For every excellent equicharacteristic local domain R of dimension ≤ d, given a rational valuation ν, there exists a quotient R̂^{(ν)} of R̂^{m} by an ideal H such that H∩R = (0) and an extension ν̂ of ν to R̂^{(ν)} inducing a scalewise birational map gr_{ν}R → grν̂R̂^{(ν)} of graded algebras such that, given a field of representatives k ⊂ R̂^{(ν)} of the residue field of R and representatives (η_{j})_{j∈J} in R̂^{(ν)} of a system of generators of the k-algebra grν̂R̂^{(ν)}, the map of k-algebras k[(w_{j})_{j∈J}] → R̂^{(ν)} determined by w_{j} ↦ η_{j} extends to a continuous surjection for the ν̃ and ν̂-adic topologies

{ \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)} with surjective associated graded map k[(W_{j})_{j∈J}] → grν̂R̂^{(ν)},

where the first ring is the scalewise completion of the polynomial ring (see the definition before Proposition 5.39). We denote by (W^{m} −λ_{mn}W^{n})_{(m,n)∈Ê} a system of generators for the kernel of the map k[(W_{j})_{j∈J}] → grν̂R̂^{(ν)}.

Corollary 14 Toric coordinatization: *(assuming that TLU(d-1) holds) Given a rational valuation ν of the excellent local domain R of dimension ≤ d, let c be a character of Φ i.e., an homomorphism c: Φ → k^{∗}; replacing R by R̂^{(ν)} and v^{ϕ} by c(ϕ) in 5.44 gives, in view of Propositions 5.32, c) and 5.7 a continuous surjection as in TC(d)

ĉ: { \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)}, w_{j} ↦ c(−γ_{i})η_{j}

where the hat means scalewise completion. The kernel of this map is the closure of the ideal generated by the elements

c(G_{mn}) = w^{m} − λ_{mn}w^{n} + ∑_{s} c_{s}^{(mn)}(c(ϕ))w^{s} (m, n) ∈ Ê.

In particular, taking the trivial character c_{1} determined by c_{1}(ϕ) = 1 ∀ϕ ∈ Φ, we have a surjective map

ĉ_{1} : { \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)} w_{j} ↦ η_{j}

whose kernel is generated up to closure by elements

c_{1}(G_{mn}) = w^{m} − λ_{mn}w^{n} + ∑_{s} c_{s}^{(mn)}(1)w^{s}.∗

As I have already noted, this may be thought of as a valuative analogue of Cohen’s theorem stating that every complete equicharacteristic nœtherian local ring is a quotient of a power series ring in finitely many variables over its residue field, remembering that the associated graded map must then be surjective. The advantage here is that, when the residue field is algebraically closed, we know that the kernel of the map is generated, up to closure, by deformations of binomials. One may summarize a part of my approach by saying that the elements η_{j} ∈ R̂^{(ν)} representing generators \overline{η}_{j} of the k-algebra grν̂R̂^{(ν)} form a system of generators of the maximal ideal of R̂^{(ν)} in which the valuation ν̂ is analytically monomial in the sense of section 3.7. The elements c(G_{mn}) may be thought of as an infinite standard, or Gröbner, basis for the kernel of the map { \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)} with respect to the the monomial order on the -countably many- variables w_{j} defined by deciding that w^{s} < w^{t} if the valuation of the image of w^{s} in R̂^{(ν)} is less than the valuation of the image of w^{t}.

Corollary 15 a) With the notations of subsection 2.4, each ideal S_{δ}(A_{ν}(R̂^{(ν)})) is generated as a k[v^{Φ_{+}}]-module, up to closure, by monomials in the η_{j}v^{−γ_{i}}.

b) Each ideal P_{ϕ}(R̂^{(ν)}) is generated as a k-vector space, up to closure, by monomials in the η_{j}.

c) If the k-algebra grν̂R̂^{(ν)} is finitely generated, say by (\overline{η}_{j})_{j∈F}, the k[v^{Φ_{+}}]-algebra

{ \atop Aν̂(R̂^{(ν)})} is a quotient of k[v^{Φ_{+}}][[(w_{j})_{j∈F}]].

Proof : a) is a direct consequence of 5.43 and Proposition 5.39. Statement b) is a consequence of a) by evaluation via a character of Φ in k^{∗}. For c), one checks by induction on the height that in the finitely generated case, the scalewise completion of k[v^{Φ_{+}}][(w_{j})_{j∈F}] coincides with k[v^{Φ_{+}}][[(w_{j})_{j∈F}]]. □

Remarks 5.46 1) Part b) of the Corollary means that the valuation ν̂ on R̂^{(ν)} is analytically monomial in the variables (w_{j})_{j∈J}.

2) Assume that the ring R contains a field of representatives. In spite of the flatness of the k[v^{Φ_{+}}]-algebra A_{ν}(R), it may happen that the k-algebra gr_{ν}R is finitely generated while the k[v^{Φ_{+}}]-algebra A_{ν}(R) is not; if we go back to example 5.5 of subsection 5.1, we can see that for each integer n, the ideal P_{n}(R) is generated by (u_{2}^{n}, u_{1} − ∑_{1}^{n−1} a_{i}u_{2}^{i}), while P_{n}(R̂^{(ν)}) = u_{2}^{n}k[[u_{2}]] and gr_{ν}R = grν̂R̂^{(ν)} = k[U_{2}].

3) It would be interesting to relate the fact that grν̂R̂^{(ν)} is finitely generated with the same condition for gr_{ν}R.

5.5 The abyssal phenomenon. Let us keep the notations of Corollary 14. The key fact now is that R̂^{(ν)} is nœtherian and complete for the m-adic topology. The images by ĉ of the w_{j} generate the maximal ideal of R̂^{(ν)}, and therefore there exists a finite subset L ⊂ J, which in the case of a valuation of height one we may assume to consist of {1, 2, …, q}, such that the images of (w_{ℓ})_{ℓ∈L} generate the maximal ideal of R̂^{(ν)}. This means that for i ∉ L and any character c: Φ_{+} → k^{∗}, we have a relation

w_{j} − ∑_{ℓ∈L} R_{ℓ}w_{ℓ} ∈ \overline{(\{w^{m} − λ_{mn}w^{n} + ∑_{s} c_{s}^{(mn)}(c(ϕ))w^{s}\}_{(m,n)})} in { \atop k[(w_{j})_{j∈J}]},

and this implies that one of the equations w^{m} − λ_{mn}w^{n} + ∑_{s} c_{s}^{(mn)}(c(ϕ))w^{s} must contain w_{j} linearly (compare with example 5.6). According to Remark 4.4, this w_{j} does not appear in the binomial term of G_{mn}.

The abyssal phenomenon is that, whenever we have among the equations defining R̂^{(ν)} as a quotient of { \atop k[(w_{j})_{j∈J}]} infinitely many consecutive indices j for which such an equation exists, these equations cannot create singularities, and at the same time they produce no decrease of the Krull dimension of R̂^{(ν)} (compare with example 4.16). The creation of singularities, as well as the decrease in dimension, are pushed away to infinity, since there is ”no equation”, but an endless sequence of substitutions. The proof of the abyssal phenomenon relies on an infinite dimensional implicit function theorem which is not proved here. It states that, using the fact that R̂^{(ν)} is henselian, we may solve this equation for w_{j} in R̂^{(ν)}. More precisely, what we need has the following form, where each variable w_{j} has weight ν(η_{j}) ∈ Φ:

Theorem 5.47 (Implicit functions Theorem ) * Let (G_{h})_{h∈H} be elements

in { \atop k[((w_{j})_{j∈J}]} of the form

G_{h} = w^{n(h)} − λ_{h}w^{m(h)} + c_{h}w_{h} + ∑_{s} c_{s}w^{s},

indexed by a subset H ⊂ J, with c_{h} ∈ k^{∗}, c_{s} ∈ k and the weight of w_{h} and all the w^{s} is greater than the weight of the terms of the initial isobaric binomial . Then, these series generate, up to closure, the same ideal as the following:

c_{h}w_{h} − (w^{n(h)} − λ_{h}w^{m(h)} + ∑_{t} d_{t}w^{t})

where the weight of all the monomials w^{t} is greater than the weight of the initial binomial and no variable w_{k} for k ∈ H appears in the monomials w^{t}.*

Proof This requires an infinite-dimensional implicit function theorem in { \atop k[((w_{j})_{j∈J}]}, somewhat similar to that of [VdH], see also ([B1], Chap. IV), to use the fact that all the ∂_{u_{h+1}}c(G_{h}) are invertible in { \atop k[((w_{j})_{j∈J}]}. This implicit function theorem should also allow us to recognize from the partial derivatives of the G_{h} when a nœtherian quotient of { \atop k[((w_{j})_{j∈J}]} by the closure of the ideal generated by elements (G_{h})_{h∈H} is regular.

Because of part b) of Proposition 5.44 and Lemma 5.26, the implicit function theorem can be proved by induction on the height, and at each step we have only a situation of height one type. □

Let us now consider a finite set of representatives (η_{j})_{j∈F} ∈ R̂^{(ν)} of generators \overline{η}_{j} of the k-algebra grν̂R̂^{(ν)} which has the following properties (by definition, as in Corollary 1 of subsection 3.1, the ordinal i + 1 is the index of the generator γ̂_{i+1} of the semigroup Γ̂ following γ̂_{i}):

a) they form a system of generators of the maximal ideal of R̂^{(ν)},

b) their valuations rationally generate the valuation group Φ of ν, c)With the usual notation for convex subgroups of Φ, whenever the set J_{t} = {j ∈ J|ν(η_{j}) ∈ Ψ_{h−t} \ Ψ_{h−t+1}} is finite, it is contained in F.

d) If i ∈ J satisfies i ∉ F, then i + 1 ∉ F.

Definition 5.48 Such sets of generators of the maximal ideal of R̂^{(ν)} will be called sufficient sets of generators.

Lemma 5.49 Let ν be a valuation of a nœtherian local ring R, and Γ = ⟨γ_{1}, …, γ_{i}, …⟩ its semigroup of values. Let Δ = (δ_{1} < … < δ_{j} < δ_{j+1} < …) be an increasing sequence of elements of Γ. Given a finite subset F of Δ, there is a finite subset F̃ of Δ containing F and such that δ_{j} ∉ F̃ implies δ_{j+1} ∉ F̃.

Proof If ν is of height one in R, it suffices to take for F̃ the set of elements of Δ smaller than or equal to the largest element of F; it is a finite set. Assume that the result is true for valuations of height ≤ h − 1, and let ν be a valuation of height h. Let λ: Φ → Φ_{1} be the map of groups corresponding to the valuation ν_{1} of height h − 1 with which ν is composed. Set Δ_{1} = λ(Δ) (as a set) and F_{1} = λ(F); by the induction hypothesis, we have a finite set F̃_{1} containing F_{1} and with the property of the Proposition with respect to Δ_{1}. Define F̃ as follows: it is the union of finite subsets F̃_{ϕ_{1}} of the λ^{−1}(ϕ_{1}) for ϕ_{1} ∈ F̃_{1}. If Δ∩λ^{−1}(ϕ_{1}) is finite (which is always the case if the center of ν_{1} is equal to the center of ν, by [Z-S], Appendix 3, Corollary to Lemma 4, or Proposition 3.14), set F̃_{ϕ_{1}} = Δ∩λ^{−1}(ϕ_{1}). If λ^{−1}(ϕ_{1})∩Δ is infinite, define F̃_{ϕ_{1}} to be the set of elements of λ^{−1}(ϕ_{1}) ∩ Δ which are smaller than or equal to the largest element of F contained in λ^{−1}(ϕ_{1}) ∩ Δ. By loc. cit., Lemma 4, or Proposition 3.14, each of these sets if finite. If δ_{j} ∉ F̃, either λ(δ_{j}) ∉ F̃_{1}, and then λ(δ_{j+1}) ∉ F̃_{1} so that δ_{j+1} ∉ F̃, or λ(δ_{j}) = ϕ_{1} ∈ F̃_{1} and λ^{−1}(ϕ_{1}) ∩ Δ is infinite, so that λ(δ_{j+1}) = ϕ_{1} and δ_{j+1} ∉ F̃ by definition of F̃_{ϕ_{1}}. □

Proposition 5.50 Sufficient sets of generators for the maximal ideal of R̂^{(ν)} exist.

Proof That condition a) can be satisfied follows from the nœtherianity of R̂^{(ν)}. For condition b) it is the finiteness of the rational rank r(ν), and for condition c) the finiteness of the height h(ν), which shows that there are finitely many sets J_{t}. The only thing left to check is that d) can be satisfied; it is a consequence of the preceding Lemma. □

Let

Γ̂ = ⟨γ̂_{1}, γ̂_{2}, …, γ̂_{i}, …⟩ ⊂ Φ_{+}

be the semigroup of the values taken on R̂^{(ν)}\{0} by the valuation ν̂. For each i ≥ 1, set Γ̂_{i} = ⟨γ̂_{1}, γ̂_{2}, …, γ̂_{i}⟩, and similarly, denote by Γ̂_{<i} the semigroup generated by elements γ̂_{k} with γ̂_{k} < γ̂_{i}; I will abbreviate this last inequality to k < i. If the height of Φ is one, by the Corollary to Proposition 3.9, there is a smallest integer p_{0} such that the group generated by Γ̂_{p_{0}} is equal to Φ. This means that for each i > p_{0} we have an expression t_{i}γ̂_{i} = ∑_{1≤j≤p_{0}} s_{j}^{(i)}γ̂_{j} with s_{j}^{(i)} ∈ Z. In the general case, we still have the following:

Proposition 5.51 a) With the notations just introduced, there is a finite set of generators (γ̂_{j})_{j∈F} which rationally generates Φ. For every i ∉ F there is a finite relation with non negative integral coefficients

n_{i}γ̂_{i} + ∑_{j∈F} n_{j}^{(i)}γ̂_{j} = ∑_{j∈F} s_{j}^{(i)}γ̂_{j},

such that n_{i} > 0 is minimal among all such relations. In addition, each relation between the γ̂_{j} is equivalent, modulo these relations, to a relation between the γ̂_{j}, j ∈ F. If the height of ν in R is one, in particular if the height of ν is one, there is an integer p_{0} such that one may choose F = {1, …, p_{0}}.

b) Given any relation between the generators γ̂_{s} of Γ̂, and denoting by γ̂_{i} the generator of highest degree which appears in it, it is equivalent, modulo the relations which exist between the (γ̂_{k})_{k<i}, to a relation of the form:

n_{i}γ̂_{i} + ∑_{k<i} n_{k}^{(i)}γ̂_{k} = ∑_{k<i} ℓ_{k}^{(i)}γ̂_{k},

such that n_{i} > 0 is minimal among all such relations.

c) We may assume that no relation beween the (γ̂_{k})_{k<i} appears in the difference ∑_{k<i} n_{k}^{(i)}γ̂_{k} − ∑_{k<i} ℓ_{k}^{(i)}γ̂_{k}, in the sense that no restriction of the sums gives a difference equal to zero.

Proof The first two statements follow from the fact that the rational rank of Φ is finite; the cardinality of F may be taken to be r(Φ). More precisely, we may take γ̂_{1} as the first element of F, then γ̂_{i_{2}} to be the smallest element of Γ̂ rationally independent of γ̂_{1}, γ̂_{i_{3}} to be the smallest element > γ̂_{i_{2}} which is rationally independent of γ̂_{1}, γ̂_{i_{2}}, and so on. If

(0) = Ψ_{h} ⊂ Ψ_{h−1} ⊂ ⋯ ⊂ Ψ_{1} ⊂ Ψ_{0} = Φ

are the convex subgroups of Φ, the first r(Ψ_{h−1}) elements of this sequence will be in Ψ_{h−1} ∩Γ̂, the next r(Ψ_{h−2})−r(Ψ_{h−1}) will be in (Ψ_{h−2} \Ψ_{h−1})∩Γ̂, and so on. In this way we build a finite sequence γ̂_{1} < γ̂_{i_{2}} < ⋯ < γ̂_{i_{r}} which rationally generates Γ̂. The notation is chosen to stress the fact that it is a subset of a generating sequence of the semigroup. For i ∉ F, if γ̂_{i_{j}} is the largest element in our sequence such that γ̂_{i_{j}} < γ̂_{i}, then γ̂_{i} is rationally dependant on (γ̂_{1}, γ̂_{i_{2}}, …, γ̂_{i_{j}}) and we may write

n_{i}γ̂_{i} + ∑_{k<i} n_{k}^{(i)}γ̂_{k} = ∑_{k<i} ℓ_{k}^{(i)}γ̂_{k}

a relation expressing the rational dependance of γ̂_{i} on the (γ̂_{k})_{k<i}; we may choose such a relation with minimal n_{i}. Let

m_{i}γ̂_{i} + ∑_{k<i} m_{k}^{(i)}γ̂_{k} = ∑_{k<i} p_{k}^{(i)}γ̂_{k}

be another relation with non negative coefficients. By the minimality of n_{i}, we have m_{i} ≥ n_{i}, and we may write m_{i} = qn_{i} + r with q ≥ 1 and 0 ≤ r < n_{i}. Adding q ∑_{k<i} n_{k}^{(i)}γ̂_{k} to both sides of the second relation, and substracting to each side of the equality q times the corresponding side of the first gives

rγ̂_{i} + ∑_{k<i} m_{k}^{(i)}γ̂_{k} + q ∑_{k<i} ℓ_{k}^{(i)}γ̂_{k} = ∑_{k<i} p_{k}^{(i)}γ̂_{k} + q ∑_{k<i} n_{k}^{(i)}γ̂_{k}.

By definition of n_{i}, we must have r = 0, so that we have proved that modulo the first relation, the second one becomes a relation between (γ̂_{k})_{k<i}. The last statement of a) follows from the fact that in the height one case, there are only finitely many γ̂_{k} smaller than any given element of Γ̂. Statement c) follows from the fact that among all relations with minimal n_{i}, we may choose one such that the number of ℓ_{k}^{(i)} which it contains is minimal. □

It is important in the sequel to remember only the fact that for i ∉ F some multiple of γ̂_{i} to which we add a sum of the form ∑_{k<i} t_{k}^{(i)}γ̂_{k} is in Γ̂_{<i}; choosing the smallest such multiple, we will then write:

(R_{i}) n_{i}γ̂_{i} + ∑_{k<i} n_{k}^{(i)}γ̂_{k} = ∑_{k<i} ℓ_{k}^{(i)}γ̂_{k}.

The coefficients of each such relation have no common divisor. The reason why it is important is that all relations between the γ̂_{i}’s appear in this way, and the coefficient n_{i} is really the smallest which can occur in a relation involving γ̂_{i} with smaller terms. If we require that these smaller terms are actually in F, the exponent n_{i} is in general larger.

With the notations of subsection 7.1, given a finite sufficient set of generators (w_{j})_{j∈F} for the maximal ideal of R̂^{(ν)}, let us choose h large enough for k[x_{1}^{(h)}, …, x_{r}^{(h)}] to contain the images in grν̂R̂^{(ν)} of the (W_{j})_{j∈F}.

Remark that the relations R_{i} just written provide us with the following binomial relations among those defining grν̂R̂^{(ν)}; we have, for i ∉ F:

(B_{i}) W_{i}^{n_{i}} ∏_{k<i} W_{k}^{n_{k}^{(i)}} − λ_{i} ∏_{k<i} W_{k}^{ℓ_{k}^{(i)}} = 0 in grν̂R̂^{(ν)} with λ_{i} ∈ k^{∗}.

Recall that the products appearing here and below are finite. Let us denote by B(F) the binomial equations between (W_{j})_{j∈F} which define the graded algebra gr(F) ; ν̂R̂^{(ν)} ⊂ grν̂R̂^{(ν)}. The ideal B in k[(W_{j})_{j∈J}] which defines grν̂R̂^{(ν)} contains the ideal B′ generated by (B(F), (B_{i})_{i∉F} ).

From now on, we take a finite subset (δ_{j})_{j∈F} of the generators of the semigroup Γ̂ of the values which ν̂ takes on R̂^{(ν)}, such that (η_{j})_{j∈F} is a sufficient set of generators for the maximal ideal of R̂^{(ν)}. Such a set exists by Proposition 5.50. Turning now to R̂^{(ν)}, we note that the result just proved provides us with a totally ordered subset of the set of equations G_{mn}; those such that their initial binomial part is the equation B(i), for i ∉ F; they can be written (G_{i})_{i∉F}.

Proposition 5.52 Let j ∉ F be the index of one of the variables w_{j}. Assume that j has a predecessor in J which is not in F and write j = i + 1; there exists an equation among the G_{mn} of Corollary 14, and in which w_{i+1} appears linearly. The equation can be chosen to be such that its initial binomial is of the form W_{i}^{n_{i}}W^{n(i)}− λ_{i}W^{m(i)}.

Proof The first observation is that in R̂^{(ν)}, by Cohen’s theorem, each of the elements η_{ℓ}; ℓ ∉ F, is a power series in the (η_{j})_{j∈F}, say η_{ℓ} = ∑_{p} d_{p}^{(ℓ)}η^{p}. Since it

involves only finitely many variables, this series makes sense in { \atop k[(w_{j})_{j∈J}]} and so we get elements w_{ℓ} − ∑_{p} d_{p}^{(ℓ)}w^{p} which belong to the kernel T of the map

{ \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)}. We may, in the series G_{mn} which generate this kernel up to closure, substitute freely the series ∑_{p} d_{p}^{(ℓ)}w^{p} for every occurrence of w_{ℓ} outside of the initial binomial, without changing this kernel. This shows that we may choose the G_{mn} corresponding to binomials containing only variables with indices in F in such a way that all the variables w_{j} which they contain have indices j ∈ F as well. Of course the initial forms of these generators of the kernel no longer necessarily generate the initial ideal (W^{m} − λ_{mn}W^{n})_{(m,n)∈Ê}. Another remark is that any system of generators, up to closure, of the ideal T must contain for each ℓ ∉ F a generator in which the variable w_{ℓ} appears linearly; otherwise there could not be a relation such as η_{ℓ} = ∑_{p} d_{p}^{(ℓ)}η^{p} in R̂^{(ν)}. Now we use again the fact that each element η_{j} for j ∉ F belongs in R̂^{(ν)} to the maximal ideal, generated by (η_{j})_{j∈F}. Let now i denote the smallest index of a generator of the semigroup Γ̂ such that i ∉ F. By the Proposition and our choice of generators for the ideal defining R^{(ν)}, we may choose a relation among those which are such that w_{i} appears for the first time in their initial (binomial) part, with minimal n_{i}; it is of the form:

G_{i} = w_{i}^{n_{i}} ∏_{k<i} w_{k}^{n_{k}^{(i)}} − λ_{i} ∏_{k<i} w_{k}^{ℓ_{k}^{(i)}} + ∑_{s} c_{s}^{(i+1)}(v^{ϕ})w^{s} = 0,

with the condition that the weight of each w^{s} must be greater than the weight

of w_{i}^{n_{i}} ∏_{k<i} w_{k}^{n_{k}^{(i)}}, which is equal to that of ∏_{k<i} w_{k}^{ℓ_{k}^{(i)}}, so that the variable w_{i+1} cannot appear in an equation G_{mn} whose initial binomial contains a variable of index > i. By what we have just seen, we may assume that w_{i+1} does not appear in the equations G_{mn} whose initial form contains only variables of index in F. We may assume that w_{i+1} appears linearly in the equation G_{i}, since it must appear in one of the equations whose initial binomial contains w_{i} and other variables of index < i. □

Proposition 5.53 (The abyssal phenomenon) *With the same notations, there exists a finite set F′ containing F such that the kernel T of the map

{ \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)}

can be generated up to closure by :

- a finite set of equations involving the variables (w_{j})_{j∈F′} and whose initial forms are binomials generating the kernel of the natural map k[(W_{j})_{j∈F′}] → grν̂R̂^{(ν)}, and - equations G_{i} = w_{i}^{n_{i}}w^{n(i)} − λ_{i}w^{m(i)} + c_{i+1}w_{i+1} + ∑_{p} c_{p}w^{p} , i ∉ F′, c_{i+1} ∈ k^{∗}.*

Proof We keep the same notations. We may also transform, without modifying the ideal, all the other equations G_{mn} whose initial binomial contains only variables of index ≤ i into equations in the (w_{j})_{j∈F}. Then we add the ordinal j to our set F and continue with its successor. By transfinite induction on j ∉ F, we build a set of equations

G_{i} = w_{i}^{n_{i}}w^{n(i)} − λ_{i}w^{m(i)} + c_{i+1}w_{i+1} + ∑_{p} c_{p}w^{p}

in bijection with the variables i ∉ F. Let us now consider the ideal F of { \atop k[(w_{j})_{j∈J}]} generated by the equations G_{mn} whose initial binomial contains only variables with index in F. By the implicit function theorem, the quotient of { \atop k[(w_{j})_{j∈J}]} by the closure of the ideal (F, (G_{i})_{i∉F}) is a quotient of k[[(w_{j})_{j∈F}]] which maps onto R̂^{(ν)}. The kernel of this map is generated by the images of the equations G_{mn} whose initial form involves some W_{j} with j ∉ F and which we have not used in the construction of the G_{i}. Since the ring k[[(w_{j})_{j∈F}]] is nœtherian, the images of finitely many of the equations G_{mn} suffice to generate the kernel. We choose such a finite set and add to our set F the variables involved in the initial forms of these equations, and consider the finite set F′ containing this new set and having the property that i ∉ F′ implies i + 1 ∉ F′. We consider the new ideal F′ obtained in this manner. The closure of the ideal generated by F′ and the equations (G_{i})_{i∉F} is now the kernel T. By construction, this set of generators of T has the property that the initial forms of those among its elements which involve only variables of index i ∈ F′ generate the prime ideal which is the kernel of the map k[(W_{j})_{j∈F′}] → grν̂R̂^{(ν)}. □

This is a fundamental fact: all but finitely many of the equations defining R̂^{(ν)} serve only to express u_{i+1}, for i ∉ F, in terms of (u_{j})_{j∈F} and the initial binomials of the finitely many equations generate a prime ideal in a polynomial ring in finitely many variables. More precisely, keeping the notations introduced at the beginning of this subsection, we have: —————————– ——————————————-

Corollary 16 Let us denote by F′ the finite set consisting of those equations among the G_{mn} whose initial forms depend only on the (W_{j})_{j∈F′}. Given an homomorphism c: Φ → k^{∗}, the kernel of the surjective map

ĉ: { \atop k[(w_{j})_{j∈J}]} → R̂^{(ν)}, w_{j} ↦ η_{j}c(−γ̂_{j})

is generated, up to closure, by the ideal (c(F′), c(G_{i})_{i∉F} ). The initial binomials of these generators which depend only on the (W_{j})_{j∈F′} generate the kernel of the map of k-algebras k[(W_{j})_{j∈F′}] → grν̂R̂^{(ν)} determined by W_{j} ↦ \overline{η}_{j}c(−γ̂_{j}).

Definition 5.54 Given an excellent local domain R with a rational valuation ν, a finite finite subset of the index set of the minimal system of generators of the semigroup Γ̂ = ν̂(R̂^{(ν)} \ {0} having the property described in Proposition 5.53 will be called a quite sufficient set.

Corollary 17 For each j ∉ F, the image η_{j+1} in R̂^{(ν)} of the element w_{j+1} can be written as the image of a series in the images of (w_{j})_{j∈F′}; such a series, viewed in { \atop k[(w_{j})_{j∈J}]}, may be called an j-th semiroot for R̂^{(ν)}; semiroots are now indexed by ordinals < ω^{h_{R}(ν)}. Its expression in R̂^{(ν)} as the image of a series in (w_{k})_{k≤j} is:

η_{j+1} = d_{j+1}(η_{j}^{n_{j}} ∏_{k<j} η_{k}^{n_{k}^{(j)}} − λ_{j} ∏_{k<j} η_{k}^{ℓ_{k}^{(j)}} + ∑_{s} d_{s}^{(j)}η^{s}),

with d_{j+1} ∈ k^{∗} and ν̂(η^{s}) > ν̂(∏_{k<j} η_{k}^{ℓ_{k}^{(j)}}) = ν̂(η_{j}^{n_{j}} ∏_{k<j} η_{k}^{n_{k}^{(j)}}), obtained by solving for the (w_{j+1})_{i∉F} the equations

(c(G_{j}) = 0)_{j∉F} .∗

This suggests a way to generalizes the theory of approximate roots, or rather of semiroots in the sense of [PP]. From a valuation theoretic viewpoint the idea of approximate roots appears already in the “key polynomials” of MacLane (see [McL 1], [McL 2] and [V2]), but in a different guise it was developed by Abhyankar-Moh, (see [A-M1], [A-M 2]), Cossart-Moreno (in preparation), Lejeune-Jalabert, (see [L]) (for curves, the idea being that of branches having “critical” contact with a given plane curve singularity), and by Spivakovsky (in [S1] for surfaces, and in [S2] for valuations of height one). I refer to [PP], especially to Corollary 1.5.4, to be compared with Proposition 5.43, and Corollary 1.5.5. The connection between the various viewpoints is explained in a special case at the end of subsection 4.4.

Corollary 18 For i ∉ F, we have the inequality

γ_{i+1} > n_{i}γ_{i}.

Remark 5.55 The choice of the finite set F made above is simple but far from economical; one could begin by taking elements whose valuations rationally generate Γ, in minimal number r(ν). Then Lemma 3.3 implies that the elements of gr_{ν}R are connected to them by binomial equations, and we can add enough such elements for their representatives to generate the maximal ideal. Whether one of the corresponding equations F_{mn} will contain a linear term then depends on the structure of Γ. The simplest exemple is the case of a plane branch.

5.6 An example: complex plane branches.

Example 5.56 Let R be the analytic algebra of a complex analytic plane branch (i.e., germ of analytically irreducible curve) (X, 0); then it is dominated by only one valuation, the t-adic valuation ν induced by the injection R ⊂ C{t} of R into its normalization, and the valuation ring is C{t}. Let Γ = ν(R\{0}) ⊂ N∪{0} be the semigroup of values of ν on R; it has g + 1 generators (\overline{β_{0}}, …, \overline{β_{g}}) and if we choose for each 0 ≤ i ≤ g an element ξ_{i}(t) ∈ R with valuation \overline{β_{i}}, then the completion Â(R) of A(R) with respect to the (t)-adic valuation (our ν_{A}) (resp. the analyticization A(R)^{an} of A(R)) is isomorphic to the (t)-adic completion Â (resp. the analyticization) of the sub C[v]-algebra of R[v, v^{−1}] generated by the elements ξ_{i}(t)v^{−\overline{β_{i}}}. This follows from ([T1]) and what we have seen above. Anyway, this is the algebra of a family M, parametrized by v, of branches, each fiber M(v) for v ≠ 0 being isomorphic to the branch (X, 0), and the special fiber being isomorphic to the monomial curve described parametrically by u_{i} = t\overline{β_{i}}; 0 ≤ i ≤ g. There is a natural map which resolves its singularities; it is the map A^{an} → C{v, t′} given by

ξ_{i}(t)v^{−\overline{β_{i}}} ↦ ξ_{i}(vt′)v^{−\overline{β_{i}}}.

Indeed, the natural resolution of singularities for SpecA(R) is SpecA(C{t}). This last ring is a genuine Rees algebra for the ideal (t) in C{t}

A(C{t}) = C{t}[v] ⨁ ⨁_{n∈N}(t)^{n}v^{−n},

and its analytization maps isomorphically to C{v, t′} by v ↦ v, t ↦ vt′. This shows that the parametric specialization described in ([T1], 1.10, p. 167) and ([G-T], §3) coincides with the specialization given by the valuation algebra. It was shown in [T1] that the binomial equations

u_{i}^{n_{i}} − u_{0}^{ℓ_{0}^{(i)}} … u_{i−1}^{ℓ_{i−1}^{(i)}} = 0 , 1 ≤ i ≤ g,

defining the monomial curve in C^{g+1} extend to equations defining the family M in C × C^{g+1}, which are of the form

\begin{aligned}u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}} + b_{2}vu_{2} + ∑_{w(s)>n_{1}\overline{β}_{1}} c_{s}^{(1)}(v)u^{s} & = 0 \\ u_{2}^{n_{2}} − u_{0}^{ℓ_{0}^{(2)}}u_{1}^{ℓ_{1}^{(2)}} + b_{3}vu_{3} + ∑_{w(s)>n_{2}\overline{β}_{2}} c_{s}^{(2)}(v)u^{s} & = 0 \\ ⋮ & ⋮ \\ u_{g−1}^{n_{g−1}} − u_{0}^{ℓ_{0}^{(g−1)}} ⋯ u_{g−2}^{ℓ_{g−2}^{(g−1)}} + b_{g}vu_{g} + ∑_{w(s)>n_{g−1}\overline{β}_{g−1}} c_{s}^{(g−1)}(v)u^{s} & = 0 \\ u_{g}^{n_{g}} − u_{0}^{ℓ_{0}^{(g)}} ⋯ u_{g−1}^{ℓ_{g−1}^{(g)}} + ∑_{w(s)>n_{g} \overline{β}_{g}} c_{s}^{(g)}(v)u^{s} & = 0 \end{aligned}

where the c_{s}^{(j)}(v) are in (v)C\{v\}, w(s) = ∑_{0}^{g} \overline{β}_{j}s_{j} is the weight of the monomial u^{s} with respect to the weight vector w = (\overline{β}_{0}, …, \overline{β}_{g}), i.e., w(s) = ⟨w, s⟩, and with all b_{j} ≠ 0 in C, which corresponds to the fact that our branch is plane. The other u^{s} appearing in the j-th equation are different from u_{j+1}. Note that since n_{i}\overline{β}_{i} < \overline{β}_{i+1} (see [T1]), we add to each binomial only terms of higher weight, one of which is linear except in the bottom equation, which “creates” the singularity. We see that we have a surjective map

π: C{v}[[u_{o}, …, u_{g}]] → Â(R),

and its kernel is generated by the equations just written, which illustrates Proposition 5.44. It was shown in [G-T] that a toric map π: Z → C^{g+1} resolving the monomial curve extends to a toric map Id_{C} × π: C × Z → C × C^{g+1} inducing a resolution of M which is a strong simultaneous embedded resolution along C × {0}. In particular the map π gives an embedded resolution for our germ of curve C ⊂ C^{g+1}, the embedding being given by the generators (ξ_{0}, …, ξ_{g}) of the maximal ideal of R, or alternatively by the equations written just above, with v = 1. In this example, the valuation ν̃ of subsection 3.2 has values in the group Z ⊕ Z ordered lexicographically, with the values ν̃(v) = (1, 0), ν̃(t) = (1, 1), this last value corresponding to ν̃(t′) = (0, 1) on C{v, t′}, and so to the natural height two valuation on this last ring adapted to the coordinates (v, t′).

By successive elimination in the equations shown above, and thanks to the usual implicit function theorem, we can express the image of u_{2} in the ring R of our branch in the form

u_{2} = b_{2}^{−1}(u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}} + ∑ d_{s}^{(2)}u_{0}^{s_{0}}u_{1}^{s_{1}}),

and then

u_{3} = b_{3}^{−1}((u_{1}^{n_{1}} − u_{0}^{ℓ_{0}^{(1)}} + ∑ d_{s}^{(2)}u_{0}^{s_{0}}u_{1}^{s_{1}})^{n_{2}} − u_{0}^{ℓ_{0}^{(2)}}u_{1}^{ℓ_{1}^{(2)}} + ∑_{s} d_{s}^{(3)}u_{0}^{s_{0}}u_{1}^{s_{1}}),

and so on. The plane branches with equations u_{2} = 0, u_{3} = 0, …, u_{k} = 0 for 2 ≤ k ≤ g correspond to semiroots in the sense of [PP] of the Weierstrass polynomial defining our plane branch. Compare with Corollary 17; the fact that no abyssal phenomenon is involved here does not prevent the fact that all variables but two appear linearly in the equations from having its consequences.

6 Toric modifications

This section begins the study of partial uniformization, by a combinatorial process, of the valuation ν_{gr} on gr_{ν}R when ν is a rational valuation of the nœtherian local ring R. I show how to resolve singularities of an irreducible variety defined by a binomial ideal in finite-dimensional space, which is the case for Specgr_{ν}R in the situation just described if in addition gr_{ν}R happens to be a finitely generated k-algebra, and is the case for a finitely generated approximation of Specgr_{ν}R in general. As far as I know, existing literature deals only with toric varieties associated to fans, which are in particular normal. I begin with an example.

6.1 Valuative motivation: an example. Assume now that k = k_{R} is algebraically closed and that ν is a valuation of rational rank and height equal to d = dimR dominating R. By Abhyankar’s inequality, the extension k → k_{ν} is trivial, so we are in the situation of Proposition 4.2, and the ordered group Φ is isomorphic to Z^{d} with the lexicographic order (see the text before Proposition 3.9).

Then we can write:

Γ = ⟨γ_{1}, γ_{2}, …, γ_{j}, …⟩ with γ_{j} ∈ Z_{≥0}^{d}.

The semigroup Γ is the image of N^{N} by a surjective map

b: Z^{N} → Z^{d}

defined by sending the i-th basis vector to γ_{i}. Passing to semigroup algebras over k givess a map

k[(U_{j})_{j∈J}] → k[t_{1}^{±1}, …, t_{d}^{±1}] determined by U_{j} ↦ t^{γ_{j}}

having k[t^{Γ}] for image. This describes Speck[t^{Γ}] as the closure of an orbit of the torus (k^{∗})^{d} in the N-dimensional affine space A^{N}(k). Note that since R and R_{ν} have the same field of fractions, the semigroup Γ generates Φ as a group. Note that it will not be the case in general for a valuation with value group Φ that there is a finite index i_{0} such that the finitely generated semigroup ⟨γ_{1}, …, γ_{i_{0}}⟩ generates Z^{d} as a group if the valuation is not of height one and Φ is not a nœtherian Z-module, for example if Φ = (Z ⊕ Q)_{lex}. Note also that even in this special case where Φ = Z^{d} we have when d is > 1 a difficulty which does not appear when Φ = Z; the γ_{j} do not have all their coordinates positive, so that our orbit closure is not in general the image of a map A^{d}(k) → A^{N}(k).

By dualizing the map b, we obtain a map

w: Ž^{d} → Ž^{N}.

If the right-hand group was finite dimensional, say Ž^{N}, we would know how to find an embedded resolution of the singularities of Speck[t^{Γ}] ⊂ Speck[U_{1}, …, U_{N}] by a single toric map, as is explained in the next section.

6.2 Embedded toric resolution of orbit closures and binomial ideals in finite dimensions. I describe here the embedded resolution of singularities of torus orbit closures (in finite dimensions) as a generalization of what is done in [G-T] for monomial curves. I assume that the reader is aware of the basics of toric geometry, and especially of the fact that any fan can be refined to a regular simplicial fan, by iterated stellar subdivisions. I refer to David Cox’s notes [Cox] and to Günter Ewald’s book ([E], VI, No.8, p. 253), and I use the description of orbit closures given by Sturmfels in [St1], [St2]; see [Cox], §4. I keep the notations introduced above, so that our orbit closure is described by a map of semigroups b: N^{N} → Z^{d}, where N is finite and the image of b generates the group Z^{d}. Choosing generators (m^{ℓ} − n^{ℓ})_{ℓ∈{1,...,L}} for Kerb gives us an exact sequence

Z^{L} → Z^{N} → Z^{d} → 0 .

Note that the generators m^{ℓ} − n^{ℓ} are necessarily primitive vectors in Z^{N}, and we may choose them in such a way that m^{ℓ} and n^{ℓ} both have non-negative coordinates (see [St]). I assume from now on that they are chosen in this way. By dualizing, we obtain an exact sequence

0 → Ž^{d} → Ž^{N} → Ž^{L},

where the image of the i − th basis vector of Ž^{d} is the vector

w^{i} = (γ_{1i}, γ_{2i}, …, γ_{Ni}).

Choosing one of the m^{ℓ} − n^{ℓ} gives us an injective map Z → Z^{N}, and a surjective Ž^{N} → Ž having kernel H_{ℓ} = {a ∈ Ž^{N} | ⟨a, m^{ℓ} − n^{ℓ}⟩ = 0}. The intersection ⋂_{ℓ∈{1,...,L}}(H_{ℓ} ⊗_{Z} R) is the R-linear span W of the w^{i}. From now on I will write H_{ℓ} indifferently for H_{ℓ}⊗_{Z}R or its integral points. The interpretation will be clear from the context. By construction, we have ⟨w^{i}, m^{ℓ} − n^{ℓ}⟩ = 0 for i = 1, 2, …, d, and ℓ ∈ {1, …, L}, hence also ⟨w, m^{ℓ}−n^{ℓ}⟩ = 0 for ℓ ∈ {1, …, L} and all vectors w ∈ W. Conversely, W is the dual of the vector space generated by the vectors m^{ℓ} − n^{ℓ} for ℓ ∈ {1, …, L}. Remark that we may assume that W contains no basis vector; if it contained the i-th base vector, the variable U_{i} would appear in none of the equations, so we might as well remove it. The complement in Ř_{+}^{N} of the union of the hyperplanes H_{ℓ} is a union of rational N-dimensional convex cones, and the union of hyperplanes itself is the reunion of rational convex cones of smaller dimensions. Note that H_{ℓ} meets the interior of Ř_{+}^{N} since the coordinates of the vectors m^{ℓ} − n^{ℓ} are not all of the same sign.

By the fundamental result on resolution of toric varieties (see [O], [Cox]), we can find a regular fan Σ with support Ř_{+}^{N} which is not only compatible with the w^{i} but also with the union of the hyperplanes H_{ℓ} in the sense that any σ ∈ Σ meets any one of the H_{ℓ} along one of its faces. Then, Σ is also compatible with the intersection W of the H_{ℓ}. In the sequel I tacitly assume Σ to have these properties.

I am going to show that for a suitable choice of such a regular fan Σ, the toric map

π(Σ): Z(Σ) → A^{N}(k)

induces an embedded resolution of the orbit closure X ⊂ A^{N}(k) corresponding to b.

Let k be an algebraically closed field and let (U^{m^{ℓ}} −U^{n^{ℓ}})_{ℓ∈{1,...,L}} be generators of the kernel of the map

k[b]: k[U_{1}, …, U_{N}] → k[t_{1}^{±1}, …, t_{d}^{±1}] given by U_{i} ↦ t^{γ_{i}},

where U^{m} = U_{1}^{m_{1}} ⋯ U_{N}^{m_{N}}. Since the ideal kerk[b] defines an orbit closure, in view of ([E-S], Corollary 2.5) and the identity

U^{m+m′} − λ_{mn}λ_{m′n′}U^{n+n′} = U^{m′}(U^{m} − λ_{mn}U^{n}) + λ_{mn}U^{n}(U^{m′} − λ_{m′n′}U^{n′}),

the ideal kerk[b] is generated by binomials U^{m} − U^{n} such that the exponents m − n belong to the lattice L ⊂ Z^{N} generated by the m^{ℓ} − n^{ℓ} for ℓ ∈ {1, …, L}, and this lattice is a direct factor in Z^{N}, which corresponds to the fact that the ideal kerk[b] is prime. If we take N − d of the m^{ℓ} − n^{ℓ}, they generate a sublattice L_{1} ⊂ L. We can choose these elements in such a way that they rationally generate the lattice L, which means that L/L_{1} is a torsion Z-module. Since L is a direct factor, it is the saturation of L_{1} in the sense of [E-S]; it is the lattice of elements of Z^{N} which possess a multiple in L_{1}.

If we take a regular simplicial cone of maximum dimension σ = ⟨a^{1}, …, a^{N}⟩ ∈ Σ, in the affine chart U_{σ} ⊂ Z(Σ) with coordinates Y_{1}, …, Y_{N} associated to σ by definition of the toric map π(Σ): Z(Σ) → A^{N}(k), we have the following expression for π(Σ)|U_{σ}: U_{σ} → A^{N}(k)

U_{i} ↦ Y_{1}^{a_{i}^{1}} ⋯ Y_{N}^{a_{i}^{N}} and so U^{m} ↦ Y_{1}^{⟨a^{1},m⟩} ⋯ Y_{N}^{⟨a^{N} ,m⟩},

where ⟨a^{i}, m⟩ = ∑_{j=1}^{N} a_{j}^{i}m_{j}. Let us compute the transform by this map of one of our binomial generators, denoted by U^{m} − U^{n}. We may assume that a^{1}, …, a^{t} are those among the a^{j} which lie on the hyperplane H dual to m − n, i.e., such that ⟨a^{j}, m − n⟩ = 0 , 1 ≤ j ≤ t. Because our fan is compatible with H, all the other ⟨a^{j}, m − n⟩ are of the same sign, say ⟨a^{j}, m − n⟩ > 0. We have then

U^{m} − U^{n} ↦ Y_{1}^{⟨a^{1},n⟩} ⋯ Y_{N}^{⟨a^{N} ,n⟩}(Y_{t+1}^{⟨a^{t+1},m−n⟩} ⋯ Y_{N}^{⟨a^{N} ,m−n⟩} − 1).

Compare with [Z1], Th. 2, p.863 and [S2]. This essentially means that in toric geometry, there is no need for Hironaka’s game (see [H1]); it is replaced by the existence of a fan compatible with the H_{ℓ} as above. Hironaka’s game comes into play when one wants to dominate our toric map by the composition of a sequence of blowing-ups with non singular centers. The exceptional divisor of the map π(σ) is the union of the y_{j} = 0 for those j such that a^{j} is not a basis vector of Ž^{N} (see [G-T]). For simplicity I will assume that none of the a^{j} is a basis vector, so that the exceptional divisor is Y_{1} … Y_{N} = 0. If t = 0, the strict transform Y_{t+1}^{⟨a^{t+1},m−n⟩} ⋯ Y_{N}^{⟨a^{N} ,m−n⟩} − 1 = 0 does not meet the exceptional divisor; therefore the strict transform of U^{m} − U^{n} = 0 meets the exceptional divisor only in those charts U_{σ} for which at least one of the primitive vectors a^{i} of σ is in H. More generally, we have.

Proposition 6.1 If the strict transform by π(Σ) of the subspace X ⊂ A^{N}(k) defined by the ideal (U^{m^{ℓ}} − U^{n^{ℓ}})_{ℓ∈{1,...,L}} is to meet the exceptional divisor in the chart π(σ): U_{σ} → A^{N}(k), where σ = ⟨a^{1}, …, a^{N}⟩, there must be a vector a^{j} such that ⟨a^{j}, m^{ℓ} − n^{ℓ}⟩ = 0 for ℓ ∈ {1, …, L}, i.e., a^{j} ∈ W.

Proof Look at the equations of the strict transform. □

This generalizes the result of [G-T], 5.2.

The R-vector space W̌ generated by the m^{ℓ} −n^{ℓ} is of dimension N −d; let us choose N − d of the vectors m^{ℓ} − n^{ℓ} which generate W̌, say m^{1} − n^{1}, …, m^{N−d} − n^{N−d}. I now have to check that whenever the strict transform of the subspace of A^{N}(k) defined by

U^{m^{1}} − U^{n^{1}} = ⋯ = U^{m^{N−d}} − U^{n^{N−d}} = 0

is not empty it is non singular and transverse to the exceptional divisor, and that the strict transforms of all the other U^{m^{ℓ}} − U^{n^{ℓ}} = 0 vanish on one of its irreducible components.

The strict transforms of the equations have the form

\begin{aligned}Y_{1}^{⟨a^{1},m^{1}−n^{1}⟩} ⋯ Y_{N}^{⟨a^{N} ,m^{1}−n^{1}⟩} − 1 & = & 0 \\ Y_{1}^{⟨a^{1},m^{2}−n^{2}⟩} ⋯ Y_{N}^{⟨a^{N} ,m^{2}−n^{2}⟩} − 1 & = & 0 \\ ⋮ & ⋮ & ⋮ \\ Y_{1}^{⟨a^{1},m^{N−d}−n^{N−d}⟩} ⋯ Y_{N}^{⟨a^{N} ,m^{N−d}−n^{N−d}⟩} − 1 & = & 0 \end{aligned}

Note that this strict transform meets the component Y_{j} = 0 of the exceptional divisor if and only if a^{j} ∈ W; let us renumber the a^{j} so that a^{1}, …, a^{t} are those which lie in W, i.e., ⟨a^{i}, m^{s} − n^{s}⟩ = 0 for 1 ≤ i ≤ t and all s; if none of the a^{i} is in W, set t = 0; in this case, the strict transform does not meet any of the coordinate hyperplanes in the chart U_{σ}. As in ([G-T], §4), we can compute the jacobian matrix J of these equations by logarithmic differentiation, and find the equality of N × (N − d) matrices

Y_{t+1} ….Y_{N}J = Y_{t+1}^{∑_{s}⟨a^{t+1},m^{s}−n^{s}⟩} … Y_{N}^{∑_{s}⟨a^{N} ,m^{s}−n^{s}⟩}(⟨a^{j}, m^{s} − n^{s}⟩).

Proposition 6.2 Given an irreducible binomial variety X ⊂ A^{N}(k), with the notations just introduced, the rank of the image in Mat_{N×L}(k) of the matrix (⟨a^{j}, m^{s}−n^{s}⟩) ∈ Mat_{N×L}(Z) is N −d. The strict transform by π(Σ) of the subspace X ⊂ A^{N}(k) defined by the ideal (U^{m^{ℓ}} − U^{n^{ℓ}})_{ℓ∈{1,...,L}} is regular and transversal to the exceptional divisor; it is also irreducible in each chart.

Proof Let σ be a cone of of maximal dimension in the fan Σ. Let us denote as above by t the dimension of σ ∩ W. Since dimW = d, we must have t ≤ d; since the vectors a^{j} form a basis of Q^{N}, and the space W̌ generated by the m^{s} − n^{s} is of dimension N − d, the rank of the matrix (⟨a^{j}, m^{s} − n^{s}⟩) is N − d, which proves the lemma if k is of characteristic zero.

To prove the general case, view our matrix (⟨a^{j}, m^{s} − n^{s}⟩) as describing the composed map

Z^{L} → Z^{N} ⟶^{M(σ)} Z^{N};

we have to check that the (N − d) × (N − d)-minors of this matrix have no common divisor. This follows from the:

Lemma 6.3 In a sequence of maps of free Z-modules as above, assume that the image of the first map is a direct factor of rank N − d and that the determinant of M(σ) is ±1. Then the (N − d) × (N − d)-minors of the matrix describing the composed map have no common divisor.

Proof Let us consider the sequence of maps obtained by taking (N − d)th exterior powers:

Λ^{N−d} Z^{L} → Λ^{N−d} Z^{N} ⟶^{Λ^{N−d}M(σ)} Λ^{N−d} Z^{N}.

Since the image of the first map is a direct factor, so is the image of its exterior power, which is of rank one and generated by the (N − d) × (N − d)-minors of the matrix defining the first map. This image is therefore a primitive vector, which implies that the minors are coprime. We have to prove that the image of this

primitive vector by Λ^{N−d} M(σ) is again a primitive vector. This will be true if the determinant of that matrix is ±1. Remark that its entries are the (N −d)×(N −d)- minors extracted from M(σ) and suitably ordered. By a result of Sylvester (see [Ai], p.92), nowadays an exercise in linear algebra, this implies that its determinant is the \binom{N−1}{d}-th power of the determinant of M(σ), and the result. □

So we see that the rank of J is N − d everywhere on the strict transform, and by Zariski’s jacobian criterion ([Z4]) this strict transform is smooth and transversal to the exceptional divisor. Note however that it is not necessarily irreducible; let us show that the strict transform of our orbit closure is one of its irreducible components. Since the differences of the exponents in the total transform and the strict transform of a binomial are the same, the lattice of exponents generated by the exponents of all the strict transforms of the binomials (U^{m^{ℓ}} − U^{n^{ℓ}})_{ℓ∈{1,...,L}} is the image M(σ)L of the lattice L by the linear map Z^{N} → Z^{N} corresponding to the matrix M(σ) with rows (a^{1}, …, a^{N}). Similarly the exponents of the strict transforms of U^{m^{1}} − U^{n^{1}}, …, U^{m^{N−d}} − U^{n^{N−d}} generate the lattice M(σ)L_{1}. The lattice M(σ)L is the saturation of M(σ)L_{1}, and so according to [E-S], since we assume that k is algebraically closed, the strict transform of our orbit closure is one of the irreducible components of the binomial variety defined by the N − d equations displayed above. The charts corresponding to regular cones σ ∈ Σ of dimension < N are open subsets of those which we have just studied, so they contribute nothing new. □

I now show that our toric map can be chosen so that it induces an isomorphism outside of the singular locus of our irreducible binomial variety, so that it is an embedded resolution and not only a pseudo-resolution in the sense of [G-T]. A part of this is the fruit of common reflexions with Pedro González-Pérez (see [GP-T]). The same argument may also be used to replace the jacobian rank argument just given, since it reduces the general toric case to the case of normal toric varieties, where the jacobian rank argument is unnecessary. See [GP-T] for a proof of the resolution of singularities of binomial varieties along these lines.

With the notations introduced at the beginning of this subsection, remembering that the image of the map b generates Z^{d}, let us denote by γ_{1}, …, γ_{N} the images by the map b of the basis vectors e_{1}, …, e_{N} of Z^{N}, and by σ̌ the cone which they generate in R^{d}. It depends only on b and we may also denote it by σ̌(b). Its convex dual σ (or σ(b)) in Ř^{d} is a strictly convex cone of dimension d whose image in Ř^{N} by the map b̌ is contained in the intersection of W with the first quadrant of Ř^{N}. Indeed, we know by basic properties of duality that the image of Ř^{d} in Ř^{N} consists of those linear forms on R^{N} which vanish on the kernel of b, and is therefore equal to W. The image of σ consists of those linear forms f̃ on R^{N} which are in W and such that f̃(e_{i}) = f(γ_{i}) ≥ 0, i = 1, …, N, where f ∈ σ. Therefore f̃ is in the dual of the first quadrant of R^{N}, which is the first quadrant of Ř^{N}.

Definition 6.4 Given a map b: N^{N} → Z^{d} of semigroups, the strictly convex cone σ(b) ⊂ W ∩ Ř_{+}^{N} is called the weight cone associated to b.

Definition 6.5 (Condition RES(b)) Let X be an orbit closure corresponding to a map b: N^{N} → Z^{d} as above. Let σ(b) be the weight cone associated to b just above. A fan Σ with support R_{+}^{N} satisfies condition RES(b) if for some system of generators (m^{ℓ} − n^{ℓ})_{ℓ∈{1,...,L}} of the kernel of the extended map b: Z^{N} → Z^{d}, it is compatible with the hyperplanes H_{ℓ} dual to the vectors (m^{ℓ} − n^{ℓ})_{ℓ∈{1,...,L}}, and in addition all the regular faces of the cone σ(b) belong to Σ.

Remark that by the classical results on resolution of normal toric varieties (see [Cox]), fans satisfying (RES) do exist.

Proposition 6.6 Assuming that the field k is algebraically closed, let X ⊂ A^{N}(k) be an orbit closure corresponding to a morphism b: Z^{N} → Z^{d}. The singular locus of X is a union of intersections of X with linear coordinate spaces. The toric map π(Σ): Z(Σ) → A^{N}(k) associated to a regular simplicial fan Σ satisfying the condition RES(b) just defined is an embedded resolution of the singularities of X.

Proof Let d be the dimension of our binomial variety X and (m^{ℓ}−n^{ℓ})_{ℓ∈{1,...,L}} and Σ be a system of generators and a fan as given by condition RES(b). A straightforward computation using logarithmic differentials shows that the jacobian determinant J_{I,L′} of rank c = N − d of the generators (U^{m^{ℓ}} − λ_{m^{ℓ}n^{ℓ}}U^{n^{ℓ}})_{ℓ∈{1,...,L}} of our prime binomial ideal P ⊂ k[U_{1}, …, U_{N}] associated to I = (i_{1}, …, i_{c}) and a subset L′ ⊆ {1, …, L} of cardinality c satisfies the congruence

U_{i_{1}} … U_{i_{c}}.J_{I,L′} ≡ ( ∏_{ℓ∈L′} U^{m^{ℓ}})Det_{I,L′}((⟨m − n⟩)) mod.P,

where (⟨m−n⟩) is the matrix of the vectors (m^{ℓ}−n^{ℓ})_{ℓ∈{1,...,L}}, and Det_{I,L′} indicates the minor in question. By Proposition 6.2, the rank of the image in k^{N×L} of the jacobian matrix is equal to c. This proves the first part of the Proposition. For example, the only possibility for the intersection of our variety with U_{i} = 0 not to be in the singular locus is that there exists such a minor such that all the m_{i}^{ℓ} (resp. n_{i}^{ℓ}) for ℓ ∈ L′ are zero except one, which is equal to one. In that case one sees that one of the equations of our binomial variety is of the form

U_{i}U′^{r} − U′^{s} = 0,

and all the others are independent of U_{i}. By ([St3], Corollary 13.6), the normalisation of our binomial variety X is X_{σ(b)} = Speck[σ̌(b) ∩ Z^{d}]. The proof is as follows: let P be the prime binomial ideal which is the kernel of the map of semigroup algebras

k[b] : k[U_{1}, …, U_{N}] → k[t_{1}^{±1}, …, t_{d}^{±1}]

corrresponding to b. By construction of σ̌(b) the image of this map is contained in k[σ̌(b) ∩ Z^{d}]. Finally we have an injection

k[U_{1}, …, U_{N}]/P → k[σ̌(b) ∩ Z^{d}]

which is birational since the image of b generates Z^{d}, and the algebra on the right is normal since the semigroup σ̌(b)∩Z^{d} is saturated. A toric resolution of singularities of X is obtained by a regular refinement of the fan in W ∩ Ř_{+}^{N} consisting of σ(b) and its faces which does not affect the regular faces of σ(b), since that toric map resolves the singularities of X_{σ(b)} and is an isomorphism outside of the singular locus. Now we can choose a regular fan supported in Ř_{+}^{N}, compatible with the hyperplanes H_{mn} and which contains the cones of our regular subdivision of σ(b). Any such fan corresponds to a toric map Z(Σ) → A^{N}(k) which is a toric embedded resolution of X, since the strict transform of X is transversal to the exceptional divisor by the argument given above, and the restriction to this strict transform is an isomorphism outside of the singular locus of X. This proves the result. □

Remarks 6.7 1) At least in characteristic zero, the equivariant resolution theorem implies that an irreducible binomial variety X as above can be resolved by a sequence of equivariant permissible blowing ups. The theorem of De Concini-Procesi ([DC-P]) implies that our toric resolution can be dominated by a sequence of equivariant blowing-ups. It would be interesting to prove directly that these blowing ups can be chosen to be permissible, in the sense that at each step the space blown up is normally flat along the center.

2) It has been remarked by P. González Pérez (see [GP]) that any fan Σ which is compatible with W and contains the regular cones of σ(b) will also provide a resolution of singularities of the orbit closure correponding to b; this description depends only on b and not on the choice of equations. This is illustrated by the fact that in [GT] a resolution of the monomial curve is obtained precisely with such fans.

Proposition 6.8 Let (u^{m_{k}})_{k∈K} be a finite collection of monomials in a polynomial ring R = k[(u_{i})_{i∈I}] endowed with a rational valuation μ. There is a birational toric extension R → R′ = R[(u^{α_{j}})_{j∈J}]_{m′} such that in R′ the ideal (u^{m_{k}})_{k∈K}R′ is principal and generated by the monomial with the least valuation.

Proof Let J ⊂ I be the finite set of the variables appearing in the monomials considered. Consider the toric modification associated to a fan of the first quadrant of R_{+}^{|J|} which is compatible with the hyperplanes dual to the vectors (m_{k}−m_{k′})_{k≠k′}; by the valuative criterion for properness the valuation μ picks a point in a chart of that modification, and the corresponding toric extension satisfies the condition of the proposition. □

Let us now suppose that we have an irreducible subspace of A^{N}(k) defined by binomials U^{m^{ℓ}} − λ_{m^{ℓ}n^{ℓ}}U^{n^{ℓ}} for ℓ ∈ {1, …, L}, where as usual U^{m} = U_{1}^{m_{1}} ⋯ U_{N}^{m_{N}} and λ_{mn} ∈ k^{∗}. By ([E-S], Cor. 2.3), at least if our field k is algebraically closed, the subspace defined by the binomials obtained by changing each λ_{mn} to 1 is a torus orbit closure, and by Proposition 6.6 has an embedded resolution by a toric map π(Σ): Z(Σ) → A^{N}(k) as above. In one of the charts of Z(Σ), the ideal (U^{m^{k}} − U^{n^{k}})_{ℓ∈{1,...,L}} becomes the ideal generated (up to permutation of the coordinates) by

Y^{e(n^{k})}(Y_{t+1}^{⟨a^{t+1},m^{k}−n^{k}⟩} ⋯ Y_{N}^{⟨a^{N} ,m^{k}−n^{k}⟩} − 1); 1 ≤ k ≤ N − d,

where e(n) = (⟨a^{1}, n⟩, …, ⟨a^{N}, n⟩), and we are assuming as above that

⟨a^{i}, m − n⟩ = 0, 1 ≤ i ≤ t and ⟨a^{i}, m − n⟩ > 0, t + 1 ≤ i ≤ N.

The nature of the computation shows that we have:

U^{m} − λ_{mn}U^{n} ↦ Y^{e(n)}(Y_{t+1}^{⟨a^{t+1},m−n⟩} ⋯ Y_{N}^{⟨a^{N} ,m−n⟩} − λ_{mn}).

The relations between the λ_{mn} associated to linear relations between the m − n, of the form (with the notations used above)

λ_{m^{ℓ}n^{ℓ}} = ∏_{k=1}^{N−d} λ_{m^{k}n^{k}}^{b_{k}^{ℓ}},

are exactly what is needed to ensure that once we have chosen a system of generators m^{k} − n^{k} as above, the transforms of the remaining binomials U^{m} − λ_{mn}U^{n} will define an irreducible component of the non singular variety defined by

Y_{t+1}^{⟨a^{t+1},m^{k}−n^{k}⟩} ⋯ Y_{N}^{⟨a^{N} ,m^{k}−n^{k}⟩} − λ_{m^{k}n^{k}}^{ϵ(k)} = 0 ; 1 ≤ k ≤ N − d,

where ϵ(k) = ±1 is the common sign which the vectors a^{i}, t + 1 ≤ i ≤ N take on m^{k} − n^{k}, so that the subspace defined by our binomial ideal is also resolved by π(Σ). Keeping the notations just introduced, we see that we have proved:

Proposition 6.9 Assuming that the field k is algebraically closed, let X ⊂ A^{N}(k) be an irreducible binomial variety, defined by the ideal of k[U_{1}, …, U_{N}] generated by (U^{m^{ℓ}} − λ_{m^{ℓ}n^{ℓ}}U^{n^{ℓ}})_{ℓ∈{1,...,L}}, λ_{m^{ℓ}n^{ℓ}} ∈ k^{∗}. For any regular simplicial fan Σ satisfying the condition RES(b) with respect to the orbit closure defined by the binomials (U^{m^{ℓ}} −U^{n^{ℓ}})_{ℓ∈{1,...,L}}, the corresponding toric map π(Σ): Z(Σ) → A^{N}(k) is an embedded resolution of the singularities of X ⊂ A^{N}(k).

Remark 6.10 After the description of the singular locus of a binomial variety given in the proof of Proposition 6.6, Zariski’s jacobian criterion implies that the (N − d) × (N − d)-minors of the matrix (m^{ℓ} − n^{ℓ}) are coprime integers if the binomial variety determined by the vectors (m^{ℓ}−n^{ℓ}) is reduced and irreducible over any algebraically closed field. The geometric fact that this is true if it is reduced and irreducible over C also follows, in view of [E-S], from the combinatorial fact expressed in Lemma 6.3. In the case where d = 1 and our (equations-wise) binomial variety is the (parametrization-wise) monomial curve corresponding to the semigroup Γ = ⟨\overline{β_{0}}, …, \overline{β_{g}}⟩ and is a complete intersection, as in the example of subsection 5.6, this Lemma 6.3 solves the minor difficulty mentioned at the end of §5 of [G-T] and shows that the minors of the matrix coincide, up to sign, with the generators of the semigroup.

6.3 An example: branches revisited. Let R be a one-dimensional excellent equicharacteristic analytically irreducible local ring, i.e., the local ring of a branch. Let us assume that the residue field k of R is algebraically closed. The ring R has only one non trivial valuation, whose valuation ring is the integral closure R_{ν} of R in its field of fractions. Since R is excellent, R_{ν} is a finite R-module ([EGA], Scholie 7.8.3, vii)). Therefore there exists an element d ∈ R, d ≠ 0, such that dR_{ν} ⊂ R and N \ Γ is finite, so that the semigroup Γ of values of ν on R \ {0} is finitely generated. Let us write Γ = ⟨γ_{1}, γ_{2}, …, γ_{N}⟩. Since k is algebraically closed, the valuation ν is rational, and the completion R̂^{m} of R is equal to R̂^{ν} and is a subring of k[[t]], the valuation ν̂ being induced by the t-adic valuation. Let us choose elements (ξ_{i})_{1≤i≤N} in R such that ν(ξ_{i}) = γ_{i}. Their initial forms for the ν-adic valuation generate the graded ring

gr_{ν}R = k[t^{γ_{1}}, …, t^{γ_{N}}] ⊂ k[t] = gr_{ν}R_{ν},

which is the affine algebra corresponding to the monomial curve C^{Γ} associated to Γ. Since the (ξ_{i})_{1≤i≤N} in R are finite in number, they form a minimal quite sufficient set of generators for the maximal ideal of R̂^{(ν)} = R̂^{m}. By Proposition 5.43, the map u_{i} ↦ ξ_{i}v^{−ν(ξ_{i})} determines a surjective map

k[v][[u_{1}, …, u_{N}]] → { \atop A_{ν}(R)} ⊂ k[v][[t]].

The kernel of this map is generated by series

F_{mn} = u^{m} − u^{n} + ∑_{s} c_{s}^{(mn)}(v)u^{s}

which are deformations of binomial generators u^{m} − u^{n} of the kernel of the map

k[u_{1}, …, u_{N}] → k[t^{γ_{1}}, …, t^{γ_{N}}], u_{i} ↦ t^{γ_{i}}.

Denoting by w = (γ_{1}, γ_{2}, …, γ_{N}) ∈ N^{N} the weight vector, for each of these equations, we add to the binomial u^{m} − u^{n} only terms c_{p}^{(mn)}(v)u^{s} such that w(s) > w(m) = w(n). If now we choose a regular fan Σ in R_{+}^{N} compatible with the dual hyperplanes of the vectors m − n, and so in particular with the vector w, we obtain a toric map

π(Σ): Z(Σ) → A^{N}(k)

which is a toric embedded resolution of the monomial curve C^{Γ} ⊂ A^{N}(k) according to what we saw in subsection 6.2. The map

Id_{A^{1}(k)} × π(Σ): A^{1}(k) × Z(Σ) → A^{1}(k) × A^{N}(k)

then induces an embedded resolution of singularities for the formal subspace defined in k[v][[u_{1}, …, u_{N}]] by the ideal generated by the F_{mn}, which is a simultaneous resolution for all the fibers of the projection to A^{1}(k). This is checked exactly as in [G-T]. It implies that the map π(Σ) gives an embedded resolution of the singularities of the image of the formal completion of our curve embedded in A^{N}(k) by using the elements ξ_{i} ∈ R ⊂ R̂^{ν} as coordinates. Now since π(Σ) is a monomial map, it corresponds to adding to R̂^{ν} certain monomials in the ξ_{i}, with some negative exponents, and then localizing the R̂^{ν}-algebra which they generate. This operation has a meaning in R itself, and builds a local ring R′ which is essentially of finite type over R and birationally equivalent to R. The local ring R′ is excellent since R is, and its completion is regular since it is the completion of the transform of R̂^{ν}, so R′ itself is regular.

To illustrate the difference between this approach and Abhyankar’s, consider, over an algebraically closed field k of characteristic p, the curve given parametrically by:

u_{0} = t^{p} + t^{p+1}, u_{1} = t^{p^{2}+1} + t^{p^{2}+p+1}.

It has no Puiseux presentation of the form u′_{0} = t^{p}, u′_{1} = u′_{1}(t). However, we may consider it as a deformation of a plane monomial curve, with an equation of the form:

u_{1}^{p} − u_{0}^{p^{2}+1} − u_{0}^{p^{2}−p+1}u_{1} + ⋯

Here we can remark that the irreducible initial binomial describes a curve which is purely inseparable over k((u_{0})), while this is not the case for our original curve. However, if we resolve this binomial, say by a toric modification of A^{2}(k) having as one of its charts

u_{0} = y_{0}^{p}y_{1}, u_{1} = y_{0}^{p^{2}+1}y_{1}^{p},

we find that the transform of the equation is

y_{0}^{p(p^{2}+1)}y_{1}^{p^{2}}(1 − y_{1} − y_{0}y_{1} + ⋯)

and therefore we have an embedded resolution of our curve.

The usual relations between Puiseux exponents and semigroup may fail in characteristic p. Consider, over an algebraically closed field k of characteristic p, the curve (see [Ca], p.114):

u_{0} = t^{p^{3}}, u_{1} = t^{p^{3}+p^{2}} + t^{p^{3}+p^{2}+p+1}.

One can check that although only three exponents are visible on the expansions, its semigroup has four generators:

Γ = ⟨p^{3}, p^{3} + p^{2}, p^{4} + p^{3} + p^{2} + p, p^{5} + p^{4} + p^{3} + p^{2} + p + 1⟩,

and the corresponding monomial curve has equations

U_{1}^{p} − U_{0}^{p+1} = 0, U_{2}^{p} − U_{0}^{p(p+1)}U_{1} = 0, U_{3}^{p} − U_{1}^{p^{3}}U_{2} = 0,

while the curve itself has, in its natural coordinates, equations

u_{1}^{p} − u_{0}^{p+1} − u_{2} = 0, u_{2}^{p} − u_{0}^{p(p+1)}u_{1} + u_{3} = 0, u_{3}^{p} − u_{1}^{p^{3}}u_{2} − u_{0}^{p+1}u_{3}^{p} = 0.

The following question and example are motivated by two problems:

1) If there is a natural toric specialization of a singularity, where are the regular points of the strict transform of the singularity by a toric resolution of the toric variety?

2) Of what nature is the complexity of the binomial relations defining gr_{ν}R? this is of course essential for the description of its initial partial toric resolution.

6.4 A question on surfaces. Let k be an algebraically closed field of characteristic p. Consider in A^{3}(k) the surface

z^{p+1} + y^{p−1}z + x^{p+1} = 0

studied by Abhyankar in ([A4], p.589); it is quasi-ordinary with respect to the projection onto the (x, y)-plane; the z-discriminant is x^{p+1} = 0. For p ≥ 5, however, its Galois group with respect to this projection is “very large and quite complicated” (loc. cit.) and not a cyclic group as one would expect in characteristic zero.

Study the rational valuations, if any, for which the associated graded ring of this surface singularity is

k[X, Y, Z, U]/(Z^{p} + Y^{p−1}, ZU + X^{p+1}),

so that k[x, y, z]/(z^{p+1} + y^{p−1}z + x^{p+1}) appears as a deformation of its associated graded ring; its equations in natural coordinates are

z^{p} + y^{p−1} − u = 0, zu + x^{p+1} = 0.

Note that the associated graded ring is not quasi-ordinary for the projection onto the (X, Y)-plane and that the field extension associated to this projection displays inseparability.

6.5 Another example, in dimension three.

Example 6.11 (partly inspired by an example of Spivakovsky) Let us give Z^{2} the lexicographic order and denote by k((t^{Z_{lex}^{2}})) the field of Puiseux series associated to the ordered group Z_{lex}^{2} and the field k as in [Ka], [B2]; it is the field of formal series with exponents forming a well ordered subset of Z_{lex}^{2}. It is naturally endowed with the t-adic valuation with values in Z_{lex}^{2}. Let us denote by k[[t^{Z_{+}^{2}}]] the corresponding valuation ring. Choose a sequence of pairs of positive integers (a_{i}, b_{i})_{i≥3} and a sequence of elements (λ_{i} ∈ k^{∗})_{i≥3} such that b_{i+1} > b_{i}, the series ∑_{i≥3} λ_{i}u_{2}^{b_{i}} is not algebraic over k[u_{2}], and the ratios \frac{a_{i+1}−a_{i}}{b_{i+1}} are positive and increases strictly with

i. Let R_{0} be the k-subalgebra of k[[t^{Z_{+}^{2}}]] generated by

u_{1} = t^{(0,1)}, u_{2} = t^{(1,0)}, u_{3} = ∑_{i≥3} λ_{i}u_{1}^{−a_{i}}u_{2}^{b_{i}}.

There cannot be an algebraic relation between u_{1}, u_{2}, and u_{3}, so the ring R_{0} = k[u_{1}, u_{2}, u_{3}] is the polynomial ring in three variables. It inherits the t-adic valuation of k[[t^{Z_{+}^{2}}]]. One checks that this valuation extends to the localization R = k[u_{1}, u_{2}, u_{3}]_{(u_{1},u_{2},u_{3})}; it is a rational valuation of height two and rational rank two. Let us try to compute the semigroup Γ of the values that it takes on R. We have γ_{1} = (0, 1), γ_{2} = (1, 0), γ_{3} = (b_{3}, −a_{3}) ∈ Γ. Set Γ_{3} = ⟨γ_{1}, γ_{2}, γ_{3}⟩. Then we have u_{1}^{a_{3}}u_{3} − λ_{3}u_{2}^{b_{3}} = ∑_{i≥4} λ_{i}u_{1}^{a_{3}−a_{i}}u_{2}^{b_{i}} ∈ R, so that γ_{4} = (b_{4}, a_{3} − a_{4}) is in Γ. It is easy to deduce from our assumptions that no multiple of γ_{4} is in Γ_{3}, and that it is the smallest element of Γ which is not in Γ_{3}. We set u_{4} = u_{1}^{a_{3}}u_{3} − λ_{3}u_{2}^{b_{3}}, and continue in the same manner: u_{1}^{a_{4}−a_{3}}u_{4} − λ_{4}u_{2}^{b_{4}} = u_{5}, ..., u_{1}^{a_{i}−a_{i−1}}u_{i} − λ_{i}u_{2}^{b_{i}} = u_{i+1},... with the generators γ_{i} = ν(u_{i}) = (b_{i}, a_{i−1} − a_{i}) for i ≥ 4. Finally we have:

Γ = ⟨γ_{1}, γ_{2}, …, γ_{i}, …⟩,

the initial forms of the u_{i} constitute a minimal system of generators of the graded k-algebra gr_{ν}R, and the equations (setting a_{2} = 0)

u_{1}^{a_{i}−a_{i−1}}u_{i} − λ_{i}u_{2}^{b_{i}} = u_{i+1}, i ≥ 3

above describe R̂^{(ν)}. In fact they even describe R; it is clear that from them we can reconstruct the value of u_{3} as a function of u_{1}, u_{2} by (infinite) elimination. The binomial equations for gr_{ν}R are the

U_{1}^{a_{i}−a_{i−1}}U_{i} − λ_{i}U_{2}^{b_{i}} = 0, i ≥ 3,

showing that all the U_{i} for i ≥ 3 are rationally dependent on U_{1}, U_{2}. Using the remark following proposition 3.7, we see that the Krull dimension of gr_{ν}R is two.

The fact that R is regular of dimension three is due to the abyssal phenomenon that we met in example 4.16. Remark that in this example gr_{ν}R is not regular. From our assumption on the growth of the ratios we see moreover that no multiple of γ_{i} is in Γ_{i−1} = ⟨γ_{1}, …, γ_{i−1}⟩. In fact γ_{i} is outside of the cone with vertex 0 generated by Γ_{i−1} in R^{2}. It is interesting to study in this case the construction of a system of regular fans which will resolve the singularities of gr_{ν}R.

Remark 6.12 Olivier Piltant has communicated to me an example of a valuation of height one on the ring R = k[u_{1}, u_{2}, u_{3}]_{(u_{1},u_{2},u_{3})} where it is also the case that no multiple of γ_{i} is in Γ_{i−1}. The valuation is induced on R from a rational height one valuation, of rational rank three, on the power series ring k[[u_{1}, u_{2}/u_{1}, u_{3}/u_{1}]] which is monomial in the variables u_{1}, u_{2}/u_{1} and u_{3}/u_{1} − (u_{2}/u_{1})^{2} − (u_{2}/u_{1})^{3}. It is related to the example of [C-G-P].

Problem. If Γ = ⟨γ_{1}, …, γ_{i}, γ_{i+1}, …⟩ is the semigroup of a valuation of a nœtherian local integral domain R, and Γ_{i} = ⟨γ_{1}, …, γ_{i}⟩, can the fact that even for large i no multiple of γ_{i} is in Γ_{i−1} occur with R of dimension two? Hint: use [A2]. Compare with subsection 6.4.

7 Local uniformization (speculative sketch)

...and he will be too dazzled by the light to look at the objects whose shadows he was seeing a moment ago...

Plato, The Republic, VII, 515

Assume in this section that the local equicharacteristic excellent integral domain R has an algebraically closed residue field k.

Given a valuation ν_{0} on the ring R, we may specialize ν_{0} to a valuation ν whose ring R_{ν} dominates R with trivial residue field extension (see subsection 3.6, Proposition 3.17). We first uniformize the valuation ν̂ on R̂^{(ν)}. In the last subsection I will sketch the descent from R̂^{(ν)} to R. After this descent, by the first Corollary to Proposition 5.32, which ensures the nice behaviour of the specialization above with respect to composition of valuations, the same toric map will also simultaneously uniformize the valuations in the fibers of the total space of the specialization of R to gr_{ν_{0}}R, which is the subspace of the preceding specialization defined by the ideal (v^{ψ} − 1)_{ψ∈Ψ_{+}}, and hence also uniformize ν_{0} on R. The proof proceeds by induction in the dimension, since in order to prove the existence of a scalewise completion for a ring of dimension d I have used local uniformization in dimension < d. The proof actually follows the scheme T LU(d − 1) ⟹ T C(d) ⟹ T LU(d). On the way we use that T LU(d − 1) ⟹ T P (d − 1) by Corollary 20 below.

Since ν is rational, by Proposition 4.2 the graded algebra gr_{ν}R is a quotient of a polynomial ring by a binomial ideal. Using Proposition 5.16, we may assume that the completion R̂^{(ν)} has a field of representatives k ⊂ R̂^{(ν)} which we take as our base field. According to Propositions 5.43 and 5.44, we have a surjective map

{ \atop k[v^{Φ_{+}}][(w_{j})_{j∈J}]} → { \atop Aν̂(R̂^{(ν)})}^{(ν_{A})},

the kernel of which is the closure in { \atop k[v^{Φ_{+}}][(w_{j})_{j∈J}]} of the ideal generated by the elements

G̃_{mn} = w^{m} − λ_{mn}w^{n} + ∑_{s} c_{s}^{(mn)}(v^{ϕ})w^{s},

where the tilda means that we have taken the varying (with coefficients in k[v^{Φ_{+}}]) version of the equation. The idea for the proof is that if the k-algebra gr_{ν}R is finitely generated, say with N generators, it is also finitely presented and one can proceed essentially as in [G-T]: we have in this case by Lemma 7.2 below dimR̂^{(ν)} = dimgr_{ν}R = r(ν), and a toric map Z → A^{N}(k) which resolves Specgr_{ν}R will also uniformize ν̂ on R̂^{(ν)}. If gr_{ν}R is not finitely generated, we have to show that it suffices to resolve a finite approximation to gr_{ν}R, the subalgebra gr_{ν}^{(h)}R of k[x_{1}^{(h)}, …, x_{r}^{(h)}] generated by the images of (U_{j})_{j∈F} for sufficiently large h, where h is a level of approximation for gr_{ν}R_{ν} in the sense of Proposition 4.11 and F is chosen according to Proposition 5.49 and the text which follows it as a quite sufficient initial generating set in Γ. Here the key point is that the abyssal phenomenon of subsection 5.5) tells us that if we consider the family defined in { \atop k[v^{Φ_{+}}][(w_{j})_{j∈J}]} by the equations (F̃, (G̃_{i})_{i∉F}) it has ”generic fiber” SpecR̂^{(ν)} and a special fiber which may be larger that Specĝrν̂R^{(ν)}, but coincides with it if one considers only the variables with index in F′. Then, because the equations G̃_{i} create no new singularity it is sufficient to resolve the space defined by F̃, and for that, it is sufficient to resolve the ideal generated by its initial binomials, which reduces us to the case of the finitely generated approximating algebra.

7.1 The approximation process. A priori, we must examine whether it makes sense to say that we resolve ν_{gr} by a single infinite dimensional toric modification of k[(U_{i})_{i∈I}], hence uniformize the completion ν̂_{gr} on ĝr_{ν}^{(ν)}R. This toric modification should then uniformize ν̂ on the completion R̂^{(ν)}. Fortunately, as we shall see below, we do not have to overcome this difficulty, because a finite initial partial toric resolution of grν̂R̂^{(ν)} suffices to uniformize ν̂ on R̂^{(ν)}. Let us keep the notations of the preceding section for a valued ring (R, ν); we will apply the results to (R̂^{(ν)}, ν̂). The variables U_{i} are ordered by the valuations γ_{i} ∈ Γ of their images in gr_{ν}R. Let us consider the nested sequence of polynomial algebras k[x^{(h)}] = k[x_{1}^{(h)}, …, x_{r}^{(h)}] approximating gr_{ν}R_{ν} according to Proposition 4.11, b). Remember that r is the rational rank of the value group Φ of ν. For each finite set F of generators of the k-algebra gr_{ν}R, there is an integer h such that F is contained in gr_{ν}R ∩ k[x^{(h)}]. The degrees of the elements of this subalgebra are in a free subsemigroup N^{r} ⊆ Φ_{+} ∪ {0} and the subalgebra gr_{ν}^{(F)}R of gr_{ν}R generated by F is the image of the map

k[(U_{j})_{j∈F}] → k[x^{(h)}]

sending U_{j} to its image in k[x^{(h)}]. The kernel of the map k[(U_{j})_{j∈F}] → gr_{ν}^{(F)}R is a prime binomial ideal, and corresponds to a torus orbit closure in A^{F} (k) characterized by the map ω(F): Z^{F} → Z^{r} sending the j-th basis element to ν(ξ_{j}) ∈ N^{r} ⊆ Φ_{+} ∪ {0}. I apply the results of subsection 6.2 to this binomial ideal in finitely many variables, obtaining a regular fan in Ř_{+}^{F}. Adding an element to F to get F′ = F ∪ {g}, we have a commutative diagram

Z^{F} ⟶ Z^{r}
↓  ↓
Z^{F′} ⟶ Z^{r}

and after dualizing

Ž^{r} ⟶ Ž^{F′}
↓  ↓^{p_{F′,F}}
Ž^{r} ⟶ Ž^{F}

Note that in the case of curves, r = 1 and, taking for F the finite set of generators of the algebra gr_{ν}R, the image of Ž in Ž^{F} = Ž^{N} is the weight vector. Although no approximation is needed, we can illustrate the procedure by setting e_{i} = gcd(γ_{1}, …, γ_{i}) and considering the nested sequence of subalgebras

k[t^{e_{1}}] ⊂ k[t^{e_{2}}] ⊂ ⋯ ⊂ k[t^{e_{N−1}}] ⊂ k[t].

The construction shown above then builds a sequence of vector spaces (R^{i} → R^{i−1})_{2≤i≤N} and a weight vector in each. In the case of plane branches, where the relations between the γ_{i} are particularly simple, this suggests a procedure to build inductively a fan compatible with the hyperplanes corresponding to these relations; see subsection 5.6 and [G-T]. For each finite set F of variables U_{j} as above, we take a fan Σ_{F} in Ř_{+}^{F} compatible with the hyperplanes dual to generators of the kernel of ω(F). For each inclusion F ⊂ F′, we consider similarly fans Σ_{F′} compatible with the generators of the kernel of ω(F′) and with the inverse image by p_{F′,F} of Σ_{F}. In this manner, letting F grow, we obtain a projective system of vector spaces indexed by the filtering system of finite subsets of I, and a subordinate system of regular fans, which is what we would like to call a regular fan for the projective system of vector spaces. We are led to the following definition of a possibly infinite dimensional fan:

Definition 7.1 Let P be a projective system of R-linear maps between finitedimensional real vector spaces, indexed by a filtering ordered set I.

P: ((p_{i,j}: R^{n(i)} → R^{n(j)})_{i,j∈I,i>j}),

where n(i) is an integer valued function. A fan Σ subordinate to P is the datum for each i ∈ I of a rational fan Σ_{i} in R^{n(i)} in such a way that for each i > j, the fan Σ_{i} is a refinement of p_{i,j}^{−1}(Σ_{j}). We say that the fan is of finite type if there exists a finite subset F ⊂ I such that for i, j ∉ F , i > j, the image of Σ_{i} is equal to Σ_{j}, and if we denote by K_{i} ⊂ R^{n(i)} the kernel of p_{i,j}, the map σ ↦ (p_{i,j}(σ), σ ∩ K_{i}) is a bijection from the set of cones in Σ_{n(i)} to the set of pairs consisting of a cone in Σ_{j} and one of the cones in Σ_{n(i)} whose image by p_{i,j} is 0. We say that Σ is simplicial (resp. regular) if all Σ_{i} are. We say that Σ is rational if there exists a finite subset F ⊂ I such that for i, j ∉ F , i > j, the map p_{i,j} sends a basis of the integral lattice of R^{n(i)} to part of a basis of the integral lattice of R^{n(j)}.

To such a system of fans is associated a system of toric maps π(Σ_{i}): Z(Σ_{i}) → A^{n(i)}(k); if we have a simple system of surjective maps p_{i}: R^{n(i+1)} → R^{n(i)}, we get inclusions A^{n(i)}(k) ⊂ A^{n(i+1)}(k), so that we get a compatible system of toric maps of an increasing sequence of affine spaces, which we may call an infinite-dimensional toric map. I leave it to the reader to interpret geometrically the various conditions defined for the system of fans.

7.2 Toric modifications and transforms of R̂^{(ν)}. Finally, in order to uniformize the valuation ν̂ on R̂^{(ν)} it should suffice to resolve the singularities of a finite number of the equations F_{mn}, so that we may consider only finitely many elements of the projective system of toric blowing-ups which “resolves” gr_{ν}R. To check this, we must now compute the effect on R̂^{(ν)} of the toric modifications associated to our regular fan.

Using the specialization of R̂^{(ν)} to ĝr_{ν}^{(ν)}R, which we can write in a very explicit form (see 5.32) as the space associated to

{ \atop k[v^{Φ_{+}}][(w_{j})_{j∈IJ}]}/\overline{(w^{m} − λ_{mn}w^{n} + ∑_{s} c_{s}^{(mn)}(v^{ϕ})w^{s})},

with weight(w^{s}) > weight(w^{n}) = weight(w^{m}), the weight of a monomial being the valuation of its image in R̂^{(ν)}. For any homomorphism c: Φ_{+} → k^{∗}, say ϕ ↦ c(ϕ), we have an isomorphism

ĉ : { \atop k[v^{Φ_{+}}][(w_{j})_{j∈J}]}/\overline{(w^{m} − λ_{mn}w^{n} + ∑_{s} c_{s}^{(mn)}(v^{ϕ})w^{s}, (v^{ϕ} − c(ϕ))_{ϕ∈Φ_{+}})} ⟶^{≃} R̂^{(ν)}.

We will denote by c(G_{mn}) the series

c(G_{mn}) = w^{m} − λ_{mn}w^{n} + ∑_{s} c_{s}^{(mn)}(c(v^{ϕ}))w^{s} ∈ { \atop k[(w_{j})_{j∈J}]}.

Let us first assume that the k-algebra gr_{ν}Rgrν̂R̂^{(ν)} is of finite type, generated say by (\overline{η}_{1}, …, \overline{η}_{N}). We have

Lemma 7.2 If the k-algebra grν̂R̂^{(ν)} is finitely generated, the equality dimR̂^{(ν)} = r(ν) holds.

Proof Using Propositions 5.43, 5.44, and their corollaries (see Corollary 15, c)), we are reduced to the case of a family of algebras which are “formally of finite type”. This shows that we can apply the semi-continuity of the dimensions of fibers to the map

Spec{ \atop Aν̂(R̂^{(ν)})}^{(ν_{A})} → Speck[v^{Φ_{+}}]

so that by Piltant’s Theorem (see 3.1), r(ν) = dimgrν̂R̂^{(ν)} ≥ dimR̂^{(ν)}. The reverse inequality is Abhyankar’s inequality applied to R̂^{(ν)}. □

Remarks 7.3 1) By classical results on curves (see [T1]) and the results of Spivakovsky in [S1], when dimR̂^{(ν)} ≤ 2, for a rational valuation ν the equality r(ν) = dimR̂^{(ν)} implies that the k-algebra gr_{ν}R is finitely generated and the equality r(ν) = dimR implies that the k[v^{Φ_{+}}]-algebra A_{ν}(R) is finitely generated.

2) There remains the problem of characterizing among rational valuations of an excellent equicharacteristic local domain the Abhyankar valuations (characterized by the equality dimgr_{ν}R = r(ν)) and the ”weakly Abhyankar” valuations (characterized by dimgrν̂R̂^{(ν)} = r(ν)) in terms of the abyssal presentation of R̂^{(ν)} and the construction of the ideal H of R̂^{m}. One may also ask whether the k-algebra grν̂R̂^{(ν)} is finitely generated if gr_{ν}R is.

Let us keep the notations of the preceding subsection, with a quite sufficient initial set of generators (γ̂_{j})_{j∈F} (see subsection 5.5) and the corresponding variables W_{j} of degree γ̂_{j}, while h is large enough for k[x_{1}^{(h)}, …, x_{r}^{(h)}] to contain the images in gr_{ν}R_{ν} of the (W_{j})_{j∈F}, and keep the notations used in the proof of Proposition 6.9 as well. I number the variables (W_{j})_{j∈F} as (W_{j_{1}}, …, W_{j_{N})}, with N = #F. We must compute the strict transforms of the equations G_{mn} under the toric map associated to a regular fan Σ as in 6.9. By assumption (γ_{j_{1}}, …, γ_{j_{N})} are in the free submonoid N^{r} ⊆ Φ_{+} and we may consider the map

Z^{N} → Z^{r}, (n_{1}, …, n_{N}) ↦ ∑_{t=1}^{N} n_{t}γ_{j_{t}} ∈ Z^{r}.

The image E ⊂ Ž^{N} of the dual map is the r-dimensional free submodule generated by the r elements e^{j} = (γ_{ij}) _{1≤i≤N} ∈ Ž^{N}. We want to show that if w^{s} has value greater than w^{n}, after a toric modification corresponding to a regular fan compatible with the hyperplanes defining W, the transform of w^{n} divides the transform of w^{s}. More generally, we have the:

Proposition 7.4 Let (w^{p_{1}}, …, w^{p_{c}}) be a finite set of monomials in { \atop k[(w_{j})_{j∈J}]}, arranged in increasing term order. There is a toric modification in the variables w_{j} appearing in these monomials such that, at the point of the strict transform of SpecR̂^{(ν)} picked by ν̂, the image of w^{p_{1}} divides the images of all the other w^{p_{j}}.

Proof By construction, and in view of Proposition 4.11, the assumption means that at least if we have taken N large enough and 1 ≤ i ≤ c − 1, ⟨w^{j}, p_{i+1} − p_{i}⟩ ≥ 0 for 1 ≤ j ≤ r (see the Corollary following Proposition 4.9). In view of the fact that U^{p_{i}−p_{1}} ∈ gr_{ν}R_{ν} is in k[x^{(h)}] for large enough h, after refining our fan Σ to a regular fan Σ′ without changing its intersection with W, we may assume that these inequalities imply for each cone σ = ⟨a^{1}, …, a^{N}⟩ ∈ Σ such that σ ∩ W ≠ ∅ the inequalities ⟨a^{k}, p_{i} − p_{1}⟩ ≥ 0, 1 ≤ k ≤ N. These inequalities precisely mean that in the chart of Z(Σ′) corresponding to σ, the image in the transform R̂′^{m,ν} of R̂^{(ν)} by the toric map corresponding to Σ′ of the transform of u^{p_{1}} divides the image of the transform of the u^{p_{i}} (compare with [Z1], Th.2, p. 863). □

Corollary 19 Let h_{1}, …, h_{p} be elements of R̂^{(ν)}. There is a toric modification in finitely many of the variables w_{j} such that at the point of the strict transform of SpecR̂^{(ν)} picked by ν̂, the image of the ideal (h_{1}, …, h_{p}) is generated by the element of least valuation.

Proof The ideal of R̂^{(ν)} generated by all the monomials appearing in the h_{k}’s is finitely generated since R̂^{(ν)} is nœtherian. It suffices to apply the Proposition to a finite set of generators of this ideal. □

Corollary 20 * TLU(d) implies TP(d): Given finitely many elements g_{k} ∈ R, there exists a birational toric extension R → R′ subordinate to ν such that in R′ the ideal generated by the elements g_{k} is equal to a monomial in ξ′_{i} times a unit. Here the ξ′_{i} ∈ R′ lift generators of gr_{ν}R′. In particular, given finitely many elements of R, we can achieve that in the ring R′ obtained from R by a birational toric extension the element with the smallest valuation divides the others.*

Proof Let us first assume that R is complete: by the preceding Corollary, it suffices to remark that in R the ideal generated by the monomials in the ξ_{i} which appear in the series representing the g_{k} is generated by finitely many of them. In the general case, assuming TLU(d) for d = dimR, we have a birational toric extension R′ of R such that R̂′ is regular, and ν extends to a valuation ν̂ of a regular quotient R̂′/H of R̂′ with the same graded ring. We may assume, since this is a matter of refining a fan to make it compatible with hyperplanes, that in this toric extension the images of the elements g_{k} in R̂′/H generate an ideal of the form ξ′^{α}u where u is a unit in R̂′/H. We need to show that u is in fact in R′. Since R̂′/H is regular we may view it as a subring of R̂′, and argue as follows: ξ′^{α}u is in R′, and so is ξ′^{α}, so u ∈ K ∩ R̂′, which is equal to R′ by the flatness of R̂′ as an R′-module ([B3], Chap. 3, §3, No. 5, Corollary 4). □

Corollary 21 If the graded k-algebra gr_{ν} R̂^{(ν)} is finitely generated, say by the initial forms (\overline{η}_{j})_{1≤j≤N} of elements η_{j} ∈ R̂^{(ν)}, a toric map Z(Σ) → A^{N}(k) corresponding to a regular fan compatible with the hyperplanes H_{m−n} associated to the binomial equations defining grν̂R̂^{(ν)} will produce, upon taking the strict transform of SpecR̂^{(ν)} embedded in A^{N}(k) via the elements η_{j}, an embedded uniformization of the valuation ν̂ on R̂^{(ν)}.

The k-algebra grν̂R̂^{(ν)} is also finitely presented and we have to deal with finitely many equations F_{mn}. We keep the fan Σ′ of Proposition 7.4 and apply this to compute the strict transform of the G_{mn}. By the same argument as in the proof of Corollary 20, we need concern ourselves with only a finite set of the monomials appearing in F_{mn}. We may assume without loss of generality that the intersection of σ with W is of dimension r, so that (a^{1}, …, a^{r}), say, are in W. Then in our monomial transformation

w_{i} = y_{1}^{a_{i}^{1}} … y_{N}^{a_{i}^{N}} , 1 ≤ i ≤ N,

we find that the strict transform of each G_{mn} in that chart has the following expression, where I write y^{⟨a,m⟩} for y_{1}^{⟨a^{1},m⟩} … y_{N}^{⟨a^{N} ,m⟩} except when I want to single out some of the y_{i}’s

\begin{matrix}G_{mn} ∘ π(σ) = \\ y_{1}^{⟨a^{1},m⟩} … y_{r}^{⟨a^{r},m⟩}(y_{r+1}^{⟨a^{r+1},n−m⟩} … y_{N}^{⟨a^{N} ,n−m⟩} − λ_{mn} + ∑_{p} c_{p}^{(mn)}(v^{ϕ})y^{⟨a,p−m⟩}). \end{matrix}

It shows that the strict transform of R̂^{(ν)} is non singular, as a deformation of the strict transform of grν̂R̂^{(ν)}. Now the refinement of Σ to Σ′ corresponds to a birational map Z(Σ′) → Z(Σ) which induces an isomorphism outside the divisor ∏_{j=r+1}^{N} y_{j} = 0, which the strict transform of our space does not meet, so that we may descend the result of simultaneous resolution from Z(Σ′) to Z(Σ). This is the same process as in [G-T].

In the general case the first problem is to show that our valuation ν̂ on R̂^{(ν)} can be uniformized after a toric modification which is sufficiently far in the projective system, but occurs after finitely many steps.

We keep the notations of subsection 5.5. Let us now consider the binomial ideal in k[(W_{j})_{j∈F′}] which is the kernel of the map to gr(h) ; ν̂R̂^{(ν)} sending W_{j} to \overline{η}_{j}. As we saw, it is a prime ideal, so that it corresponds to a torus orbit closure since our residue field is algebraically closed ([E-S]). We apply to it Proposition 6.9, and obtain a regular fan Σ of R_{+}^{F′} compatible with the hyperplanes dual to the m − n where (W^{m} − λ_{mn}W^{n})_{mn} is a system of generators of the binomial ideal. We may choose a regular simplicial cone σ ∈ Σ whose intersection with the space of weights E ⊂ R_{+}^{F′} is of the largest possible dimension, that is, the rational rank r of Φ. This corresponds to a monomial map

w_{i} ↦ y_{1}^{a_{i}^{1}} … y_{N}^{a_{i}^{N}} , 1 ≤ i ≤ N = ♯F′,

which we may extend by setting

w_{k} = y_{k}, k ∉ F′.

Now we compute the transforms of the equations of R̂^{(ν)} under this transformation. By construction, using the same argument as we used in the case where the graded algebra is finitely generated, the equations G_{mn} whose initial forms involve only (W_{i})_{i∈F′}, become

\begin{aligned}y^{⟨a,m^{1}−n^{1}⟩}(y^{⟨a,n^{1}⟩} − λ_{m^{1}n^{1}} + ⋯) & = 0 \\ ⋮ & ⋮ \\ y^{⟨a,m^{t}−n^{t}⟩}(y^{⟨a,n^{t}⟩} − λ_{m^{t}n^{t}} + ⋯) & = 0, \end{aligned}

while the G_{i} become

w_{i}^{n_{i}} ∏_{k∉F′,k<i} w_{k}^{n_{k}^{(i)}}y^{t(n^{(i)})} − λ_{i} ∏_{k∉F,k<i} w_{k}^{ℓ_{k}^{(i)}}y^{t(ℓ^{(i)})} + ⋯ + c^{(i)}(v^{ϕ})w_{i+1} + ⋯ = 0

where y^{t(n^{(i)})} (resp. y^{t(ℓ^{(i)})}) is a monomial which is the transform of the monomial in the variables (w_{j})_{j∈F′} which affected w_{i}^{n_{i}} (resp. appeared in the other term) in the original relations. Note that one can compute the valuations of the y_{k} from those of the w_{j}. Some of the y_{k}, say for k ∈ T, will have valuation zero, so that for k ∈ T there is a constant c_{k} ∈ k_{R}^{∗} such that y_{k} − c_{k} has positive valuation; the equations (y_{k} − c_{k} = 0)_{k∈T} , (y_{k} = 0)_{k∈F \T} , (w_{j} = 0)_{j∉F} define the closed point which is the center of the valuation in the space obtained by the toric modification. By openness of transversality, if our toric map has resolved the gr(h) ; ν̂R̂^{(ν)}, the first set of equations defines a regular ν-complete local ring at that point. By the nature of the equations, and the implicit function theorem mentioned in subsection 5.5, it will be the case for the whole set of equations and the strict transform of R̂^{(ν)} will be regular.

Remarks 7.5 1) If one compares with the case where grν̂R̂^{(ν)} is finitely generated, as in the example of plane branches, one see that the effect of the “abyssal” phenomenon is also to send singularities away to infinity: if we stopped at any finite step, replacing u_{i+1} by 0 or a polynomial in (u_{k})_{k≤i} of higher weight than the binomial initial form, the new equation thus obtained from F_{i} would be singular.

b) The last equation written underlines the role of the (u_{i})_{i∈I} as semiroots in the sense of [PP]; after our toric modification the (u_{k})_{k∉F} are still part of a natural coordinate system on the transformed ring for our valuation.

c) Note that there are at least r(ν) independent variables, and possibly more if dimR̂^{(ν)} > r(ν), and that after substitution of the expressions for the u_{j}, j ∉ F, some of the equations may become identities.

At this point we have proved local uniformization by a finite dimensional toric map for the rational valuation ν̂ on R̂^{(ν)}.

To deduce from this local uniformization for R, the fact that R is excellent is again crucial. In [S2] Spivakovsky used in a similar situation the following

Proposition 7.6 (see [Ma], §33, Spivakovsky [S2], [L-M], Chap. 15, Proof of Th. 15.7, p. 202 ) If R is an excellent local integral domain, for any prime ideal q of R and any prime ideal H of R̂^{q} such that H ∩ R = (0), the localization R̂_{H}^{q} is regular.

Proof Since we have H ∩ R = (0), the ring R̂_{H}^{q} is a localization of the ring R̂^{q} ⊗_{R} K, where K is the field of fractions of R. It suffices to prove that this tensor product is regular. By ([EGA], Scholie 7.8.3, (v)), if R is excellent, the map SpecR̂^{q} → SpecR is regular. Our tensor product corresponds to the generic fiber of this map, so it is regular; it is even a geometrically regular K-algebra. □

Since the ν̂-initial forms of the elements η_{j} which we used to generate topologically the k-algebra R̂^{(ν)} are Laurent monomials in the \overline{ξ}_{i}, with ξ_{i} ∈ R, we may perform in R the trace of the toric modification described above; it consists in adding to R the monomials (with some negative exponents) in the ξ_{i} which are the y_{i} and localizing at the point specified by the values of the y_{i}: this gives us a local ring R′ with maximal ideal m′. If we denote by R̂′^{(ν)} the scalewise completion of R′, we have maps of R-algebras

R′ → R′ ⊗_{R} R̂^{m} → R̂′^{m′} → R̂′^{(ν)}.

Following through constructions, we see that the image in R̂′^{m′} of the strict transform in R′ ⊗_{R} R̂^{m} of the ideal H is contained in the ideal H′ corresponding to ν in

R̂′^{m′}, so that we have

dimR̂′^{(ν)} ≤ dimR̂^{(ν)}.

That strict inequality may occur has been shown by Spivakovsky, who discovered this “subanalytic” phenomenon in the case of valuations of height one (see [S2]). From this inequality follows the

Proposition 7.7 * After replacing R by a toric transform, me may assume that the map

R′ ⊗_{R} R̂^{(ν)} → R̂′^{(ν)}

is the completion with respect to the maximal ideal m′ ⊗ 1 + 1 ⊗ m̂, and R̂′^{(ν)} is the

completion of the transform of R̂^{(ν)}. In that case, the ideal H′ of R̂′^{m′} is the strict transform of H.*

Proof Let R → R′ be a toric modification, and consider the transform R̂^{(ν)}′ of R̂^{(ν)}, which is R′ ⊗_{R} R̂^{(ν)} localized at the prime ideal n = m′ ⊗ 1 + 1 ⊗ mR̂^{(ν)}. The extension R̂^{(ν)} → R̂^{(ν)}′ is birational since R is a subring of R̂^{(ν)}, and the graded ring grν̂R̂^{(ν)}′ is the strict transform of grν̂R^{(ν)} by the toric map defining our transform. We may choose as representatives in R̂^{(ν)}′ of generators of grν̂R̂^{(ν)}′ the transforms of positive valuation by our toric modification, as in subsection 5.2, of elements η_{j} ∈ R^{(ν)} whose initial forms generate grν̂R^{(ν)}. Now the scalewise completion (Proposition 5.16) of R̂^{(ν)}′ may be a nontrivial quotient of its n-adic completion, and in that case dimR′^{(ν)} < dimR̂^{(ν)}, but the remarks above show that given a toric transform R′ of R, and another one R^{∗} such that the dimension of R̂^{∗}^{(ν)} is minimal among all toric modifications of R, there is a toric modification of R′ which dominates R^{∗} by a toric map and therefore also reaches the minimum dimension of its scalewise completion. □

This implies that if the transform R̂^{(ν)}′ of R̂^{(ν)} by a toric map is regular, so is R̂′^{(ν)}. Let R̂^{(ν)} be the scalewise ν-adic completion of a local equicharacteristic excellent integral domain. We know by 5.1 that R̂^{(ν)} is a quotient of the completion R̂^{m} by an ideal H. By Proposition 7.6, the assumption that R is excellent implies that the localization R̂_{H}^{m} is regular. If we have equality in Abhyankar’s inequality, the kernel of the map R̂^{m} → R̂^{(ν)} is a minimal prime and the same is true for the map

R̂′^{m′} → R̂′^{(ν)}. We conclude by case 1) of the general case below. For the general case, where Abhyankar’s inequality is not assumed to be an equality, I use a variant of the method described by Spivakovsky ([S2], 1997, Lemma 6.4). We are reduced to the case where R is such that R̂_{H}^{m} and R̂^{m}/H are both regular, the first one by excellence and the second one because we have uniformized ν̂ on R̂^{(ν)}. We want to show that we can make R̂^{m} (i.e., its strict transform) non singular by a toric modification defined in R. Then R itself will be made non singular. As we have seen in Proposition 7.7, we may assume that the strict transform of H is equal to H′. -case 1) If the ideal H contains only zero divisors of R̂^{m}, it is a minimal prime ideal of R̂^{m}, say q̂_{0}. The blowing up in SpecR̂^{m} of the sum J of the minimal primes of R̂^{m} separates the strict transforms of the irreducible components. Let J_{0} be the ideal of R̂^{m}/H which is the image of the intersection of the other minimal primes of R̂^{m}. According to toric principalization, which we may apply in R̂^{m}/H because whe know that it is a quotient of { \atop k[(w_{j})_{j∈J}]}, there is a toric modification in the (ξ_{i})_{i∈I} which principalizes J_{0}. The corresponding toric map dominates the strict transform of SpecR̂^{m}/H by the blowing up of J. Therefore the completion of the transform R′ of R by this toric map is an integral domain, the ideal H′ associated

to ν in R̂′^{m′} is zero and the kernel of the map R̂^{m} → R̂′^{m′} is the ideal H. Now R′ has to be regular. -case 2) If the ideal H contains non zero divisors, the regularity of R̂^{m}/H and of R̂_{H}^{m} implies the regularity of R̂^{m} if we know that H is generated by a regular sequence in R̂^{m}. This is equivalent to saying that H/H^{2} is a locally free R̂^{m′}/H-module. It is enough to produce a toric modification of R such that the strict transform

of H/H^{2} is equal to H′/H′^{2} and is locally free on SpecR̂′^{m′}/H′. Note that since

we may assume that H′ is the strict transform of H by the map R̂^{m} → R̂′^{m′}, the

R̂′^{m′}/H′-module H′/H′^{2} is equal to H/H^{2} ⊗ R̂′^{m′}/H′ divided by its torsion. Now

the module H′/H′^{2} will be locally free if the map R̂^{m} → R̂′^{m′} makes the Fitting ideal of the R̂^{(ν)}-module H/H^{2} principal at the point picked by the valuation in the toric modification R → R′. Using the Corollary to Proposition 7.4, we see that after a toric modification of R we can obtain that the Fitting ideal is principal. Thus, we reach the situation where R′ is regular and we are done.

There will remain to translate the trace of the toric modification on the completion R̂ (or R̂^{(ν)}) into a finite sequence of blowing ups with non singular centers coming from R. But one can prove a torus-equivariant version of Gruson-Raynaud’s ([G-R]) and Hironaka’s ([H2]) proper flattening theorem and use it to show that a toric modification is dominated by a sequence of toric blowing ups, that is, blowing ups with a torus-invariant non singular center. In fact a more general result is proved in [DC-P]; see also [O], Chap. 1, §1.7, p.39. We will also have to show that the centers of these blowing-ups can be chosen contained in the singular locus of SpecR and its transforms, whereas all we know so far is that these centers are contained in the union of the coordinate hyperplanes (and their transforms), and that this union contains the singular locus of SpecR̂^{(ν)} (see Proposition 6.6).

Corollary 22 ^{∗}Given a rational valuation ν of a complete equicharacteristic nœtherian local ring R with algebraically closed residue field, there exists a rational valuation ν_{0} of R such that gr_{ν_{0}}R is finitely generated, ν_{0}(x) ≤ ν(x) for x ∈ R, and there exist uniformizations of ν_{0} which also uniformize ν on R.^{∗}

Proof Use the abyssal description of Proposition 5.53 and choose for ν_{0} the monomial valuation of R, with respect to (ξ_{i})_{i∈F′}, determined by ν_{0}(ξ_{i}) = ν(ξ_{i}) = γ_{i} for i ∈ F′. □

It remains to see whether a similar result holds for any excellent equicharacteristic local ring with an algebraically closed residue field.

A last example. An example where gr_{ν}R is finitely generated is obtained by taking the first example after Proposition 5.3. The completion R̂^{(ν)} is the ring of a branch, the associated graded ring is that of the corresponding monomial curve which may be desingularized as in subsections 5.6 and 6.3 by a toric map. The transform R′ of R by this map has the property that it is a two-dimensional local ring with maximal ideal m′ such that the maximal ideal of R̂′ contains an element f′ such that R̂′/(f′) is a regular one-dimensional local ring and (R̂′)_{(f′)} is regular. So R̂′ is regular and therefore so is R′. Essentially the same argument will work whenever dimR̂^{(ν)} = 1, since once the singularity of this ring is resolved we deal with a discrete valuation ring, a case where the argument to bring the number of generators of H down to its height presents no difficulty.

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