SEMIGROUPS OF VALUATIONS ON LOCAL RINGS

STEVEN DALE CUTKOSKY AND BERNARD TEISSIER

Dedicated to Mel Hochster on the occasion of his sixty-fifth birthday

Introduction

In recent years the progress and applications of valuation theory have brought to light the importance of understanding the semigroups of values which a Krull valuation ν of some field takes on a noetherian local ring (R, m) contained in the ring (R_{ν}, m_{ν}) of the valuation.

Two general facts about valuations dominating a noetherian local domain are proven by Zariski and Samuel in their book on commutative algebra [ZS]. The first is that these semigroups are well ordered subsets of the positive part of the value group, of ordinal type at most ω^{h}, where ω is the ordinal type of the well ordered set N and h is the rank of the valuation (see [17], Appendix 3, Proposition 2). Being well ordered, each value semigroup of a noetherian ring has a unique minimal system of generators, indexed by an ordinal no greater than ω^{h}. The second is, in the case where m_{ν} ⋂ R = m and R and R_{ν} have the same field of fractions, the Abhyankar inequality

rr(ν) + tr_{k}k_{ν} ≤ dimR,

between the rational rank of the group of the valuation, the transcendence degree of the residue field of R_{ν} as an extension of the residue field of R, and the dimension of R. If equality holds the group of the valuation is isomorphic to Z^{rr(ν)} (see [17], Vol. 2, Appendix 2). If in addition the rank h is equal to one, the ν-adic and m-adic topologies of R coincide (see [14], Proposition 5-1).

These two conditions on ordinal type and rational rank do not characterize value semigroups of noetherian rings. In [5] a third condition is given, which concerns the rate of growth of the number of generators of the semigroup of a valuation dominating an equicharacteristic noetherian local domain. This condition implies that there are subsemigroups of Q_{+} which are well ordered of ordinal type ω, so they satisfy the first two conditions, but which are not semigroups of valuations dominating an equicharacteristic noetherian local domain.

Examples, starting with plane branches (see [16] and [15]), and continuing with quasiordinary hypersurfaces (see [8]) suggest that the structure of the semigroup contains important information on the process of local uniformization.

In this paper we shall consider mostly valued noetherian local rings (R, m) ⊂ (R_{ν}, m_{ν}) such that R and R_{ν} have the same field of fractions, the equality m_{ν} ⋂ R = m holds and the residual extension R/m → R_{ν}/m_{ν} is trivial. We call such valuations rational valuations of R. In this case, the finite generation of the semigroup is equivalent to the finite generation over k_{ν} = R_{ν}/m_{ν} of the graded algebra associated to the valuation

The first author was partially supported by NSF and the second by the Miller Scholar in residence program of the University of Missouri at Columbia.

gr_{ν}R = ⨁_{γ∈Γ} P_{γ}/P_{γ}^{+},

where

P_{γ} = {f ∈ R|ν(f) ≥ γ}, P_{γ}^{+} = {f ∈ R|ν(f) > γ}

and Γ is the value group of the valuation. This equivalence is true because each homogeneous component P_{γ}/P_{γ}^{+} is non zero exactly if γ is in the value semigroup S = ν(R\{0}), and then it is a one-dimensional vector space over R/m = R_{ν}/m_{ν} (see [15], section 4).

If the semigroup is finitely generated, it is not difficult to find a valuation of a local noetherian domain which has it as semigroup: the semigroup algebra with coefficients in a field K is then a finitely generated algebra. We can define a weight on its monomials by giving to each generator as weight the generator of the semigroup to which it corresponds. Then we define a valuation by deciding that the valuation of a polynomial is the smallest weight of its monomials.

The simplest example is that of a subsemigroup of N; it is necessarily finitely generated (see [13], Th. 83 p. 203) and by dividing by the greatest common divisor we may assume that its generators γ_{1}, …, γ_{g} are coprime, so that it generates the group Z. It is the semigroup of values of the t-adic valuation on the ring K[t^{γ_{1}}, …, t^{γ_{g}}] ⊂ K[t] of the monomial curve corresponding to the semigroup.

In section 2 we give an example of a semigroup of values of a valuation on a polynomial ring L[x, y, z] over a field which generates the group Z^{2} but is not finitely generated as a semigroup. Moreover, the smallest closed cone in R^{2} with vertex 0 containing the semigroup is rational.

If the semigroup is not finitely generated, very little is known about its structure. While we know that all totally ordered abelian groups of finite rational rank appear as value groups of valuations of rational function fields centered in a polynomial ring, we do not know any general condition implying that a well ordered sub-semigroup of the positive part of a totally ordered group of finite rational rank appears as semigroup of values of a noetherian ring. We shall see in section 4 a characterization of those well-ordered subsemigroups of Q_{+} which are the semigroups of values of a K-valuation on K[X, Y]_{(X,Y)}, but no general result is known for subsemigroups contained in the positive part Q_{≽0}^{2} of Q^{2} for some total ordering ≽.

We show in section 3 that a sub-semigroup S of the positive part of a totally ordered abelian group of finite rational rank, such that S has ordinal type < ω^{ω}, has no accumulation points, and S has ordinal type ≤ ω^{h}, where h is the rank of the group generated by S. These are properties which are held by the semigroup of a valuation on a noetherian ring.

Even if one could solve the problem of determining which semigroups of a totally ordered group of (real) rank one come from noetherian rings, an induction on the rank would meet serious difficulties. A naturally occurring question concerns the position of elements of R whose values are the generators of the semigroup in regard to the valuation ideals of the valuations with which the given valuation is composed.

More precisely, let ν be a valuation taking non negative values on the noetherian local domain R, let Ψ be the convex subgroup of real rank one in the group Γ of the valuation, and let P be the center in R of the valuation ν_{1} with values in Γ/Ψ with which ν is composed. The valuation ν induces a residual valuation \overline{ν} on the quotient R/P and when the image q(γ) of γ in Γ/Ψ by the canonical quotient map q: Γ → Γ/Ψ is γ_{1} we have inclusions of the valuation ideals corresponding to ν and ν_{1}:

P_{γ_{1}}^{+} ⊂ P_{γ} ⊂ P_{γ_{1}},

Since R is noetherian, the quotients P_{γ_{1}}/P_{γ_{1}}^{+} are finitely generated R/P -modules for all γ_{1} ∈ Γ/Ψ. Each is endowed by the sequence (F_{γ} = P_{γ}/P_{γ_{1}}^{+})_{γ∈q^{−1}(γ_{1})}, which we denote by F(γ_{1}), with the structure of a filtered (R/P)-module with respect to the filtration F(0) of R/P by its valuation ideals (\overline{P_{δ}})_{δ∈Ψ}, where \overline{P_{δ}} = P_{δ}R/P = P_{δ}/P.

One could hope (see [15]) that the associated graded module

gr_{F(γ_{1})}P_{γ_{1}}/P_{γ_{1}}^{+} = ⨁_{γ∈q^{−1}(γ_{1})} P_{γ}/P_{γ}^{+}

is finitely generated over the associated graded ring

gr\overline{ν}R/P = ⨁_{δ∈Ψ_{+}} P_{δ}/P_{δ}^{+}.

For rational valuations, this would be equivalent to the fact that for each γ_{1} > 0 only finitely many new generators of the semigroup appear in P_{γ_{1}} \ P_{γ_{1}}^{+}. It would set a drastic restriction on the ordinal type of the minimal set of generators of the semigroup ν(R\{0}).

In sections 7 and 8, we give examples showing that this is not at all the case, and that in fact the cardinality of the set of new generators may vary very much with the value of γ_{1}. One might have hoped that this lack of finiteness is due to some transcendence in the residual extension from R/P to R_{ν}/m_{ν_{1}} and disappears after some birational extension of R to another noetherian local ring contained in R_{ν} which absorbs the transcendence. In one of the examples (section 8) there is no residual extension.

1. A criterion for finite generation

Given a commutative semigroup S, a set M is endowed with a structure of an S-module by an operation S × M → M written (s, m) ↦ s + m such that (s + s′) + m = s + (s′ + m). Recall that M is then said to be a finitely generated S-module if there exist finitely many elements m_{1}, …, m_{k} in M such that M = ⋃_{i=1}^{k}(S + m_{i}).

Proposition 1.1. Keeping the notations of the previous section, suppose that K is a field and R is a local domain with residue field K dominated by a valuation ring R_{ν} of the field of fractions of R. Assume that the residual extension R/m → R_{ν}/m_{ν} is trivial. Then for γ_{1} ∈ Γ/Ψ, The associated graded module

gr_{F(γ_{1})}P_{γ_{1}}/P_{γ_{1}}^{+}

is a finitely generated gr\overline{ν}R/P -module if and only if

F_{γ_{1}} = {ν(f) | f ∈ R and ν_{1}(f) = γ_{1}}

is a finitely generated module over the semigroup \overline{F} = \overline{ν}(R/P \ \{0\}).

Proof. Let us first remark that since P_{γ_{1}}/P_{γ_{1}}^{+} is an R/P -module, the set F_{γ_{1}} is an \overline{F} - module.

We know from ([15], 3.3-3.5, 4.1) that the non zero homogeneous components of the graded algebra gr_{ν}R = ⨁_{γ∈Γ} P_{γ}/P_{γ}^{+} are one-dimensional K-vector spaces whose degrees are in bijection with F = ν(R \ \{0\}), and that \overline{F} = F ⋂ Ψ.

The gr\overline{ν}R/P -module gr_{F(γ_{1})}P_{γ_{1}}/P_{γ_{1}}^{+} is nothing but the sum of the components of gr_{ν}R whose degree is in q^{−1}(γ_{1}). Since we are dealing with graded modules whose homogeneous components are one-dimensional K-vector spaces, this module is finitely generated if and only if there exist finitely many homogeneous elements e_{1}, …, e_{r} ∈ gr_{F(γ_{1})}P_{γ_{1}}/P_{γ_{1}}^{+} such that each homogeneous element of gr_{F(γ_{1})}P_{γ_{1}}/P_{γ_{1}}^{+} can be written \overline{x}e_{i} with \overline{x} ∈ R/P; this is equivalent to an assertion on degrees, which is exactly that the \overline{F} -module F_{γ_{1}} is finitely generated. □

2. An example with value group Z^{2} and non finitely generated semigroup.

Let K be a field. Let r be a positive natural number such that the characteristic of K does not divide r, and let a_{ij} for 0 ≤ i ≤ 2 and 1 ≤ j ≤ r be algebraically independent over K. Let M be the field

M = K ({a_{ij} | 0 ≤ i ≤ 2 and 1 ≤ j ≤ r}) .

Let G ≅ Z_{r} be the subgroup of the permutation group S_{r} generated by the cycle (1, 2, …, r). For σ ∈ G, define a K automorphism of M by σ(a_{ij}) = a_{i,σ(j)}. Let L = M^{G} be the fixed field of G. We have an étale, Galois morphism

Λ : P_{M}^{2} → P_{L}^{2} ≅ P_{M}^{2}/G.

Let x_{0}, x_{1}, x_{2} be homogeneous coordinates on P_{K}^{2}. Define p_{i} = (a_{0i}, a_{1i}, a_{2i}) ∈ P_{M}^{2} for 1 ≤ i ≤ r, and let q = Λ(p_{1}). Since {p_{1}, …, p_{r}} is an orbit of Λ under the action of G, we have that q = Λ(p_{i}) for 1 ≤ i ≤ r, and I_{q}O_{P_{M}^{2}} = I_{p_{1}} ⋯ I_{p_{r}}.

Let \overline{ν} be the m-adic valuation of O_{P_{L}^{2},q}.

(1) Γ(P_{L}^{2}, O_{P^{2}}(d) ⊗ I_{q}^{n}) = \{F (x_{0}, x_{1}, x_{2}) ∈ Γ(P_{L}^{2}, O_{P^{2}}(d)) | \overline{ν}(F (1, \frac{x_{1}}{x_{0}} , \frac{x_{2}}{x_{0}} )) ≥ n\}.

Since M is flat over L,

(2)  Γ(P_{L}^{2}, O_{P^{2}}(d) ⊗ I_{q}^{n}) ⊗_{L} M = Γ(P_{M}^{2}, O_{P^{2}}(d) ⊗ ((I_{p_{1}} ⋯ I_{p_{r}})^{n})).

Let ν_{1} be the m-adic valuation on R = L[x_{0}, x_{1}, x_{2}]_{(x_{0},x_{1},x_{2})}. The valuation ring R_{ν_{1}} of ν_{1} is L[x_{0}, \frac{x_{1}}{x_{0}} , \frac{x_{2}}{x_{0}} ]_{(x_{0})}, with residue field R_{ν_{1}}/m_{ν_{1}} ≅ L(\frac{x_{1}}{x_{0}} , \frac{x_{2}}{x_{0}} ). The inclusion of the valuation ring R\overline{ν} of \overline{ν} into its quotient field L(\frac{x_{1}}{x_{0}} , \frac{x_{2}}{x_{0}} ) determines a composite valuation ν = ν_{1} ∘ \overline{ν} on the field L(x_{0}, x_{1}, x_{2}), which dominates R. The valuation ν is rational and its value group of ν is Z × Z with the lex order.

For f ∈ L[x_{0}, x_{1}, x_{2}], let L_{f}(x_{0}, x_{1}, x_{2}) be the leading form of f. L_{f} is a homogeneous form whose degree is the order ord(f) of f at the homogeneous maximal ideal of k[x_{0}, x_{1}, x_{2}]. The value of f is

ν(f) = (ord(f), \overline{ν}(L_{f}(1, \frac{x_{1}}{x_{0}} , \frac{x_{2}}{x_{0}} ))) ∈ N × N.

For (d, n) ∈ N × N, we have L module isomorphisms

P_{(d,n)} ∩ L[x_{0}, x_{1}, x_{2}] ≅ Γ(P_{L}^{2}, O_{P^{2}}(d) ⊗ I_{q}^{n}) ⨁ (⨁_{m>d} Γ(P_{L}^{2}, O_{P^{2}}(m))) ,

and

(3)  P_{(d,n)}/P_{(d,n)}^{+} ≅ Γ(P_{L}^{2}, O_{P^{2}}(d) ⊗ I_{q}^{n})/Γ(P_{L}^{2}, O_{P^{2}}(d) ⊗ I_{q}^{n+1}).

Proposition 2.1. Suppose that r = s^{2} where s ≥ 4 is a natural number. Then for n ≠ 0

(4)  P_{(d,n)}/P_{(d,n)}^{+} = 0 if d ≤ ns

and if s′ is a real number such that s′ > s, then there exist natural numbers d, n such that d < ns′ and

(5)  P_{(d,n)}/P_{(d,n)}^{+} ≠ 0.

Proof. We have that Γ(P_{M}^{2}, O_{P^{2}}(d) ⊗ ((I_{p_{1}} ⋯ I_{p_{r}})^{n})) = 0 if d ≤ ns by (1) of Proposition 1 of [11]. Thus (4) is true by formulas (2) and (3).

Suppose that s′ is a real number greater than s. Using the Riemann Roch Theorem and Serre Duality on the blow up of P_{L}^{2} at q, as in the proof of (2) of Proposition 1 of [11], we compute that

dim_{L}(Γ(P_{L}^{2}, O_{P^{2}}(d) ⊗ I_{q}^{n}) ≥ \frac{d(d + 3)}{2} − r \frac{n(n + 1)}{2} + 1 > 0

if

\frac{d(d + 3)}{2} ≥ r \frac{n(n + 1)}{2}.

Fixing a rational number λ with s′ > λ > s, we see that we can find natural numbers d and m with \frac{d}{m} = λ and

dim_{L}(Γ(P_{L}^{2}, O_{P^{2}}(d) ⊗ I_{q}^{m}) ≠ 0.

Let n ≥ m be the largest natural number such that

dim_{L}(Γ(P_{L}^{2}, O_{P^{2}}(d) ⊗ I_{q}^{n})) ≠ 0.

Then d < ns′ and P_{(d,n)}/P_{(d,n)}^{+} ≠ 0 by (3). Thus (5) holds. □

The following result follows from Proposition 2.1

Proposition 2.2. Suppose that s ≥ 4 is a natural number. Then there exist a field M and a rank 2 valuation ν, with value group Z^{2} ordered lexicographically, of the rational function field M(x_{0}, x_{1}, x_{2}) in three variables, which dominates the regular local ring

R = M[x_{0}, x_{1}, x_{2}]_{(x_{0},x_{1},x_{2})}

and is such that

⨁_{(d,n)∈N×N} P_{(d,n)}/P_{(d,n)}^{+}

is not a finitely generated R/m_{R} ≅ M algebra.

The semigroup Γ = ν(R \ {0}) is not finitely generated as a semigroup. Further, the closed rational cone generated by Γ in R^{2} is the rational polyhedron

{(d, n) ∈ R^{2} | n ≥ 0 and d ≥ ns}.

3. semigroups of ordinal type ω^{h}

Suppose that G is an ordered abelian group of finite rank n. Then G is order isomorphic to a subgroup of R^{n}, where R^{n} has the lex order (see Proposition 2.10 [1]). If G is the value group of a valuation ν dominating a Noetherian local ring R, and S is the semigroup of values attained by ν on R, then it can be shown that S has no accumulation points in R^{n}.

In this section we prove that this property is held by any well ordered semigroup S of ordinal type ≤ ω^{h}, which is contained in the nonnegative part of R^{n}. The proof relies on the following lemma. The heuristic idea of the proof of the lemma is that the semigroup generated by a set with an accumulation point has many accumulation points, accumulations of accumulation points, and so on.

Lemma 3.1. Let A be a well ordered set, which is contained in the nonnegative part of R. Suppose that A has an accumulation point in R, for the Euclidean topology. For m a positive integer, let

mA = {x_{1} + x_{2} + ⋯ + x_{m} | x_{1}, …, x_{m} ∈ A}.

Then mA contains a well ordered subset of ordinal type ω^{m}.

Proof. Let \overline{λ} be an accumulation point of A. Since A is well ordered, there exist λ_{i} ∈ A for i ∈ N, such that λ_{i} < λ_{j} for i < j and lim_{i↦∞} λ_{i} = \overline{λ}. Let T_{1} = \{λ_{i}\}_{i∈N}. T_{1} ⊂ A is well ordered of ordinal type ω.

For m ≥ 2 we will construct a well ordered subset T_{m} of mA which has ordinal type ω^{m}.

For a_{1} ∈ N, choose σ_{1}^{m}(a_{1}) ∈ N such that

λ_{a_{1}} + \overline{λ} < λ_{a_{1}+1} + λ_{σ_{1}^{m}(a_{1})}.

Now for a_{2} ∈ N, choose σ_{2}^{m}(a_{1}, a_{2}) ∈ N such that

λ_{σ_{1}^{m}(a_{1})+a_{2}} + \overline{λ} < λ_{σ_{1}^{m}(a_{1})+a_{2}+1} + λ_{σ_{2}^{m}(a_{1},a_{2})}.

We iterate for 2 ≤ i ≤ m to choose for each a_{i} ∈ N, σ_{i}^{m}(a_{1}, a_{2}, …, a_{i}) ∈ N such that

λ_{σ_{i−1}^{m}(a_{1},...,a_{i−1})+a_{i}} + \overline{λ} < λ_{σ_{i−1}^{m}(a_{1},...,a_{i−1})+a_{i}+1} + λ_{σ_{i}^{m}(a_{1},...,a_{i})}.

Let T_{m} be the well ordered subset of mA which is the image of the order preserving inclusion N^{m} → mA where N^{m} has the reverse lex order, defined by

(a_{m}, …, a_{1}) ↦ λ_{a_{1}+1} + (∑_{i=2}^{m} λ_{σ_{i−1}^{m}(a_{1},...,a_{i−1})+a_{i}+1}) .

By its construction, T_{m} has ordinal type ω^{m}. □

Theorem 3.2. Let S be a well ordered subsemigroup of the nonnegative part of R^{n} (with the lex order) for some n ∈ N. Suppose that S has ordinal type ≤ ω^{h} for some h ∈ N. Then S does not have an accumulation point in R^{n} for the Euclidean topology.

Proof. We prove the theorem by induction on n. For n = 1, the theorem follows from Lemma 3.1. Suppose that the theorem is true for subsemigroups of R^{n−1}.

Suppose that S ⊂ R^{n} has an accumulation point α.

Let π : R^{n} → R^{n−1} be projection onto the first n − 1 factors. For x, y ∈ R^{n}, we have that x ≤ y implies π(x) ≤ π(y). Thus the set π(S) is a well ordered semigroup, which has ordinal type ≤ ω^{h} and is contained in the nonnegative part of R^{n−1}. Let \overline{α} = π(α). By the induction assumption, π(S) has no accumulation points. Thus \overline{α} ∈ π(S), and there exists an open neighborhood U of \overline{α} in R^{n−1} such that U ∩ π(S) = \{\overline{α}\}. Thus π^{−1}(U) ∩ S = π^{−1}(\{\overline{α}\}) ∩ S, and α is an accumulation point of π^{−1}(\{\overline{α}\}) ∩ S.

Projection on the last factor is a natural homeomorphism of π^{−1}(\{\overline{α}\}) to R which is order preserving. Let A ⊂ R be the image of π^{−1}(\{\overline{α}\}) ∩ S. Since A is a well ordered set which has an accumulation point, by Lemma 3.1, mA has a subset which has ordinal type ω^{m} for all m ≥ 1. Now projection onto the last factor identifies a subset of π^{−1}(\{m\overline{α}\}) ∩ S with mA for all m ≥ 1. Thus the ordinal type of S is ≥ ω^{m} for all m ∈ N, a contradiction. □

A variation of the proof of Theorem 3.2 proves the following corollary.

Corollary 3.3. Let S be a well ordered subsemigroup of the nonnegative part of R^{n} (with the lex order) for some n ∈ N. Suppose that S has ordinal type ≤ ω^{h} for some h ∈ N. Then S has ordinal type ≤ ω^{n}.

Proof. We prove the theorem by induction on n.

We first establish the theorem for n = 1. Suppose that S ⊂ R has ordinal type > ω. To the ordinal number ω there corresponds an element a of S. There are then infinitely many elements of S in the closed interval [0, a], so that S has an accumulation point. By Theorem 3.2, this is impossible.

Now suppose that S ⊂ R^{n} and the theorem is true for subsemigroups of R^{n−1}. By Theorem 3.2, S has no accumulation points.

Let π : R^{n} → R^{n−1} be projection onto the first n − 1 factors. The set π(S) is a well ordered semigroup, which has ordinal type ≤ ω^{h} and is contained in the nonnegative part of R^{n−1}. By the induction assumption, π(S) has ordinal type ≤ ω^{n−1}. For \overline{x} ∈ π(X), π^{−1}(\{\overline{x}\}) ∩ S is a well ordered subset of R which has no accumulation points, and thus has ordinal type ≤ ω. Since this is true for all \overline{x} ∈ π(S), the ordinal type of S is ≤ ω^{n−1}ω = ω^{n}. □

4. Subsemigroups of Q_{+}

Let S be a well ordered subsemigroup of Q_{+}. Let (γ_{i})_{i∈I} be its minimal system of generators. The set of the γ_{i} may or may not be of ordinal type ω.

For example let us choose two prime numbers p, q and consider the positive rational numbers γ_{i} = 1 − \frac{1}{p^{i}} for 1 ≤ i < ω and γ_{i} = 2 − \frac{1}{q^{i−ω+1}} for ω ≤ i < ω2 = ω + ω. These numbers form a well ordered subset of Q_{+} of ordinal type ω2 and generate a certain semigroup S_{p,q} which in turn is well ordered by a result of B.H. Neumann (see [12] and [14], Theorem 3.4). Because of the way their denominators grow with i, the γ_{i} are a minimal system of generators of S_{p,q}. Using k different prime numbers one can build in the same way well ordered semigroups in Q_{+} with minimal systems of generators of ordinal type ωk for any k < ω.

Remark: Since we assume S ⊂ Q_{+}, if it is a semigroup of values for some valuation, that valuation is of rank one and by the result quoted in the Introduction, if the semigroup S comes from a noetherian ring it is of ordinal type ≤ ω. The semigroup S_{p,q} is therefore an example of a well ordered subsemigroup of a totally ordered group of finite rank which cannot be realized as the semigroup of values of a noetherian ring. Using Theorem 3.2 we see that the semigroup S_{p} generated by the γ_{i} = 1 − \frac{1}{p^{i}} , 0 < i < ω cannot be realized either since it has an accumulation point. Note that by Corollary 3.3 the ordinal type of S_{p} or S_{p,q} is ≥ ω^{ω}. One can ask what is the relationship between the ordinal type of the minimal set of generators of a well ordered subsemigroup S of Q_{+} and the ordinal type of S.

We shall from now on in this section consider only semigroups S ⊂ Q_{+} whose minimal system of generators (γ_{1}, …, γ_{i}, …) is of ordinal type ≤ ω. If the γ_{i} have a common denominator, S is isomorphic to a subsemigroup of N, and finitely generated (for a short proof see [13], Th. 83 p. 203).

Let us therefore assume that there is no common denominator. Let us denote by S_{i} the semigroup generated by γ_{1}, …, γ_{i} and by G_{i} the subgroup of Q which it generates. We have S = ⋃_{i=1}^{∞} S_{i}. Set also n_{i} = [G_{i}: G_{i−1}] for i ≥ 2. It is convenient to set n_{1} = 1. The products Π_{i=1}^{k}n_{i} tend to infinity with k.

By definition we have n_{i}γ_{i} ∈ G_{i−1} and the image of γ_{i} is an element of order n_{i} in G_{i}/G_{i−1}. For each i ≥ 1 let r_{i} be the positive rational number such that r_{i}γ_{1}, r_{i}γ_{2}, …, r_{i}γ_{i} generate the group Z. Since the semigroup r_{i−1}S_{i−1} generates Z as a group, it has a finite complement (see [13], Th. 82) and some positive multiple of r_{i−1}n_{i}γ_{i} is contained in it. Thus, we know that there exists a smallest integer s_{i} such that s_{i}γ_{i} ∈ S_{i−1} and that s_{i} is an integral multiple of n_{i}.

Definition 4.1. Let S be a well ordered subsemigroup of the semigroup of positive elements of a totally ordered abelian group of finite rank. Let (γ_{1}, …, γ_{i}, …) be a minimal system of generators of S indexed by some ordinal α and for each ordinal β ≤ α define S_{β} to be the semigroup generated by the (γ_{i})_{i≤β}. Given an integer d we say that S has stable asymptotic embedding dimension ≤ d if for each β < α the semigroup S_{β} is isomorphic to the semigroup of values which a valuation takes on an equicharacteristic noetherian local domain of embedding dimension ≤ d whose residue field is algebraically closed. We say that S has stable embedding dimension ≤ d if it is isomorphic to the semigroup of values of a valuation on an equicharacteristic noetherian local domain of embedding dimension ≤ d whose residue field is algebraically closed.

Proposition 4.2. Let S be a well ordered subsemigroup of Q_{+} which is not isomorphic to N and whose minimal system of generators (γ_{1}, …, γ_{i}, …) is of ordinal type ≤ ω. The following are equivalent:

1) For each i ≥ 2 we have s_{i} = n_{i} and γ_{i} > s_{i−1}γ_{i−1}.

2) The stable asymptotic embedding dimension of S is two.

3) The stable embedding dimension of S is two.

Proof. The conditions of 1) are known to be equivalent to the fact that each semigroup S_{i} is the semigroup of values of the natural valuation of a plane branch, which is of embedding dimension 2 since S ≠ N (see [16], Appendix, for characteristic zero, and [2] for what is necessary to extend to positive characteristic). This shows that 1) is equivalent to 2).

Given a sequence of γ_{i} satisfying 1), we can associate to it a sequence of key polynomials (SKP) as in ([7], Chapter 2, Definition 2.1) over any algebraically closed field K. That is, a sequence P_{0} = x, P_{1} = y, …, P_{i}, … of polynomials in K[x, y] such that the conditions ν(P_{i}) = γ_{i} for all i determine a unique valuation ν of the regular local ring K[x, y]_{(x,y)} or, if the sequence of γ_{i} is finite, of a one-dimensional quotient K[x, y]_{(x,y)}/(Q(x, y)), which is of embedding dimension 2 since S ≠ N. The semigroup of values of ν is the semigroup generated by the γ_{i} (see [7], Theorem 2.28). So we see that 1) implies 3). Finally, if the semigroup S comes from a noetherian local ring of embedding dimension 2, since for valuations of rank one the semigroup does not change under m-adic completion (see [15], §5), we may assune that this ring corresponds either to a branch or to a two-dimensional complete equicharacteristic regular local local ring. If S is the semigroup of values of a plane branch, condition 1) is satisfied, as we have seen above and if it comes from a valuation ν of K[[x, y]], by ([7], Theorem 2.29), there exists a SKP associated to ν, and again condition 1) is satisfied. One should note that [7] is written over the complex numbers, but the results of Chapter 2 are valid in any characteristic. □

Remark: One may ask whether a subsemigroup of Q_{+} of ordinal type ω is always of bounded stable asymptotic embedding dimension or of bounded stable embedding dimension.

5. The semigroups of noetherian rings are not always rationally finitely generated

Using the results of [7], Chapter 2, one can check that for the semigroup S = ⟨γ_{1}, …, γ_{i}, …⟩ of a valuation of the ring K[x, y]_{(x,y)} there always exist a finite set of generators γ_{1}, … γ_{ℓ} which rationally generate the semigroup S in the sense that for any generator γ_{j} there is a positive integer s_{j} such that s_{j}γ_{j} ∈ ⟨γ_{1}, …, γ_{ℓ}⟩. This fails for polynomial rings of dimension ≥ 3, as is shown by the following example taken from [15]. Let us give Z^{2} the lexicographic order and consider the field K((t^{Z_{lex}^{2}})) 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 rank two and rational rank two. Let us compute the semigroup S of the values which ν takes on R. We have γ_{1} = (0, 1), γ_{2} = (1, 0), γ_{3} = (b_{3}, −a_{3}) ∈ S. Set S_{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 S. It is easy to deduce from our assumptions that no multiple of γ_{4} is in S_{3}, and that γ_{4} is the smallest element of S which is not in S_{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:

S = ⟨γ_{1}, γ_{2}, …, γ_{i}, …⟩,

the initial forms of the u_{i} constitute a minimal system of generators of the graded Kalgebra 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 R_{0} as a quotient of K[(u_{i})_{i≥1}]; 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 defining gr_{ν}R = gr_{ν}R_{0} as a quotient of K[(U_{i})_{i≥1}] 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}. From our assumption on the growth of the ratios we see moreover that no multiple of γ_{i} is in S_{i−1} = ⟨γ_{1}, …, γ_{i−1}⟩. In fact γ_{i} is outside of the cone with vertex 0 generated by S_{i−1} in R^{2}.

6. A criterion for a series \underline{z} to be transcendental

In this section, we will use the following notation. Let K be a field, and Γ a totally ordered abelian group. Let K((t^{Γ})) be the field of formal power series with a well ordered set of exponents in Γ, and coefficients in K. Let ν be the t-adic valuation of K((t^{Γ})). Suppose that R ⊂ K((t^{Γ})) is a local ring which is essentially of finite type over K (a localization of a finitely generated K-algebra). and that ν dominates R.

Let d = dim(R). Write R = A_{P} where A ⊂ R_{ν} is of finite type over K, and R = A_{P} where P is the center of ν on A.

By Noether’s normalization theorem (Theorem 24, Section 7, Chapter VIII [17]), there exist x_{1}, …, x_{d} ∈ A, which are algebraically independent over K, such that A is a finite module over the polynomial ring B = K[x_{1}, …, x_{d}]. Thus there exist b_{1}, …, b_{r} ∈ A for some finite r, such that A = Bb_{1} + ⋯ + Bb_{r}.

Let A[z] be a polynomial ring over A. For n ∈ N, define a finite dimensional K vector space D_{n} by

D_{n} = {f ∈ A[z] | f = f_{1}b_{1}+⋯+f_{r}b_{r} where f_{1}, …, f_{r} ∈ K[x_{1}, …, x_{d}, z] have total degree ≤ n}.

Lemma 6.1. Suppose that w ∈ K((t^{Γ})) has positive value and n ∈ N. Then the set of values

E_{n} = {ν(f(w)) | f ∈ D_{n} and f(w) ≠ 0}

is finite.

Proof. For τ ∈ Γ_{+}, let

C_{τ} = {f ∈ D_{n} | ν(f(w)) ≥ τ}.

C_{τ} is a K subspace of D_{n}. Since C_{τ_{1}} ⊂ C_{τ_{2}} if τ_{2} ≤ τ_{1}, E_{n} must be a finite set. □

Lemma 6.2. Suppose that w ∈ K((t^{Γ})) has positive value. Let

τ = max{ν(f(w)) | f ∈ D_{n} and f(w) ≠ 0}.

Choose λ ∈ Γ such that λ > τ and h ∈ K((t^{Γ})) such that ν(h) = λ.

Suppose that 0 ≠ f ∈ D_{n}. Then f(w + h) ≠ 0.

Proof. Suppose that 0 ≠ f ∈ D_{n}. Let m = deg_{z}(f). We have 0 < m ≤ n (the case m = 0 is trivial). Write

f = a_{m}z^{m} + a_{m−1}z^{m−1} + ⋯ + a_{0}

where a_{m} ≠ 0 and each a_{i} has an expression

a_{i} = c_{i1}b_{1} + ⋯ + c_{ir}b_{r}

where c_{ij} ∈ K[x_{1}, …, x_{d}] is a polynomial of degree ≤ n for all i, j. Subsituting w + h for z, we have

f(w + h) = h^{m}d_{m}(w) + h^{m−1}d_{m−1}(w) + ⋯ + d_{0}(w)

where d_{i}(z) ∈ D_{n} for all i and d_{m}(z) = a_{m}, so that h^{m}d_{m}(w) ≠ 0.

Suppose that i < j, h^{i}d_{i}(w) ≠ 0, h^{j}d_{j}(w) ≠ 0 and ν(h^{i}d_{i}(w)) = ν(h^{j}d_{j}(w)). Then

iλ + ν(d_{i}(w)) = jλ + ν(d_{j}(w)),

which yields

(j − i)λ = ν(d_{i}(w)) − ν(d_{j}(w)).

But

ν(d_{i}(w)) − ν(d_{j}(w)) ≤ τ < λ ≤ (j − i)λ,

a contradiction. Thus all nonzero terms h^{i}d_{i}(w) of f(w + h) have distinct values. Since at least one of these terms was shown to be non zero, f(w + h) has finite value, so f(w + h) ≠ 0. □

Theorem 6.3. Suppose that K is a field, Γ is a totally ordered abelian group, R ⊂ K((t^{Γ})) is a local ring which is essentially of finite type over K and which is dominated by the t-adic valuation ν of K((t^{Γ})).

Suppose that z_{i} ∈ K((t^{Γ})) are defined as follows:

Let

(6)  τ_{1} = max{ν(f) | f ∈ D_{1} and f(0) ≠ 0}.

Choose α_{1} ∈ Γ_{+} with α_{1} > τ_{1} and h_{1} ∈ K((t^{Γ})) such that ν(h_{1}) = α_{1}. Set z_{1} = h_{1}.

Inductively define α_{i} ∈ Γ_{+}, h_{i} ∈ K((t^{Γ})) and z_{i} = z_{i−1} + h_{i} with ν(h_{i}) = α_{i} for 2 ≤ i so that if

(7)  τ_{i} = max{ν(f(z_{i−1})) | f ∈ D_{i} and f(z_{i−1}) ≠ 0},

then

α_{i} > τ_{i}.

Then \underline{z} = lim_{i↦∞} z_{i} ∈ K((t^{Γ})) is transcendental over the quotient field L of R.

Proof. Since {α_{i}} is an increasing sequence in Γ with ν(h_{j} − h_{i}) = α_{i+1} for j > i, we have that the limit \underline{z} = lim_{i↦∞} z_{i} exists in K((t^{Γ})).

Assume that \underline{z} is not transcendental over L. Then there exists a nonzero polynomial g(z) ∈ A[z] such that g(\underline{z}) = 0. Let m = deg_{z}(g) ≥ 1. Expand

g(z) = a_{m}z^{m} + a_{m−1}z^{m−1} + ⋯ + a_{0}

with a_{i} ∈ A for all i. Each a_{i} has an expansion

a_{i} = c_{i1}b_{1} + ⋯ + c_{ir}b_{r}

where c_{ij} ∈ K[x_{1}, …, x_{d}] for all i, j. Let n ∈ N be such that n ≥ deg(c_{ij}) for 1 ≤ i ≤ m and 1 ≤ j ≤ r, and n ≥ m.

By our construction, we have

z_{n} = z_{n−1} + h_{n} with ν(h_{n}) = α_{n} > τ_{n}.

By Lemma 6.2, g(z_{n}) ≠ 0. In K((t^{Γ})), we compute

g(\underline{z}) = g(z_{n} + \underline{z} − z_{n}) = g(z_{n}) + (\underline{z} − z_{n})e

with ν(e) ≥ 0. We have

ν(\underline{z} − z_{n}) = ν(h_{n+1}) = α_{n+1} > τ_{n+1} ≥ ν(g(z_{n})).

Thus ν(g(\underline{z})) ≤ τ_{n+1} < ∞, so that g(\underline{z}) ≠ 0, a contradiction. □

Corollary 6.4. Suppose that K is a field, Γ is a totally ordered abelian group, R ⊂ K((t^{Γ})) is a local ring which is essentially of finite type over K and which is dominated by the t-adic valuation ν of K((t^{Γ})). Then there exists \underline{z} ∈ K((t^{Γ})) such that \underline{z} is transcendental over the quotient field of R.

Remark 6.5. The conclusions of the theorem may fail if R is Noetherian, but not essentially of finite type over K. A simple example is Γ = Z and R = K[[t]], since K((t^{Γ})) is the quotient field of R.

7. An example where all gr_{F(γ_{1})}P_{γ_{1}}/P_{γ_{1}}^{+} are not finitely generated gr\overline{ν}R/P modules

Let K be an algebraically closed field. Let K(x, y) be a two dimensional rational function field over K. Let A = K[x, y]_{(x,y)}. In Example 3 of Chapter VI, Section 18 of [17] a construction is given of a valuation \overline{ν} of K(x, y) which dominates A, and such that the value group of \overline{ν} is Q. By Theorem [9, Thm. 6], since K is algebraically closed and Q is divisible, there is an embedding K(x, y) ⊂ K((t^{Q})) of K algebras, where K((t^{Q})) is the formal power series field with a well ordered set of exponents in Q, and coefficients in K.

Let Q_{+} denote the positive rational numbers.

Let

(8)  F_{0} = \{\overline{ν}(f) | f ∈ A and f ≠ 0\}

be the semigroup of the valuation \overline{ν} on A.

We will use the criterion of Theorem 6.3 to construct a limit \underline{z} = lim_{i→∞} z_{i} in K((t^{Q})) which is transcendental over the quotient field of A.

Let z be a transcendental element over K[x, y] and let

D_{i} = {f ∈ K[x, y, z] | the total degree of f is ≤ i}.

In our construction, we inductively define z_{i} ∈ K((t^{Q})). Then for n ∈ N, we may define F_{0} = \overline{ν}(A \ \{0\}) modules M_{i}^{n} by

M_{i}^{n} = {ν(a_{0} + a_{1}z_{i} + ⋯ + a_{n}z_{i}^{n}) | a_{0}, …, a_{n} ∈ K[x, y]}.

Let

A_{i} = A[z_{i}]_{(x,y,z_{i})}.

The local ring A_{i} is dominated by \overline{ν}, so the semigroup Γ_{i} of \overline{ν} on A_{i} is topologically discrete. It follows that there are arbitrarily large elements of Q_{+} which are not in Γ_{i}.

We first choose λ_{1} ∈ Q_{+} such that λ_{1} ∉ F_{0} and λ_{1} > τ_{1} where τ_{1} is defined by (6). Set α_{1} = λ_{1}. Choose f_{1}, g_{1} ∈ K[x, y] such that

\overline{ν} (\frac{f_{1}}{g_{1}} ) = α_{1}.

We then inductively construct λ_{i} ∈ Q_{+}, α_{i} ∈ Q_{+}, f_{i}, g_{i} ∈ K[x, y] and τ_{i} ∈ Q_{+} such that λ_{i} ∉ Γ_{i−1}, τ_{i} is defined by (7), and

λ_{i} > max\{λ_{i−1} + \overline{ν}(g_{1} ⋯ g_{i−1}), τ_{i} + \overline{ν}(g_{1} ⋯ g_{i−1})\}.

We define

α_{i} = λ_{i} − \overline{ν}(g_{1} ⋯ g_{i−1})

and choose f_{i}, g_{i} ∈ K[x, y] so that

\overline{ν} (\frac{f_{i}}{g_{i}} ) = α_{i}.

Let

z_{i} = z_{i−1} + \frac{f_{i}}{g_{i}} .

The resulting series \underline{z} = lim_{i→∞} z_{i} is transcendental over K(x, y) by Theorem 6.3, since α_{i} > τ_{i} for all i > 1.

Let B = A[\underline{z}]_{(x,y,\underline{z})}. Since \underline{z} ∈ K((t^{Q})) is transcendental over K(x, y) the embedding A ⊂ K((t^{Q})) which appears at the beginning of the section extends to an embedding B ⊂ K((t^{Q})) so that the three dimensional local ring B is dominated by the t-adic valuation \overline{ν} of K((t^{Q})).

Lemma 7.1. a) With the above notation, for n ∈ N, define a F_{0} module

T^{n} = \{\overline{ν}(a_{0} + a_{1}\underline{z} + ⋯ + a_{n}\underline{z}^{n}) | a_{0}, a_{1}, ⋯, a_{n} ∈ K[x, y]\}.

Then for all n > 0, T^{n} is not finitely generated as a F_{0} module.

b) With the notations just introduced, let T^{∞} = \overline{ν}(B \ \{0\}) be the semigroup of the valuation \overline{ν} on B. Then T^{∞} is not a finitely generated F_{0} module.

Proof. Suppose that n ≥ 1. We will show that T^{n} is not finitely generated as a F_{0} module. We compute

\overline{ν}(\underline{z}) = \overline{ν} (\frac{f_{1}}{g_{1}} ) = λ_{1},

and for i ≥ 2,

\overline{ν} (g_{1} ⋯ g_{i−1}\underline{z} − (f_{1}g_{2} ⋯ g_{i−1} + f_{2}g_{1}g_{3} ⋯ g_{i−1} + ⋯ + f_{i−1}g_{1} ⋯ g_{i−2}))
= \overline{ν} (g_{1} ⋯ g_{i−1} \frac{f_{i}}{g_{i}} )
= \overline{ν} (\frac{f_{i}}{g_{i}} ) + \overline{ν}(g_{1} ⋯ g_{i−1})
= λ_{i}.

Thus λ_{i} ∈ T^{n} for all i.

The F_{0} modules M_{i}^{n} introduced above are subsets of Q for all i. We compare the intersections of the M_{i}^{n} with various intervals [0, σ) in Q. Since \overline{ν}(z_{1}) = α_{1}, we have that

M_{1}^{n} ∩ [0, α_{1}) = F_{0} ∩ [0, α_{1}),

and since

z_{i} = z_{i−1} + \frac{f_{i}}{g_{i}}

with

\overline{ν} (\frac{f_{i}}{g_{i}} ) = α_{i},

we have that

M_{i}^{n} ∩ [0, α_{i}) = M_{i−1}^{n} ∩ [0, α_{i})

for all i ≥ 2.

We further have that

(9)  T^{n} ∩ [0, α_{i}) = M_{i}^{n} ∩ [0, α_{i})

for i ≥ 1.

Suppose that n ≥ 1 and T^{n} is a finitely generated F_{0} module. We will derive a contradiction. With this assumption, there exist x_{1}, …, x_{m} ∈ T^{n} such that every element v ∈ T^{n} has an expression v = y + x_{j} for some y ∈ F_{0}, and for some x_{j} with 1 ≤ j ≤ m.

There exists a positive integer l such that x_{j} < α_{l} for 1 ≤ j ≤ m. Thus x_{1} …, x_{m} ∈ M_{l}^{n} by (9). It follows that T^{n} ⊂ M_{l}^{n} since M_{l}^{n} is a F_{0} module. But λ_{n+1} ∉ M_{l}^{n} by our construction, as M_{l}^{n} ⊂ Γ_{l}. This gives a contradiction, since we have shown that λ_{l+1} ∈ T^{n}.

The proof that T^{∞} = ⋃_{n=0}^{∞} T^{n} is not a finitely generated F_{0} module is similar. □

Proposition 7.2. Suppose that K is an algebraically closed field, and K(x, y, u, v) is a rational function field in 4 variables. Then there exists a rank 2 valuation ν = ν_{1} ∘ \overline{ν} of K[x, y, u, v] with value group Z × Q with the lex order, which dominates the regular local ring

R = K[x, y, u, v]_{(x,y,u,v)},

with R/P ≅ K[x, y]_{(x,y)} where P is the center of ν_{1} on R, such that the associated graded module

gr_{F(n)}P_{n}/P_{n}^{+}

is not a finitely generated gr\overline{ν}R/P module for all positive integers n.

Proof. Since the \underline{z} we have just constructed is transcendental over K(x, y), the association z → \underline{z} defines an embedding of K algebras K(x, y, z) → K((t^{Q})) which extends our embedding K(x, y) → K((t^{Q})). We identify \overline{ν} with the induced valuation on K(x, y, z), which by our construction has value group Q and residue field K.

Let R be the localization

R = K[x, y, u, v]_{(x,y,u,v)}

of a polynomial ring in four variables. Let ν_{1} be the (u, v)-adic valuation of R. The valuation ring of the discrete, rank 1 valuation ν_{1} is

R_{ν_{1}} = K [x, y, u, \frac{v}{u}]_{(u)} .

The residue field of R_{ν_{1}} is the rational function field in three variables

R_{ν_{1}}/m_{ν_{1}} = K(x, y, z)

where z = \frac{v}{u}. Let ν be the composite valuation ν_{1} ∘ \overline{ν} on K(x, y, u, v).

For i ∈ N, we have

F_{i} = {ν(f) | f ∈ R and ν_{1}(f) = i}.

F_{0} is the semigroup F_{0} = \overline{ν}(R/P \ \{0\}), and F_{i} are F_{0} modules for all i.

We have that m_{ν_{1}} ∩ R = (u, v), so that F_{0} = Γ_{0}, the semigroup of (8). From our construction of \overline{ν}, we have that F_{n} is isomorphic to T^{n} as a Γ_{0} module. Thus for all n ≥ 1, F_{n} is not finitely generated as a F_{0} module.

By Proposition 1.1,

gr_{F(n)}P_{n}/P_{n}^{+}

is not a finitely generated gr\overline{ν}R/P module for all positive integers n.

□

Remark 7.3. In the example of Proposition 7.2, the residue field R_{ν_{1}}/m_{ν_{1}} is transcendental over the quotient field of R/P, a fact which is used in the construction. In Proposition 8.4 of the following section, an example is given where R_{ν_{1}}/m_{ν_{1}} is equal to the quotient field of R/P.

Remark 7.4. We can easily construct a series \underline{z} ∈ K((t^{Q})) such that the modules T^{n} and T^{∞} of the conclusions of Lemma 7.1 are all finitely generated Γ_{0} modules, and thus the modules

gr_{F(n)}P_{n}/P_{n}^{+}

are finitely generated gr\overline{ν}R/P module for all positive integers n. To make the construction, just take \underline{z} ∈ K[[x, y]] to be any transcendental series.

8. An example with no residue field extension

Suppose that K is an algebraically closed field, and A = K[x, y]_{(x,y)}. We will define a valuation \overline{ν} on L = K(x, y) that dominates A and has value group ⋃_{i=0}^{∞} \frac{1}{2^{i}} Z.

Define \overline{β}_{0} = 1 and \overline{β}_{i+1} = 2\overline{β}_{i} + \frac{1}{2^{i+1}} for i ≥ 0. We have

\overline{β}_{i} = \frac{1}{3}(2^{i+2} − \frac{1}{2^{i}})

for i ≥ 0, and

2\overline{β}_{i} = \overline{β}_{i−1} + 2^{i+1}\overline{β}_{0}

for i ≥ 1.

Define groups

Γ_{i} = ∑_{j=0}^{i} Z\overline{β}_{j} = \frac{1}{2^{i}} Z

for i ≥ 0.

For i ≥ 1, let

x_{i} = \frac{2^{i+1}\overline{β}_{i−1} + 1}{2^{i−1}},

and let m_{i} = 2.

Since 2^{i+1}\overline{β}_{i−1} is an even integer for all i, x_{i} has order m_{i} = 2 in Γ_{i−1}/m_{i}Γ_{i−1} and \overline{β}_{i} = \frac{x_{i}}{m_{i}}.

By our construction, for i ≥ 1, \overline{β}_{i+1} > m_{i}\overline{β}_{i}. By the irreducibility criterion of [3], Remark 7.17 [6], or Theorem 2.22 [7], there exists a valuation \overline{ν} of L dominating A and a (minimal) generating sequence P_{0}, P_{1}, ⋯, P_{i}, ⋯ for \overline{ν} in K[x, y] of the form

P_{0} = x
P_{1} = y
P_{2} = y^{2} − x^{5}
P_{3} = P_{2}^{2} − x^{8}y
⋮
P_{i+1} = P_{i}^{2} − P_{0}^{2^{i+1}}P_{i−1}
⋮

with \overline{β}_{i} = ν(P_{i}) for all i. The semigroup M_{0} of \overline{ν} on A is

M_{0} = ∑_{i=0}^{∞} N\overline{β}_{i}.

Let z = \frac{y}{x} ∈ L, and define for n ∈ N,

W_{n} = {a_{0} + a_{1}z + ⋯ + a_{n}z^{n} | a_{0}, …, a_{n} ∈ K[x, y]}.

Define M_{0} modules M_{n} by

M_{n} = \{\overline{ν}(f) | 0 ≠ f ∈ W_{n}\}.

Lemma 8.1. For all i ≥ 0. we have an expression P_{i} = x^{i}h_{i} with h_{i} ∈ W_{i}.

Proof. The statement is clear for i = 0 and i = 1. Suppose, by induction, that the statement is true for j ≤ i, so that P_{j} = x^{j}h_{j} with h_{j} ∈ W_{j} for j ≤ i. Write

P_{i+1} = P_{i}^{2} − P_{0}^{2^{i+1}}P_{i−1}.

We have that

P_{i}^{2} = x^{i+1} [(\frac{P_{i}}{x^{i}} ) (\frac{P_{i}}{x^{i}} ) x^{i−1}]

with

(\frac{P_{i}}{x^{i}} ) (\frac{P_{i}}{x^{i}} ) x^{i−1} ∈ W_{2i−(i−1)} = W_{i+1}.

Further,

P_{0}^{2^{i+1}}P_{i−1} = x^{2^{i+1}}x^{i−1}h_{i−1} ∈ x^{i+1}W_{i−1} ⊂ x^{i+1}W_{i+1}.

Thus P_{i+1} = x^{i+1}h_{i+1} with h_{i+1} ∈ W_{i+1}. □

For n ∈ N, let

U_{n} = \{λ ∈ M_{n} | λ = ∑_{j=0}^{n−1} l_{j}\overline{β}_{j} for some l_{i} ∈ Z\}.

Lemma 8.2. For n ≥ 1,

M_{n} = U_{n} ⋃ ( ⋃_{j≥n}((\overline{β}_{j} − n) + M_{0})) .

Proof. \frac{P_{j}}{x^{j}} ∈ W_{j} for all j ≥ 0 implies

\frac{P_{j}}{x^{n}} = (\frac{P_{j}}{x^{j}} ) x^{j−n} ∈ W_{n}

for j ≥ n. Thus \overline{β}_{j} − n ∈ M_{n} for j ≥ n.

Now suppose that λ ∈ M_{n}. λ = \overline{ν}(a_{0} + a_{1}z + ⋯ + a_{n}z^{n}) for some a_{0}, …, a_{n} ∈ K[x, y]. Set τ = \overline{ν}(a_{0}x^{n} +a_{1}x^{n−1}y +⋯+a_{n}y^{n}) ∈ M_{0}. We have λ = τ −n. τ = ∑ l_{j}\overline{β}_{j} = \overline{ν}(∏ P_{j}^{l_{j}}) for some l_{j} ∈ N. Suppose l_{k} ≠ 0 for some k ≥ n. Then

λ = \overline{ν} ((∏_{j≠k} P_{j}^{l_{j}} ) P_{k}^{l_{k}−1} \frac{P_{k}}{x^{n}} ) = (\overline{β}_{k} − n) + \overline{ν} ((∏_{j≠k} P_{j}^{l_{j}} ) P_{k}^{l_{k}−1}) .

Now suppose that l_{k} = 0 for k < n. Then

λ = ∑_{j=1}^{n−1} l_{j}\overline{β}_{j} + (l_{0} − n)\overline{β}_{0} ∈ U_{n}.

□

Lemma 8.3. Suppose that n ≥ 1. Then M_{n} is not a finitely generated M_{0} module.

Let B = K[x, \frac{y}{x}]_{(x, \frac{y}{x})}. B is a regular local ring that birational dominates A = K[x, y]_{(x,y)}. Let M_{∞} = \overline{ν}(B \ \{0\}). Then M_{∞} is not a finitely generated module over Γ_{0} = \overline{ν}(A \ \{0\}).

Proof. For i ≥ 0, define Ψ_{i} to be the M_{0} module generated by U_{n} and \{\overline{β}_{j} −n | i ≥ j ≥ n\}. For i ≥ n we have

Ψ_{i} = U_{n} ⋃ ( ⋃_{i≥j≥n}((\overline{β}_{j} − n) + M_{0})) ⊂ \frac{1}{2^{i}} N.

Thus \overline{β}_{i+1} − 1 ∉ Ψ_{i} for i ≥ n.

We will now show that M_{n} is not a finitely generated M_{0} module.

Suppose that M_{n} is finitely generated as an M_{0} module. Then M_{n} is generated by a set a_{1}, …, a_{r}, b_{1}, …, b_{s}, where

a_{i} = (\overline{β}_{σ(i)} − n) + λ_{i}

with σ(i) ≥ n and λ_{i} ∈ M_{0} for 1 ≤ i ≤ r and b_{i} ∈ U_{n} for 1 ≤ i ≤ s.

Let m = max\{σ(i)\}. Then M_{1} ⊂ Ψ_{m}, which is impossible since \overline{β}_{m+1} − n ∉ Ψ_{m}.

The proof that M_{∞} is not a finitely generated Γ_{0} module is similar. □

Proposition 8.4. Suppose that K is an algebraically closed field, and K(x, y, u, v) is a rational function field in 4 variables.

Suppose that Γ is a totally ordered Abelian group and α ∈ Γ is such that 1 and α are rationally independent and 1 < α.

Then there exists a rank 2 valuation ν = ν_{1} ∘ \overline{ν} of K(x, y, u, v) with value group

(Z + αZ) × (⋃_{i=0}^{∞} \frac{1}{2^{i}} Z)

in the lex order, which dominates the regular local ring

R = K[x, y, u, v]_{(x,y,u,v)}

such that (1) R_{ν_{1}}/m_{ν_{1}} ≅ (R/P)_{P} ≅ K(x, y) where P is the center of ν_{1} on R.

(2) The associated graded module

gr_{F(n)}P_{n}/P_{n}^{+}

is not a finitely generated gr\overline{ν}R/P module for n ∈ N.

(3) The associated graded module

gr_{F(nα)}P_{nα}/P_{nα}^{+}

is a finitely generated gr\overline{ν}R/P module for n ∈ N.

Proof. We use the notation developed earlier in this section. Define a valuation ν_{1} on the rational function field L(u, v) in 2 variables by the embedding of L algebras

L(u, v) → L((t^{Γ}))

induced by

u ↦ t, v ↦ v(t) = \frac{y}{x}t + t^{α}.

If ν = ν_{1}∘\overline{ν} be the composite valuation on K(x, y, u, v), then ν dominates R = K[x, y, u, v]_{(x,y,u,v)}. The center of ν_{1} on R is the prime ideal P = (u, v). We have L = (R/P)_{P} = R_{ν_{1}}/m_{ν_{1}} and K = R_{ν}/m_{ν}, proving part 1) of the proposition.

For i ∈ N,

F_{i} = {ν(f) | f ∈ R and ν_{1}(f) = i}.

Suppose that f ∈ K[x, y, u, v]. Expand

f = ∑ a_{ij}u^{i}v^{j}

with a_{ij} ∈ K[x, y]. We have

f(t, v(t)) = a_{00} + (a_{10} + a_{01} \frac{y}{x})t + higher order terms in t.

We see that ν_{1}(f) = 0 if and only if a_{00} ≠ 0. Thus F_{0} ≅ M_{0} as semigroups.

ν_{1}(f) = 1 if and only if a_{00} = 0 and a_{10} + a_{01} \frac{y}{x} ≠ 0. Thus F_{1} ≅ M_{1} as F_{0} modules, so that F_{1} is not finitely generated as an F_{0} module.

To see that F_{n} ≅ M_{n} as an F_{0} module for all n ≥ 0, we expand

\begin{aligned}f(t, v(t)) & = & ∑_{k=0}^{∞} ∑_{i+j=k} a_{ij}t^{i}v(t)^{j} \\ & = & ∑_{k=0}^{∞} ∑_{j=0}^{k} φ_{jk}t^{(k−j)+jα} \end{aligned}

where

φ_{jk} = ∑_{i=j}^{k} a_{k−i,i}\binom{i}{j} (\frac{y}{x})^{i−j} .

We have (k_{1} − j_{1}) + j_{1}α = (k_{2} − j_{2}) + j_{2}α implies j_{1} = j_{2} and k_{1} = k_{2}. Thus

ν_{1}(f) = min{(k − j) + jα | φ_{jk} ≠ 0}.

We further have for fixed k, j_{1} < j_{2} implies

(10)  (k − j_{1}) + j_{1}α < (k − j_{2}) + j_{2}α.

Suppose that ν_{1}(f) = n ∈ N. Then

f(t, v(t)) = φ_{0n}t^{n} + higher order terms in t

with

φ_{0n} = a_{0n} + a_{n−1,1} \frac{y}{x} + ⋯ + a_{0n} (\frac{y}{x})^{n} ≠ 0.

Further, by (10), we see that for n ∈ N and a_{n0}, a_{n−1,1}, …, a_{0n} ∈ K[x, y],

ν_{1}(a_{n0}u^{n} + a_{n−1,1}u^{n−1}v + ⋯ + a_{0n}v^{n}) = n

if and only if

a_{n0} + a_{n−1,1} \frac{y}{x} + ⋯ + a_{0n} (\frac{y}{x})^{n} ≠ 0.

Thus for n ∈ N,

F_{n} ≅ \{\overline{ν}(h) | h ∈ W_{n} and h ≠ 0\}

which is isomorphic to M_{n} as an M_{0} module. Since M_{n} is not finitely generated as an M_{0} module, we obtain (2) of the conclusions of the Proposition from Proposition 1.1.

Let us now consider the polynomials

Φ_{jk}(W) = ∑_{i=j}^{k} a_{k−i,i}\binom{i}{j}W^{i−j},

and remark that we have the equalities

\frac{∂^{j}Φ_{0k}(W)}{∂W^{j}} = j!Φ_{jk}(W).

In order for the series f(t, v(t)) to be of order nα, all the φ_{jk} must be zero for (k−j)+jα < nα, while φ_{nn} = a_{0n} must be non zero. In particular, φ_{jn} must be zero for j < n.

In view of the equalities we have just seen, the n conditions φ_{jn} = 0, 0 ≤ j ≤ n − 1 are equivalent to the fact that \frac{y}{x} is a root of order n of the polynomial Φ_{0n}(W), so that Φ_{0n}(W) = a_{0n}(W − \frac{y}{x})^{n}. From this it follows that the a_{n−j,j} for j < n are determined and the only condition on a_{0n} is that x^{n} divides a_{0n}. The elements of F_{nα} coincide with the values of ν on K[x, y] translated by n, so that for each n ∈ N we have F_{nα} = n + F_{0} ≅ F_{0}. Thus (3) of the conclusions of the Proposition follows from Proposition 1.1. □

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Dale Cutkosky, Bernard Teissier, 202 Mathematical Sciences Bldg, Institut Mathématique de Jussieu

University of Missouri, UMR 7586 du CNRS, Columbia, MO 65211 USA 175 Rue du Chevaleret, 75013 Paris,France cutkoskys@missouri.edu teissier@math.jussieu.fr