Three facts of valuation theory

Bernard Teissier

February 1, 2000

Corrected version of the paper which appeared in the Proceedings of the Japan Academy, Vol. LXXIV, Ser. A, No. 7, September 1988. The statement on the ν-adic completion depended on two corollaries in [Z-S] which, as I learned later from reading [Ve], are false. The correct statement is below.

1 Statement of results

Let k be a field and (R, m) a local integral k-algebra with field of fractions K. We study k-valuations ν of K with a center in R, that is, such that their valuation ring (R_{ν}, m_{ν}) contains R and m_{ν} ∩ k = (0). We denote by Φ the totally ordered group of the valuation and set Γ = ν(R \ {0}) ⊂ Φ_{+} ∪ {0}. The valuation determines on R filtrations defined by the ideals

P_{ϕ}(R) = {x ∈ R/ν(x) ≥ ϕ} or P_{ϕ}^{+}(R) = {x ∈ R/ν(x) > ϕ},

and the associated graded ring introduced by Spivakovsky ([S], see also [L-T], [T1]):

gr_{ν}R = ⨁_{ϕ∈Γ} P_{ϕ}/P_{ϕ}^{+},

which is a Γ (or Φ_{+})-graded (R/m_{ν} ∩R)-algebra. We assume throughout that Φ has finite rational rank r(Φ) (and therefore is countable) and finite height (or rank) h(Φ). The three facts, extracted from [T2], are the following:

1) A connexion between valuation theory and toric geometry

Proposition 1.1 For any valuation ν_{0} with a center in R and such that m_{ν_{0}} ∩ R = m and the residue field extension k_{R} → k_{ν_{0}} = R_{ν_{0}}/m_{ν_{0}} induced by the inclusion R ⊂ R_{ν_{0}} is trivial, the algebra gr_{ν_{0}}R is isomorphic to a quotient of a polynomial ring k_{R}[(U_{i})_{i∈I}] with coefficients in k_{R} and possibly countably many indeterminates by a binomial ideal, i.e, an ideal with (possibly countably many) generators of the form U^{m} − λ_{mn}U^{n} where U^{m} = U_{1}^{m_{1}} … U_{s}^{m_{s}} and λ_{mn} ∈ k_{R}^{∗}. It means that it is a deformation of a (non normal) toric variety (see [E-S]), possibly of infinite embedding dimension, which is nothing but Speck_{R}[t^{Γ}], where k_{R}[t^{Γ}] is the semigroup algebra of Γ, obtained by replacing all λ_{mn} by 1.

2) Structure of valuation semigroup algebras and regularity of gr_{ν}R_{ν}

Proposition 1.2 The graded k_{ν}-algebra gr_{ν}R_{ν} is a filtering direct limit of termic maps (i.e mapping a variable to a term, of the form constant times a monomial) between polynomial subalgebras in r(Φ) variables. The semigroup algebra k_{ν}[t^{Φ_{+}}] is the direct limit of the corresponding system of toric (or monomial) maps, obtained by replacing all the constants by 1.

3) Nœtherianity of certain ν-adic completions

Proposition 1.3 Assume that R is an analytically irreducible nœtherian local ring. If ν_{1} denotes the valuation of height one with which ν is composed, and p = m_{ν_{1}} ∩ R the center of ν_{1} on R, then the completion R̂^{ν} of the ring R with respect to the topology defined by the (P_{ϕ})_{ϕ∈Φ_{+}}-filtration, is isomorphic as topological ring to a quotient of the symbolic p-adic completion R̂^{(p)} of R; if R is regular, it is nœtherian.

In particular, if R is excellent and regular, the ring R̂^{ν} is excellent since R̂^{(p)} is excellent in this case by ([Ro], [Ve]).

2 Ideas of proofs

1) Since R_{ν} is a valuation ring, the Φ_{+}-graded k_{ν}-algebra gr_{ν}R_{ν} has the property that each of its homogeneous components is a 1-dimensional vector space over k_{ν}. If the residual extension is trivial, the same is true over k_{R}, and since the k_{R}-algebra gr_{ν}R is a graded subalgebra of gr_{ν}R_{ν}, each of its homogeneous components is a k_{R}-vector space of dimension ≤ 1. By an observation of Korkina ([Ko], see also [E-S]), this implies the result: taking a (possibly countable) system of homogeneous generators of the algebra gives a graded surjection k_{R}[(U_{i})_{i∈I}] → gr_{ν}R once U_{i} is given the degree of its image. The kernel is generated by homogeneous polynomials, but any two terms of such a polynomial have non zero k_{R}-proportional images, which shows that the kernel is generated by binomials.

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

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, 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 (becoming longer and longer) 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 [E], V, 2, p. 149) is contained in the half space ∑_{j=1}^{m} a_{j}τ_{j} ≥ 0, the integral points of which form the semigroup Φ_{+}. For any commutative ring A, the A-algebra of the semigroup σ̌^{(h)} ∩ Z^{m} is a polynomial algebra A[x_{1}^{(h)}, …, x_{m}^{(h)}] (loc.cit , VI,2) contained in A[t^{Φ_{+}}], and since by assumption there are no integral points on the hyperplane ∑_{j=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 A[t^{Φ_{+}}] is the union, or direct limit, of these polynomial subalgebras.

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 ([Z], B. I, p. 862) that the new weight vector w = (τ_{1}, …, τ_{m}, τ_{m+1}) ∈ R^{m+1} is con tained in a rational simplicial cone σ ⊂ R^{m+1} generated by m integral vectors of the first quadrant forming part of a basis of the integral lattice. Indeed w is contained in a unique rational hyperplane. The dual cone σ̌ ⊂ Ř^{m+1} is the product of an m dimensional strictly convex cone by a 1-dimensional vector space (see [E], V, 2) generated by an integral vector which is the dual of the rational hyperplane containing w. By refining as above by the Perron algorithm for w inside the linear span < σ > of σ, starting with the barycentric coordinates of w in σ, we find a sequence of regular simplicial cones σ^{(h)} ⊂ σ whose duals σ̌^{(h)} ⊂ Φ_{+} correspond ([E], VI, Th. 2.12) to algebras of the form A[x_{1}^{(h)}, …, x_{m}^{(h)}, x_{m+1}^{±1}] ⊂ A[t^{Φ_{+}}]. 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 corresponding to the variable x_{m+1} is in the kernel. The σ̌^{(h)} ∩ Z^{m+1} fill up f^{−1}(Φ_{+} ∪ {0}) as h → ∞ since the only rational points of the hyperplane ∑_{j=1}^{m+1} a_{j}τ_{j} = 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 A[t^{f}]: A[x_{1}^{(h)}, …, x_{m}^{(h)}, x_{m+1}^{±1}] → A[t^{Φ_{+}}] is A[t^{Φ_{+}}]. But these images are isomorphic to A[x_{1}^{(h)}, …, x_{m}^{(h)}, x_{m+1}^{±1}]/(x_{m+1} − 1) so that they are again polynomial rings A[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. A group Φ ⊂ R of finite rational rank is a direct limit of such extensions.

Let now Φ be a totally ordered group of finite height h > 1. We have a surjective ordered 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^{r(Φ_{1})}, 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_{+} 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, ϕ + ψ ∈ Φ_{+} ∪ {0}, 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}. The F̃_{i}(n) ⊕ G_{j} are of rank equal to the rational rank of Φ and fill up Φ_{+}. This proves the result.

Corollary 2.1 The semigroup algebra k_{ν}[t^{Φ_{+}}] is a quotient of a polynomial ring over k_{ν} in countably many indeterminates k_{ν}[(U_{j})_{j∈J}] by a binomial ideal with generators of the form

(U_{j} − U^{m(j)})_{j∈J′}

for some J′ ⊂ J, with |m(j)| ≥ 2.

The proof above shows that k_{ν}[t^{Φ_{+}}] is a union of polynomial algebras with monomial maps. Each member of a system of homogeneous generators of the k_{ν}-algebra k_{ν}[t^{Φ_{+}}] must appear in one of the polynomial subalgebras, and we may assume that it appears as one of the generators of that algebra. The only relations between these generators are those corresponding to the toric inclusions k_{ν}[x_{1}, …, x_{m}] → k_{ν}[y_{1}, …, y_{m}], and they are of the announced type.

Corollary 2.2 . Given the ring R_{ν} of a valuation of finite rational rank, the graded algebra gr_{ν}R_{ν} is a quotient of a polynomial ring k_{ν}[(U_{j})_{j∈J}] in countably many indeterminates over the residue field k_{ν} by a binomial ideal of the form

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

for some J′ ⊂ J, with |m(j)| ≥ 2.

it suffices to observe that setting all constants equal to 1 in the binomial relations given by fact 1) for gr_{ν}R_{ν} must give k_{ν}[t^{Φ_{+}}].

3) We note that since Φ is countable, the ν-adic topology has a countable basis of neighborhoods for every point. The quotients R/P_{ϕ} form a projective system since the semigroup Φ_{+} is 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 valuation ν has a canonical extension ν̂ to a valuation of R̂^{ν} with values in Φ. Let us assume that R is a nœtherian local ring and that ν is a valuation of height one. 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}. Note that since by definition of the symbolic powers p^{(n)} = p^{n}R_{p} ∩ R of p we have

p^{(n)} = {x ∈ R/∃s ∉ p : sx ∈ p^{n}},

we have the inclusion p^{(N(ϕ))} ⊂ P_{ϕ}. Then we have a morphism of completions R̂^{(p)} → R̂^{ν}, where R̂^{(p)} is the completion of R for the topology defined by the symbolic powers. Since R̂^{ν} is an integral domain, the kernel of this map is a prime ideal H ⊂ R̂^{(p)}. By the definition of completions, this kernel is ⋂_{ϕ∈Φ_{+}} P_{ϕ}R̂^{(p)}. Let us show that the natural injection

R̂^{(p)}/H → R̂^{ν}

is an isomorphism. The ring R̂^{(p)}/H is local with maximal ideal m̂ = mR̂^{(p)}/H. Moreover, the injection

R̂^{(p)}/H ⊂ R̂^{ν}

implies that the valuation ν extends to R̂^{(p)}/H as a valuation ν̂ of height one. Recall that when p is the maximal ideal, the symbolic powers coincide with the ordinary ones. The valuation ν is still ≥ 0 on the localization R_{p}, and so we have a morphism of completions R̂_{p}^{pR_{p}}/H_{p} ⊂ R̂_{p}^{ν}. 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 and have intersection zero. We 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}.

When the center of ν is not necessarily m, this shows that on R we have the inclusions P_{j} ⊂ p^{(s(j))}. This gives the result in the general case.

It now suffices to note that whenever a valuation ν is composed with a valuation ν_{1}, the topologies which they define are equivalent. Finally, if R is regular, by [Ve] the symbolic and usual topologies coincide.

Bibliography

[E] G. Ewald, Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Math., No. 168, Springer 1996. [E-S] D. Eisenbud and B. Sturmfels, Binomial ideals, Duke Math. J. 84, I, 1996, 1-45.

[Ko] E. Korkina, classification of A-graded algebras with 3 generators, Indag. Math., New Series, 3, 1992, 27-40. [L-T] M. Lejeune-Jalabert et B. Teissier, Cloture intégrale des idéaux et équisingularité. Séminaire Lejeune-Teissier 1974, Publ. Institut Fourier, Grenoble.

[Ro] C. Rotthaus, Kompletierung semilokaler quasiausgezeichneter ringe, Nagoya Math. J. 76, 1980, 173-180.

[S] M. Spivakovsky, Valuations in function fields of surfaces, Amer. J. of Math. bf 112 (1990), 107-156.

[T1] B. Teissier, ”Appendice”, in ”Le problème des modules pour les branches planes” by O. Zariski. Centre de Mathématiques de l’Ecole polytechnique, 1975, republished by Hermann, Paris 1986.

[T2] B. Teissier, Valuations, deformations, and toric geometry, in preparation.

[Ve] J-K. Verma, On the symbolic topology of an ideal, J. of Algebra, 112 (1988), 416-429.

[Z] O. Zariski, Local uniformization on algebraic varieties, Annals of Math. Vol. 41, No. 4, October 1940, 852-886. [Z-S] O. Zariski and P. Samuel, Commutative Algebra, Van Nostrand 1960.

Bernard Teissier

D.M.I, E.N.S, URA No. 762 du CNRS.

45 Rue d’Ulm, F75005 Paris email: teissier@dmi.ens.fr \begin{matrix}c \\ ◯ \end{matrix}Bernard Teissier 1999

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