Let ν_{1} and ν_{2} be valuations on the noetherian local ring (R, m), with centers p_{2} ⊂_{≠} p_{1} and such that ν_{1} is composed with ν_{2} and their ranks differ by one. Let λ: Γ_{1} → Γ_{2} be the corresponding map of value groups. Set t_{1} = ν_{1}(p_{1}), t_{2} = ν_{2}(p_{2}).

Given φ_{2} ∈ Γ_{2}, denote by φ̃_{2} ∈ Γ_{1} the minimum of {ν_{1}(f); f ∈ R, ν_{2}(f) = φ_{2}}. Note that λ(φ̃_{2}) = φ_{2} and that t_{1} ∈ Kerλ. Since the ranks of the valuations differ by one, the ordered group Kerλ is of rank one, and we identify it with a subgroup of the ordered group R.

We remark that for φ_{1} ∈ [φ̃_{2}, φ̃_{2} + y_{1}t_{1}] we have the inclusions

p_{1}^{y_{1}}P_{φ_{2}} ⊂ P_{φ_{1}} ⊂ P_{φ_{2}},

and since the valuation \overline{ν}_{1} which is the image of ν_{1} in R/p_{2} is of rank one, the number of elements in the interval [φ̃_{2}, φ̃_{2} + y_{1}t_{1}] is finite.

The elements of the group Γ_{i} will be denoted by φ_{i} or γ_{i}, and it is understood that P_{φ_{i}} = {x ∈ R; ν_{i}(x) ≥ φ_{i}}.

Lemma 0.1. For any additive function A on R-modules we have

(1)  ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+y_{1}t_{1}]} A(P_{φ_{1}}/P_{φ_{1}}^{+}) ≤ A(M_{φ_{2}}/p_{1}^{y_{1}}M_{φ_{2}}),

where M_{φ_{2}} = P_{φ_{2}}/P_{φ_{2}}^{+}, which is a finitely generated torsion free R/p_{2}-module.

Indeed, by additivity the left hand side is equal to A(\frac{M_{φ_{2}}}{P_{φ_{2}}/P_{φ̃_{2}+ty_{1}}} ) which is a quotient of M_{φ_{2}}/p_{1}^{y_{1}}M_{φ_{2}}. □

In the special case where ν_{1} is a valuation of rank one and ν_{2} is the trivial valuation, we have Γ_{2} = 0, φ̃_{2} = 0 ∈ Γ_{1} and the inequality reduces to:

(2)  ∑_{φ∈[0,y_{1}t_{1}]} A(P_{φ}/P_{φ}^{+}) ≤ A(R/p_{1}^{y_{1}}R),

We can take as additive function A on R-modules of dimension ≤ dimR/p_{1} the multiplicity in dimension dimR/p_{1} with respect to the maximal ideal. The inequality (1) then becomes

(3)  ∑_{φ_{1}∈[φ̃_{2},φ̃_{2}+y_{1}t_{1}]} e_{m}(P_{φ_{1}}/P_{φ_{1}}^{+}) ≤ e_{m}(M_{φ_{2}}/p_{1}^{y_{1}}M_{φ_{2}}),

It is possible to evaluate the right hand side thanks to the:

Proposition 0.2. (See ([B], §7, No.1, Prop 3)Given a finitely generated R-module M of dimension d ≥ 0, denote by B the set of minimal prime ideals of Supp(M) such that dim(R/p) = d. Let q be an ideal of R contained in its radical and such that M/qM has finite length. Then we have

e_{q}(M) = ∑_{p∈B} length(M_{p})e_{q}(R/p).

□

We shall also use the following

Corollary 0.3. Let (R, m) be a noetherian local ring and let M be a finitely generated module whose annihilator is a prime ideal p. Let q be a prime ideal of R containing p. Then we have the inequality

e_{qR_{q}}(M_{q}) ≤ e_{m}(M).

By Proposition 0.2, we have the following two equalities:

e_{qR_{q}}(M_{q}) = length_{(R_{q})_{pRq}} ((M_{q})_{pR_{q}})e_{qR_{q}}(R_{q}/pR_{q}) and e_{m}(M) = length_{R_{p}}(M_{p})e_{m}(R/p).

Now it suffices to use the fact that (R_{q})_{pR_{q}} = R_{p}, (R/p)_{q/p} = R_{q}/pR_{q} and (M_{q})_{pR_{q}} = M_{p}, and the semicontinuity of mutiplicity (see [L]) which tells us that e_{qR_{q}}(R_{q}/pR_{q}) ≤ e_{m}(R/p). □

If we apply Proposition 0.2 to M_{φ_{2}}/p_{1}^{y_{1}}M_{φ_{2}} we get:

e_{m}(M_{φ_{2}}/p_{1}^{y_{1}}M_{φ_{2}}) = length_{R_{p_{1}}} ((M_{φ_{2}})_{p_{1}}/m_{1}^{y_{1}}(M_{φ_{2}})_{p_{1}})e_{m}(R/p_{1}),

where m_{1} = p_{1}R_{p_{1}} and the support of the R-module M_{φ_{2}} is the ideal p_{2}. By the main result on Hilbert-Samuel polynomials, in view of the fact that the support of M_{φ_{2}})_{p_{1}} is the ideal p_{2}R_{p_{1}}, we have an asymptotic estimate:

length_{R_{p_{1}}} ((M_{φ_{2}})_{p_{1}}/m_{1}^{y_{1}}(M_{φ_{2}})_{p_{1}}) ≍ \frac{e_{m_{1}}((M_{φ_{2}})_{p_{1}}))}{(dim(R_{p_{1}}/p_{2}R_{p_{1}})!}y_{1}^{dim(R_{p_{1}}/p_{2}R_{p_{1}})}

Let

(0) = p_{n+1} ⊂ p_{n} ⊂ p_{n−1} ⊂ ⋯ ⊂ p_{1} ⊂ m

be the sequence of the centers of valuations ν_{i} with which a given valuation ν of rank n is composed.

Let

Γ ⟶^{λ_{1}} Γ_{1} ⟶^{λ_{2}} ⋯ ⟶^{λ_{n}} Γ_{n} ⟶ 0

be the corresponding sequence of morphisms of value groups. For each φ_{j+1} ∈ ν_{j+1}(R \ {0}) ⊂ Γ_{j+1}, define φ̃_{j+1} = min{ν_{j}(f); f ∈ R, ν_{j+1}(f) = φ_{j+1}}, and remark the equality of valuation ideals P_{φ_{j+1}} = Pφ̃_{j+1} with respect to the two valuations ν_{j+1} and ν_{j}.

Define also t_{j} = ν_{j}(p_{j}) and remark that it is in the subgroup of rank one Kerλ_{j+1} of Γ_{j}, so that we may identify it with a real number. We assume chosen from the beginning an identification of all the Kerλ_{j} with subgroups of R.

Theorem 0.4. For each j ∈ {1, …, n} and every ε > 0 the sum

∑_{φ_{n}∈[0,t_{n}y_{n}]} ∑_{φ_{n−1}∈[φ̃_{n},φ̃_{n}+t_{n−1}y_{n−1}]} ⋯ ∑_{φ_{j}∈[φ̃_{j+1},φ̃_{j+1}+t_{j}y_{j}]} e_{m}(P_{φ_{j}}/P_{φ_{j}}^{+})

is bounded for y_{j}, …, y_{n} large enough by

(1 + ε)\frac{∏_{i=j−1}^{n} e_{m_{i}}(R_{p_{i}}/p_{i+1}R_{p_{i}})}{∏_{i=j}^{n}(dimR_{p_{i}}/p_{i+1}R_{p_{i}})!} ∏_{i=j}^{n} y_{i}^{dimR_{p_{i}}/p_{i+1}R_{p_{i}}} ,

where m_{i} = p_{i}R_{p_{i}}.

The proof is by descending induction on j. The first step of the induction concerns the valuation ν_{n}, which is of rank one and is given by considering the special case of Lemma 0.1 where the valuation is of height one, and taking as additive function the multiplicity in dimension dimR/p_{1}. Applying Proposition 0.2 to the right hand side of the inequality gives:

∑_{φ∈[0,y_{1}t_{1}]} A(P_{φ}/P_{φ}^{+}) ≤ H_{R_{p_{1}}} (y_{1})e_{m}(R/p_{1}),

where H_{R_{p_{1}}} (y_{1}) is the Hilbert-Samuel function of the local ring R_{p_{1}} with respect to its maximal ideal m_{1} = p_{1}R_{p_{1}}. It is for large y_{1} a polynomial in y_{1} whose highest degree term

is \frac{e_{m_{1}}(R_{p_{1}})}{(dimR_{p_{1}})!}y_{1}^{dimR_{p_{1}}} , which gives the result in this case.

Define now t_{j} = ν_{j}(p_{j}) ∈ Γ_{j} for each j and remark that it is in the kernel of the map Γ_{j} → Γ_{j+1}, which is of rank one. We can therefore think of it as a real number. We are going to use the following

Lemma 0.5. For each j we have p_{j}^{y_{j}}P_{φ_{j−1}} ⊂ P_{φ̃_{j−1}+t_{j}y_{j}} and the interval [φ̃_{j−1}, φ̃_{j−1}+t_{j}y_{j}] is finite.

This follows directly from [ZS], Vol. II, appendix 3, Lemma 4. □

Assume that the result is known for j and prove it for j − 1. Let us apply the result above to

In view of the lemma, we have the inequality

∑_{φ_{j}∈[φ̃_{j+1},φ̃_{j+1}+t_{j}y_{j}]} e_{m}(P_{φ_{j}}/P_{φ_{j}}^{+}) ≤ e_{m}(M_{φ_{j+1}}/p_{j}^{y_{j}}M_{φ_{j+1}}).

Now we apply Proposition 0.2 and the properties of the Hilbert-Samuel polynomial of the R_{p_{j}}-module (M_{φ_{j+1}})_{p_{j}}, whose annihilator is p_{j+1}R_{p_{j}}, to obtain an asymptotic estimate

e_{m}(M_{φ_{j+1}}/p_{j}^{y_{j}}M_{φ_{j+1}}) ≍ \frac{e_{m_{j}}((M_{φ_{j+1}})_{p_{j}})}{(dimR_{p_{j}}/p_{j+1}R_{p_{j}})!}y_{j}^{dimR_{p_{j}} /p_{j+1}R_{p_{j}}} ,

so that for each φ_{j+1} ∈ Γ_{j+1} we have for any ε > 0 for y_{j} sufficiently large (in a way which depends on ε and φ_{j}) an inequality

∑_{φ_{j}∈[φ̃_{j+1},φ̃_{j+1}+t_{j}y_{j}]} e_{m}(P_{φ_{j}}/P_{φ_{j}}^{+}) ≤ (1 + ε)\frac{e_{m_{j}}((M_{φ_{j+1}})_{p_{j}})}{(dimR_{p_{j}}/p_{j+1}R_{p_{j}})!}y_{j}^{dimR_{p_{j}} /p_{j+1}R_{p_{j}}} .

Now we can use Corollary 0.3 to see that

e_{m_{j}}((M_{φ_{j+1}})_{p_{j}}) ≤ e_{m}(M_{φ_{j+1}}) = e_{m}(P_{φ_{j}}/P_{φ_{j+1}}).

Summing up over the φ_{j+1} ∈ [φ̃_{j+2}, φ̃_{j+2} + t_{j+1}y_{j+1}], which are finite in number, then over the φ_{j+2} ∈ [φ̃_{j+3}, φ̃_{j+3} + t_{j+2}y_{j+2}] and so on, we obtain the statement. □ Remark If the centers of ν_{j} and ν_{j+1} coincide, the variable y_{j} does not appear in the bounding expression; this corresponds to the fact that there are only finitely many elements of ν_{j}(R \ {0}) in the union of all intervals [φ̃_{j+1}, φ̃_{j+1} + t_{j}y_{j}] (see [ZS], loc.cit.).

Corollary 0.6. For each ε > 0, the number of elements of the semigroup ν(R \ {0}) contained in the union of all the intervals appearing in the theorem is bounded, for y_{j} large enough, by

(1 + ε)\frac{∏_{i=0}^{n} e_{m_{i}}(R_{p_{i}}/p_{i+1}R_{p_{i}})}{∏_{i=1}^{n}(dimR_{p_{i}}/p_{i+1}R_{p_{i}})!} ∏_{i=1}^{n} y_{i}^{dimR_{p_{i}}/p_{i+1}R_{p_{i}}} ,

with the convention that p_{0} = m.

Indeed, for the valuation ν itself, the multiplicity of P_{φ}/P_{φ}^{+} is equal to its length, because it is a finite dimensional vector space over k = R/m, and it is nonzero if and only if φ is in ν(R \ {0}). □

References

[1] N. Bourbaki, Commutative Algebra, Chap. 8.

[2] J. Lipman,

[3] M. Lejeune-Jalabert and B. Teissier, Normal cones and sheaves of relative jets, Compositio Math. 28 (1974), 305–331.

[4] B. Teissier, Valuations, deformations, and toric geometry, Proceedings of the Saskatoon Conference and Workshop on valuation theory (second volume), F-V. Kuhlmann, S. Kuhlmann, M. Marshall, editors, Fields Institute Communications, 33, 2003, 361-459.

[5] O. Zariski, P. Samuel Commutative Algebra, Vol. II, Springer-Verlag (1960).