RESOLUTION OF SINGULARITIES OF GERMS IN CHARACTERISTIC POSITIVE ASSOCIATED WITH VALUATION RINGS OF ITERATED DIVISOR TYPE

TOHSUKE URABE

Abstract. In this paper we show that any hypersurface singularities of germs of varieties in positive characteristic can be resolved by iterated monoidal transformations with centers in smooth subvarieties, if we have a valuation ring of iterated divisor type associated with the germ. Besides, we introduce fundamental concepts for the study of resolution of singularities of germs such as space germs, iterated analytic monoidal transforms with a normal crossing, Weierstrass representations, reduction sequences, and so forth.

1. Introduction

In this paper we consider the problem of resolution of singularities in the case of positive characteristic. Since Hironaka’s success in resolution of singularities in characteristic zero (Hironaka [5].), the solution of this problem has long been expected. However, it is still open.

This is perhaps because the problem is hard by nature.

Therefore, we consider the problem in the category of germs of varieties. Besides, we assume that the singularity is a hypersurface.

Recently very clear exposition was given in Appendix of Abhyankar [2] for resolution of hypersurface singularities in characteristic zero. I have noticed that most of Abhyankar’s ideas are effective even in characteristic positive. I supply the missing part to his ideas, and develop the theory of positive characteristic in this paper.

The main result is the following: The algebraically closed ground field of characteristic positive is denoted by k. By B and B′ we denote rings isomorphic to the formal power series ring k[[x_{0}, x_{1}, …, x_{r}]] over k with r + 1 variables x_{0}, x_{1}, …, x_{r} for some non-negative integer r.

Theorem 1.1. Let (B, W) be a space germ over k. Assume that the valuation ring W is of iterated divisor type. Let f_{1}, f_{2}, …, f_{ℓ} be a finite number of arbitrary non-zero elements in the ring B. Then, there exists an iterated analytic monoidal transform (B′, W′) of (B, W) such that the product f′_{1}f′_{2} ⋯ f′_{ℓ} ∈ B′ has a normal crossing, and the ideal in B′ generated by any non-empty subset of {f′_{1}, f′_{2}, …, f′_{ℓ}} is a principal ideal, where for every 1 ≤ i ≤ ℓ, f′_{i} ∈ B′ denotes the image of f_{i} ∈ B by the associated homomorphism B → B′.

Terminologies will be defined in this paper. I can give only brief explanation for them here. By W we denote a valuation ring of the quotient field of B. We assume that W dominates B and dim_{k} W = 0. The pair (B, W) is called a space germ over k. The valuation ring W is said to be of iterated divisor type, if dim B = rank W holds.

1991 Mathematics Subject Classification. Primary 14B05; Secondary 32S05, 13H99.

Key words and phrases. singularity, resolution of singularities, monoidal transformation, normal crossing, Weierstrass preparation theorem, characteristic positive, valuation.

Let σ : Σ → Spec(B) be a blowing-up of the scheme Spec(B) with center in some non-singular closed subscheme. It is known that the morphism Spec(W) → Spec(B) induced by the inclusion B ⊂ W of rings has a unique lift Spec(W) → Σ such that the composition Spec(W) → Σ → Spec(B) coincides with Spec(W) → Spec(B). Let s ∈ Σ be the image of the unique closed point of Spec(W) by Spec(W) → Σ, B_{1} be the completion of the local ring O_{Σ,s}, and W_{1} be a valuation ring of the quotient field of B_{1} such that dim_{k} W_{1} = 0 and W_{1} dominates W. The pair (B_{1}, W_{1}) is called an elementary monoidal transform of (B, W). Iterating this procedure, we define an iterated analytic monoidal transform.

Now, applying a blowing-up to a germ, the obtained exceptional variety is a global object. To treat resolution of singularities in the category of germs, we have to choose a closed point lying over the origin on the exceptional variety at random. The valuation ring W in the second term of a space germ plays the role of being who chooses a closed point on the exceptional variety at random.

Our formulation of space germs is similar to Zariski’s uniformization. (Abhyankar [1], Zariski [9], [10].) However, it differs from uniformization in that the ring B is complete. There are two reasons. First, treating complete rings, we can make it clear that we treat the problem locally in the category of germs. Second, we apply the Weierstrass preparation theorem in the essential part of our theory. If the ring is not complete, Weierstrass’ theorem does not hold.

The above theorem solves only a small part of the problem. However, it is the first result on resolution of singularities in positive characteristic treating arbitrary dimensions, using monoidal transformations with smooth centers and giving concrete descriptions for the choice of centers of monoidal transformations. I hope that this paper together with several related results in a few years (Kuhlmann [6], Spivakovsky [8].) will play a key role to open the door toward the final solution.

Throughout this paper we fix the ground field k. We assume that k is an algebraically closed field and the characteristic number p of k is positive. The prime number p is used to denote the characteristic number of k.

Let Γ be a totally ordered abelian group. Often in this paper we associate a special element ∞ called the infinity with Γ. We assume that the element ∞ satisfies the following two conditions:

1. ∞ ≥ ∞, ∞ + ∞ = ∞, and n∞ = ∞n = ∞ for every positive integer n.

2. For every element γ ∈ Γ γ ≠ ∞, γ < ∞ and γ + ∞ = ∞ + γ = ∞.

Any positive integer m can be written uniquely in the form m = lq where l is an integer prime to p and q = p^{δ} is a power of p. The exponent δ is a non-negative integer. The pair (l, q) is called the p-decomposition of m. For simplicity we say that m = lq is the p-decomposition of m, or that m = lp^{δ} is the p-decomposition of m.

By ℤ, ℚ and ℝ we denote the ring of integers, the rational number field and the real number field respectively. By ℤ_{0} we denote the set of non-negative integers.

The number of elements in a finite set X is denoted by ♯ X.

Section 1 is the introduction. Sections 2, 3, 4 are preliminary parts. We collect results on commutative rings, totally ordered abelian groups and valuations in respective sections. In Section 5 we give definitions for fundamental objects in our formulation such as a space germ, a framed space germ, a basic space germ, an elementary monoidal transform, an iterated elementary monoidal transform (an IEMT, in short), an iterated analytic monoidal transform (an IAMT, in short), and so forth. The concept of Weierstrass representations is introduced in Section 6. This concept plays the central role in our theory. The Abhyankar condition, the strong Abhyankar condition and the condition “reductive” are defined for Weierstrass representations. The condition “not reductive” is essential in characteristic positive. Let G_{1}, G_{2}, …, G_{m} ∈ Z = k[[x_{1}, x_{2}, …, x_{n}]] and g = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j}. We define w(g, t) = min{ord(G_{j})/j | 1 ≤ j ≤ m} ∈ ℚ ∪ {∞}. Note that for any H ∈ Z there are elements G′_{1}, G′_{2}, …, G′_{m} ∈ Z with g = (t + H)^{m} + ∑_{j=1}^{m} G′_{j}(t + H)^{m−j}. Thus w(g, t + H) = min{ord(G′_{j})/j | 1 ≤ j ≤ m} is defined. The pair (g, t) is said to be reductive, if there is an element H ∈ Z with w(g, t) < w(g, t + H). In Section 7 we show that if a Weierstrass representation satisfies both the Abhyankar condition and the condition “not reductive”, then the singularity can be resolved even in characteristic positive. By this result our goal becomes to modify a given Weierstrass representation to satisfy both conditions. In Section 8 properties of the collection C_{0} of all framed space germs of iterated divisor type are studied. An effective tool called a reduction sequence is introduced in Section 9 to treat both the Abhyankar condition and the condition “not reductive” simultaneously. Central idea of a reduction sequence is like the following: Let Z_{0} = Z. We consider a pair (g, t_{0}) = (g, t) as above. Assume that we have already obtained rings Z_{0}, Z_{1}, …, Z_{ν−1} and pairs (g, t_{0}), (g, t_{1}), …, (g, t_{ν−1}). If w(g, t_{ν−1}) = ∞, then g = t_{ν−1}^{m} is a function with a normal crossing. We can treminate our procedure. We consider the case w(g, t_{ν−1}) < ∞ below. By the induction hypothesis on dimensions there exists an IAMT Z_{ν} of Z_{ν−1} such that the pair (g, t_{ν−1}) satisfies the Abhyankar condition, if we regard the coefficient function sequence G_{ν−1,1}, G_{ν−1,2}, …, G_{ν−1,m} ∈ Z_{ν−1} of g as elements in Z_{ν} by the associated homomorphism Z_{ν−1} → Z_{ν}. If (g, t_{ν−1}) is not reductive when we regard G_{ν−1,j}’s as elements in Z_{ν}, then we terminate our procedure. Otherwise, we choose an element H_{ν} ∈ Z_{ν} with w(g, t_{ν−1}) < w(g, t_{ν−1} + H_{ν}) and put (g, t_{ν}) = (g, t_{ν−1} + H_{ν}). The sequence obtained by repeating this procedure is called a reduction sequence. We will see that any member G_{ν,j} of the coefficient function sequence of (g, t_{ν}) has a special form. (By this form it seems intuitively that lim_{ν→∞} G_{ν,j} = 0 holds for every 1 ≤ j ≤ m. However, it is difficult to give a meaning to this limit. We apply the valuation theory for this purpose.) In Section 10 we show essential results in the ring theory for the final goal. These results are necessary to avoid purely inseparable extensions of the function field. In the final section Section 11 we show that it is possible to modify a given Weierstrass representation to satisfy both the Abhyankar condition and the condition “not reductive”, if we assume that the corresponding valuation ring is of iterated divisor type. In other words, we show that there exists a reduction sequence with finite length under the assumption

After I sent the first draft of this paper to the AMS preprint server, Joseph Lipman kindly informed me that Section 4 of Bierstone and Milman [3] (and Section 3 of Bierstone and Milman [4]) contains the same results as ones in Appendix of Abhyankar [2], and I received envelope containing a lot of results by Bierstone and Milman from Bierstone. By Lemma 4.7 in Bierstone and Milman [3] (or Lemma 3.12 in Bierstone and Milman [4]) I could improve some descriptions in Section 5 and Section 6 of my manuscript. Because of difference on how to choose centers of monoidal transformations among in [2], in [3] and in [4], I take Abhyankar [2] as the basis of our theory in this paper. (The methods in [2] and [3] are simpler, but they choose centers even outside the zero-locus of the function under consideration. Two methods are slightly different. In the third method in [4] they choose centers inside the singular sets of the zero-locus of the function. The third can be regarded as the improvement of the second one in [3].)

I express thanks to Max-Planck-Institut für Mathematik for warm hospitality. I thank also Herwig Hauser for hospitality and interesting discussions we held at Innsbruck University.

2. Ring theory

In this section we review several important points in the ring theory and explain non-standard concepts.

By a ring we mean a commutative ring with the identity. We use standard terminologies in the commutative ring theory. ( Matsumura [7]. Zariski-Samuel [11].) We shall denote the identity by the symbol 1, while the element zero will be denoted by 0. A ring homomorphism φ : A → B from a ring A to a ring B is always assumed to satisfy φ(1_{A}) = 1_{B}, where 1_{A} and 1_{B} denote the identities of A and B respectively. When a ring homomorphism φ : A → B is given, we say that B is an A-algebra.

Let A be a ring. By A^{∗} we denote the set of invertible elements of A, i.e.,

A^{∗} = {a ∈ A | There exists an element b ∈ A with ab = 1}.

Note that A^{∗} is an abelian group with respect to the multiplication. Note also that E^{∗} = E − {0} for a field E.

A ring with a unique maximal ideal is called a local ring, while a ring with a finite number of maximal ideals is called a semi-local ring. For a local ring A by M(A) we denote the maximal ideal of A. It is easy to see that a local ring A is the disjoint union of A^{∗} and M(A).

Let A and B be local rings and φ : A → B be a ring homomorphism. We can check that φ(M(A)) ⊂ M(B) if and only if φ^{−1}(M(B)) = M(A). We say that φ is a local homomorphism, or it is a homomorphism of local rings, if these equivalent conditions are satisfied.

Assume that local rings A and B are subrings of a field E. We say that B dominates A, if A ⊂ B and if M(A) = M(B) ∩ A. The ring B dominates A, if and only if the inclusion homomorphism A ↪ B is defined and the inclusion homomorphism is local.

A subset S of a ring A is said to be a multiplicative subset, if 1 ∈ S and 0 ∉ S and if the product of any two elements in S is again in S.

Any A-submodule 𝔦 of a ring A with 𝔦 ≠ A is called an ideal of A. An ideal 𝔭 of a ring A is a prime ideal, if the complement A − 𝔭 is a multiplicative subset. The set of prime ideals of a ring A is denoted by Spec(A). Any homomorphism φ : A → B of rings induces a map φ^{∗} : Spec(B) → Spec(A) in the opposite direction. The image of 𝔮 ∈ Spec(B) is defined to be φ^{−1}(𝔮) ∈ Spec(A). Note that φ^{−1}(𝔮) = 𝔮 ∩ A if φ is injective. Obviously id_{A}^{∗} = id_{Spec(A)} and (ψφ)^{∗} = φ^{∗}ψ^{∗}, where id_{X} denotes the identity map of a set X, and φ : A → B, ψ : B → C are homomorphisms of rings.

For any multiplicative subset S of a ring A and for any A-module M, we can define the localization S^{−1}M of M with respect to S. (Matsumura [7] p.6–p.7 (1.F).) If S = A − 𝔭 for some prime ideal 𝔭, we write M_{𝔭} for S^{−1}M.

A finite sequence 𝔭_{0}, 𝔭_{1}, …, 𝔭_{n} of n + 1 prime ideals of a ring A is called a prime chain of length n, if 𝔭_{0} ⊂ 𝔭_{1} ⊂ ⋯ ⊂ 𝔭_{n} and if any two of 𝔭_{0}, 𝔭_{1}, …, 𝔭_{n} are different. The prime ideal 𝔭_{0} is called the bottom member of the chain, while 𝔭_{n} is called the top member of the chain.

Let 𝔭 ∈ Spec(A) be a prime ideal. The supremum of the length of prime chains with the top member 𝔭 is called the height of a prime ideal 𝔭. Let 𝔦 be a ideal of A. The height of the ideal 𝔦 is defined to be the minimum of the heights of prime ideals containing 𝔦, and it is denoted by ht(𝔦). In case where 𝔦 is prime, two definitions of height of 𝔦 coincide.

The dimension of a ring A is defined to be the supremum of heights of the prime ideals in A, and is denoted by dim A. If A is a noetherian semi-local ring, dim A is finite. (Matsumura [7], p.74, (12.E), Lemma 1.) Since A_{𝔭} is a local ring for any 𝔭 ∈ Spec(A), and since ht(𝔭) = dim A_{𝔭} by definition, ht(𝔦) is a finite integer for any ideal 𝔦 of a noetherian ring A.

A prime chain 𝔭_{0}, 𝔭_{1}, …, 𝔭_{n} is said to be maximal, if there is no prime chain 𝔭′_{0}, 𝔭′_{1}, …, 𝔭′_{m} such that 𝔭_{0} = 𝔭′_{0}, 𝔭_{n} = 𝔭′_{m}, n < m and {𝔭_{0}, 𝔭_{1}, …, 𝔭_{n}} ⊂ {𝔭′_{0}, 𝔭′_{1}, …, 𝔭′_{m}}. For any pair of ideals 𝔦, 𝔧 with 𝔦 ⊃ 𝔧, the height of the ideal 𝔦/𝔧 in the residue ring A/𝔧 is denoted by ht(𝔦/𝔧).

A ring A is said to be catenary, if for each pair of prime ideals 𝔭, 𝔮 with 𝔭 ⊃ 𝔮, the length of any maximal chain of prime ideals with top member 𝔭 and with the bottom member 𝔮 is finite, and if it does not depend on the choice of the maximal chain satisfying the conditions.

A ring without a zero divisor is called an integral domain or a domain in short. A noetherian domain A is catenary, if and only if, for any pair of prime ideals 𝔭,

𝔮 with 𝔭 ⊃ 𝔮, we have ht(𝔭) = ht(𝔮) + ht(𝔭/𝔮).

A ring A is said to be universally catenary, if A is noetherian and if every finitely generated A-algebra is catenary. (Matsumura [7], p.84, (14.B))

Let B be a domain, and A be a subdomain. The quotient field L of B contains the quotient field K of A. The transcendence degree tr.deg_{K}L of L over K is defined. By putting tr.deg_{A}B = tr.deg_{K}L we define the transcendence degree tr.deg_{A}B of B over A.

The following theorem plays an important role in this paper.

Theorem 2.1. Let B be a domain and A be a subdomain. We assume that A is noetherian and B is a finitely generated A-algebra. Let 𝔮 ∈ Spec(B) be a prime ideal in B. Then we have

ht(𝔮) ≤ ht(𝔮 ∩ A) + tr.deg_{A}B − tr.deg_{(A/A∩𝔮)}(B/𝔮).

And the equality holds, if A is universally catenary, or if B is a polynomial ring A[x_{1}, x_{2}, …, x_{n}] over A with a finite number of algebraically independent elements x_{1}, x_{2}, …, x_{n}.

Proof. See Matsumura [7], p.84–p.86, (14.C), Theorem 23, and Zariski-Samuel [11], Vol. II, p.321–p.329, Appendix 1.

We have a class of rings called excellent rings. The exact characterization of excellent rings is complicated. (Matsumura [7], p.259, (34.A).) We will apply only the properties of excellent rings in the theorem below.

Theorem 2.2. 1. Any ring isomorphic to an excellent ring is excellent. 2. An excellent ring is noetherian.

3. A noetherian complete semi-local ring is excellent. In particular, a field is excellent.

4. If a ring A is excellent, then for any finite number of algebraically independent elements x_{1}, x_{2}, …, x_{n} over A and for any ideal 𝔦 in the polynomial ring A[x_{1}, x_{2}, …, x_{n}] over A, the residue ring A[x_{1}, x_{2}, …, x_{n}]/𝔦 is excellent.

5. If a ring A is excellent, then for any multiplicative subset S of A the localization S^{−1}A of A with respect to S is excellent.

6. An excellent ring is universally catenary.

7. Let A be an excellent domain with the quotient field K. Let L be a field containing K. If L is a finite algebraic extension of K, then the integral closure of A in L is a finitely generated A-module.

8. The completion of an excellent local domain is reduced.

9. The completion of an excellent normal local domain is a normal local domain.

10. An excellent ring A is a G-ring. In other words, A has the following property: for any two prime ideals 𝔭, 𝔮 of A with 𝔭 ⊃ 𝔮, for any field L such that L contains the quotient field of A/𝔮 and L is finitely generated over the quotient field of A/𝔮 and for any prime ideal 𝔔 of the ring Â_{𝔭} ⊗_{A} L the localization (Â_{𝔭} ⊗_{A} L)_{𝔔} is a regular local ring, where Â_{𝔭} denotes the completion of the local ring A_{𝔭}.

Proof. See Matsumura [7], p.259–p.260, (34.A) and (34.B).

By ℤ_{0} we denote the set of non-negative integers.

Let A be a ring. The binomial coefficient \binom{r}{j} is the mapping (r, j) ∈ ℤ_{0}^{2} ↦

\binom{r}{j} ∈ A satisfying the following two conditions:

1. If 0 ≤ r < j, then \binom{r}{j} = 0.

2. For every non-negative integer r the equality

(t + 1)^{r} = ∑_{j=0}^{r} \binom{r}{j}t^{j}

holds in the ring A[t], where t is an algebraically independent element over A.

Lemma 2.3. 1.

\binom{r}{0} = \binom{r}{r} = 1.

2.

\binom{r + 1}{j + 1} = \binom{r}{j} + \binom{r}{j + 1}.

3. If 0 ≤ j ≤ r, then

\binom{r}{j} = \binom{r}{r − j}.

4.

\binom{r}{1} = r,

where the image of r ∈ ℤ by the canonical map ℤ → A is denoted by the same symbol r ∈ A in the right-hand side.

5.

\binom{r + s}{j} = ∑_{i=0}^{j} \binom{r}{j − i}\binom{s}{j}.

6. Let B be a ring containing A. For every x, y ∈ B

(x + y)^{r} = ∑_{j=0}^{r} \binom{r}{j}x^{r−j}y^{j}.

7. If 0 ≤ a ≤ b and a ≤ r, then

\binom{b}{a}\binom{r}{b} = \binom{r}{a}\binom{r − a}{b − a}.

Proof. Let t, u be algebraically independent elements over A. Computing (tu+u+ 1)^{r} in two different ways we obtain claim 7.

Lemma 2.4. Assume that a ring A contains a field of characteristic p. 1. For 1 ≤ j ≤ p − 1

\binom{r}{j} = \frac{r(r − 1) ⋯ (r − j + 1)}{j!}.

2. Let δ be a non-negative integer. If 1 ≤ j ≤ p^{δ} − 1, then

\binom{p^{δ}}{j} = 0.

Let t be an algebraically independent element over A.

(t + 1)^{p^{δ}} = t^{p^{δ}} + 1.

3. Let δ and l be non-negative integers. If a non-negative integer j is not a multiple of p^{δ}, then

\binom{lp^{δ}}{j} = 0.

If there is a non-negative integer i with j = ip^{δ}, then

\binom{lp^{δ}}{j} = \binom{lp^{δ}}{ip^{δ}} = \binom{l}{i}.

4. Let δ, a and b be non-negative integers, and c and d be integers with 0 ≤ c ≤ p^{δ} − 1, 0 ≤ d ≤ p^{δ} − 1.

\binom{ap^{δ} + c}{bp^{δ} + d} = \binom{a}{b}\binom{c}{d}.

Proof. Not difficult.

Definition 2.5. Let A be a k-algebra and δ be a non-negative integer. The map

F_{δ} : A → A

defined by F_{δ}(a) = a^{p^{δ}} for a ∈ A is called the Frobenius homomorphism of degree δ over A. The Frobenius homomorphism of degree 1 is called the Frobenius homomorphism and is denoted by F.

Lemma 2.6. 1. The Frobenius homomorphism F_{δ} : A → A of degree δ is a ring

homomorphism, but is not a homomorphism of k-algebras. F_{δ} = F^{δ}.

2. For every homomorphism φ : A → B of k-algebras we have F_{δ}φ = φF_{δ}.

3. If A is a domain, then F_{δ} is injective.

4. If A is a topological ring with 𝔦-adic topology defined by an ideal 𝔦, then F_{δ} is continuous.

Let n be a non-negative integer, and x_{0}, x_{1}, …, x_{n} be n + 1 of variables. We consider the formal power series ring k[[x]] = k[[x_{0}, x_{1}, …, x_{n}]] over k. Any element f ∈ k[[x]] can be written uniquely in the form

f = ∑_{α∈ℤ_{0}^{n+1}} f_{α}x^{α},

where α = (α_{0}, α_{1}, …, α_{n}) ∈ ℤ_{0}^{n+1}, α_{0}, α_{1}, …, α_{n} are non-negative integers, f_{α} ∈ k for any α ∈ ℤ_{0}^{n+1}, and x^{α} = x_{0}^{α_{0}}x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}.

Lemma 2.7. 1. The ring k[[x]] is a noetherian complete regular local ring of dimension n + 1, and is a unique factorization domain (UFD in short.) with the unique maximal ideal M(k[[x]]) = {f ∈ k[[x]] | f_{(0,0,... ,0)} = 0}. The set {x_{0}, x_{1}, …, x_{n}} is a parameter system of k[[x]].

2. Let y_{0}, y_{1}, …, y_{n} be n + 1 elements in k[[x]]. The following three conditions are equivalent:

(a) The set {y_{0}, y_{1}, …, y_{n}} is a parameter system of k[[x]].

(b) There exists a unique continuous isomorphism φ : k[[x]] → k[[x]] of kalgebras satisfying φ(x_{i}) = y_{i} for every 0 ≤ i ≤ n.

(c) y_{0}, y_{1}, …, y_{n} ∈ M(k[[x]]) and det(∂y_{i}/∂x_{j}) ∉ M(k[[x]]).

Definition 2.8. 1. Let Q = {x_{0}, x_{1}, …, x_{n}} be a parameter system of the ring k[[x]] and f ∈ k[[x]]. We say that f has a normal crossing with respect to Q, if there are non-negative integers β_{0}, β_{1}, …, β_{n} with

\frac{f}{x_{0}^{β_{0}}x_{1}^{β_{1}} ⋯ x_{n}^{β_{n}}} ∈ k[[x]]^{∗}.

2. An element f ∈ k[[x]] is said to have a normal crossing, if there is a parameter system Q of k[[x]] such that f has a normal crossing with respect to Q.

Lemma 2.9. Let Q be a parameter system of k[[x]], and f ∈ k[[x]], g ∈ k[[x]].

1. the following two conditions are equivalent:

(a) The product fg has a normal crossing with respect to Q.

(b) Both of f and g have a normal crossing with respect to Q.

2. If the product fg has a normal crossing, then both of f and g have a normal crossing.

Proof. It follows from the uniqueness of the irreducible decomposition.

Let w = (w_{0}, w_{1}, …, w_{n}) ∈ ℤ_{0}^{n+1}, and Q = {x_{0}, x_{1}, …, x_{n}} be a parameter system of k[[x]]. We will define a map

ord_{w} : k[[x]] − {0} → ℤ_{0}

associated with w and Q. The value ord_{w}(f) ∈ ℤ_{0} for f ∈ k[[x]] with f ≠ 0 is called the order of f with weight w. Let f = ∑_{α∈ℤ_{0}^{n+1}} f_{α}x^{α} ∈ k[[x]] be an element with f ≠ 0. Let

ex(f) = {α ∈ ℤ_{0}^{n+1} | f_{α} ≠ 0}.

Since f ≠ 0, the set ex(f) is not empty. We define

ord_{w}(f) = min{∑_{i=0}^{n} w_{i}α_{i} | (α_{0}, α_{1}, …, α_{n}) ∈ ex(f)}.

Lemma 2.10. 1. If f, g ∈ k[[x]] − {0}, then ord_{w}(fg) = ord_{w}(f) + ord_{w}(g).

2. If f, g ∈ k[[x]] − {0} and if f + g ∈ k[[x]] − {0}, then ord_{w}(f + g) ≥ min{ord_{w}(f), ord_{w}(g)}.

3. Assume that w_{i} = 0 for 0 ≤ i ≤ m and w_{i} > 0 for m + 1 ≤ i ≤ n. We have ord_{w}(f) = 0 if and only if f_{(α_{0},α_{1},... ,α_{m},0,0,... ,0)} ≠ 0 for some m + 1 non-negative integers α_{0}, α_{1}, …, α_{m}.

4. Assume that w_{i} > for every 0 ≤ i ≤ n. We have ord_{w}(f) = 0 if and only if f ∈ k[[x]]^{∗}.

Let E denote the quotient field of k[[x]]. We can define an extended map

ord_{w} : E^{∗} → ℤ

by putting ord_{w}(h) = ord_{w}(f) − ord_{w}(g), where h ∈ E^{∗} and f, g ∈ k[[x]] − {0} are elements with h = f/g. By Lemma 2.10.1 ord_{w}(h) does not depend on the choice of elements f, g ∈ k[[x]] − {0} satisfying h = f/g.

Besides, by putting ord_{w}(0) = ∞ we have the extension

ord_{w} : E → ℤ ∪ {∞}.

If w_{i} = 1 for every 0 ≤ i ≤ n, we write ord instead of ord_{w}. The value ord(f) is called the order of f.

Definition 2.11. Let E be a field and Γ be a totally ordered abelian group. A map v : E → Γ∪{∞} is called valuation of E, if it satisfies the following conditions:

1. v(h) = ∞ if and only if h = 0.

2. For every f, g ∈ E v(fg) = v(f) + v(g).

3. For every f, g ∈ E v(f + g) ≥ min{v(f), v(g)}. The group Γ is called the value group of the valuation v.

Lemma 2.12. 1. Let E be the quotient field of k[[x]]. The map ord_{w} : E → ℤ ∪ {∞} associated with an element w ∈ ℤ_{0}^{n+1} and a parameter system Q is a valuation of E with the value group ℤ.

2. ord_{w}(f) ≥ 0 for every f ∈ k[[x]].

3. Assume that w_{i} > 0 for every 0 ≤ i ≤ n. If f ∈ k[[x]] and ord_{w}(f) = 0, then f ∈ k[[x]]^{∗}.

4. Assume w_{i} = 1 for every 0 ≤ i ≤ n. Then, ord_{w} = ord does not depend on the choice of the parameter system Q.

Proof. We consider claim 4. Under the assumption we have ord_{w}(f) = max{ν | f ∈ M(k[[x]])^{ν}}, which shows the claim.

Definition 2.13. Let f ∈ k[[x]] be an element with f ≠ 0. Let f = ug_{1}^{β_{1}}g_{2}^{β_{2}} ⋯ g_{r}^{β_{r}} be the irreducible decomposition of f, where g_{1}, g_{2}, …, g_{r} ∈ k[[x]] are irreducible elements such that any two of them are coprime, u ∈ k[[x]]^{∗} and β_{1}, β_{2}, …, β_{r} are positive integers. We assume that ord(g_{i}) > 1 for 1 ≤ i ≤ s and ord(g_{i}) = 1 for s + 1 ≤ i ≤ r.

1. A triplet (v, h_{s}, h_{n}) satisfying the following conditions are called the singular non-singular decomposition of f.

(a) f = vh_{s}h_{n}. v ∈ k[[x]]^{∗}. h_{s} ∈ k[[x]]. h_{n} ∈ k[[x]].

(b)

\frac{h_{s}}{g_{1}^{β_{1}}g_{2}^{β_{2}} ⋯ g_{s}^{β_{s}}} ∈ k[[x]]^{∗}.

(c)

\frac{h_{n}}{g_{s+1}^{β_{s+1}}g_{s+2}^{β_{s+2}} ⋯ g_{r}^{β_{r}}} ∈ k[[x]]^{∗}.

2. Let (v, h_{s}, h_{n}) be the singular non-singular decomposition of f. We call ord(h_{s}) the singular order of f, and denote it by sord(f). We call ord(h_{n}) the non-singular order of f, and denote it by nord(f).

Lemma 2.14. 1. The singular order sord(f) and the non-singular order nord(f) of an element f ∈ k[[x]] − {0} are well-defined. They do not depend on the choice of the singular non-singular decomposition (v, h_{s}, h_{n}) of f in the definition.

2. sord(f) and nord(f) are non-negative integers. ord(f) = sord(f) + nord(f). sord(f) ≠ 1. sord(fg) = sord(f) + sord(g). nord(fg) = nord(f) + nord(g).

Let w = (w_{0}, w_{1}, …, w_{n}) ∈ ℤ_{0}^{n+1}, and Q = {x_{0}, x_{1}, …, x_{n}} be a parameter system of k[[x]].

Let f = ∑_{α∈ℤ_{0}^{n+1}} f_{α}x^{α} ∈ k[[x]] be an element with f ≠ 0. Let iex(f) = {α ∈ ex(f) | ∑_{i=0}^{n} w_{i}α_{i} = ord_{w}(f)}. Putting

in_{w}(f) = ∑_{α∈iex(f)} f_{α}x^{α},

we call in_{w}(f) the w-initial polynomial of f. For f = 0 ∈ k[[x]] we define in_{w}(0) = 0. We can check that in_{w}(fg) = in_{w}(f)in_{w}(g) for any f, g ∈ k[[x]]. Assume that w_{i} = 0 for 0 ≤ i ≤ m and w_{i} > 0 for m + 1 ≤ i ≤ n. One sees that

in_{w}(f) ∈ k[[x_{0}, x_{1}, …, x_{m}]][x_{m+1}, x_{m+2}, …, x_{n}].

We say that f is a w-homogeneous polynomial, if f = in_{w}(f). If w_{i} > 0 for every 0 ≤ i ≤ n, then any w-homogeneous polynomial contains only a finite number of terms, and it is a polynomial with coefficients in k.

Consider the case where w_{i} = 1 for every 0 ≤ i ≤ n in particular. In this case in_{w}(f) does not depend on the choice of the parameter system Q. We write in(f) instead of in_{w}(f), and call in(f) the initial polynomial of f. It has only a finite number of terms. Also in this case a w-homogeneous polynomial is called a homogeneous polynomial.

In the theorem below one of n + 1 variables plays a special role. Putting t = x_{0}, and y_{i} = x_{i} for 1 ≤ i ≤ n, we write

\begin{aligned}k[[t, y]] & = & k[[t, y_{1}, y_{2}, …, y_{n}]] & = & k[[x]], \\ k[[y]] & = & k[[y_{1}, y_{2}, …, y_{n}]]. \end{aligned}

Theorem 2.15 (Weierstrass division theorem). Let w = (w_{0}, w_{1}, …, w_{n}) ∈ ℤ_{0}^{n+1}. We assume that w_{i} > 0 for every 0 ≤ i ≤ n. Let f ∈ k[[t, y]]. Assume that in_{w}(f)(1, 0, 0, …, 0) ≠ 0, where in_{w}(f)(1, 0, 0, …, 0) denotes the result of substitution t = 1, y_{1} = y_{2} = ⋯ = y_{n} = 0 in the w-initial polynomial of f.

1. The quotient m = ord_{w}(f)/ord_{w}(t) is a non-negative integer.

2. For every h ∈ k[[t, y]] there exists a unique set of an element e ∈ k[[t, y]] and m numbered elements H_{1}, H_{2}, …, H_{m} ∈ k[[y]] satisfying

h = ef + ∑_{j=1}^{m} H_{j}t^{m−j}.

Proof. Not difficult.

Corollary 2.16 (Weierstrass preparation theorem). Let w = (w_{0}, w_{1}, …, w_{n}) ∈ ℤ_{0}^{n+1}. We assume that w_{i} > 0 for every 0 ≤ i ≤ n. Let f ∈ k[[t, y]]. Assume that in_{w}(f)(1, 0, 0, …, 0) ≠ 0. Let m = ord_{w}(f)/ord_{w}(t), which is a non-negative integer.

1. There exists a unique set of an element e ∈ k[[t, y]] and m numbered elements G_{1}, G_{2}, …, G_{m} ∈ k[[y]] satisfying

ef = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j}.

2. e ∈ k[[t, y]]^{∗}, and ord_{w}(G_{j}) ≥ j ord_{w}(t) for every 1 ≤ j ≤ m.

3. The element g = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j} ∈ k[[t, y]] belongs to the subring k[[y]][t], or the ring of polynomials with variable t with coefficients in k[[y]]. m ord_{w}(t) = ord_{w}(g).

Proof. It follows from the special case of Theorem 2.15 where h = t^{m}.

Lemma 2.17. Let f ∈ k[[x]] = k[[x_{0}, x_{1}, …, x_{n}]] be an element with f ≠ 0.

1. There exist n elements a_{1}, a_{2}, …, a_{n} ∈ k with in(f)(1, a_{1}, a_{2}, ⋯, a_{n}) ≠ 0, where in(f)(1, a_{1}, a_{2}, ⋯, a_{n}) denotes the result of substitution x_{0} = 1, x_{1} = a_{1}, x_{2} = a_{2}, …, x_{n} = a_{n} in in(f).

2. Assume that in(f)(1, a_{1}, a_{2}, ⋯, a_{n}) ≠ 0 for a_{1}, a_{2}, …, a_{n} ∈ k. Let t = x_{0} and y_{i} = x_{i} − a_{i}x_{0} for 1 ≤ i ≤ n. Then t, y_{1}, y_{2}, …, y_{n} is a parameter system of k[[x]], and the result of substitution t = 1, y_{1} = y_{2} = ⋯ = y_{n} = 0 in in(f) is not zero.

Proposition 2.18. Let f_{1}, f_{2}, …, f_{ℓ} ∈ k[[x]] = k[[x_{0}, x_{1}, …, x_{n}]] be a finite number of elements with f = f_{1}f_{2} ⋯ f_{ℓ} ≠ 0. Let m_{ν} = ord(f_{ν}) for 1 ≤ ν ≤ ℓ.

Let a_{1}, a_{2}, …, a_{n} ∈ k be elements with in(f)(1, a_{1}, a_{2}, …, a_{n}) ≠ 0.

Let t, y_{1}, y_{2}, …, y_{n} be the parameter system of k[[x]] satisfying t = x_{0} and y_{i} = x_{i} − a_{i}x_{0} for 1 ≤ i ≤ n. We denote k[[t, y]] = k[[t, y_{1}, y_{2}, …, y_{n}]] = k[[x]] and k[[y]] = k[[y_{1}, y_{2}, …, y_{n}]]. For every 1 ≤ ν ≤ ℓ the following 1–3 hold:

1. The result of substitution t = 1, y_{1} = y_{2} = ⋯ = y_{n} = 0 in the initial polynomial of f_{ν} is not equal to 0.

2. There exists a unique set of an element e_{ν} ∈ k[[t, y]] and m_{ν} numbered elements G_{ν,1}, G_{ν,2}, …, G_{ν,m_{ν}} ∈ k[[y]] satisfying e_{ν}f_{ν} = t^{m_{ν}} + ∑_{j=1}^{m_{ν}} G_{ν,j}t^{m_{ν}−j}.

3. e_{ν} ∈ k[[t, y]]^{∗}, and ord(G_{ν,j}) ≥ j for every 1 ≤ j ≤ m_{ν}.

Proof. It follows from Lemma 2.17 and Corollary 2.16.

Remark . Let f = ug_{1}^{β_{1}}g_{2}^{β_{2}} ⋯ g_{r}^{β_{r}} be the irreducible decomposition of above f. For every 1 ≤ ν ≤ r we define a k-linear map

λ_{ν} : k^{n+1} → k[[x]]/g_{ν}k[[x]]

by λ_{ν}(a_{0}, a_{1}, a_{2}, …, a_{n}) = ∑_{i=0}^{n} a_{i}(∂g_{ν}/∂x_{i}) mod g_{ν}k[[x]]. We can show λ_{ν} ≠ 0 for every 1 ≤ ν ≤ r. For each 1 ≤ ν ≤ r we choose a homogeneous polynomial h_{ν} of degree 1 whose zero-locus contains the kernel of λ_{ν}. Let Ψ = ∏_{ν=1}^{r} in(g_{ν})h_{ν}.

We can show that if elements a_{1}, a_{2}, …, a_{n} ∈ k satisfy a stronger condition Ψ(1, a_{1}, a_{2}, …, a_{n}) ≠ 0, then for every 1 ≤ ν ≤ r any irreducible component of the polynomial t^{m_{ν}} +∑_{j=1}^{m_{ν}} G_{ν,j}t^{m_{ν}−j} in the ring D[t] is a separable polynomial, where D denotes the quotient field of k[[y]].

Definition 2.19. 1. Assume that a parameter system Q of the ring k[[x]] is given. We call a pair (g, t) of elements g ∈ k[[x]] and t ∈ Q a semi-Weierstrass pair of k[[x]] under Q, if there exist a non-negative integer m and m elements G_{1}, G_{2}, …, G_{m} ∈ k[[y]] = k[[y_{1}, y_{2}, …, y_{n}]] satisfying g = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j}, where y_{1}, y_{2}, …, y_{n} are elements in Q except t.

2. A non-empty subset of a parameter system of a regular local ring is called a partial parameter system.

3. Assume that a partial parameter system P of the ring k[[x]] with ♯ P = dim k[[x]]−1 is given. We call a pair (g, t) of elements g ∈ k[[x]] and t ∈ k[[x]] a semi-Weierstrass pair of k[[x]] over P, if t ∉ P and the union P ∪ {t} is a parameter system of k[[x]], and if there exist a non-negative integer m and m elements G_{1}, G_{2}, …, G_{m} ∈ k[[y]] = k[[y_{1}, y_{2}, …, y_{n}]] satisfying g = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j}, where P = {y_{1}, y_{2}, …, y_{n}}.

4. The integer m is called the degree of (g, t), and is denoted by deg(g, t).

5. The m-tuple (G_{1}, G_{2}, …, G_{m}) of elements in k[[y]] is called the coefficient function sequence of (g, t).

6. If m = deg(g, t) = 0, we put w(g, t) = ∞. If m = deg(g, t) > 0, we put w(g, t) = min{ord(G_{j})/j | 1 ≤ j ≤ m}. We call w(g, t) ∈ ℚ ∪ {∞} the weight of (g, t).

7. For 1 ≤ j ≤ m = deg(g, t), let

\overline{G_{j}} = \{ \begin{matrix}0 & if ord(G_{j}) > j w(g, t) \\ in(G_{j}) & if ord(G_{j}) = j w(g, t). \end{matrix}

The polynomial t^{m} + ∑_{j=1}^{m} \overline{G_{j}}t^{m−j} is called the initial polynomial of (g, t), and is denoted by in(g, t).

8. If the coefficient function sequence (G_{1}, G_{2}, …, G_{m}) of a semi-Weierstrass pair (g, t) satisfies ord(G_{j}) ≥ j for every 1 ≤ j ≤ m, then we say that the pair (g, t) is a Weierstrass pair.

Lemma 2.20. Assume either that a parameter system Q of the ring k[[x]] is given, or that a partial parameter system P of the ring k[[x]] with ♯ P = dim k[[x]] − 1 is given. Let (g, t) be a semi-Weierstrass pair either under Q or over P.

1. The degree deg(g, t), the coefficient function sequence (G_{1}, G_{2}, …, G_{m}), the weight w(g, t) and the initial polynomial in(g, t) of (g, t) are uniquely defined depending on the pair (g, t).

2. deg(g, t) ≥ ord(g).

3. Let m = deg(g, t). The following three conditions are equivalent:

(a) w(g, t) = ∞.

(b) G_{j} = 0 for every 1 ≤ j ≤ m.

(c) in(g, t) = t^{m}.

4. The following three conditions are equivalent:

(a) The pair (g, t) is a Weierstrass pair.

(b) deg(g, t) = ord(g).

(c) w(g, t) ≥ 1.

Lemma 2.21. Let Q be a parameter system of the ring k[[x]], and (g, t) and (g′, t) be semi-Weierstrass pairs under Q with the common second term t ∈ Q. Let D denote the quotient field of k[[y]] = k[[y_{1}, y_{2}, …, y_{n}]] where y_{1}, y_{2}, …, y_{n} are elements in Q except t. Let g = g_{1}^{β_{1}}g_{2}^{β_{2}} ⋯ g_{r}^{β_{r}} be the irreducible decomposition in the ring D[t]. We assume that the coefficient of the term with the highest degree in g_{ν} ∈ D[t] is equal to 1 for every 1 ≤ ν ≤ r.

1. The pair (gg′, t) is a semi-Weierstrass pair. deg(gg′, t) = deg(g, t)+deg(g′, t). The pair (gg′, t) is a Weierstrass pair, if and only if, both of (g, t) and (g′, t) are Weierstrass pairs.

2. For every 1 ≤ ν ≤ r, g_{ν} ∈ k[[y]][t] ⊂ k[[x]].

3. Assume that (g, t) is a Weierstrass pair. For every 1 ≤ ν ≤ r, (g_{ν}, t) is a Weierstrass pair, and the decomposition g = g_{1}^{β_{1}}g_{2}^{β_{2}} ⋯ g_{r}^{β_{r}} gives the irreducible decomposition in the ring k[[x]]

Proof. The claim 2 follows from Lemma of Gauss. (Zariski-Samuel[11], Vol.I, p.32.)

3. Totally ordered abelian groups

In this section we consider totally ordered abelian groups, or abelian groups with total order compatible with the group structure. We prepare for treating the valuation theory in the next section.

Let Γ be an abelian group. For a subset Λ ⊂ Γ we write −Λ = {−γ | γ ∈ Λ}. We assume that total order ≥ is defined on Γ. Either α ≥ β or β ≥ α holds for any two elements α, β. We write also β ≤ α instead of α ≥ β. We denote α > β or β < α, if α ≥ β and α ≠ β.

If the following condition toag is satisfied, then we call Γ a totally ordered abelian group.

toag: If α, β, γ ∈ Γ and α ≥ β, then α + γ ≥ β + γ. By Γ we denote a totally ordered abelian group below.

Let Γ′ be another totally ordered abelian group, and φ : Γ → Γ′ be a map. If φ(α) ≥ φ(β) for every element α, β ∈ Γ with α ≥ β, then we say that φ preserves order. A homomorphism of abelian groups preserving order is called a homomorphism of totally ordered abelian groups.

Lemma 3.1. Let α, β be elements in Γ.

1. α ≥ β if and only if α − β ≥ 0.

2. α > 0 if and only if −α < 0.

3. If α ≥ 0 and β ≥ 0, then α + β ≥ 0.

4. A totally ordered abelian group Γ contains no elements of finite order except 0.

5. Let φ : Γ → Γ′ be a homomorphism of abelian groups. The homomorphism φ preserves order, if and only if, we have φ(γ) ≥ 0 for every element γ ∈ Γ with γ ≥ 0.

Lemma 3.2. Let G be an abelian group, and G_{+} be a subset such that G_{+} ∪ (−G_{+}) = G, G_{+} ∩ (−G_{+}) = {0}, and α + β ∈ G_{+} for every α ∈ G_{+} and β ∈ G_{+}.

Then, we can define total order ≥ on G such that α ≥ β for α, β ∈ G if α − β ∈ G_{+}. With this order ≥ the group G is a totally ordered abelian group, and G_{+} coincides with the set {γ ∈ G | γ ≥ 0} of non-negative elements.

Definition 3.3. 1. A subgroup Δ of Γ is said to be an isolated subgroup of Γ,

if for any elements α ∈ Δ, β ∈ Γ with α ≥ β ≥ 0 it follows that β ∈ Δ.

2. The cardinal number of the set of all isolated subgroups Δ of Γ with Δ ≠ Γ is called the rank of Γ, and is denoted by rank Γ.

3. The dimension of the ℚ-vector space Γ ⊗_{ℤ} ℚ is called the rational rank of Γ, and is denoted by r.rank Γ.

4. We say that Γ is archimedean, if for every element α, β ∈ Γ with α ≥ 0 and β > 0 we have a positive integer m with mβ ≥ α.

Lemma 3.4. 1. Let Δ and Δ′ be isolated subgroups of Γ. Either Δ ⊂ Δ′ or

Δ ⊃ Δ′ holds.

2. Let α ∈ Γ be an element with α > 0. The set of elements γ ∈ Γ such that there is a positive integer m with mα ≥ γ ≥ −mα is a non-zero isolated subgroup of Γ.

3. If r.rank Γ is finite, then

rank Γ ≤ r.rank Γ.

4. Let Δ be an isolated subgroup of Γ. We have unique total order on the residue group Γ/Δ such that Γ/Δ is a totally ordered abelian group and the canonical surjective homomorphism π : Γ → Γ/Δ preserves order.

rank Γ = rank Δ + rank Γ/Δ, r.rank Γ = r.rank Δ + r.rank Γ/Δ.

5. Let φ : Γ → Γ′ be a homomorphism of totally ordered abelian groups. The kernel Δ = φ^{−1}(0) is an isolated subgroup of Γ. The induced map Γ/Δ → Γ′ is an injective homomorphism of totally ordered abelian groups.

6. Let φ : Γ → Γ′ be an injective homomorphism of totally ordered abelian groups. The inverse image φ^{−1}(Δ′) = Δ′ ∩ Γ is an isolated subgroup of Γ for any isolated subgroup Δ′ of Γ′. For every isolated subgroup Δ of Γ there exists an isolated subgroup Δ′ of Γ′ with Δ = Δ′ ∩ Γ.

rank Γ ≤ rank Γ′, r.rank Γ ≤ r.rank Γ′.

Proof. 3. See Zariski-Samuel [11], Vol. II, p.50, Note.

4. Apply Lemma 3.2.

6. The existence of an isolated subgroup Δ′ of Γ′ with Δ = Δ′ ∩ Γ follows from Zorn’s lemma and 2 above.

Let Γ′ and Γ′′ be totally ordered abelian groups. We can define lexicographic order ≥ on the direct sum Γ′ + Γ′′. Let α′, β′ ∈ Γ′ and α′′, β′′ ∈ Γ′′. We define (α′, α′′) ≥ (β′, β′′) with respect to the lexicographic order, if either α′ > β′, or α′ = β′ and α′′ ≥ β′′.

Lemma 3.5. Let Δ be an isolated subgroup of Γ. Assume that there exists a homomorphism σ : Γ/Δ → Γ of abelian groups such that the composition πσ is the identity map of Γ/Δ, where π : Γ → Γ/Δ denotes the canonical surjective homomorphism. Then, Γ is isomorphic to the direct sum (Γ/Δ)+Δ with the lexicographic order as totally ordered abelian groups.

Lemma 3.6. The following three conditions are equivalent:

1. Γ is archimedean.

2. rank Γ ≤ 1.

3. There exists an injective homomorphism Γ → ℝ of totally ordered abelian groups.

Proof. The implication 1⇒3 follows from the theory of Dedekind cut. (Zariski-Samuel [11], Vol. II, p.45.)

Let n be a positive integer. We can define the lexicographic order ≥ on the abelian group ℤ^{n}. Let α = (α_{1}, α_{2}, …, α_{n}), β = (β_{1}, β_{2}, …, β_{n}) ∈ ℤ^{n}, where α_{1}, α_{2}, …, α_{n}, β_{1}, β_{2}, …, β_{n} are integers. If there is a number i with 1 ≤ i ≤ n such that α_{j} = β_{j} for 1 ≤ j < i and α_{i} > β_{i}, then we write α > β. We define α ≥ β, if α > β or α = β. We can check that with the lexicographic order the group ℤ^{n} is a totally ordered abelian group with rank ℤ^{n} = n. Let Δ_{i} = {(α_{1}, α_{2}, …, α_{n}) ∈ ℤ^{n} | α_{j} = 0 for 1 ≤ j ≤ i.} for 0 ≤ i ≤ n. We have

ℤ^{n} = Δ_{0} ⊃ Δ_{1} ⊃ ⋯ ⊃ Δ_{n−1} ⊃ Δ_{n} = {0},

and Δ_{i}’s are isolated subgroups of ℤ^{n}.

Unless it is mentioned otherwise, we assume below that ℤ^{n} has the lexicographic order.

Remark . We can define total order on ℤ^{n} essentially different from the lexicographic order, and with it ℤ^{n} is a totally ordered abelian group.

Let θ_{1} = 1, θ_{2}, θ_{3}, …, θ_{n} ∈ ℝ be real numbers linearly independent over ℚ. Giving the restriction of the order on ℝ, the subgroup Γ_{0} = ∑_{i=1}^{n} ℤθ_{i} of ℝ is a totally ordered abelian group. The group Γ_{0} is isomorphic to ℤ^{n} as abelian groups.

However, by Lemma 3.6 we have rank Γ_{0} = 1.

Thus, even if a totally ordered abelian group Γ is isomorphic to ℤ^{n} as abelian groups, Γ is not necessarily isomorphic to ℤ^{n} as totally ordered abelian groups.

4. Valuation theory

We treat the valuation theory in this section. (Zariski-Samuel [11], Vol.II, Chapter VI and Appendix 2.) By E we denote a fixed field.

We have defined the concept of valuations in Definition 2.11.

Lemma 4.1. Let v : E → Γ ∪ {∞} be a valuation.

1. v(f) = v(−f).

2. Let f_{1}, f_{2}, …, f_{r} ∈ E. We have

v(∑_{i=1}^{r} f_{i}) ≥ min{v(f_{i}) | 1 ≤ i ≤ r}.

If, moreover, v(f_{1}) < v(f_{i}) for 2 ≤ i ≤ r, then we have

v(∑_{i=1}^{r} f_{i}) = v(f_{1}).

3. Let R_{v} = {f ∈ E | v(f) ≥ 0}. The set R_{v} is a normal local subdomain of E with the maximal ideal M(R_{v}) = {f ∈ E | v(f) > 0}. If f ∈ E and f ∉ R_{v}, then 1/f ∈ R_{v}.

4. Let D be a subfield of E. The restricted map v | D : D → Γ ∪ {∞} is a valuation of D.

5. The image v(E^{∗}) is a subgroup of Γ.

Proof. We consider the normality in claim 3. Assume that an element f ∈ E is integral over R_{v}. We have a positive integer m and m elements a_{1}, a_{2}, …, a_{m} ∈ R_{v} with f^{m} + ∑_{i=1}^{m} a_{i}f^{m−i} = 0. By 1 and 2 above, one obtains mv(f) ≥ min{(m − i)v(f) + v(a_{i}) | 1 ≤ i ≤ m}. Thus, for some integer i with 1 ≤ i ≤ m we have mv(f) ≥ (m−i)v(f)+v(a_{i}) ≥ (m−i)v(f), which implies v(f) ≥ 0 and f ∈ R_{v}.

Definition 4.2. Let v : E → Γ ∪ {∞} be a valuation.

1. The local domain R_{v} = {f ∈ E | v(f) ≥ 0} is called the valuation ring of v.

2. Putting Γ_{1} = v(E^{∗}) we have a surjective induced map by v

v_{1} : E → Γ_{1} ∪ {∞}.

Obviously also v_{1} is a valuation of E. We call v_{1} the surjective reduction of v.

Below by v_{1} : E → Γ_{1} ∪ {∞} we denote the surjective reduction of v.

3. If Γ_{1} = {0}, then we say that the valuation v is trivial.

4. Let D be a subfield of E. If the restricted map v | D : D → Γ ∪ {∞} is a trivial valuation, then we say that v is a valuation over D.

5. The rank rank Γ_{1} of the totally ordered abelian group Γ_{1} is called the rank of v, and is denoted by rank v.

6. The rational rank r.rank Γ_{1} of the totally ordered abelian group Γ_{1} is called the rational rank of v, and is denoted by r.rank v.

7. Let v′ : E → Γ′ ∪ {∞} be another valuation of the field E. Let v′_{1} : E → Γ′_{1}∪{∞} be the surjective reduction of v′. We say that v and v′ are equivalent, if there exists an isomorphism φ : Γ_{1} → Γ′_{1} of totally ordered abelian groups such that φv_{1} = v′_{1}, where we denote the extension Γ_{1} ∪ {∞} → Γ′_{1} ∪ {∞} of φ satisfying φ(∞) = ∞ by the same symbol φ.

Lemma 4.3. Equivalent valuations have the same valuation ring, the same rank, and the same rational rank.

Definition 4.4. Let v : E → Γ ∪ {∞} and v′ : D → Γ′ ∪ {∞} be valuations. We say that the valuation v is an extension of v′, or v′ is a restriction of v, if D is a subfield of a field E and if there is an injective homomorphism φ : Γ′ → Γ of totally ordered abelian groups such that φv′ = v | D, where we denote the extension Γ′ ∪ {∞} → Γ ∪ {∞} of φ satisfying φ(∞) = ∞ by the same symbol φ.

Lemma 4.5. Assume that a valuation v′ of a field D is a restriction of a valuation v. We have R_{v′} = R_{v} ∩ D, where R_{v′} and R_{v} denote the valuation rings of v′ and v respectively.

Lemma 4.6. 1. A valuation v of a field E is trivial, if and only if, the valuation

ring R_{v} coincides with E.

2. A valuation v is one over a subfield D, if and only if, the valuation ring R_{v} contains D.

Definition 4.7. A subdomain V of E is said to be a valuation ring of E, if for every element f ∈ V with f ∉ V it follows that 1/f ∈ V.

Remark . By the above condition E coincides with the quotient field of V.

Below in this section until Theorem 4.24, by V we denote a valuation ring of the field E.

Lemma 4.8. 1. Let Γ = E^{∗}/V^{∗} and Γ_{+} = (V − {0})/V^{∗}. We can define total order on Γ such that Γ_{+} coincides with the set of non-negative elements. With this total order Γ is a totally ordered abelian group.

2. Let v : E^{∗} → E^{∗}/V^{∗} = Γ be the canonical surjective homomorphism as abelian groups. By putting v(0) = ∞ we define the extension v : E → Γ ∪ {∞}, which is denoted by the same symbol. Then v is a valuation of E.

Proof. Claim 1 follows from Lemma 3.2.

The above map v : E → (E^{∗}/V^{∗}) ∪ {∞} is called the valuation defined by a valuation ring V.

Lemma 4.9. 1. Let v : E → Γ∪{∞} be a valuation of E, R_{v} be a valuation ring of v, and v_{1} : E → (E^{∗}/R_{v}^{∗}) ∪ {∞} be the valuation defined by the valuation ring R_{v}. Then, v and v_{1} are equivalent.

2. Let V be a valuation ring of E, and v : E → (E^{∗}/V^{∗}) ∪ {∞} be the valuation defined by V. Then, we have V = R_{v}.

Corollary 4.10. 1. The set of equivalence classes of valuations of E and the

set of valuation rings of E have one-to-one correspondence described above.

2. Any valuation ring of E is a normal local domain.

Definition 4.11. Let Γ = E^{∗}/V^{∗}.

1. The rank rank Γ of the totally ordered abelian group Γ is called the rank of V, and is denoted by rank V.

2. The rational rank r.rank Γ of the totally ordered abelian group Γ is called the rational rank of V, and is denoted by r.rank V.

3. Assume that V contains a subfield D. We identify D and the image of D by the canonical surjective homomorphism V → V/M(V). The transcendence degree tr.deg_{D} V/M(V) of the residue field V/M(V) over D is called the dimension of V over D, and it is denoted by dim_{D} V.

Lemma 4.12. Let v : E → Γ ∪ {∞} be the valuation defined by V.

1. rank V = rank v = rank Γ, r.rank V = r.rank v = r.rank Γ.

2. If r.rank V is finite, then rank V ≤ r.rank V.

Proof. Claim 2 follows from Lemma 3.4.3.

Lemma 4.13. Let D be a subfield of E.

1. The intersection V′ = V ∩ D is a valuation ring of D. V′^{∗} = V^{∗} ∩ D. M(V′) = M(V) ∩ D.

2. The valuation v : E → (E^{∗}/V^{∗}) ∪ {∞} defined by V is an extension of the valuation v′ : D → (D^{∗}/V′^{∗}) ∪ {∞} defined by V′.

3. rank V′ ≤ rank V, r.rank V′ ≤ r.rank V.

Proof. Claim 3 follows from Lemma 3.4.6.

Lemma 4.14. Let v : E → Γ ∪ {∞} be the valuation defined by V.

1. Let Δ be an isolated subgroup of Γ. Then, 𝔭 = {f ∈ V | v(f) ∉ Δ} is a prime ideal in V. Besides, we have V − 𝔭 = {f ∈ V | v(f) ∈ Δ}.

2. Let 𝔭 be a prime ideal of V. Then, Δ = v(V_{𝔭}^{∗}) is an isolated subgroup of Γ. Besides, we have V_{𝔭}^{∗} = v^{−1}(Δ).

3. By the correspondence in 1 and 2 above, isolated subgroups of Γ and prime ideals of V have one-to-one correspondence, under which inclusion relations are reversed.

Corollary 4.15. 1. The rank rank V of a valuation ring V is equal to the car-

dinal number of the set of all non-zero prime ideals of V.

2. Let 𝔭 and 𝔭′ be prime ideals of a valuation ring V. Either 𝔭 ⊃ 𝔭′ or 𝔭 ⊂ 𝔭′ holds.

3. Let D be a subfield of E, and V′ = V ∩ D. The map Spec(V) → Spec(V′) induced by the inclusion homomorphism V′ ↪ V is surjective.

Proof. Claim 2 follows from Lemma 3.4.1. Claim 3 follows from Lemma 3.4.6.

Remark . By the above corollary rank V coincides with the dimension of V for a valuation ring V. Therefore, we should write dim V instead of rank V for consistency. However, for valuation rings the word “dimension” is used for the different meaning as in Definition 4.11.3. This is because of a historical reason.

Lemma 4.16. Let v : E → Γ ∪ {∞} be the valuation defined by V, and 𝔭 be a prime ideal of V.

1. The residue ring \overline{V} = V/𝔭 of V by 𝔭 is a valuation ring of the field V_{𝔭}/𝔭V_{𝔭}. Let \overline{v} : V_{𝔭}/𝔭V_{𝔭} → \overline{Γ} ∪ \{∞\} denote the valuation defined by \overline{V}. Let Δ = v(V_{𝔭}^{∗}). There exists an isomorphism Δ ⟶^{∼} \overline{Γ} of totally ordered abelian groups which makes the following diagram commutative:

V_{𝔭}^{∗} ⟶ (V_{𝔭}/𝔭V_{𝔭})^{∗}
↓^{v}  \sqrt{v}
Δ ⟶^{∼}  \overline{Γ}

The map V_{𝔭}^{∗} → (V_{𝔭}/𝔭V_{𝔭})^{∗} is the one induced by the canonical surjective homomorphism V_{𝔭} → V_{𝔭}/𝔭V_{𝔭}. Note that it is surjective.

If V contains a subfield D, then we have dim_{D} V = dim_{D} \overline{V}.

2. The localization V_{1} = V_{𝔭} of V by 𝔭 is a valuation ring of E. Let v_{1} : E → Γ_{1} ∪ {∞} denote the valuation defined by V_{1}. Let Δ = v(V_{𝔭}^{∗}). There exists an isomorphism Γ/Δ →^{∼} Γ_{1} of totally ordered abelian groups which makes the following diagram commutative:

E^{∗}  E^{∗}
↓^{v}  ↓^{v_{1}}
Γ ⟶ Γ_{1}

The map Γ → Γ_{1} is the composition of the canonical surjective homomorphism Γ → Γ/Δ and the isomorphism Γ/Δ →^{∼} Γ_{1}.

Corollary 4.17. Let 𝔭 be a prime ideal of V. We have

rank V = rank (V/𝔭) + rank V_{𝔭}, r.rank V = r.rank (V/𝔭) + r.rank V_{𝔭}.

Proof. It follows from Lemma 3.4.4.

Theorem 4.18 (Zariski-Samuel [11], Vol.II. p.8, Theorem 3.). Let V′ be a subring of E with V ⊂ V′ ⊂ E. Then, we have V′ = V_{𝔭} for some prime ideal 𝔭 of V.

Corollary 4.19. Let V′ be a local subring of E dominating V. Then, we have V′ = V.

Remark . Recall that V′ dominates V, if V ⊂ V′ and M(V′) ∩ V = M(V). (Refer to the beginning part of Section 2.)

Theorem 4.20 (Zariski-Samuel [11], Vol.II, p.12, Theorem 5.). Let A be a local subring of E. Then, there exists a valuation ring of E dominating A.

Theorem 4.21. 1. (Zariski-Samuel [11], Vol.II, p.331, Proposition 2.) Let A be a noetherian local domain with the quotient field E, and assume that V dominates A. We have

r.rank V + tr.deg_{(A/M(A))}(V/M(V)) ≤ dim A.

2. (Zariski-Samuel [11], Vol.II, p.335 Proposition 3.) If the equality holds in the above, then the value group of the valuation defined by V is isomorphic to ℤ^{r} as abelian groups (not necessarily as totally ordered abelian groups) where r = r.rank V, and the residue field V/M(V) is finitely generated over A/M(A).

Corollary 4.22. 1. Under the same assumptions as in the theorem we have

rank V + tr.deg_{(A/M(A))}(V/M(V)) ≤ dim A.

2. If the equality holds in the above, then the value group of the valuation defined by V is isomorphic to ℤ^{r} as totally ordered abelian groups where r = rank V, and the residue field V/M(V) is finitely generated over A/M(A).

Proof. Claim 1 follows from Theorem 4.21.1 and Lemma 4.12.2. Claim 2 follows from Theorem 4.21.2 and Lemma 3.5.

Lemma 4.23. Let D be a subfield of E. Let U = V ∩ D.

1. (Zariski-Samuel [11], Vol.II, p.25, Corollary 1.)

tr.deg_{(U/M(U))}(V/M(V)) ≤ tr.deg_{D}E.

2. (Zariski-Samuel [11], Vol.II, p.333, Lemma 1.) Assume that tr.deg_{D}E is finite. We have

r.rank V − r.rank U ≤ tr.deg_{D}E − tr.deg_{(U/M(U))}(V/M(V)).

Theorem 4.24 (Zariski-Samuel [11], Vol.II, p.41, Theorem 16.). A valuation ring V is noetherian, if and only if, either V = E or the value group of the valuation defined by V is isomorphic to ℤ.

Proposition 4.25. Let V be a ring. The following three conditions are equivalent: 1. V is a noetherian normal local ring of dimension one.

2. V is a regular local ring of dimension one.

3. V is a domain, it is a valuation ring of the quotient field, and the value group of the valuation defined by V is isomorphic to ℤ.

Proof. 1⇔2. See Matsumura [7], p.124, (17.H), Theorem 37. 2⇔3. Easy exercise in the ring theory.

Lemma 4.26. Let V be a regular local ring of dimension one with the quotient field E. Let v : E → Γ ∪ {∞} be the valuation defined by V. Let π ∈ M(V) be a generator of the maximal ideal M(V). Then, Γ = ℤv(π), and v(f) = νv(π) for f ∈ E^{∗} and ν ∈ ℤ, if and only if, there exists an element u ∈ V^{∗} with f = uπ^{ν}.

Proof. Since V is a UFD, (Matsumura [7], p. 142, (19.B), Theorem 48.), every element f ∈ E^{∗} can be written uniquely in the form f = uπ^{ν} with u ∈ V^{∗} and ν ∈ ℤ. Thus the correspondence f ∈ E^{∗} ↦ ν ∈ ℤ induces an isomorphism Γ = E^{∗}/V^{∗} →^{∼} ℤ of abelian groups.

Definition 4.27. Let A be a noetherian local domain with the quotient field E, and V be a valuation ring of E dominating A. We say that the pair (A, V) is of iterated divisor type, if the equality rank V = dim A holds.

If a given ring A is obvious, we say that a valuation ring V is of iterated divisor type, if the pair (A, V) is so.

Remark . Assume that a pair (A, V) is of iterated divisor type. Let n = rank V = dim A. V has n + 1 prime ideals 𝔭_{i}, 1 ≤ i ≤ n with

{0} = 𝔭_{0} ⊂ 𝔭_{1} ⊂ ⋯ ⊂ 𝔭_{n−1} ⊂ 𝔭_{n} = M(V).

By Corollary 4.22, and Theorem 4.24 and Proposition 4.25 for every 1 ≤ i ≤ n the ring V_{𝔭_{i}}/𝔭_{i−1}V_{𝔭_{i}} is a regular local ring of dimension one, and the residue field V/M(V) is a finite algebraic extension of A/M(A).

We can show that if A is excellent, then the set of all valuation rings V of iterated divisor type is dense in the Zariski-Riemann space of A. (Zariski-Samuel [11], Vol.II. p.110–p.123, §17.) We call what Zariski called “the Riemann surface” the Zariski-Riemann space, since the word “surface” is strange. In Zariski’s notation E(A) is our Zariski-Riemann space of A.

The converse of Lemma 4.16 holds.

Lemma 4.28. Let V_{1} be a valuation ring of E and \overline{V} be a valuation ring of the residue field V_{1}/M(V_{1}). Let V be the inverse image of \overline{V} by the canonical surjective homomorphism V_{1} → V_{1}/M(V_{1}). Then, V is a valuation ring of E with M(V_{1}) ⊂ V ⊂ V_{1} ⊂ E. Besides, 𝔭 = M(V_{1}) ⊂ V is a prime ideal of V, V_{1} = V_{𝔭}, \overline{V} ≅ V/𝔭, and V/M(V) ≅ \overline{V} /M(\overline{V}).

Remark . If V_{1} ≠ E and \overline{V} ≠ V_{1}/M(V_{1}), then the valuation ring V is called the composite with V_{1} and \overline{V}.

Corollary 4.29. Let V be a valuation ring, and 𝔭 and 𝔮 be prime ideals of V with 𝔭 ⊂ 𝔮. Then, 𝔭 = 𝔭V_{𝔮} as sets.

Proof. Obviously 𝔭 ⊂ 𝔭V_{𝔮} ⊂ 𝔭V_{𝔭}. Let V_{1} = V_{𝔭}. By Lemma 4.28 we have M(V_{1}) = 𝔭V_{𝔭} = 𝔭V_{𝔭} ∩ V = 𝔭.

Definition 4.30. Let v : E → Γ ∪ {∞} be a valuation of E with the valuation ring V = R_{v}, and D be a subfield of E. We assume that E is a finite algebraic extension of D.

1. The index of the subgroup v(D^{∗}) in v(E^{∗}) is called the ramification index of v (or of V) over D.

2. The relative degree [V/M(V) : (V ∩ D)/M(V ∩ D)] of residue fields is called the relative degree of v (or of V) over D.

Theorem 4.31. Let D be a subfield of E. Assume that E is a finite algebraic extension of D. Let V′ be a valuation ring of D. Let V_{1}, V_{2}, …, V_{g} be valuation rings of E with V_{i} ∩ D = V′ for 1 ≤ i ≤ g. We assume V_{i} ≠ V_{j} if i ≠ j. Let e_{i} and f_{i} denote the ramification index and the relative degree of V_{i} over D respectively for 1 ≤ i ≤ g.

1. (Zariski-Samuel [11], Vol. II, p.55, Theorem 19.) We have

∑_{i=1}^{g} e_{i}f_{i} ≤ [E : D].

2. (Zariski-Samuel [11], Vol. II, p.17, Theorem 8 and p.60-p.61, Theorem 20.) Assume that any valuation ring V of E with V ∩ D = V′ coincides with one of V_{i}’s. Then, \overline{V} = ∩_{i=1}^{g}V_{i} coincides with the integral closure of V′ in E. If \overline{V} is a finitely generated V′-module, then the equality holds in the above.

3. (Zariski-Samuel [11], Vol. I, p.285, Theorem 21.) Assume moreover that V′ is a regular local ring of dimension one. If the equality above holds, then the integral closure of V′ in E is a finitely generated V′-module.

Lemma 4.32. Let A be a noetherian local domain with quotient field E, and V be a valuation ring of E dominating A. Assume that the completion Â of A is a domain. Then, there exists a valuation ring V̂ of the quotient field Ê of Â such that V̂ dominates Â and V̂ ∩ E = V.

Proof. This is Lemma 13 of Abhyankar [1] p.513.

Lemma 4.33. In above Lemma 4.32, if A contains the ground field k and if dim_{k} V = 0, then there exists a valuation ring V̂ of the quotient field Ê of Â such that V̂ dominates Â, dim_{k} V̂ = 0 and V̂ ∩ E = V.

Proof. First, note that A/M(A) ≅ Â/M(Â) ≅ V/M(V) ≅ k.

Since tr.deg_{k}(V/M(V)) = dim_{k} V = 0 and since k is algebraically closed, we have V/M(V) ≅ k. Since V dominates A, we have k ⊂ A/M(A) ⊂ V/M(V) ≅ k. Since A/M(A) ≅ Â/M(Â) by definition, we obtain the desired claim.

By Lemma 4.32 there exists a valuation ring V̂_{0} of Ê dominating Â with V = V̂_{0} ∩ E. Let L = V̂_{0}/M(V̂_{0}). By Theorem 4.21 we have r = tr.deg_{k}L ≤ dim Â. Let x_{1}, x_{2}, …, x_{r} ∈ L be algebraically independent elements over k. The field k(x_{1}, x_{2}, …, x_{r}) is isomorphic to the rational function field over k with r variables, and there is a valuation ring \overline{V}_{1} of k(x_{1}, x_{2}, …, x_{r}) such that k ⊂ \overline{V}_{1} and dim_{k} \overline{V}_{1} = 0. By Theorem 4.20 there is a valuation ring \overline{V} of L dominating \overline{V}_{1}. By Corollary 4.19 one knows \overline{V}_{1} = \overline{V} ∩ k(x_{1}, x_{2}, …, x_{r}). By Lemma 4.23.1 one has 0 ≤ tr.deg_{(\overline{V}_{1}/M(\overline{V}_{1}))}(\overline{V} /M(\overline{V})) ≤ tr.deg_{k(x_{1},x_{2},... ,x_{r})}L = 0 and dim_{k} \overline{V} = 0.

Let π : V̂_{0} → L = V̂_{0}/M(V̂_{0}) denote the canonical surjective homomorphism. Let V̂ = π^{−1}(\overline{V}) be the inverse image of \overline{V} ⊂ L by π. By Lemma 4.28 V̂ is a valuation ring of E with dim_{k} V̂ = dim_{k} \overline{V} = 0.

π(Â) ≅ Â/(M(V̂_{0}) ∩ Â) ≅ Â/M(Â) ≅ k. Thus V̂ = π^{−1}(\overline{V}) ⊃ π^{−1}(k) ⊃ Â. Since M(V̂) ⊃ M(V̂_{0}) ⊃ M(Â) by Corollary 4.29, one knows that V̂ dominates Â.

Since π(V) ≅ V/(M(V̂_{0}) ∩ V) = V/M(V) ≅ k, one has V̂ = π^{−1}(\overline{V}) ⊃ π^{−1}(k) ⊃ V. Putting V′ = V̂ ∩ E, one has V ⊂ V′. Since M(V) ⊂ M(V̂_{0}) ⊂ M(V̂), we have M(V) ⊂ M(V̂) ∩ E = M(V′), and V′ dominates V. By Corollary 4.19 one concludes V̂ ∩ E = V′ = V.

5. Space germ

In this section we give the definition of our main objects, a space germ, a framed space germ, a basic space germ, an elementary monoidal transform, an iterated elementary monoidal transform (an IEMT, in short), an iterated analytic monoidal transform (an IAMT, in short) and so on.

Let k[[x]] = k[[x_{0}, x_{1}, …, x_{r}]] denote the formal power series ring over k with r + 1 variables x_{0}, x_{1}, …, x_{r}. We give the M(k[[x]])-adic topology to the ring k[[x]]. Recall that a non-empty subset of a parameter system of a regular local ring is called a partial parameter system. (Definition 2.19.2.)

Definition 5.1. 1. A pair (B, W) is called a space germ over k, if it satisfies the following two conditions:

(a) The first term B is a topological local ring. Ring B is isomorphic to k[[x]] = k[[x_{0}, x_{1}, …, x_{r}]] for some non-negative integer r.

(b) The second term W is a valuation ring of the quotient field of B. We

assume that W dominates B, and dim_{k} W = 0.

2. A partial parameter system Q of B is called a frame of (B, W).

3. For any partial parameter system Q = {y_{0}, y_{1}, …, y_{n}} of B by k[[Q]] we denote the minimal closed k-subalgebra k[[y_{0}, y_{1}, …, y_{n}]] of B generated by y_{0}, y_{1}, …, y_{n}.

4. A triplet

(B, W, Q),

where the pair (B, W) is a space germ over k, and Q is a frame of (B, W), is called a framed space germ over k.

5. The number dim B = r + 1 is called the dimension of a space germ (B, W) or a framed space germ (B, W, Q).

6. The number ♯ Q of elements in Q is called the basic dimension of a framed space germ (B, W, Q).

Lemma 5.2. Let (B, W) be a space germ.

1. W/M(W) ≅ k.

2. Let Q be a frame of (B, W), A = k[[Q]], and V be the intersection of W and the quotient field of A. The triplet (A, V, Q) is a framed space germ over k such that Q is a parameter system of A.

Proof. 1. Since tr.deg_{k}(W/M(W)) = dim_{k} W = 0, and since k is an algebraically closed field, we have W/M(W) ≅ k.

2. Obviously the pair (A, V) satisfies the first condition of a space germ, and Q is a parameter system of A. By Lemma 4.13, V is a valuation ring of the quotient field E of A. Since M(A) ⊂ M(B) ⊂ M(W), we have M(A) ⊂ M(W) ∩ E = M(V). Thus, V dominates A. Since k ⊂ V/M(V) ⊂ W/M(W), we have 0 ≤ dim_{k} V ≤ dim_{k} W = 0.

Definition 5.3. 1. Let (B, W, Q) be a framed space germ. The space germ (B, W) without a frame is called the underlying space germ of (B, W, Q). We write (B, W) = (B, W, Q)_{•} to represent that (B, W) is the underlying space germ of (B, W, Q).

2. We call the framed space germ (A, V, Q) described in Lemma 5.2.2 the basic framed space germ of a framed space germ (B, W, Q), or the basic framed space germ of the space germ (B, W) with respect to the frame Q. We write (A, V, Q) = \underline{(B, W, Q)} to represent that (A, V, Q) is the basic framed space germ of (B, W, Q).

3. Let (B, W) and (B′, W′) be space germs. A morphism φ : F → F′ of kalgebras from the quotient field F of B to the quotient field F′ of B′ satisfying the following conditions is called a morphism of space germs from (B, W) to (B′, W′)

(a) φ(B) ⊂ B′ and φ(W) ⊂ W′.

(b) The induced homomorphisms φ : B → B′ and φ : W → W′ are local.

(c) The homomorphism φ : B → B′ is continuous. We denote φ : (B, W) → (B′, W′).

4. Let (B, W, Q) and (B′, W′, Q′) be framed space germs. A morphism φ : F → F′ of k-algebras from the quotient field F of B to the quotient field F′ of B′ satisfying the following conditions is called a morphism of framed space germs from (B, W, Q) to (B′, W′, Q′)

(a) φ(B) ⊂ B′, φ(W) ⊂ W′, and φ(k[[Q]]) ⊂ k[[Q′]].

(b) The induced homomorphisms φ : B → B′, φ : W → W′ and φ : k[[Q]] → k[[Q′]] are local.

(c) The homomorphisms φ : B → B′ φ : k[[Q]] → k[[Q′]] are continuous.

(d) There exists a parameter system R of B such that Q ⊂ R, φ(R−Q)∩Q′ = ∅, and φ(R − Q) ∪ Q′ is a parameter system of B′.

We denote φ : (B, W, Q) → (B′, W′, Q′).

The above notation (A, V, Q) = \underline{(B, W, Q)} in Definition 5.3.2 will be used frequently.

Lemma 5.4. Let (B, W, Q) be a framed space germ, and (A, V, Q) = \underline{(B, W, Q)}.

1. For the framed space germ (A, V, Q), Q is a parameter system of A.

2. The extension of the inclusion homomorphism A → B to the quotient fields defines a morphism (A, V) → (B, W) of space germs.

3. Let R = {x_{0}, x_{1}, …, x_{r}} be a parameter system of B containing the parameter system Q = {x_{0}, x_{1}, …, x_{n}} of A. Let R′ = {y_{0}, y_{1}, …, y_{s}} be a finite subset of B containing a parameter system Q′ = {y_{0}, y_{1}, …, y_{n}} of A. Then, R′ is a parameter system of B, if and only if, s = r, R′ ⊂ M(B) and det(∂y_{j}/∂x_{i})_{n<i, j≤r} ∉ M(B).

We call the morphism (A, V) → (B, W) in Lemma 5.4.2 the inclusion morphism.

Lemma 5.5. Let φ : (B, W, Q) → (B′, W′, Q′) be a morphism of framed space germs. Let (A, V, Q) = \underline{(B, W, Q)}, and (A′, V′, Q′) = \underline{(B′, W′, Q′)}.

1. The morphism φ induces an injective homomorphism from the quotient field F of B to the quotient field F′ of B′ such that the image of the quotient field E of A is contained in the quotient field E′ of A′. We can regard F as a subfield of F′, and E as a subfield of E′. We have W = W′ ∩ F and V = V′ ∩ E.

2. For any parameter system R_{1} of B containing a parameter system Q_{1} of A and for any parameter system Q′_{1} of A′, the union φ(R_{1} − Q_{1}) ∪ Q′_{1} is a parameter system of B′.

3. The morphism φ induces a morphism φ_{•} : (B, W) → (B′, W′) of underlying space germs.

4. The morphism φ induces a morphism \underline{φ} : (A, V, Q) → (A′, V′, Q′) of basic framed space germs.

5. As the composition of above two correspondences, the morphism φ induces a morphism \underline{φ}_{•} : (A, V) → (A′, V′).

6. The following diagram is commutative:

(A, V) ⟶ (B, W)
↓\underline{φ}_{•}  ↓^{φ_{•}}
(A′, V′) ⟶ (B′, W′)

Horizontal arrows are inclusion morphisms.

Proof. 1. The equalities W = W′ ∩ F and V = V′ ∩ E follow from Corollary 4.19.

2. It follows from Lemma 5.4.3.

We call φ_{•} the underlying morphism of φ, while we call \underline{φ} the basic morphism of φ.

Lemma 5.6. Let φ : (B, W, Q) → (B′, W′, Q′) be a morphism of framed space germs. Let Q_{1} be a partial parameter system of B with Q ⊂ Q_{1}.

Let Q′_{1} = Q′ ∪ φ(Q_{1} − Q). The set Q′_{1} is a partial parameter system of B′. It satisfies Q′ ⊂ Q′_{1} and ♯ (Q_{1} − Q) = ♯ (Q′_{1} − Q′). There exists a unique morphism φ_{1} : (B, W, Q_{1}) → (B′, W′, Q′_{1}) of framed space germs such that φ_{•} = (φ_{1})_{•} and the following diagram is commutative: We denote the basic framed space germs of (B, W, Q), (B, W, Q_{1}), (B′, W′, Q′), and (B′, W′, Q′_{1}) by (A, V, Q), (A_{1}, V_{1}, Q_{1}), (A′, V′, Q′), and (A′_{1}, V′_{1}, Q′_{1}) respectively.

(A, V) ⟶ (A_{1}, V_{1}) ⟶ (B, W)
↓\underline{φ}_{•}  ↓^{(\underline{φ}_{1})_{•}}  ↓^{φ_{•}=(φ_{1})_{•}}
(A′, V′) ⟶ (A′_{1}, V′_{1}) ⟶ (B′, W′)

Proof. The claim on Q′_{1} follows from Lemma 5.5.2. The remaining parts are easy.

Definition 5.7. Let (B, W, Q) be a framed space germ, and Q′ be an arbitrary parameter system of k[[Q]]. The triplet (B, W, Q′) is a framed space germ, and we call it a coordinate transform of (B, W, Q).

We associate a morphism (B, W, Q) → (B, W, Q′) of space germs induced by the identity homomorphisms of the quotient field of B.

Note that the associated morphism (B, W, Q) → (B, W, Q′) can be defined above, since for any parameter system R of B containing Q, the union (R − Q) ∪ Q′ is a parameter system of B containing Q′. (Lemma 5.4.3.)

Also note (B, W, Q)_{•} = (B, W, Q′)_{•} and \underline{(B, W, Q)}_{•} = \underline{(B, W, Q′)}_{•}, if (B, W, Q′) is a coordinate transform of (B, W, Q).

We will define an elementary monoidal transform of a framed space germ.

Let (B, W, Q) be a framed space germ, and C ⊂ Q be a subset containing at least two elements. Let C = {x_{0}, x_{1}, …, x_{t}} with t ≥ 1. By F we denote the quotient field of B. Let v : F → Γ ∪ {∞} be the valuation defined by W. Since Q ⊂ M(B) ⊂ M(W), we have v(x) > 0 for every x ∈ Q. We have at least one number j with 0 ≤ j ≤ t satisfying v(x_{j}) = min{v(x_{i}) | 0 ≤ i ≤ t}. By exchanging the numbers of x_{i}’s, we can assume v(x_{0}) = min{v(x_{i}) | 0 ≤ i ≤ t}. Consider a subring

B_{1} = B[\frac{x_{1}}{x_{0}} , \frac{x_{2}}{x_{0}} , ⋯, \frac{x_{t}}{x_{0}} ]

of F generated by B and t elements x_{1}/x_{0}, x_{2}/x_{0}, ⋯, x_{t}/x_{0}. We have v(x_{i}/x_{0}) ≥ 0 for 1 ≤ i ≤ t, and B_{1} ⊂ W. Let 𝔪 = M(W)∩B_{1}. Since k ⊂ B_{1}/𝔪 ⊂ W/M(W) ≅ k, B_{1}/𝔪 ≅ k and 𝔪 is a maximal ideal of B_{1}.

Besides, since W/M(W) ≅ k, for every number i with 1 ≤ i ≤ t, we have a unique element λ_{i} ∈ k with v((x_{i}/x_{0}) − λ_{i}) > 0. Let y_{0} = x_{0} and y_{i} = (x_{i}/x_{0}) − λ_{i} for 1 ≤ i ≤ t. Let B_{2} = (B_{1})_{𝔪}, and Q′ = (Q − C) ∪ {y_{0}, y_{1}, …, y_{t}}. One sees by Theorem 2.2.4 and 5 that B_{2} is a excellent regular local ring, Q′ is a partial parameter system of B_{2}, and W dominates B_{2}.

Let B′ be the completion of B_{2}, and F′ be the quotient field of B′. Ring B′ is a complete regular local ring and Q′ is a partial parameter system of B′. Ring B′ coincides with the completion of B_{1} with respect to the ideal 𝔪. The inclusion homomorphism B ⊂ B_{2} and the canonical homomorphism B_{2} → B′ to the completion define an injective homomorphism B → B′ of rings. Since B and B_{2} have common quotient field F, an injective homomorphism F → F′ of fields is induced. We can regard F as a subfield of F′. By Lemma 4.33 there exists a valuation ring W′ of F′ such that W′ dominates B′, dim_{k} W′ = 0 and W′ ∩F = W.

The triplet (B′, W′, Q′) is a framed space germ over k.

Let A = k[[Q]], A′ = k[[Q′]] A_{1} = A[x_{1}/x_{0}, x_{2}/x_{0}, …, x_{t}/x_{0}], and 𝔩 = A_{1} ∩ M(W). One sees easily that the completion of A_{1} with respect to 𝔩 coincides with A′. Thus composing the inclusion homomorphism A → A_{1} and the canonical homomorphism A_{1} → A′ to the completion, we obtain an injective ring homomorphism A → A′. Obviously A → A′ coincides with the restriction of the homomorphism F → F′ to A.

The homomorphism F → F′ induces a morphism (B, W, Q) → (B′, W′, Q′) of framed space germs.

Definition 5.8. Let (B, W, Q) be a framed space germ over k, F be a quotient field of B, and v : F → Γ ∪ {∞} be the valuation defined by W.

1. A triplet (C, x_{0}, W′) satisfying the following three conditions is called a datum of a monoidal transformation over (B, W, Q).

(a) C ⊂ Q. C contains at least two elements.

(b) x_{0} ∈ C. v(x_{0}) = min{v(x) | x ∈ C}.

(c) Let B_{1} be a subring of F generated by B and the set {x/x_{0} | x ∈ C}, B′ denote the completion of B_{1} with respect to the ideal B_{1} ∩ M(W), and F′ denote the quotient field of B′. W′ is a valuation ring of F′ such that W′ dominates B′, dim_{k} W′ = 0 and W′ ∩ F = W.

2. Let (C, x_{0}, W′) be a datum of a monoidal transformation over (B, W, Q). Let B′ denote the ring in the above 1.(c). For every x ∈ C we choose an element λ_{x} ∈ k with v((x/x_{0}) − λ_{x}) > 0, and put y_{x} = (x/x_{0}) − λ_{x}. Let Q′ = (Q − C) ∪ {x_{0}} ∪ {y_{x} | x ∈ C, x ≠ x_{0}}. The framed space germ (B′, W′, Q′) is called the elementary monoidal transform of (B, W, Q) associated with a datum (C, x_{0}, W′).

3. The induced morphism (B, W, Q) → (B′, W′, Q′) described above is called the associated morphism with the elementary monoidal transform (B′, W′, Q′).

4. A space germ (B′, W′) without a frame is called an elementary monoidal transform of (B, W), if there exist a frame Q of B and a datum (C, x_{0}, W′) of a monoidal transformation over (B, W, Q) such that the underlying space germ of the elementary monoidal transform (B′, W′, Q′) of (B, W, Q) associated with (C, x_{0}, W′) coincides with (B′, W′). The underlying morphism (B, W) → (B′, W′) of the associated morphism (B, W, Q) → (B′, W′, Q′) is called the associated morphism with the elementary monoidal transform (B′, W′) of (B, W).

Definition 5.9. Let (B, W, Q) and (B′, W′, Q′) be framed space germs over k. Assume that there exists a finite sequence {(B_{ν}, W_{ν}, Q_{ν}) | 0 ≤ ν ≤ ω} of framed space germs satisfying the following two conditions:

1. (B, W, Q) = (B_{0}, W_{0}, Q_{0}), (B′, W′, Q′) = (B_{ω}, W_{ω}, Q_{ω}).

2. For every number ν with 1 ≤ ν ≤ ω, (B_{ν}, W_{ν}, Q_{ν}) is an elementary monoidal transform of (B_{ν−1}, W_{ν−1}, Q_{ν−1}) associated with some datum. (resp. an elementary monoidal transform of (B_{ν−1}, W_{ν−1}, Q_{ν−1}) associated with some datum, or a coordinate transform of (B_{ν−1}, W_{ν−1}, Q_{ν−1}).)

Then, we say that (B′, W′, Q′) is an iterated elementary monoidal transform, or an IEMT in short (resp. an iterated analytic monoidal transform, or an IAMT in short) of (B, W, Q). The composition (B, W, Q) = (B_{0}, W_{0}, Q_{0}) → (B_{ω}, W_{ω}, Q_{ω}) = (B′, W′, Q′) of ω associated morphisms (B_{ν−1}, W_{ν−1}, Q_{ν−1}) → (B_{ν}, W_{ν}, Q_{ν}), 1 ≤ ν ≤ ω is called the associated morphism with the IEMT (resp. IAMT) (B′, W′, Q′) of (B, W, Q).

A space germ (B′, W′) without a frame is called an IAMT of (B, W), if there exist a frame Q of B and an IAMT (B′, W′, Q′) of (B, W, Q) such that the underlying space germ of (B′, W′, Q′) coincides with (B′, W′). The underlying morphism (B, W) → (B′, W′) of the associated morphism (B, W, Q) → (B′, W′, Q′) is called the associated morphism with the IAMT (B′, W′) of (B, W).

Definition 5.10. 1. Let φ : (B, W, Q) → (B′, W′, Q′) be a morphism of framed space germs. If, for every x ∈ Q, the image φ(x) ∈ k[[Q′]] has a normal crossing with respect to Q′, then, we say that the morphism φ preserves normal crossings, or φ is a morphism preserving normal crossings.

2. Let (B′, W′, Q′) be an IAMT of a framed space germ (B, W, Q). We say that (B′, W′, Q′) is an IAMT with a normal crossing of (B, W, Q), if the associated morphism (B, W, Q) → (B′, W′, Q′) preserves normal crossings.

Lemma 5.11. 1. If a morphism φ : (B, W, Q) → (B′, W′, Q′) preserves normal crossings, then, for any parameter system R of B containing Q and for every x ∈ R, φ(x) ∈ B′ has a normal crossing with respect to the parameter system φ(R − Q) ∪ Q′.

2. If two morphisms φ : (B, W, Q) → (B′, W′, Q′) and ψ : (B′, W′, Q′) → (B′′, W′′, Q′′) of framed space germs preserve normal crossings, then the composition ψφ : (B, W, Q) → (B′′, W′′, Q′′) also preserves normal crossings.

3. A morphism associated with an IEMT preserves normal crossings.

Proof. 1, 2. Easy.

3. By 2 we can consider only the case of an elementary monoidal transform. Assume that y_{0}, y_{1}, …, y_{t} is a subset of a parameter system R′ of some complete

regular local ring. Assume moreover that elements x_{0}, x_{1}, …, x_{t} satisfy relations y_{0} = x_{0} and y_{i} = (x_{i}/x_{0}) − λ_{i} for 1 ≤ i ≤ t, where λ_{i}’s are elements in k.

Then, we have x_{0} = y_{0} and x_{i} = y_{0}(λ_{i} + y_{i}) for 1 ≤ i ≤ t. One sees that every x_{i} has a normal crossing with respect to R′.

The case of an elementary monoidal transform follows from this observation.

We can explain our goal of this paper here. Let (B, W, Q) be a framed space germ over k. We consider the following statement RS(B, W, Q) on (B, W, Q):

RS(B, W, Q) : Assume that a finite number of arbitrary non-zero elements f_{1}, f_{2}, …, f_{ℓ} in k[[Q]] are given. Then, there exists an IAMT (B′, W′, Q′) of (B, W, Q) such that for every 1 ≤ i ≤ ℓ the image f′_{i} ∈ k[[Q′]] of f_{i} ∈ k[[Q]] by the associated homomorphism k[[Q]] → k[[Q′]] has a normal crossing with respect to Q′, and the ideal in k[[Q′]] generated by any non-empty subset of {f′_{1}, f′_{2}, …, f′_{ℓ}} is a principal ideal.

Also we consider another statement RW (B, W, Q) on (B, W, Q):

RW(B, W, Q) : Let f ∈ k[[Q]] be a non-zero element. Then, there exists an IAMT (B′, W′, Q′) of (B, W, Q) such that the image f′ ∈ k[[Q′]] of f by the induced homomorphism k[[Q]] → k[[Q′]] has a normal crossing with respect to Q′.

Obviously RS(B, W, Q) implies RW (B, W, Q). Also the converse holds. The following lemma can be shown easily:

Lemma 5.12 (Bierstone and Milman [3], p.25, Lemma 4.7). Let (B′, W′, Q′) be a framed space germ over k, and Q′ = {y_{0}, y_{1}, …, y_{n}}. Let α_{0}, α_{1}, …, α_{n}, β_{0}, β_{1}, …, β_{n}, γ_{0}, γ_{1}, …, γ_{n} be non-negative integers, and a, b, c ∈ k[[Q′]]^{∗}. If

ay_{0}^{α_{0}}y_{1}^{α_{1}} ⋯ y_{n}^{α_{n}} − by_{0}^{β_{0}}y_{1}^{β_{1}} ⋯ y_{n}^{β_{n}} = cy_{0}^{γ_{0}}y_{1}^{γ_{1}} ⋯ y_{n}^{γ_{n}},

then either α_{i} ≤ β_{i} for every 0 ≤ i ≤ n, or α_{i} ≥ β_{i} for every 0 ≤ i ≤ n.

Let f_{1}, f_{2}, …, f_{ℓ} ∈ k[[Q]] be finite elements in RS(B, W, Q). We can assume that any two of them are different. Let f = (∏_{i=1}^{ℓ} f_{i}) (∏_{1≤i<j≤ℓ}(f_{i} − f_{j})) ∈ k[[Q]]. By Lemma 2.9 and Lemma 5.12 one sees that RW (B, W, Q) implies RS(B, W, Q).

Corollary 5.13. Let (B, W, Q) be a framed space germ over k. Two statements on (B, W, Q), RS(B, W, Q) and RW (B, W, Q) are equivalent.

We hope to show that for any framed space germ (B, W, Q) the statement RW (B, W, Q) is true.

Note that if ♯ Q = dim k[[Q]] = 1, then every non-zero element in k[[Q]] has a normal crossing with respect to Q.

Lemma 5.14. If the basic dimension ♯ Q of a framed space germ (B, W, Q) is equal to one, then RW (B, W, Q) is true.

6. Weierstrass representation

We hope to apply induction on the basic dimension ♯ Q to show the statement RW (B, W, Q). The Weierstrass preparation theorem (Corollary 2.16) provides us a method for induction.

Definition 6.1. Let (B, W, Q) be a framed space germ with ♯ Q ≥ 2, and A = k[[Q]]. A triplet (h, g, t) satisfying the following four conditions is called a Weierstrass representation under (B, W, Q).

1. h ∈ A, g ∈ A and t ∈ Q.

We write Q = {t, x_{1}, x_{2}, …, x_{n}}, and Z = k[[Q − {t}]] below. Note that A = k[[t, x_{1}, x_{2}, …, x_{n}]] and Z = k[[x_{1}, x_{2}, …, x_{n}]].

2. There exist non-negative integers s_{1}, s_{2}, …, s_{n} with

\frac{h}{x_{1}^{s_{1}}x_{2}^{s_{2}} ⋯ x_{n}^{s_{n}}} ∈ A^{∗}.

3. The pair (g, t) is a semi-Weierstrass pair of the ring A under the parameter system Q. In other words, there are a non-negative integer m and m elements G_{1}, G_{2}, …, G_{m} ∈ Z with

g = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j}.

4. For every integer j with 1 ≤ j ≤ m

ord (G_{j}) ≥ j.

If (h, g, t) satisfies only conditions 1–3, then we call (h, g, t) a semi-Weierstrass representation. The frame Q of the space germ (B, W) is called the upper frame of (h, g, t), while the frame P = Q − {t} of the space germ (B, W) is called the lower frame of (h, g, t). The framed space germ (B, W, Q) is called the upper framed space germ of (h, g, t), and (B, W, Q − {t}) is called the lower framed space germ of (h, g, t).

Let P = Q − {t}. We say also that (h, g, t) is a semi-Weierstrass representation over (B, W, P), referring to the lower framed space germ (B, W, P), not to the upper framed space germ (B, W, Q). If (h, g, t) is one over (B, W, P), then (B, W, P) is the lower framed space germ of (h, g, t), t ∉ P, and the union P ∪ {t} is a partial parameter system of B. If it is one under (B, W, Q), then (B, W, Q) is the upper framed space germ of (h, g, t) and t ∈ Q.

Let f ∈ A be a non-zero element. If a semi-Weierstrass representation (h, g, t) satisfies f = hg, then we say that (h, g, t) is a semi-Weierstrass representation of f.

If a semi-Weierstrass pair (g, t) of A under Q has a certain property X, then, for simplicity, we say that the semi-Weierstrass representation (h, g, t) under (B, W, Q) has a property X.

Lemma 6.2. Let (B, W, Q) be a framed space germ with ♯ Q ≥ 2, and f ∈ k[[Q]] be a non-zero element.

1. There exists a coordinate transform (B, W, Q′) of (B, W, Q) such that there exists a Weierstrass representation (h, g, t) of f under (B, W, Q′) satisfying h ∈ k[[Q]]^{∗}.

2. Any semi-Weierstrass representation (h, g, t) of f satisfies ord(f) = ord(g) + ord(h), sord(h) = 0, sord(f) = sord(g), and nord(f) = nord(g) + nord(h).

3. Any Weierstrass representation (h, g, t) of f with h ∈ k[[Q]]^{∗} satisfies ord(f) = ord(g) = deg(g, t), sord(f) = sord(g), and nord(f) = nord(g).

Proof. 1. Note that \underline{(B, W, Q′)}_{•} = \underline{(B, W, Q)}_{•} by definition of a coordinate transform. It follows from Proposition 2.18.

2. Refer to Definition 2.13 and Lemma 2.14 for definition of sord and nord.

3. It follows from 2 and Lemma 2.20.4.

Lemma 6.3. Let (h, g, t) be a semi-Weierstrass representation over a framed space germ (B, W, P). Let φ : (B, W, P) → (B′, W′, P′) be a morphism preserving normal crossings from the lower framed space germ.

1. Triplet (φ(h), φ(g), φ(t)) is a semi-Weierstrass representation over (B′, W′, P′) and under (B′, W′, Q′), where Q′ = P′ ∪ {φ(t)}.

2. w(g, t) ≤ w(φ(g), φ(t)), ord(g) ≤ ord(φ(g)), deg(g, t) = deg(φ(g), φ(t)).

3. Assume moreover that (h, g, t) is a Weierstrass representation. Also (φ(h), φ(g), φ(t)) is a Weierstrass representation. ord(g) = ord(φ(g)), nord(g) ≤ nord(φ(g)), sord(g) ≥ sord(φ(g)).

Proof. 1. It is easy to check the first and the third condition in Definition 6.1. Since φ preserves normal crossings, also the second condition is satisfied.

2. For every f ∈ k[[P ∪ {t}]] we have ord(f) ≤ ord(φ(f)).

3. By Lemma 2.20.4 we have deg(g, t) = ord(g). By 2 and Lemma 2.20.2 we have ord(g) ≤ ord(φ(g)) ≤ deg(φ(g), φ(t)) = deg(g, t) = ord(g). We have deg(φ(g), φ(t)) = ord(φ(g)), and again by Lemma 2.20.4 one concludes that (φ(h), φ(g), φ(t)) is a Weierstrass representation. The equality ord(g) = ord(φ(g)) follows from the above. Let D denote the quotient field of k[[P]], and D′ denote the quotient field of k[[P′]]. Let g = g_{1}^{β_{1}}g_{2}^{β_{2}} ⋯ g_{r}^{β_{r}} be the irreducible decomposition of g in D[t]. We assume that the coefficient of the term with the highest degree in g_{ν} ∈ D[t] is equal to 1 for every 1 ≤ ν ≤ r, and that for 1 ≤ ν ≤ s deg(g_{ν}, t) = 1 and for s < ν ≤ r deg(g_{ν}, t) > 1. Since deg(g, t) = ord(g), by Lemma 2.21.3 we have deg(g_{ν}, t) = ord(g_{ν}) for every 1 ≤ ν ≤ r, and g = g_{1}^{β_{1}}g_{2}^{β_{2}} ⋯ g_{r}^{β_{r}} gives the irreducible decomposition in k[[P ∪{t}]]. Thus nord(g) = ∑_{ν=1}^{s} β_{ν}. On the other hand, we have a decomposition φ(g) = φ(g_{1})^{β_{1}}φ(g_{2})^{β_{2}} ⋯ φ(g_{r})^{β_{r}} in D′[φ(t)]. Since deg(φ(g), φ(t)) = ord(φ(g)), by Lemma 2.21.1 we have deg(φ(g_{ν}), φ(t)) = ord(φ(g_{ν})) for every 1 ≤ ν ≤ r. Thus ord(φ(g_{ν})) = deg(φ(g_{ν}), φ(t)) = deg(g_{ν}, t) = 1 for 1 ≤ ν ≤ s. One concludes nord(φ(g)) ≥ ∑_{ν=1}^{s} β_{ν} = nord(g).

The last equality follows from Lemma 2.14.2.

The semi-Weierstrass representation (φ(h), φ(g), φ(t)) is called the pull-back of (h, g, t) by φ. If (h, g, t) is a semi-Weierstrass (resp. Weierstrass) representation of f ∈ k[[P ∪ {t}]], then the pull-back (φ(h), φ(g), φ(t)) is a semi-Weierstrass (resp. Weierstrass) representation of φ(f) ∈ k[[P′ ∪ {φ(t)}]].

We denote the integral part of a real number r by [r] (The Gauss symbol), and the part under the decimal point by ⟨r⟩.

[r] = max{i ∈ ℤ | i ≤ r},
⟨r⟩ = r − [r].

Definition 6.4. Let (h, g, t) be a semi-Weierstrass representation over a space germ (B, W, P). We write P = {x_{1}, x_{2}, …, x_{n}}, Z = k[[P]]. Let m = deg(g, t) and G_{1}, G_{2}, …, G_{m} ∈ Z be the coefficient function sequence of (g, t).

g = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j}.

1. We say that a semi-Weierstrass pair (g, t) satisfies the Abhyankar condition, if the following two conditions are simultaneously satisfied:

(a) m = deg(g, t) ≥ 1.

(b) There exist n non-negative integers b_{1}, b_{2}, …, b_{n} such that (i)

\frac{G_{j}}{(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}} ∈ Z

for every 1 ≤ j ≤ m, and (ii)

min\{\frac{1}{j} ord(\frac{G_{j}}{(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}}) | 1 ≤ j ≤ m\} < 1.

The non-negative integers b_{1}, b_{2}, …, b_{n} in (b) are called the characteristic numbers of (g, t).

2. We say that (g, t) satisfies the strong Abhyankar condition, if the following three conditions are simultaneously satisfied:

(c) Let J = {j ∈ ℤ | 1 ≤ j ≤ m, G_{j} ≠ 0}. For every j ∈ J there exist n integers b_{j1}, b_{j2}, …, b_{jn} with

\frac{G_{j}}{x_{1}^{b_{j1}}x_{2}^{b_{j2}} ⋯ x_{n}^{b_{jn}}} ∈ Z^{∗}.

(d) The subset

E = \{e ∈ J | \frac{G_{j}^{m!/j}}{G_{e}^{m!/e}} ∈ Z for every j ∈ J\}

of J is not empty.

(e) There exists an element e ∈ E with

∑_{i=1}^{n}⟨\frac{b_{ei}}{e} ⟩ < 1.

Remark . The strong Abhyankar condition is due to Abhyankar. (Abhyankar [2], p.285–p.299, Appendix.)

Lemma 6.5. We consider conditions in Definition 6.4.

1. If (g, t) satisfies the Abhyankar condition, then characteristic numbers b_{1}, b_{2}, …, b_{n} are uniquely defined depending on (g, t). In other words, n numbered non-negative integers b_{1}, b_{2}, …, b_{n} satisfying (b) are unique, if they exist.

2. If (g, t) satisfies the Abhyankar condition, and if (g, t) is a Weierstrass pair, then at least one of characteristic numbers b_{1}, b_{2}, …, b_{n} is not zero.

3. If (g, t) satisfies the Abhyankar condition, then w(g, t) < ∞.

4. Non-negative integers b_{ji}, j ∈ J, 1 ≤ i ≤ n in (c) are uniquely determined depending on (g, t), if they exist. Therefore, if (c) is satisfied, then (e) has definite meaning.

5. If (c) and (d) are satisfied, then for a fixed number i with 1 ≤ i ≤ n, the

rational number

\frac{b_{ei}}{e}

does not depend on the choice of an element e ∈ E. Therefore, the inequality in (e) holds for every element e ∈ E.

Proposition 6.6. Let (g, t) be a semi-Weierstrass pair of a ring k[[x]] over a partial parameter space P. If (g, t) satisfies the strong Abhyankar condition, then (g, t) satisfies the Abhyankar condition. In other words, the strong Abhyankar condition implies the Abhyankar condition.

Theorem 6.7 (Abhyankar [2], p.285–p.299, Appendix.). Let (h, g, t) be a Weierstrass representation over a framed space germ (B, W, P) satisfying w(g, t) < ∞.

1. Assume that RW (B, W, P) is true. Then, there exist an IAMT (B′, W′, P′) with a normal crossing of (B, W, P) such that the pull-back of (h, g, t) by (B, W, P) → (B′, W′, P′) is a Weierstrass representation over (B′, W′, P′) satisfying the condition (c) and (d) in Definition 6.4.

2. Assume that (h, g, t) satisfies (c) and (d) in Definition 6.4. Then, there exist an IEMT (B′, W′, P′) of (B, W, P) such that the pull-back of (h, g, t) by the associated morphism (B, W, P) → (B′, W′, P′) is a Weierstrass representation over (B′, W′, P′) satisfying the strong Abhyankar condition.

Proof. Let m = deg(g, t), G_{1}, G_{2}, …, G_{m} ∈ k[[P]] be the coefficient function sequence of (g, t), P = {x_{1}, x_{2}, …, x_{n}}, and J = {j ∈ ℤ | 1 ≤ j ≤ m, G_{j} ≠ 0}. Since w(g, t) < ∞, we have J ≠ ∅.

1. Since RW (B, W, P) is true, also RS(B, W, P) is true by Corollary 5.13. Consider finite elements G_{j}^{m!/j}, j ∈ J and x_{1}, x_{2}, …, x_{n}. Claim 1 follows from RS(B, W, P).

2. It follows from Lemma 6.8 below.

Lemma 6.8 (Abhyankar [2], p.290, Reduction Lemma). Let (B, W, P) be a framed space germ over k, G_{1}, G_{2}, …, G_{m} ∈ k[[P]] be a finite number of non-zero elements, and e be a positive integer. Assume that the product G = G_{1}G_{2} ⋯ G_{m} has a normal crossing with respect to P.

Then, there exists an IEMT (B′, W′, P′) of (B, W, P) such that the non-negative integers b_{11}, b_{12}, …, b_{1n} defined by the condition

\frac{G′_{1}}{y_{1}^{b_{11}}y_{2}^{b_{12}} ⋯ y_{n}^{b_{1n}}} ∈ k[[P′]]^{∗}

satisfy

∑_{i=1}^{n}⟨\frac{b_{1i}}{e} ⟩ < 1,

where we write P′ = {y_{1}, y_{2}, …, y_{n}}, and for 1 ≤ i ≤ m by G′_{i} ∈ k[[P′]] we denote the image of G_{i} ∈ k[[P]] by k[[P]] → k[[P′]].

Remark . Since (B′, W′, P′) is an IEMT of (B, W, P), the associated homomorphism k[[P]] → k[[P′]] preserves normal crossings (Lemma 5.11.3) and the image G′ = G′_{1}G′_{2} ⋯ G′_{m} ∈ k[[P′]] of G by k[[P]] → k[[P′]] has a normal crossing with respect to P′.

Let α_{1}, α_{2}, …, α_{m} be positive integers. Assume, moreover, that the ideal in k[[P]] generated by G_{1}^{α_{1}}, G_{2}^{α_{2}}, …, G_{m}^{α_{m}} is a principal ideal generated by G_{1}^{α_{1}}. Then, obviously, the ideal in k[[P′]] generated by (G′_{1})^{α_{1}}, (G′_{2})^{α_{2}}, …, (G′_{m})^{α_{m}} is a principal ideal generated by (G′_{1})^{α_{1}}, since there is a ring homomorphism k[[P]] → k[[P′]].

We can check that Abhyankar’s proof is valid under our formulation with framed space germs.

Corollary 6.9. Consider the same situation as in Theorem 6.7. We assume that RW (B, W, P) is true and w(g, t) < ∞. Then, there exist an IAMT (B′, W′, P′) with a normal crossing of (B, W, P) such that the pull-back of (h, g, t) by (B, W, P) → (B′, W′, P′) is a Weierstrass representation over (B′, W′, P′) satisfying the strong Abhyankar condition.

We introduce another condition “reductive” for semi-Weierstrass pairs, which is essential in the theory of positive characteristic.

Lemma 6.10. Let (h, g, t) be a semi-Weierstrass representation over a framed space germ (B, W, P), m = deg(g, t), and G_{1}, G_{2}, …, G_{m} ∈ k[[P]] be the coefficient function sequence of the semi-Weierstrass pair (g, t).

1. For every element H ∈ k[[P]] the triplet (h, g, t + H) (resp. the pair (g, t + H)) is also a semi-Weierstrass representation over the framed space germ (B, W, P) (resp. a semi-Weierstrass pair of k[[P ∪ {t}]] over P). deg(g, t) = deg(g, t + H).

2. Let H ∈ k[[P]], and G′_{1}, G′_{2}, …, G′_{m} ∈ k[[P]] be the coefficient function sequence of the semi-Weierstrass pair (g, t + H). Let G_{0} = G′_{0} = 1. We have

G′_{j} = ∑_{i=0}^{j} \binom{m − i}{m − j}(−H)^{j−i}G_{i},

G_{j} = ∑_{i=0}^{j} \binom{m − i}{m − j}H^{j−i}G′_{i},

for every 1 ≤ j ≤ m.

Definition 6.11. Let (B, W, P) be a framed space germ over k.

1. Let t ∈ B be an element such that t ∉ P and P ∪ {t} is a partial parameter system of B. The coordinate transformation (B, W, P ∪ {t}) ↦ (B, W, P ∪ {t + H}) defined for any element H ∈ k[[P]] is called the parallel coordinate transformation of (B, W, P ∪ {t}) along H.

2. Let (h, g, t) be a semi-Weierstrass representation over (B, W, P), and H ∈ k[[P]]. The semi-Weierstrass representation (h, g, t + H) (resp. the semi-Weierstrass pair (g, t + H)) is called the parallel coordinate transform of (h, g, t) (resp. (g, t)) along H.

3. An element H ∈ k[[P]] with w(g, t) < w(g, t+H) is called a reducing element of (h, g, t) or of (g, t).

4. If (h, g, t) or (g, t) has a reducing element H ∈ k[[P]], then we say that (h, g, t) or (g, t) is reductive.

Lemma 6.12. Let n be a positive integer and A be a ring isomorphic to the formal power series ring over k with n + 1 variables. Let P = {x_{1}, x_{2}, …, x_{n}} be a partial parameter system of A with ♯ P = n, and (g, t) be a semi-Weierstrass pair of A over P. Let m = deg(g, t) and G_{1}, G_{2}, …, G_{m} ∈ k[[P]] be the coefficient function sequence of (g, t).

1. For any reducing element H ∈ k[[P]] of (g, t), we have ord(H) = w(g, t) < ∞.

2. For an element H ∈ k[[P]] the following two conditions are equivalent: (a) H is a reducing element of (g, t).

(b) m ≥ 1, H ≠ 0 and in(g, t) = (t + in(H))^{m}.

3. The following four conditions are equivalent:

(a) Pair (g, t) is reductive.

(b) Pair (g, t) is reductive, w(g, t) < ∞ and m ≥ 1.

(c) m ≥ 1, there exists an element H_{0} ∈ k[[P]] such that H_{0} ≠ 0 and in(g, t) = (t + H_{0})^{m}.

(d) There exists a unique element H_{0} ∈ k[[P]] satisfying in(g, t) = (t+H_{0})^{m}, and H_{0} is a non-zero homogeneous polynomial.

4. One and only one of following two claims holds:

(a) Pair (g, t) is not reductive.

(b) Pair (g, t) is reductive, and there exists an element H ∈ k[[P]] such that

(g, t + H) is not reductive.

Below we assume that (g, t) is reductive. Let m = lq be the p-decomposition of m. (Refer to the latter half of Introduction for the definition.)

5. Weight w(g, t) is an integer.

6. For an element H ∈ k[[P]] the following two conditions are equivalent: (a) H is a reducing element of (g, t).

(b) H ≠ 0 and in(G_{q}) = l in(H)^{q}.

Proof. Note Lemma 6.10.2 and Lemma 2.4.3. For every integer j with 1 ≤ j ≤ m

and j ≢ 0 mod q we have \binom{m}{j} = 0. For an integer j with 1 ≤ j ≤ m and j ≡ 0

mod q we have \binom{m}{j} = \binom{lq}{iq} = \binom{l}{i}, where i = j/q. In particular, \binom{m}{q} = \binom{l}{1} =

l ≠ 0.

By Lemma 2.3.7, we obtain implication 2.(b)⇒(a).

7. The first reduction

In this section following Abhyankar’s ideas in [2], p.285–p.299, Appendix, we add our concept “reductive” to his ideas. We will show that the Abhyankar condition plus the condition “not reductive” implies that the singularity can be resolved even in characteristic positive.

Theorem 7.1 (Abhyankar [2], p.292–p.293, Good Point Lemma.). Let (B, W, Q) be a framed space germ, f ∈ k[[Q]] be a non-zero element, and (h, g, t) be a Weierstrass representation of f under (B, W, Q).

We write Q − \{t\} = P = \{x_{1}, x_{2}, …, x_{n}\}, and (Z, U, P) = \underline{(B, W, P)}. Let F denote the quotient field of B, v : F → Γ ∪ {∞} be the valuation defined by W, m = deg(g, t), and G_{1}, G_{2}, …, G_{m} ∈ Z be the coefficient function sequence of (g, t).

Assume the following two conditions: ⟨A⟩ For every integer j with 1 ≤ j ≤ m G_{j}/x_{1}^{j} ∈ Z. ⟨B⟩ v(t) ≥ v(x_{1}).

Let (B′, W′, Q′) be the elementary monoidal transform of (B, W, Q) associated with the datum ({t, x_{1}}, x_{1}, W′). We identify elements in F and their images in the quotient field of B′ by the associated homomorphism.

1. There exists a unique element λ ∈ k satisfying Q′ = {(t/x_{1}) − λ} ∪ P. λ = 0 if and only if v(t) > v(x_{1}).

2. (Z, U, P) = \underline{(B′, W′, P)}.

3. Let

t′ = \frac{t}{x_{1}} − λ, g′ = \frac{g}{x_{1}^{m}} , h′ = hx_{1}^{m}.

The triplet (h′, g′, t′) is a semi-Weierstrass representation of f over (B′, W′, P′). deg(g′, t′) = deg(g, t).

4. ord(g′) ≤ ord(g). Besides, ord(g′) = ord(g) if and only if (h′, g′, t′) is a Weierstrass representation of f.

5. If (h, g, t) is not reductive and if ord(g′) = ord(g) > 0, then λ = 0 and (h′, g′, t′) is not reductive, either.

6. If λ = 0 and if (h, g, t) satisfies the Abhyankar condition, then also (h′, g′, t′) satisfies the Abhyankar condition.

7. If λ = 0 and if w(g, t) < ∞, then w(g′, t′) = w(g, t) − 1. If λ = 0 and if w(g, t) = ∞, then w(g′, t′) = ∞.

8. sord(g′) ≤ sord(g).

Proof. We have k[[Q′]] = k[[t′, x_{1}, x_{2}, …, x_{n}]] ⊃ k[[Q]] = k[[t, x_{1}, x_{2}, …, x_{n}]] ⊃ Z = k[[x_{1}, x_{2}, …, x_{n}]].

3. Obviously f = h′g′. Since g = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j} and t = (t′ + λ)x_{1}, we have

g′ = \frac{g}{x_{1}^{m}} = (t′ + λ)^{m} + ∑_{j=1}^{m} \frac{G_{j}}{x_{1}^{j}} (t′ + λ)^{m−j}.

By assumption ⟨A⟩ one knows g′ ∈ k[[Q′]].

Since (h, g, t) is a Weierstrass representation, we have non-negative integers s_{1}, s_{2}, …, s_{n} with

\frac{h′}{x_{1}^{s_{1}+m}x_{2}^{s_{2}} ⋯ x_{n}^{s_{n}}} = \frac{h}{x_{1}^{s_{1}}x_{2}^{s_{2}} ⋯ x_{n}^{s_{n}}} ∈ k[[Q]]^{∗} ⊂ k[[Q′]]^{∗}.

Let

G′_{j} = \binom{m}{m − j}λ^{m} + ∑_{i=1}^{j} \binom{m − i}{m − j}λ^{j−i} \frac{G_{j}}{x_{1}^{i}}

for 1 ≤ j ≤ m. By ⟨A⟩ one has G′_{j} ∈ Z for 1 ≤ j ≤ m. One can check g′ = (t′)^{m} + ∑_{i=1}^{m} G′_{j}(t′)^{m−j}.

4. By 3 and by Lemma 2.20.2 ord(g′) ≤ deg(g′, t′) = m = ord(g).

5. Since m = ord(g′) = ord(g) > 0, for every 1 ≤ j ≤ m we have ord(G′_{j}) ≥ j, and G′_{j} ∈ M(Z).

On the other hand, we have

\frac{G_{j}}{x_{1}^{j}} = \binom{m}{m − j}(−λ)^{j} + ∑_{i=1}^{j} \binom{m − i}{m − j}(−λ)^{j−i}G′_{i}.

Thus, for every 1 ≤ j ≤ m there is an element R_{j} ∈ M(Z) with

G_{j} = \binom{m}{m − j}(−λ)^{j}x_{1}^{j} + x_{1}^{j}R_{j}.

Assume λ ≠ 0. We have in(g, t) = (t − λx_{1})^{m}. By Lemma 6.12.3 (g, t) is reductive, which contradicts the assumption.

Assume λ = 0. For every 1 ≤ j ≤ m we have

G′_{j} = \frac{G_{j}}{x_{1}^{j}} . (*)

Let φ(t) = in(g, t). By (*) one has φ(x_{1}t′) = x_{1}^{m}in(g′, t′).

Assume that (h′, g′, t′) is reductive. By Lemma 6.12.3 there is an element H ∈ Z with H ≠ 0 and in(g′, t′) = (t′ + H)^{m}. We have in(g, t) = (t + x_{1}H)^{m}. Since 0 ≠ x_{1}H ∈ Z, we conclude (h, g, t) is reductive, which contradicts the assumption.

8. We can consider the order ord_{w} : k[[Q]] − {0} → ℤ_{0} associated with the weight w = (1, 1, 0, 0, …, 0) and the parameter system Q = {x_{0} = t, x_{1}, x_{2}, …, x_{n}} of k[[Q]]. Since G_{j}/x_{1}^{j} ∈ Z for 1 ≤ j ≤ m, we have ord_{w}(g) = deg(g, t) = m.

Let δ = nord(g). By Lemma 2.21.3 we can write

g = (∏_{ν=1}^{δ}(t − H_{ν})) g_{s}, g_{s} = t^{m−δ} + ∑_{j=1}^{m−δ} L_{j}t^{m−δ−j}

with some elements H_{1}, H_{2}, …, H_{δ}, L_{1}, L_{2}, …, L_{m−δ} ∈ Z. We have ∑_{ν=1}^{δ} ord_{w}(t− H_{ν}) + ord_{w}(g_{s}) = ord_{w}(g) = deg(g, t) = ∑_{ν=1}^{δ} deg(t − H_{ν}, t) + deg(g_{s}, t). Since ord_{w}(t − H_{ν}) ≤ deg(t − H_{ν}, t) and ord_{w}(g_{s}) ≤ deg(g_{s}, t) by definition, we know ord_{w}(t−H_{ν}) = deg(t−H_{ν}, t) = 1 for 1 ≤ ν ≤ δ and ord_{w}(g_{s}) = deg(g_{s}, t) = m−δ. The last claim is equivalent to the following two conditions hold:

1. H′_{ν} = H_{ν}/x_{1} ∈ Z for every 1 ≤ ν ≤ δ.

2. L′_{j} = L_{j}/x_{1}^{j} ∈ Z for every 1 ≤ j ≤ m − δ. Thus we can write

g = (∏_{ν=1}^{δ}(t + x_{1}H′_{ν})) g_{s}, g_{s} = t^{m−δ} + ∑_{j=1}^{m−δ} x_{1}^{j}L′_{j}t^{m−δ−j}.

Substituting t = (t′ + λ)x_{1}, we obtain

g′ = (∏_{ν=1}^{δ}(t′ + λ + H′_{ν})) g′_{s}, g′_{s} = (t′ + λ)^{m−δ} + ∑_{j=1}^{m−δ} L′_{j}(t′ + λ)^{m−δ−j}.

Since ord(t′ + λ + H′_{ν}) = 0 or 1 for every 1 ≤ ν ≤ δ, we have sord(g′) = sord(g′_{s}) ≤ ord(g′_{s}) ≤ m − δ = ord(g) − nord(g) = sord(g).

Theorem 7.2. Let (B, W, Q) be a framed space germ, f ∈ k[[Q]] be a non-zero element, and (h, g, t) be a Weierstrass representation of f under (B, W, Q).

Assume that (h, g, t) satisfies the Abhyankar condition and is not reductive. Then, for some elementary monoidal transform (B′, W′, Q′) of (B, W, Q) and

for some elements h′ ∈ k[[Q′]], g′ ∈ k[[Q′]] and t′ ∈ Q′ one of the following four conditions holds: By f′ ∈ k[[Q′]] we denote the image of f ∈ k[[Q]] by the associated homomorphism k[[Q]] → k[[Q′]]. ⟨α⟩ The element f′ has a normal crossing. ⟨β⟩ sord(f′) < sord(f). ⟨γ⟩ sord(f′) = sord(f), nord(g′) < nord(g) and (h′, g′, t′) is a Weierstrass repre-

sentation of f′ under (B′, W′, Q′). ⟨δ⟩ sord(f′) = sord(f), nord(g′) = nord(g) and (h′, g′, t′) is a Weierstrass repre-

sentation of f′ under (B′, W′, Q′). Besides, (h′, g′, t′) satisfies the Abhyankar condition, (h′, g′, t′) is not reductive and 1 ≤ w(g′, t′) = w(g, t) − 1.

Remark . By assumption and by Lemma 6.2.2 sord(f) = sord(g). If (h′, g′, t′) is a semi-Weierstrass representation of f′, then sord(f′) = sord(g′) by Lemma 6.2.2. Therefore, in condition ⟨γ⟩ and ⟨δ⟩, the condition sord(f′) = sord(f) is equivalent to sord(g′) = sord(g).

Proof. We write Q = {t, x_{1}, x_{2}, …, x_{n}}, Z = k[[x_{1}, x_{2}, …, x_{n}]], m = deg(g, t) = ord(g). Let G_{1}, G_{2}, …, G_{m} ∈ Z be the coefficient function sequence of (g, t), F be the quotient field of B, and v : F → Γ ∪ {∞} be the valuation defined by W.

First, note that m ≥ 1 and w(g, t) < ∞ by the Abhyankar condition.

By Lemma 6.5.2 there is an integer i with 1 ≤ i ≤ n such that G_{j}/x_{i}^{j} ∈ Z for every 1 ≤ j ≤ m.

Exchanging the numbering of x_{i}’s, we can assume that G_{j}/x_{1}^{j} ∈ Z for every 1 ≤ j ≤ m. The assumption ⟨A⟩ in Theorem 7.1 is satisfied.

Case v(t) < v(x_{1}). Let (B′, W′, Q′) be the elementary monoidal transform of (B, W, Q) associated with the datum ({t, x_{1}}, t, W′). We identify elements in F and their images in the quotient field of B′ by the associated morphism. Let t′ = t, x′_{1} = x_{1}/t and x′_{i} = x_{i} for 2 ≤ i ≤ n. We have Q′ = {t′, x′_{1}, x′_{2}, …, x′_{n}}, and k[[Q′]] = k[[t′, x′_{1}, x′_{2}, …, x′_{n}]] ⊃ k[[Q]] = k[[t, x_{1}, x_{2}, …, x_{n}]] ⊃ Z.

g = t^{m} + ∑_{j=1}^{m} \frac{G_{j}}{x_{1}^{j}} · x_{1}^{j}t^{m−j} = (t′)^{m} \{1 + ∑_{j=1}^{m} \frac{G_{j}}{x_{1}^{j}} (x′_{1})^{j}\}

Thus g/(t′)^{m} ∈ k[[Q′]]^{∗}. On the other hand,

\frac{h}{x_{1}^{s_{1}}x_{2}^{s_{2}} ⋯ x_{n}^{s_{n}}} = \frac{h}{(t′)^{s_{1}}(x′_{1})^{s_{1}}(x′_{2})^{s_{2}} ⋯ (x′_{n})^{s_{n}}} ∈ k[[Q]]^{∗} ⊂ k[[Q′]]^{∗},

where s_{1}, s_{2}, …, s_{n} are non-negative integers in Definition 6.1.2. Thus f′ = f = hg has a normal crossing in k[[Q′]]. The condition ⟨α⟩ holds in this case.

Case v(t) ≥ v(x_{1}). The assumption ⟨B⟩ in Theorem 7.1 is satisfied in this case, and we can apply Theorem 7.1. Let (B′, W′, Q′) be the elementary monoidal transform of (B, W, Q) associated with the datum ({t, x_{1}}, x_{1}, W′). Let t′, g′, and h′ be the elements in Theorem 7.1.3. By Theorem 7.1.3 (h′, g′, t′) is a semi-Weierstrass representation of f′. We have sord(f′) = sord(g′).

First, we show that in the case ord(g′) = ord(g) one of ⟨α⟩, ⟨β⟩, ⟨δ⟩ holds.

By Theorem 7.1.4 (h′, g′, t′) is a Weierstrass representation of f′. By Theorem 7.1.5 and by the assumption we have λ = 0. By Theorem 7.1.5, 6, 7, (h′, g′, t′) satisfies the Abhyankar condition, (h′, g′, t′) is not reductive, and w(g′, t′) = w(g, t)− 1. By Lemma 2.20.4 we have w(g′, t′) ≥ 1.

Assume that ⟨α⟩ and ⟨β⟩ do not hold. Since ⟨β⟩ does not hold, we have sord(f′) ≥ sord(f), which is equivalent to sord(g′) ≥ sord(g). On the other hand, by Theorem 7.1.8, we have sord(g′) ≤ sord(g). Thus sord(g′) = sord(g), which is equivalent to sord(f′) = sord(f). By assumption and Lemma 2.14.2 we obtain nord(g′) = nord(g).

Second, we show that if ord(g′) ≠ ord(g), then one of ⟨α⟩, ⟨β⟩, ⟨γ⟩ holds. By Theorem 7.1.4 we have ord(g′) < ord(g)

Assume that ⟨α⟩ and ⟨β⟩ do not hold. By the same argument as above, we obtain sord(f′) = sord(g′) = sord(f) = sord(g) by Theorem 7.1.8. Since ord(g′) < ord(g), we have nord(g′) = ord(g′) − sord(g′) < ord(g) − sord(g) = nord(g).

Below, we will show that there are elements h′_{1}, g′_{1} ∈ k[[Q′]] such that the triplet (h′_{1}, g′_{1}, t′) is a Weierstrass representation of f′ under (B′, W′, P′) and nord(g′_{1}) ≤ nord(g′). The triplet (h′_{1}, g′_{1}, t′) satisfies ⟨γ⟩.

Let δ = nord(g). By the proof of Theorem 7.1.8 we know that there are elements H′′_{1}, H′′_{2}, …, H′′_{δ}, L′′_{1}, L′′_{2}, …, L′′_{m−δ} ∈ Z satisfying

g′ = (∏_{ν=1}^{δ}(t′ + H′′_{ν})) g′_{s}, g′_{s} = (t′)^{m−δ} + ∑_{j=1}^{m−δ} L′′_{j}(t′)^{m−δ−j}.

We have sord(g′) ≤ sord(g′_{s}) ≤ ord(g′_{s}) ≤ m − δ = sord(g). Since sord(g′) = sord(g) under our assumption, we have sord(g′_{s}) = ord(g′_{s}) = m − δ = deg(g′_{s}, t′). In particular, (g′_{s}, t′) is a Weierstrass pair.

Let

Δ = {ν ∈ ℤ | 1 ≤ ν ≤ δ, H′′_{ν} ∉ M(Z)}
Γ = {ν ∈ ℤ | 1 ≤ ν ≤ δ, H′′_{ν} ∈ M(Z)}

h′_{1} = h′ ( ∏_{ν∈Δ}(t′ + H′′_{ν})) , g′_{1} = (∏_{ν∈Γ}(t′ + H′′_{ν})) g′_{s}.

One can check that (h′_{1}, g′_{1}, t′) is a Weierstrass representation of f′ under (B′, W′, P′). We have nord(g′) = nord(g′_{1}) + ∑_{ν∈Δ} nord(t′ + H′′_{ν}) = nord(g′_{1}).

Definition 7.3. A collection C of framed space germs (B, W, Q) over k is called an admissible class of framed space germs over k, if C satisfies the following two conditions:

1. If (B, W, Q) ∈ C, then for every non-zero element f ∈ k[[Q]] there exist a coordinate transform (B, W, Q′) of (B, W, Q) and a Weierstrass representation (h, g, t) of f under (B, W, Q′) such that (B, W, Q′ − {t}) ∈ C.

2. If (B, W, Q) ∈ C, and if (B′, W′, Q′) is an IAMT of (B, W, Q), then (B′, W′, Q′) ∈ C.

The collection of all framed space germs over k is, of course, an admissible class. Let C_{0} be the collection of framed space germs (B, W, Q) over k such that the

underlying space germ (A, V) = \underline{(B, W, Q)}_{•} of the basic framed space germ of it is of iterated divisor type. Recall Definition 4.27 here. A space germ (A, V) is said to be of iterated divisor type, if it satisfies dim A = rank V. In the next section we will show that the collection C_{0} is an admissible class. (Corollary 8.3.)

Let (B, W, P) be a framed space germ over k. We consider the following statement AN(B, W, P) on (B, W, P).

AN(B, W, P) : For every Weierstrass representation (h, g, t) over (B, W, P), there exist an IAMT (B′, W′, P′) with a normal crossing of (B, W, P) and an element H′ ∈ k[[P′]] satisfying the following condition: Let (h′, g′, t′) denote the pull-back of (h, g, t) by (B, W, P) → (B′, W′, P′). The triplet (h′, g′, t′ + H′) is a Weierstrass representation over (B′, W′, P′). Besides, if w(g′, t′ +H′) < ∞, then (h′, g′, t′ +H′) satisfies the Abyankar condition and moreover (h′, g′, t′ + H′) is not reductive.

Recall the statement RW (B, W, Q) we gave before Lemma 5.12.

Theorem 7.4. Let C be an admissible class of framed space germs over k, and r be a positive integer. Assume that AN(B, W, P) is true for every framed space germ (B, W, P) such that (B, W, P) ∈ C and ♯ P ≤ r. Then, RW (B, W, Q) is true for every framed space germ (B, W, Q) such that (B, W, Q) ∈ C and ♯ Q ≤ r + 1.

Proof. Assume that there is a framed space germ (B, W, Q) such that (B, W, Q) ∈ C, ♯ Q ≤ r + 1, and RW (B, W, Q) is false. There is a non-zero element f ∈ k[[Q]] such that for any IAMT (B′, W′, Q′) of (B, W, Q) the image f′ ∈ k[[Q′]] by the associated homomorphism k[[Q]] → k[[Q′]] does not have a normal crossing.

Note that the following three claims hold under these assumptions:

1. For any IAMT (B′, W′, Q′) of (B, W, Q) and for any Weierstrass representation (h′, g′, t′) of f′ under (B′, W′, Q′), we have w(g′, t′) < ∞.

2. For any IAMT (B′, W′, Q′) of (B, W, Q), there exist a coordinate transform (B′, W′, Q′_{1}) of (B′, W′, Q′) and a Weierstrass representation (h′, g′, t′) of f′ under (B′, W′, Q′_{1}) such that (B′, W′, Q′_{1} − {t′}) ∈ C.

3. For any IAMT (B′, W′, Q′) of (B, W, Q) and for any Weierstrass representation (h′, g′, t′) of f′ under (B′, W′, Q′) such that (B′, W′, Q′ −{t′}) ∈ C, there exist an IAMT (B′′, W′′, Q′′) of (B′, W′, Q′) and a Weierstrass representation (h′′, g′′, t′′) of f′′ under (B′′, W′′, Q′′) such that (B′′, W′′, Q′′ − {t′′}) ∈ C, (h′′, g′′, t′′) satisfies the Abhyankar condition, (h′′, g′′, t′′) is not reductive, ord(g′′) = ord(g′), and sord(g′′) ≤ sord(g′), where by f′′ ∈ k[[Q′′]] we denote the image of f′ ∈ k[[Q′]] by the associated homomorphism k[[Q′]] → k[[Q′′]].

We will show claim 1. Assume that for some IAMT (B′, W′, Q′) of (B, W, Q) and for some Weierstrass representation (h′, g′, t′) of f′ under (B′, W′, Q′), we have w(g′, t′) = ∞. Then, g′ = (t′)^{m} for some non-negative integer m. Let Q′ = {t′, x′_{1}, x′_{2}, …, x′_{n}}. Since (h′, g′, t′) is a Weierstrass representation, there are nonnegative integers s_{1}, s_{2}, …, s_{n} with

\frac{h′g′}{(t′)^{m}(x′_{1})^{s_{1}}(x′_{2})^{s_{2}} ⋯ (x′_{n})^{s_{n}}} ∈ k[[Q′]]^{∗},

Thus f′ = h′g′ ∈ k[[Q′]] has a normal crossing, which contradicts the assumption. Claim 2 follows from Definition 7.3. We proceed to claim 3. Let P′ = Q′ − {t′}. Since (B′, W′, P′) ∈ C and since

♯ P′ = ♯ Q − 1 ≤ r, AN(B′, W, P′) is true. By AN(B′, W, P′) and by 1 above, there are an IAMT (B′′, W′′, P′′) with a normal crossing of (B′, W′, P′) and an element H′′ ∈ k[[P′′]] such that the triplet (h′′, g′′, t′′_{1} + H′′) is a Weierstrass representation over (B′′, W′′, P′′), (h′′, g′′, t′′_{1} + H′′) satisfies the Abhyankar condition, and (h′′, g′′, t′′_{1} + H′′) is not reductive, where (h′′, g′′, t′′_{1}) denotes the pull-back of (h′, g′, t′) by the associated morphism (B′, W′, P′) → (B′′, W′′, P′′). Since C is an admissible class, we have (B′′, W′′, P′′) ∈ C.

Let t′′ = t′′_{1} + H′′ and Q′′ = P′′ ∪ {t′′}. The triplet (B′′, W′′, Q′′) is an IAMT of (B′, W′, Q′), and (B′′, W′′, Q′′ − {t′′}) = (B′′, W′′, P′′) ∈ C. One sees that the triplet (h′′, g′′, t′′) satisfies the desired conditions by Lemma 6.3.3.

Let X be the set of pairs ((B′, W′, Q′), (h′, g′, t′)) satisfying the following conditions:

(a) The first term (B′, W′, Q′) is an IAMT of (B, W, Q).

(b) The second term (h′, g′, t′) is a Weierstrass representation of f′ under (B′, W′, Q′), where f′ ∈ k[[Q′]] denotes the image of f ∈ k[[Q]] by the associated morphism k[[Q]] → k[[Q′]]. Moreover, we assume that (B′, W′, Q′ −{t′}) ∈ C, (h′, g′, t′) satisfies the Abhyankar condition, and (h′, g′, t′) is not reductive.

By 2 and 3 above one knows X ≠ ∅.

We consider the totally ordered abelian group ℤ^{3} with the lexicographic order. By 1 with every X = ((B′, W′, Q′), (h′, g′, t′)) ∈ X we can associate the element

c(X) = (sord(g′), nord(g′), [w(g′, t′)]) ∈ ℤ^{3},

where [ ] denotes the Gauss symbol. By definition of X we have sord(g′) ≥ 0, nord(g′) ≥ 0 and [w(g′, t′)] ≥ 1. Since ℤ is discrete, there exists an element X_{0} = ((B_{0}, W_{0}, Q_{0}), (h_{0}, g_{0}, t_{0})) ∈ X such that c(X) ≥ c(X_{0}) for every X ∈ X. Let f_{0} = h_{0}g_{0} ∈ k[[Q_{0}]] denote the image of f ∈ k[[Q]].

By Theorem 7.2, there is an elementary monoidal transform (B_{1}, W_{1}, Q_{1}) of (B_{0}, W_{0}, Q_{0}) and an element h_{1} ∈ k[[Q_{1}]], g_{1} ∈ k[[Q_{1}]] and t_{1} ∈ Q_{1} and one of the four cases ⟨α⟩, ⟨β⟩, ⟨γ⟩, ⟨δ⟩ holds. Let f_{1} ∈ k[[Q_{1}]] denote the image of f_{0} ∈ k[[Q_{0}]].

Assume that case ⟨α⟩ holds. Then f_{1} has a normal crossing, which contradicts our assumption.

Assume that ⟨β⟩ holds. We have sord(f_{1}) < sord(f_{0}) = sord(g_{0}). By 2 and 3 there are an element g′_{1} ∈ k[[Q_{1}]] with sord(g′_{1}) = sord(f_{1}) and an element X_{2} = ((B_{2}, W_{2}, Q_{2}), (h_{2}, g_{2}, t_{2})) ∈ X such that (B_{2}, W_{2}, Q_{2}) is an IAMT of (B_{1}, W_{1}, Q_{1}), f_{2} = h_{2}g_{2} ∈ k[[Q_{2}]] coincides the image of f_{1} ∈ k[[Q_{1}]] by k[[Q_{1}]] → k[[Q_{2}]], and sord(g_{2}) ≤ sord(g′_{1}). We have sord(g_{2}) < sord(g_{0}), and c(X_{2}) < c(X_{0}), which contradicts the property of X_{0}.

Assume that ⟨γ⟩ holds. The triplet (h_{1}, g_{1}, t_{1}) is a Weierstrass representation of f_{1}, sord(g_{1}) = sord(g_{0}), and nord(g_{1}) < nord(g_{0}). By 3 there is an element X_{2} = ((B_{2}, W_{2}, Q_{2}), (h_{2}, g_{2}, t_{2})) ∈ X such that (B_{2}, W_{2}, Q_{2}) is an IAMT of (B_{1}, W_{1}, Q_{1}), f_{2} = h_{2}g_{2} ∈ k[[Q_{2}]] coincides the image of f_{1} ∈ k[[Q_{1}]] by k[[Q_{1}]] → k[[Q_{2}]], ord(g_{2}) = ord(g_{1}). and sord(g_{2}) ≤ sord(g_{1}). If sord(g_{2}) < sord(g_{1}), then we have sord(g_{2}) < nord(g_{0}), and c(X_{2}) < c(X_{0}). If sord(g_{2}) = sord(g_{1}), then we have nord(g_{2}) = ord(g_{2}) − sord(g_{2}) = ord(g_{1}) − sord(g_{1}) = nord(g_{1}) < nord(g_{0}). Thus we have c(X_{2}) < c(X_{0}) also in this case. Anyway we obtain c(X_{2}) < c(X_{0}), which contradicts the property of X_{0}.

Assume that ⟨δ⟩ holds. We have X_{1} = ((B_{1}, W_{1}, Q_{1}), (h_{1}, g_{1}, t_{1})) ∈ X and sord(g_{1}) = sord(g_{0}), nord(g_{1}) = nord(g_{0}), and [w(g_{1}, t_{1})] = [w(g_{0}, t_{0})] − 1. Thus we have c(X_{1}) < c(X_{0}), which contradicts the property of X_{0}.

In every case we obtained a contradiction. Thus our theorem is true.

Let C be an admissible class of framed space germs over k. By Theorem 7.4 it turns out that the problem below depending on C is essential.

Problem(C): Let r be a positive integer. Show that AN(B, W, P) is true for every framed space germ (B, W, P) such that (B, W, P) ∈ C and ♯ P ≤ r, assuming that RW (B, W, P) is true for every framed space germ (B, W, P) such that (B, W, P) ∈ C and ♯ P ≤ r.

If we could solve Problem(C), then RW (B, W, Q) would be true for every (B, W, Q) ∈ C. Note that if we drop the last condition “(h′, g′, t′ + H′) is not reductive” from AN(B, W, P), then Corollary 6.9 solves Problem(C) for the admissible class of all framed space germs.

In the remaining part of this paper, we solve the above problem for the admissible class C_{0}.

Remark . In the essential parts until here, we have never used the assumption that the characteristic number p of the ground field k is positive.

If the characteristic number of the ground field k is zero, then we can easily solve Problem(C) for the admissible class of all framed space germs.

Let (g, t) be a Weierstrass pair with w(g, t) < ∞, and G_{1}, G_{2}, …, G_{m} be the coefficient function sequence of (g, t). If the characteristic number is zero, then we can apply the parallel coordinate transformation along H = G_{1}/m to (g, t). As the result, we can assume that (g, t) is a Weierstrass pair with G_{1} = 0 from the beginning. It is easy to see that if G_{1} = 0 and if the characteristic number is zero, then any pull-back (g′, t′) of (g, t) is not reductive. Thus, by Corollary 6.9 Problem(C) can be solved.

Any hypersurface singularity of a germ of a variety in characteristic zero can be resolved by iterated monoidal transformations.

In Abhyankar [2] the parallel coordinate transformation along H = G_{1}/m is called the Shreedharacharya transformation.

8. The admissible class C_{0}

Recall that by C_{0} we denoted the collection of all framed space germs (B, W, Q) over k such that the equality dim A = rank V holds for (A, V) = \underline{(B, W, Q)}_{•}. (a paragraph after Definition 7.3.)

Lemma 8.1. Let (B, W, Q) be a framed space germ with (B, W, Q) ∈ C_{0}, and f ∈ k[[Q]] be a non-zero element. There are a coordinate transform (B, W, Q′) and a Weierstrass representation (h, g, t) of f under (B, W, Q′) such that h ∈ k[[Q′]]^{∗} and (B, W, Q′ − {t}) ∈ C_{0}.

Proof. Let (A, V) = \underline{(B, W, Q)}_{•} and n + 1 = dim A = rank V. A = k[[Q]]. By 1 and 2 of Corollary 4.15 the valuation ring V has n + 2 prime ideals 𝔭_{i}, and we can assume that they satisfy

{0} = 𝔭_{0} ⊂ 𝔭_{1} ⊂ ⋯ ⊂ 𝔭_{i−1} ⊂ 𝔭_{i} ⊂ ⋯ ⊂ 𝔭_{n+1} = M(V).

For every integer i with 1 ≤ i ≤ n+1 we choose an element a_{i} ∈ 𝔭_{i}−𝔭_{i−1}. Let b ∈ A be a non-zero element with ba_{i} ∈ A for every 1 ≤ i ≤ n + 1. Since A and V have the common quotient field, there exists b satisfying this condition. Let c_{i} = ba_{i} ∈ A for every 1 ≤ i ≤ n + 1. We have a_{i} ≠ 0 and c_{i} ≠ 0, since a_{i} ∉ 𝔭_{0} = {0}.

By Proposition 2.18 there exist a parameter system Q′ = {t, x_{1}, x_{2}, …, x_{n}} and elements h, g, ϵ, b̃, δ_{1}, c̃_{1}, δ_{2}, c̃_{2}, …, δ_{n+1}, c̃_{n+1} satisfying the following three conditions:

1. h, ϵ, δ_{1}, δ_{2}, …, δ_{n+1} ∈ A^{∗}.

2. f = hg, b = ϵb̃, and c_{i} = δ_{i}c̃_{i} for every 1 ≤ i ≤ n + 1.

3. (g, t), (b̃, t) and (c̃_{i}, t) for 1 ≤ i ≤ n + 1 are Weierstrass pairs of A under Q′. The triplet (h, g, t) is a Weierstrass representation of f under (B, W, Q′) such that h ∈ A^{∗} = k[[Q′]]^{∗}.

Let P = Q′ − \{t\}. Let (Z, U, P) = \underline{(B, W, P)}, and D denote the quotient field of Z. Below we will show (B, W, P) ∈ C_{0}, or in other words, dim Z = rank U.

Let Ã = Z[t], Ẽ = D(t) be the quotient field of Ã, and Ṽ = V ∩ Ẽ. Ã is a finitely generated Z-subalgebra of A, Ẽ is a subfield of the quotient field of A, D ⊂ Ẽ, and tr.deg_{D}Ẽ = 1. By Lemma 4.13.1 Ṽ is a valuation ring of the field Ẽ. g, b̃, c̃_{1}, c̃_{2}, …, c̃_{n+1} ∈ Ã ⊂ Ṽ.

We will show rank Ṽ = n + 1.

Let 𝔭̃_{i} = 𝔭_{i} ∩ Ṽ for every integer i with 0 ≤ i ≤ n + 1. For every i 𝔭̃_{i} is a prime ideal of Ṽ, and they satisfy

{0} = 𝔭̃_{0} ⊂ 𝔭̃_{1} ⊂ ⋯ ⊂ 𝔭̃_{i−1} ⊂ 𝔭̃_{i} ⊂ ⋯ ⊂ 𝔭̃_{n+1}.

Since k ⊂ Ṽ /𝔭̃_{n+1} ⊂ V/𝔭_{n+1} = V/M(V) ≅ k, one knows 𝔭̃_{n+1} = M(Ṽ), and Ṽ /𝔭̃_{n+1} ≅ k. b̃ ≠ 0, since ϵb̃ = b ≠ 0. Let ã_{i} = c̃_{i}/b̃ ∈ Ẽ for every 1 ≤ i ≤ n + 1. We have ϵ/δ_{i} ∈ A^{∗} ⊂ V^{∗}, since ϵ ∈ A^{∗} and δ_{i} ∈ A^{∗}. Thus ã_{i} = (ϵ/δ_{i})a_{i} ∈ (𝔭_{i} − 𝔭_{i−1}) ∩ Ẽ = 𝔭̃_{i} − 𝔭̃_{i−1} for every 1 ≤ i ≤ n + 1. One knows 𝔭̃_{i} ≠ 𝔭̃_{i−1} for every 1 ≤ i ≤ n + 1. Since Ṽ contains mutually different n + 2 prime ideals 𝔭̃_{i}, one concludes rank Ṽ ≥ n + 1.

Let 𝔮 = M(Ṽ) ∩ Ã. We apply Theorem 2.1 to Ã ⊃ Z and a prime ideal 𝔮 ∈ Spec(Ã). Note that Z is universally catenary by Theorem 2.2. Since k ⊂ Z/𝔮∩Z ⊂ Ã/𝔮 ⊂ Ṽ /M(Ṽ) ≅ k, one has Z/𝔮 ∩ Z ≅ Ã/𝔮 ≅ k. tr.deg_{Z} Ã = tr.deg_{D}Ẽ = 1. Since Z/𝔮∩Z ≅ k, we have 𝔮∩Z = M(Z) and ht(𝔮∩Z) = ht(M(Z)) = dim Z = n.

By Theorem 2.1 one concludes ht(𝔮) = ht(𝔮∩Z)+tr.deg_{Z} Ã−tr.deg_{(Z/𝔮∩Z)}(Ã/𝔮) = n + 1 and dim Ã_{𝔮} = n + 1. Since the valuation ring Ṽ dominates the noetherian local domain Ã_{𝔮}, we have rank Ṽ ≤ n + 1 by Corollary 4.22. Thus rank Ṽ = n + 1.

By Lemma 4.12.2 and Theorem 4.21 n+1 = rank Ṽ ≤ r.rank Ṽ ≤ dim Ã_{𝔮} = n+1, and rank U ≤ r.rank U ≤ dim Z = n.

On the other hand, since k ⊂ U/M(U) ⊂ Ṽ /M(Ṽ) ≅ k, we have U/M(U) ≅ Ṽ /M(Ṽ) ≅ k and tr.deg_{(U/M(U))}(Ṽ /M(Ṽ)) = 0. Since tr.deg_{D}Ẽ = 1, we obtain n + 1 − r.rank U = r.rank Ṽ − r.rank U ≤ tr.deg_{D}Ẽ − tr.deg_{(U/M(U))}(Ṽ /M(Ṽ)) = 1 by Lemma 4.23.2. We obtain rank U ≤ r.rank U = n.

Assume that r = rank U < n.

Let ṽ : Ẽ → Γ ∪ {∞} and u : D → Γ̄ ∪ {∞} denote the valuations defined by valuation rings Ṽ and U respectively. By Lemma 4.13.2 we can regard Γ̄ as a subgroup of Γ and ṽ | D = u.

By Theorem 4.21.2 there exists an isomorphism Γ̄ ≅ ℤ^{n} as abelian groups. By Corollary 4.22.2 there exists an isomorphism Γ ≅ ℤ^{n+1} as totally ordered abelian groups. Below by rank_{ℤ}L we denote the rank over ℤ of a free ℤ-module L.

Let

Γ̄ = Δ̄_{0} ⊃ Δ̄_{1} ⊃ ⋯ ⊃ Δ̄_{i−1} ⊃ Δ̄_{i} ⊃ ⋯ ⊃ Δ̄_{r} = {0}

be the sequence of isolated subgroups of Γ̄. Since ∑_{i=1}^{r} rank_{ℤ}(Δ̄_{i−1}/Δ̄_{i}) = rank_{ℤ}Γ̄ = n, there is an integer i_{0} such that 1 ≤ i_{0} ≤ r and rank_{ℤ}(Δ̄_{i_{0}−1}/Δ̄_{i_{0}}) ≥ 2.

By Lemma 4.14.2 for every integer j with 0 ≤ j ≤ n + 1, Δ_{j} = ṽ(Ṽ ∗ ; 𝔭̃_{j}) is an isolated subgroup of Γ.

Γ = Δ_{0} ⊃ Δ_{1} ⊃ ⋯ ⊃ Δ_{i−1} ⊃ Δ_{i} ⊃ ⋯ ⊃ Δ_{n+1} = {0}.

For every integer j with 1 ≤ j ≤ n + 1 we have rank_{ℤ}(Δ_{j−1}/Δ_{j}) = 1.

For every 0 ≤ i ≤ r the set {j | 0 ≤ j ≤ n + 1, Δ_{j} ∩ Γ̄ = Δ̄_{i}} is not empty by Corollary 3.4.6. Let

j_{1} = min{j | 0 ≤ j ≤ n + 1, Δ_{j} ∩ Γ̄ = Δ̄_{i_{0}}}, and
j_{2} = max{j | 0 ≤ j ≤ n + 1, Δ_{j} ∩ Γ̄ = Δ̄_{i_{0}−1}}.

We can easily show j_{2} = j_{1} − 1.

Since Δ̄_{i_{0}} = Δ_{j_{1}} ∩ Γ̄, there exists an injective homomorphism

Δ̄_{i_{0}−1}/Δ̄_{i_{0}} → Δ_{j_{2}}/Δ_{j_{1}} = Δ_{j_{1}−1}/Δ_{j_{1}} ≅ ℤ

of abelian groups. We have 2 ≤ rank_{ℤ}(Δ̄_{i_{0}−1}/Δ̄_{i_{0}}) ≤ 1, which is a contradiction. We conclude rank U = n = dim Z and (B, W, P) ∈ C_{0}.

Lemma 8.2. Let (B, W, P) be a space germ over k. Let (Z, U, P) = \underline{(B, W, P)}, P = {x_{1}, x_{2}, …, x_{n}}, r = rank U and

{0} = 𝔮_{0} ⊂ 𝔮_{1} ⊂ ⋯ ⊂ 𝔮_{i−1} ⊂ 𝔮_{i} ⊂ ⋯ ⊂ 𝔮_{r} = M(U)

be the sequence of prime ideals of U. Consider the following three conditions on (B, W, P).

⟨α⟩ dim Z = rank U. ⟨β⟩ The morphism Spec(U) → Spec(Z) induced by the inclusion homomorphism

Z ⊂ U is injective. ⟨γ⟩ Under an appropriate numbering of x_{i}’s, for every integer i with 1 ≤ i ≤ r

the prime ideal 𝔮_{i} ∩ Z of Z is generated by i elements x_{1}, x_{2}, …, x_{i}.

1. Assume that (B, W, P) satisfies ⟨α⟩. Let (B′, W′, P′) be an IAMT of (B, W, P). Also (B′, W′, P′) satisfies ⟨α⟩. Let (Z′, U′, P′) = \underline{(B′, W′, P′)}, and

{0} = 𝔮′_{0} ⊂ 𝔮′_{1} ⊂ ⋯ ⊂ 𝔮′_{i−1} ⊂ 𝔮′_{i} ⊂ ⋯ ⊂ 𝔮′_{n} = M(U′)

denote the sequence of prime ideals of U′. For every 0 ≤ i ≤ n we have 𝔮_{i} = 𝔮′_{i} ∩ U and (𝔮′_{i} ∩ Z′) ∩ Z = 𝔮_{i} ∩ Z.

2. Assume that (B, W, P) satisfies ⟨α⟩ and ⟨β⟩. Let (B′, W′, P′) be an IAMT of (B, W, P). Also (B′, W′, P′) satisfies ⟨α⟩ and ⟨β⟩. For every 0 ≤ i ≤ n we have ht(𝔮′_{i} ∩ Z′) = i, where 𝔮′_{i} and Z′ denote the same as above.

3. If (B, W, P) satisfies ⟨α⟩, ⟨β⟩ and ⟨γ⟩, then any IAMT (B′, W′, P′) with a normal crossing of (B, W, P) satisfies ⟨α⟩, ⟨β⟩ and ⟨γ⟩.

Proof. 1. By Lemma 5.5.1 and Lemma 4.13.3 we have rank U′ ≥ rank U. By Corollary 4.22.1 rank U′ ≤ dim Z′ = dim Z = rank U. Thus rank U′ = rank U = dim Z = dim Z′ = n.

By Lemma 5.5.1 and Corollary 4.15.3 the morphism Spec(U′) → Spec(U) is bijective. Since the morphism Spec(U′) → Spec(U) preserves inclusion relations, we have 𝔮_{i} = 𝔮′_{i}∩U for every 0 ≤ i ≤ n. Besides, (𝔮′_{i}∩Z′)∩Z = (𝔮′_{i}∩U)∩Z = 𝔮_{i}∩Z.

2. By 1 (B′, W′, P′) satisfies ⟨α⟩. The following diagram is commutative:

Spec(U′) ⟶ Spec(U)
↓  ↓
Spec(Z′) ⟶ Spec(Z)

By 1 the morphism Spec(U′) → Spec(U) is bijective. By ⟨β⟩ for (B, W, P) the morphism Spec(U) → Spec(Z) is injective. Thus Spec(U′) → Spec(Z′) is injective.

Obviously we have inclusion relations

{0} = 𝔮′_{0} ∩ Z′ ⊂ 𝔮′_{1} ∩ Z′ ⊂ ⋯ ⊂ 𝔮′_{i−1} ∩ Z′ ⊂ 𝔮′_{i} ∩ Z′ ⊂ ⋯ ⊂ 𝔮′_{n} ∩ Z′.

By ⟨β⟩ for (B′, W′, P′), which we have shown above, any two of n + 1 prime ideals 𝔮′_{i} ∩ Z′ never coincide. Since dim Z′ = n, one knows that ht(𝔮′_{i} ∩ Z′) = i for every 0 ≤ i ≤ n.

3. By 1 and 2 (B′, W′, P′) satisfies ⟨α⟩ and ⟨β⟩.

We will show ⟨γ⟩ for (B′, W′, P′). Let Z′, U′ and 𝔮′_{i} be the same as in 1. We write P′ = {y_{1}, y_{2}, ⋯, y_{n}}. By ⟨γ⟩ for (B, W, P) we can assume that for every 1 ≤ i ≤ n the prime ideal 𝔮_{i} ∩ Z is generated by x_{1}, x_{2}, …, x_{i}. We assume this condition.

We apply induction. Let j be an integer with 1 ≤ j ≤ n. We assume that under an appropriate renumbering of y_{i}’s, for every integer i with 0 ≤ i < j the prime ideal 𝔮′_{i} ∩ Z′ is generated by y_{1}, y_{2}, …, y_{i}. In case i = 0, the ideal generated by the empty set is the ideal {0} containing only one element 0.

Since (B′, W′, P′) is an IAMT with a normal crossing of (B, W, P), there are an element u ∈ (Z′)^{∗} and non-negative integers s_{1}, s_{2}, …, s_{n} with x_{j} = uy_{1}^{s_{1}}y_{2}^{s_{2}} ⋯ y_{n}^{s_{n}}.

We infer s_{1} = s_{2} = ⋯ = s_{j−1} = 0. Assume the contrary. Since 𝔮′_{j−1} ∩ Z′ is generated by y_{1}, y_{2}, …, y_{j−1}, we have x_{j} ∈ (𝔮′_{j−1} ∩ Z′) ∩ Z = 𝔮_{j−1} ∩ Z. We used 1 here. The ideal 𝔮_{j−1} ∩ Z is generated by x_{1}, x_{2}, …, x_{j−1}. In the ring Z = k[[x_{1}, x_{2}, …, x_{n}]] the element x_{j} belongs to the ideal generated x_{1}, x_{2}, …, x_{j−1}, which is a contradiction.

By the above we can write x_{j} = uy_{j}^{s_{j}}y_{j+1}^{s_{j+1}} ⋯ y_{n}^{s_{n}}. Since x_{j} ∈ 𝔮_{j} ∩Z = (𝔮′_{j} ∩Z′)∩ Z ⊂ 𝔮′_{j} ∩ Z′, and since 𝔮′_{j} ∩ Z′ is a prime ideal of Z′, one of y_{j}, y_{j+1}, …, y_{n} belongs to 𝔮′_{j} ∩ Z′. Fixing the numbers of y_{1}, y_{2}, …, y_{j−1} and exchanging the numbers y_{j}, y_{j+1}, …, y_{n}, we can assume that y_{j} ∈ 𝔮′_{j} ∩ Z′.

We consider the residue ring Z̄ = Z′/(𝔮′_{j−1} ∩ Z′) and the prime ideal 𝔮̄ = (𝔮′_{j} ∩ Z′)/(𝔮′_{j−1} ∩ Z′) of Z̄. By ē ∈ Z̄ we denote the image of an element e ∈ Z′ by the canonical surjective homomorphism Z′ → Z̄.

ȳ_{j} ∈ 𝔮̄. Since 𝔮′_{j−1} ∩ Z′ is the ideal in Z′ = k[[y_{1}, y_{2}, …, y_{n}]] generated by y_{1}, y_{2}, …, y_{j−1}, we have y_{j} ∉ 𝔮′_{j−1} ∩ Z′ and ȳ_{j} ≠ 0. Thus {0} ≠ ȳ_{j}Z̄ ⊂ 𝔮̄.

Since Z̄/ȳ_{j}Z̄ ≅ k[[y_{j+1}, y_{j+2}, …, y_{n}]], ȳ_{j}Z̄ is a prime ideal of Z̄.

By 3 and 6 of Theorem 2.2 Z′ is a catenary noetherian domain. Thus by 2 we have ht(𝔮̄) = ht(𝔮′_{j} ∩ Z′) − ht(𝔮′_{j−1} ∩ Z′) = j − (j − 1) = 1, and we can conclude ȳ_{j}Z̄ = 𝔮̄. Thus 𝔮′_{j} ∩ Z′ is generated by y_{1}, y_{2}, …, y_{j}.

In the above renumbering procedure, we fixed the numbers of y_{1}, y_{2}, …, y_{j−1}. Thus one can conclude that under an appropriate renumbering of y_{i}’s, for every integer i with 0 ≤ i < j + 1 the prime ideal 𝔮′_{i} ∩ Z′ is generated by y_{1}, y_{2}, …, y_{i}.

For j = 1 the claim is true, since 𝔮′_{0} ∩Z′ = {0}. Thus by induction one concludes that ⟨γ⟩ holds for (B′, W′, P′).

Corollary 8.3. The collection C_{0} of all framed space germs (B, W, Q) over k such that (A, V) = \underline{(B, W, Q)}_{•} satisfies dim A = rank V is an admissible class.

Proof. It follows Lemma 8.1 and Lemma 8.2.1.

Recall again the statement RW (B, W, P) we gave before Lemma 5.12.

Lemma 8.4. Let r be a positive integer. Assume that RW (B, W, P) is true for every space germ (B, W, P) such that (B, W, P) ∈ C_{0} and ♯ P ≤ r.

Then, for every (B, W, P) ∈ C_{0} satisfying ♯ P ≤ r, there exists an IAMT (B′, W′, P′) with a normal crossing of (B, W, P) such that (B′, W′, P′) satisfies the three conditions ⟨α⟩, ⟨β⟩ and ⟨γ⟩ in Lemma 8.2.

Proof. Let (B, W, P) be a framed space germ over k such that (B, W, P) ∈ C_{0} and ♯ P ≤ r. By assumption RW (B, W, P) is true. Let (Z, U, P) = \underline{(B, W, P)} and n = dim Z = rank U. Let

{0} = 𝔮_{0} ⊂ 𝔮_{1} ⊂ ⋯ ⊂ 𝔮_{i−1} ⊂ 𝔮_{i} ⊂ ⋯ ⊂ 𝔮_{n} = M(U)

be the sequence of prime ideals of U.

For every 1 ≤ i ≤ n we choose an element a_{i} ∈ 𝔮_{i} − 𝔮_{i−1}. We have a_{i} ≠ 0, since a_{i} ∉ 𝔮_{0} = {0}.

Let (B′, W′, P′) be an arbitrary IAMT of (B, W, P). Let (Z′, U′, P′) = \underline{(B′, W′,} \underline{P′)}. We identify a_{i} ∈ U and the image of a_{i} by the associated homomorphism U → U′. Define

β(B′, W′, P′) = ♯ {a_{i} | 1 ≤ i ≤ n, a_{i} ∉ Z′}.

By definition β(B′, W′, P′) is a non-negative integer.

First, by induction on β(B′, W′, P′) we will show that there exists an IAMT (B′, W′, P′) with a normal crossing of (B, W, P) such that β(B′, W′, P′) = 0.

If β(B, W, P) = 0, then (B′, W′, P′) = (B, W, P) satisfies the condition.

Assume β(B, W, P) > 0. There exists an integer i such that 1 ≤ i ≤ n and a_{i} ∉ Z. Since Z and U have the common quotient field, there are elements b, c ∈ Z with a_{i} = c/b, c ≠ 0 and b ≠ 0. By Corollary 5.13 RS(B, W, P) is true. Thus there exists an IAMT (B_{1}, W_{1}, P_{1}) with a normal crossing of (B, W, P) such that one of the following (a), (b) holds: Let (Z_{1}, U_{1}, P_{1}) = \underline{(B_{1}, W_{1}, P_{1})}.

(a) There exists an element e ∈ Z_{1} with c = eb.

(b) There exists an element e ∈ M(Z_{1}) with b = ec. Assume that (b) holds. e ∈ M(Z_{1}) ⊂ M(U_{1}). 1/e = c/b = a_{i} ∈ U ⊂ U_{1}. Thus we have 1 = e · (1/e) ∈ M(U_{1}), which is a contradiction. Thus (a) holds, and a_{i} = c/b = e ∈ Z_{1}. Since Z ⊂ Z_{1}, we have β(B, W, P) > β(B_{1}, W_{1}, P_{1}). By Lemma 8.2.1 (B_{1}, W_{1}, P_{1}) ∈ C_{0} and ♯ P_{1} = ♯ P ≤ r. By assumption RW (B_{1}, W_{1}, P_{1}) is true. By induction hypothesis there exists an IAMT (B′, W′, P′) with a normal crossing of (B_{1}, W_{1}, P_{1}) with β(B′, W′, P′) = 0.

Obviously (B′, W′, P′) is an IAMT with a normal crossing of (B, W, P). By Lemma 8.2.1 (B′, W′, P′) satisfies ⟨α⟩. We will show that (B′, W′, P′) satisfies ⟨β⟩, too.

Let (Z′, U′, P′) = \underline{(B′, W′, P′)}, and

{0} = 𝔮′_{0} ⊂ 𝔮′_{1} ⊂ ⋯ ⊂ 𝔮′_{i−1} ⊂ 𝔮′_{i} ⊂ ⋯ ⊂ 𝔮′_{n} = M(U′)

be the sequence of prime ideals of U′. By Lemma 8.2.1 we have 𝔮_{i} = 𝔮′_{i} ∩U for every 0 ≤ i ≤ n. Obviously 𝔮′_{i−1} ∩Z′ ⊂ 𝔮′_{i} ∩Z′ for every 1 ≤ i ≤ n. Since β(B′, W′, P′) = 0, we have a_{i} ∈ Z′ for every 1 ≤ i ≤ n. Thus a_{i} ∈ 𝔮_{i} ∩ Z′ = (𝔮′_{i} ∩ U) ∩ Z′ ⊂ 𝔮′_{i} ∩ Z′. Assume a_{i} ∈ 𝔮′_{i−1} ∩ Z′. We have a_{i} ∈ 𝔮′_{i−1} ∩ U = 𝔮_{i−1}, which contradicts a_{i} ∉ 𝔮_{i−1}. Thus, for every 1 ≤ i ≤ n we have a_{i} ∉ 𝔮′_{i−1} ∩ Z′ and 𝔮′_{i−1} ∩ Z′ ≠ 𝔮′_{i} ∩ Z′. We conclude that the morphism Spec(U′) → Spec(Z′) is injective. (B′, W′, P′) satisfies ⟨β⟩.

An IAMT (B′′, W′′, P′′) with a normal crossing of (B′, W′, P′) is an IAMT with a normal crossing of (B, W, P). Besides, (B′, W′, P′) ∈ C_{0} and ♯ P′ = ♯ P ≤ r. Thus, exchanging (B, W, P) and (B′, W′, P′), we can assume that (B, W, P) satisfies ⟨α⟩ and ⟨β⟩. By induction of the basic dimension ♯ P, we will show that there exists an IAMT (B′, W′, P′) with a normal crossing of (B, W, P) satisfying ⟨α⟩, ⟨β⟩, and ⟨γ⟩ under this stronger assumption.

If ♯ P = 1, then (B′, W′, P′) = (B, W, P) satisfies ⟨α⟩, ⟨β⟩, and ⟨γ⟩.

Assume r ≥ ♯ P ≥ 2. By Lemma 8.2.2 we have ht(𝔮_{1} ∩ Z) = 1 and there is an element f with 0 ≠ f ∈ 𝔮_{1} ∩ Z. By assumption RW (B, W, P) is true, and there exists an IAMT (B_{1}, W_{1}, P_{1}) with a normal crossing of (B, W, P) such that the image of f in Z_{1} = k[[P_{1}]] has a normal crossing with respect to P_{1}. Let (Z_{1}, U_{1}, P_{1}) = \underline{(B_{1}, W_{1}, P_{1})}, and P_{1} = \{y_{1}, y_{2}, …, y_{n}\}. We have an element u ∈ Z_{1}^{∗} and non-negative integers s_{1}, s_{2}, …, s_{n} with f = uy_{1}^{s_{1}}y_{2}^{s_{2}} ⋯ y_{n}^{s_{n}}. Let

{0} = 𝔮_{1,0} ⊂ 𝔮_{1,1} ⊂ ⋯ ⊂ 𝔮_{1,i−1} ⊂ 𝔮_{1,i} ⊂ ⋯ ⊂ 𝔮_{1,n} = M(U_{1})

be the sequence of prime ideals of U_{1}. By Lemma 8.2.1 we have f = uy_{1}^{s_{1}}y_{2}^{s_{2}} ⋯ y_{n}^{s_{n}} ∈ 𝔮_{1} ∩ Z = (𝔮_{1,1} ∩ Z_{1}) ∩ Z ⊂ 𝔮_{1,1} ∩ Z_{1}. Since 𝔮_{1,1} ∩ Z_{1} is a prime ideal of Z_{1}, there is an integer i such that 1 ≤ i ≤ n and y_{i} ∈ 𝔮_{1,1} ∩ Z_{1}. By exchanging numbering of y_{i}’s, we can assume that y_{1} ∈ 𝔮_{1,1} ∩ Z_{1}. y_{1}Z_{1} is a non-zero prime ideal of Z_{1}, and y_{1}Z_{1} ⊂ 𝔮_{1,1} ∩ Z_{1}. By Lemma 8.2.2 we know ht(𝔮_{1,1} ∩ Z_{1}) = 1. Thus we conclude y_{1}Z_{1} = 𝔮_{1,1} ∩ Z_{1}.

Let O_{1} = P_{1} − {y_{1}} = {y_{2}, y_{3}, …, y_{n}}, and we consider the framed space germ (B_{1}, W_{1}, O_{1}).

Let (B′, W′, O′) be an arbitrary IAMT of (B_{1}, W_{1}, O_{1}), (Z′, U′, O′ ∪ {y_{1}}) = \underline{(B′, W′, O′ ∪ \{y_{1}\})}, and (Y′, T′, O′) = \underline{(B′, W′, O′)}. We identify y_{1} ∈ B_{1} and the image in B′ of y_{1} here. Let

{0} = 𝔮′_{0} ⊂ 𝔮′_{1} ⊂ ⋯ ⊂ 𝔮′_{i−1} ⊂ 𝔮′_{i} ⊂ ⋯ ⊂ 𝔮′_{n} = M(U′)

be the sequence of prime ideals of U′.

The following five claims hold:

1. 𝔮′_{1} ∩ Z′ = y_{1}Z′.

2. The prime ideal 𝔮′_{i}∩Z′ of Z′ is generated by y_{1} and 𝔮′_{i}∩Y′ for every 1 ≤ i ≤ n.

3. 𝔮_{i−1} ∩ Y′ ≠ 𝔮_{i} ∩ Y′ for every 2 ≤ i ≤ n.

4. rank T′ = dim Y′ = n − 1. Spec(T′) = {𝔮′_{i} ∩ T′ | 1 ≤ i ≤ n}.

5. The morphism Spec(T′) → Spec(Y′) is injective. We will show claims 1–5.

By Lemma 8.2.1 y_{1} ∈ 𝔮_{1,1} ∩ Z_{1} = (𝔮′_{1} ∩ Z′) ∩ Z_{1} ⊂ 𝔮′_{1} ∩ Z′. Thus y_{1}Z′ ⊂ 𝔮′_{1} ∩ Z′. Since y_{1} is a member of a parameter system of a regular local ring Z′, y_{1}Z′ is a non-zero prime ideal of Z′. By Lemma 8.2.2 ht(𝔮′_{1}∩Z′) = 1. We have y_{1}Z = 𝔮′_{1}∩Z′. We obtain claim 1.

Let i be an integer with 1 ≤ i ≤ n. y_{1} ∈ 𝔮′_{1} ∩ Z′ ⊂ 𝔮′_{i} ∩ Z′. 𝔮′_{i} ∩ Y′ ⊂ 𝔮′_{i} ∩ Z′. Thus the prime ideal 𝔮′_{i} ∩ Z′ contains the ideal generated by y_{1} and 𝔮′_{i} ∩ Y′.

Let g ∈ 𝔮′_{i}∩Z′ be an arbitrary element. Since Y′ = k[[O′]] and Z′ = k[[O′∪{y_{1}}]], we have elements g′, g′′ such that

g = g′y_{1} + g′′, g′ ∈ Z′, g′′ ∈ Y′.

We have g′′ = g − g′y_{1} ∈ (𝔮′_{i} ∩ Z′) ∩ Y′ = 𝔮′_{i} ∩ Y′. Thus 𝔮′_{i} ∩ Z′ is contained in the ideal generated by y_{1} and 𝔮′_{i} ∩ Y′. We obtain claim 2.

By Lemma 8.2.1 Any two of n prime ideals 𝔮′_{i} ∩ Z′, 1 ≤ i ≤ n never coincide. We obtain claim 3 by claim 2.

Since 𝔮′_{i} ∩ Y′ = (𝔮′_{i} ∩ T′) ∩ Y′, any two of n prime ideals 𝔮′_{i} ∩ T′, 1 ≤ i ≤ n of T′ never coincide. Thus rank T′ ≥ n − 1 = dim Y′. On the other hand, by Corollary 4.22.1 we have rank T′ ≤ dim Y′. We obtain claim 4.

Claim 5 follows from claim 3 and claim 4.

We go back to the consideration of (B_{1}, W_{1}, P_{1}) and O_{1} = P_{1} − {y_{1}}. Applying above claim 4 and claim 5 for (B′, W′, O′) = (B_{1}, W_{1}, O_{1}), one knows that (B_{1}, W_{1}, O_{1}) satisfies ⟨α⟩ and ⟨β⟩. Since ♯ O_{1} = ♯ P − 1, we can apply the induction hypothesis.

Let (B′, W′, O′) be an appropriate IAMT with a normal crossing of (B_{1}, W_{1}, O_{1}). Let (Y′, T′, O′) = \underline{(B′, W′, O′)}, O′ = \{z_{2}, z_{3}, …, z_{n}\}, and

{0} = 𝔮̄′_{1} ⊂ 𝔮̄′_{2} ⊂ ⋯ ⊂ 𝔮̄′_{i−1} ⊂ 𝔮̄′_{i} ⊂ ⋯ ⊂ 𝔮̄′_{n} = M(T′)

be the sequence of prime ideals of T′. By the induction hypothesis, the prime ideal 𝔮̄′_{i} ∩ Y′ of Y′ is generated by z_{2}, z_{3}, …, z_{i} for every 2 ≤ i ≤ n. Let z_{1} ∈ B′ be the image of y_{1} ∈ B_{1} by the associated homomorphism B_{1} → B′. By Lemma 5.6 O′ ∪ \{z_{1}\} is a frame of B′. Let (Z′, U′, O′ ∪ \{z_{1}\}) = \underline{(B′, W′, O′ ∪ \{z_{1}\})}. (B′, W′, O′ ∪ {z_{1}}) is an IAMT with a normal crossing of (B_{1}, W_{1}, O_{1} ∪ {y_{1}}). Denoting the sequence of prime ideals of U′ by 𝔮′_{i}, 0 ≤ i ≤ n, by claim 4 above one knows that 𝔮̄′_{i} = 𝔮′_{i} ∩ T′ for every 1 ≤ i ≤ n.

By claim 2, the prime ideal 𝔮′_{i} ∩ Z′ of Z′ is generated by z_{1} and 𝔮′_{i} ∩ Y′ for every 1 ≤ i ≤ n. Besides, 𝔮′_{i} ∩ Y′ = (𝔮′_{i} ∩ T′) ∩ Y′ = 𝔮̄′_{i} ∩ Y′.

Thus, for every 2 ≤ i ≤ n, 𝔮′_{i} ∩ Z′ is generated by z_{1} and 𝔮̄′_{i} ∩ Y′. Since 𝔮̄′_{i} ∩ Y′ is generated by z_{2}, z_{3}, …, z_{i}, 𝔮′_{i} ∩ Z′ is generated by z_{1}, z_{2}, …, z_{i}.

By claim 1 𝔮′_{1} ∩ Z′ is generated by z_{1} = y_{1}.

We conclude that (B′, W′, O′ ∪{z_{1}}) satisfies ⟨γ⟩. By Lemma 8.2.2 (B′, W′, O′ ∪ {z_{1}}) satisfies also ⟨α⟩ and ⟨β⟩. (B′, W′, O′ ∪ {z_{1}}) is an IAMT with a normal crossing of (B, W, P).

Definition 8.5. Let (B, W, P) be a framed space germ over k. Let (Z, U, P) = \underline{(B, W, P)}, P = \{x_{1}, x_{2}, …, x_{n}\} and

{0} = 𝔮_{0} ⊂ 𝔮_{1} ⊂ ⋯ ⊂ 𝔮_{i−1} ⊂ 𝔮_{i} ⊂ ⋯ ⊂ 𝔮_{n} = M(U)

be the sequence of prime ideals of U. We call U the basic valuation ring of (B, W, P). If (B, W, P) satisfies the following ⟨A⟩ and ⟨B⟩, then we say that (B, W, P) has the basic valuation ring of normal crossing type, or U is of normal crossing type. ⟨A⟩ rank U = dim Z. ⟨B⟩ Under an appropriate numbering of x_{i}’s, the prime ideal 𝔮_{i} ∩ Z of Z is gen-

erated by x_{1}, x_{2}, …, x_{i} for every 1 ≤ i ≤ n.

Obviously (B, W, P) has the basic valuation ring of normal crossing type, if and only if, (B, W, P) satisfies conditions ⟨α⟩, ⟨β⟩ and ⟨γ⟩ in Lemma 8.2.

Corollary 8.6. Let r be a positive integer. Assume that RW (B, W, P) is true for every space germ (B, W, P) such that (B, W, P) ∈ C_{0} and ♯ P ≤ r.

Then, for every (B, W, P) ∈ C_{0} satisfying ♯ P ≤ r, there exists an IAMT (B′, W′, P′) with a normal crossing of (B, W, P) such that (B′, W′, P′) has the basic valuation ring of normal crossing type.

Lemma 8.7. Let (B, W, P) be a framed space germ over k. Assume that (B, W, P) has the basic valuation ring of normal crossing type. Let (Z, U, P) = \underline{(B, W, P)}, P = {x_{1}, x_{2}, …, x_{n}} and

{0} = 𝔮_{0} ⊂ 𝔮_{1} ⊂ ⋯ ⊂ 𝔮_{i−1} ⊂ 𝔮_{i} ⊂ ⋯ ⊂ 𝔮_{n} = M(U)

be the sequence of prime ideals of the basic valuation ring U.

1. For every 0 ≤ i ≤ n, the residue ring Z/𝔮_{i} ∩ Z is a normal domain, and the quotient field of Z/𝔮_{i} ∩ Z and the quotient field of U/𝔮_{i} coincide.

2. For every 1 ≤ i ≤ n,

Z_{(𝔮_{i}∩Z)}/(𝔮_{i−1} ∩ Z)Z_{(𝔮_{i}∩Z)} ≅ U_{𝔮_{i}}/𝔮_{i−1}U_{𝔮_{i}}.

3. Let (B′, W′, P′) be an IAMT with a normal crossing of (B, W, P). Then, also (B′, W′, P′) has the basic valuation ring of normal crossing type.

Proof. 1. Z = Z/𝔮_{0} ∩ Z = k[[x_{1}, x_{2}, …, x_{n}]]. For every integer i with 1 ≤ i ≤ n we have Z/𝔮_{i} ∩ Z ≅ k[[x_{i+1}, x_{i+2}, …, x_{n}]] by definition of normal crossing type. Thus, for every 0 ≤ i ≤ n, Z/𝔮_{i} ∩ Z is a regular local ring, and in particular it is a normal domain.

By induction on i we will show that Z/𝔮_{i} ∩ Z and U/𝔮_{i} have a common quotient field.

In case i = 0, by definition of space germs, the quotient field of Z = Z/𝔮_{0} ∩ Z and the quotient field of U = U/𝔮_{0} coincide.

Assume that 1 ≤ i ≤ n and the quotient field of Z/𝔮_{i−1} ∩ Z and the quotient field of U/𝔮_{i−1} coincide. The valuation ring U_{𝔮_{i}}/𝔮_{i−1}U_{𝔮_{i}} dominates the local ring Z̄_{i} = Z_{(𝔮_{i}∩Z)}/(𝔮_{i−1} ∩ Z)Z_{(𝔮_{i}∩Z)} and these two rings have the same quotient field. On the other hand, by the assumption of normal crossing type, we have Z̄_{i} ≅ k[[x_{i}, x_{i+1}, …, x_{n}]]_{(x_{i})}, where (x_{i}) denotes the prime ideal in k[[x_{i}, x_{i+1}, …, x_{n}]] generated by one element x_{i}. Thus, Z̄_{i} is a regular local ring of dimension one. By Proposition 4.25 Z̄_{i} is a valuation ring of the quotient field. By Corollary 4.19 we have Z̄_{i} = U_{𝔮_{i}}/𝔮_{i−1}U_{𝔮_{i}}. In particular, the residue field of the local ring Z̄_{i} coincides with the residue field of the local ring U_{𝔮_{i}}/𝔮_{i−1}U_{𝔮_{i}}. The quotient field of the ring Z/𝔮_{i} ∩ Z coincides with the residue field of Z̄_{i}, The residue field of U_{𝔮_{i}}/𝔮_{i−1}U_{𝔮_{i}} coincides with the quotient field of U/𝔮_{i}. Thus the quotient fields of Z/𝔮_{i} ∩ Z and U/𝔮_{i} coincide.

2. It has already been shown in the above proof of the latter half of 1.

3. It is equivalent to Lemma 8.2.3.

9. Reduction sequence

We introduce an effective tool to treat the Abhyankar condition and the condition “not reductive” simultaneously for a Weierstrass representation.

Definition 9.1. Let (B, W, P) be a framed space germ over k, (h, g, t) be a Weierstrass representation over (B, W, P), and ω be either a positive integer or symbol ∞. We consider a sequence

S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν < ω + 1}

of quadruplets (B_{ν}, W_{ν}, P_{ν}, H_{ν}) numbered by integers ν with 1 ≤ ν < ω + 1. Let (B_{0}, W_{0}, P_{0}) = (B, W, P) and (h_{0}, g_{0}, t_{0}) = (h, g, t). If S satisfies the following conditions 0–4 for every integer ν with 1 ≤ ν < ω + 1, then we call S a reduction sequence of (h, g, t). We call ω the length of S.

0. (B_{ν}, W_{ν}, P_{ν}) is a framed space germ over k. H_{ν} ∈ k[[P_{ν}]].

1. (B_{ν}, W_{ν}, P_{ν}) is an IAMT with a normal crossing of (B_{ν−1}, W_{ν−1}, P_{ν−1}). To state conditions 2–4 we define two Weierstrass representations (h′_{ν−1}, g′_{ν−1}, t′_{ν−1}) and (h_{ν}, g_{ν}, t_{ν}) over (B_{ν}, W_{ν}, P_{ν}) by the following conditions ⟨A⟩ and ⟨B⟩: ⟨A⟩ (h′_{ν−1}, g′_{ν−1}, t′_{ν−1}) is the pull-back of the Weierstrass representation (h_{ν−1},

g_{ν−1}, t_{ν−1}) over (B_{ν−1}, W_{ν−1}, P_{ν−1}) by the associated morphism (B_{ν−1}, W_{ν−1},

P_{ν−1}) → (B_{ν}, W_{ν}, P_{ν}).

⟨B⟩ (h_{ν}, g_{ν}, t_{ν}) = (h′_{ν−1}, g′_{ν−1}, t′_{ν−1} + H_{ν}). 2. (h′_{ν−1}, g′_{ν−1}, t′_{ν−1}) satisfies the Abhyankar condition.

3. If ν ≠ ω, then w(g′_{ν−1}, t′_{ν−1}) < w(g_{ν}, t_{ν}).

4. If ν = ω and (h′_{ν−1}, g′_{ν−1}, t′_{ν−1}) is not reductive, then H_{ν} = 0. If ν = ω and

(h′_{ν−1}, g′_{ν−1}, t′_{ν−1}) is reductive, then w(g_{ν}, t_{ν}) = ∞. Besides, If ω is a positive integer and if S satisfies the conditions 0, 1, 2, 3 plus the following 4* for every integer ν with 1 ≤ ν < ω + 1, then we say that S is a semi-reduction sequence of (h, g, t) of length ω. 4*. If ν = ω, then w(g′_{ν−1}, t′_{ν−1}) < w(g_{ν}, t_{ν}) < ∞. We call the empty set S = ∅ = {} semi-reduction sequence of length 0.

We call the above sequence {(h_{ν}, g_{ν}, t_{ν}) | 0 ≤ ν < ω + 1} of Weierstrass representations the transform Weierstrass representation sequence of (h, g, t) along S. We call the sequence {(B_{ν}, W_{ν}, P_{ν}) | 0 ≤ ν < ω + 1} the lower framed space germ sequence along S, while we call {(B_{ν}, W_{ν}, P_{ν} ∪ {t_{ν}}) | 0 ≤ ν < ω + 1} the upper framed space germ sequence along S.

If ω = ∞, then we say that S has infinite length. If ω < ∞, then we say that S has finite length.

Remark . 1. The length of a reduction sequence is either a positive integer or

∞. The length of a semi-reduction sequence is either a positive integer or 0.

2. For simplicity, sometimes we identify an element in the quotient field of B_{μ} and the image of it in the quotient field of B_{ν} for any pair μ, ν of integers with 0 ≤ μ ≤ ν < ω + 1.

3. Under the identification in 2, h = h_{ν}, g = g_{ν} and t_{ν} = t + H_{1} + H_{2} + ⋯ + H_{ν} for any 0 ≤ ν < ω + 1.

4. By condition 2 of Definition 9.1 we have w(g_{ν−1}, t_{ν−1}) < ∞. Thus for every 0 ≤ ν < ω we have w(g_{ν}, t_{ν}) < ∞. If S is a semi-reduction sequence, then by 4* of Definition 9.1 we have w(g_{ν}, t_{ν}) < ∞ even for ν = ω. In particular, if either (h, g, t) has a reduction sequence, or (h, g, t) has a semi-reduction sequence of positive length, then w(g, t) < ∞.

5. Assume that S ia a reduction sequence of (h, g, t) with finite length ω. The following four conditions are equivalent:

(a) w(g_{ω}, t_{ω}) < ∞.

(b) H_{ω} = 0

(c) (h_{ω}, g_{ω}, t_{ω}) = (h′_{ω−1}, g′_{ω−1}, t′_{ω−1})

(d) (h_{ω}, g_{ω}, t_{ω}) satisfies the Abhyankar condition, and is not reductive.

6. By condition 3 of Definition 9.1, for every 1 ≤ ν < ω (h′_{ν−1}, g′_{ν−1}, t′_{ν−1}) is reductive. If S is a semi-reduction sequence, then by 4* of Definition 9.1 (h′_{ν−1}, g′_{ν−1}, t′_{ν−1}) is reductive even for ν = ω.

7. If S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν < ω + 1} is a reduction sequence, then for every integer λ with 1 ≤ λ < ω, the subsequence S_{λ} = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν ≤ λ} of S is a semi-reduction sequence.

For a given infinite sequence of quadruples S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν < ∞}, S is a reduction sequence of (h, g, t) with infinite length, if and only if, for any positive integer λ the subsequence S_{λ} = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν ≤ λ} of S is a semi-reduction sequence of (h, g, t).

Recall the statement RW (B, W, P) we gave before Lemma 5.12.

Lemma 9.2. Let C be an admissible class of framed space germs over k, and r be a positive integer.

Assume that RW (B, W, P) is true for every framed space germ (B, W, P) such that (B, W, P) ∈ C and ♯ P ≤ r.

Let (B, W, P) be a framed space germ over k such that (B, W, P) ∈ C and ♯ P ≤ r, and (h, g, t) be a Weierstrass representation over (B, W, P).

Then, there exists a reduction sequence of (h, g, t), if and only if, w(g, t) < ∞.

Proof. By definition, if there exists a reduction sequence of (h, g, t), then w(g, t) < ∞.

Assume w(g, t) < ∞. The empty set S_{0} = {} is a semi-reduction sequence of (h, g, t) with length 0. Thus by induction, we can assume that a semi-reduction sequence S_{λ} = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν ≤ λ} of (h, g, t) with length λ has been already constructed for some non-negative integer λ. Let (B_{0}, W_{0}, P_{0}) = (B, W, P), (h_{0}, g_{0}, t_{0}) = (h, g, t), and {(h_{ν}, g_{ν}, t_{ν}) | 0 ≤ ν ≤ λ} be the transform Weierstrass representation sequence along S_{λ}. If λ > 0, then by condition 4^{∗} of Definition 9.1 we have w(g_{λ}, t_{λ}) < ∞. If λ = 0, then by assumption we have w(g_{λ}, t_{λ}) = w(g, t) < ∞. Since (B_{λ}, W_{λ}, P_{λ}) is an IAMT of (B_{0}, W_{0}, P_{0}) = (B, W, P) ∈ C, and since C is an admissible class, we have (B_{λ}, W_{λ}, P_{λ}) ∈ C and ♯ P_{λ} = ♯ P ≤ r. Thus RW (B_{λ}, W_{λ}, P_{λ}) is true by assumption.

By Corollary 6.9 there exists an IAMT (B_{λ+1}, W_{λ+1}, P_{λ+1}) with a normal crossing of (B_{λ}, W_{λ}, P_{λ}) such that the pull-back (h′_{λ}, g′_{λ}, t′_{λ}) of (h_{λ}, g_{λ}, t_{λ}) by the associated morphism (B_{λ}, W_{λ}, P_{λ}) → (B_{λ+1}, W_{λ+1}, P_{λ+1}) satisfies the Abhyankar condition.

In case where (h′_{λ}, g′_{λ}, t′_{λ}) is not reductive, letting H_{λ+1} = 0, S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν ≤ λ + 1}, we have a reduction sequence S of (h, g, t) with length λ + 1.

In case where (h′_{λ}, g′_{λ}, t′_{λ}) is reductive, there exists an element H_{λ+1} ∈ k[[P_{λ+1}]] satisfying w(g′_{λ}, t′_{λ}) < w(g′_{λ}, t′_{λ} + H_{λ+1}).

If w(g′_{λ}, t′_{λ} + H_{λ+1}) = ∞, letting S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν ≤ λ + 1}, we have a reduction sequence S of (h, g, t) with length λ + 1.

In case where w(g′_{λ}, t′_{λ} +H_{λ+1}) < ∞, S_{λ+1} = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν ≤ λ+1} is a semi-reduction sequence of (h, g, t) with length λ + 1, and S_{λ+1} is an extension of S_{λ}. We can ascend an induction step.

Assume that for any non-negative integer λ a reduction sequence S of (h, g, t) with length λ + 1 is not constructed, when we repeat the above procedure. Then, for any non-negative integer λ a semi-reduction sequence S_{λ} of (h, g, t) with length λ is constructed. For any pair μ, λ of integers with 0 ≤ μ < λ S_{λ} is an extension of S_{μ}. In this case S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν < ∞} is a reduction sequence of (h, g, t) with infinite length.

We here extend the Abhyankar condition for m-tuples of functions.

Definition 9.3. Let (B, W, P) be a framed space germ over k, m be a non-negative integer, and G_{1}, G_{2}, …, G_{m} ∈ Z = k[[P]] be m numbered elements. We write P = {x_{1}, x_{2}, …, x_{n}}.

1. We say that a m-tuple (G_{1}, G_{2}, …, G_{m}) satisfies the Abhyankar condition, if the following two conditions are simultaneously satisfied:

(a) m ≥ 1.

(b) There exist n non-negative integers b_{1}, b_{2}, …, b_{n} such that (i)

\frac{G_{j}}{(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}} ∈ Z

for every 1 ≤ j ≤ m, and (ii)

min\{\frac{1}{j} ord(\frac{G_{j}}{(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}}) | 1 ≤ j ≤ m\} < 1.

The non-negative integers b_{1}, b_{2}, …, b_{n} in (b) are called the characteristic numbers of (G_{1}, G_{2}, …, G_{m}).

2. If m = 0, we put w() = ∞. If m > 0, we put w(G_{1}, G_{2}, …, G_{m}) = min{ord(G_{j})/j | 1 ≤ j ≤ m}. We call w(G_{1}, G_{2}, …, G_{m}) ∈ ℚ ∪ {∞} the weight of (G_{1}, G_{2}, …, G_{m}).

Lemma 9.4. 1. If (G_{1}, G_{2}, …, G_{m}) satisfies the Abhyankar condition, then characteristic numbers b_{1}, b_{2}, …, b_{n} are uniquely defined depending on (G_{1}, G_{2}, …, G_{m}).

2. Let t ∈ B be an element such that t ∉ P and P ∪ {t} is a partial parameter system of B. Let g = t^{m} + ∑_{j=1}^{m} G_{j}t^{m−j}.

(a) w(G_{1}, G_{2}, …, G_{m}) = w(g, t).

(b) The m-tuple (G_{1}, G_{2}, …, G_{m}) satisfies the Abhyankar condition, if and only if, the semi-Weierstrass pair (g, t) satisfies the Abhyankar condition.

(c) If the equivalent conditions in (b) are satisfied, then the characteristic numbers of (G_{1}, G_{2}, …, G_{m}) and the characteristic numbers of (g, t) coincide.

3. Let v ∈ Z^{∗}.

(a) w(G_{1}, G_{2}, …, G_{m}) = w(vG_{1}, v^{2}G_{2}, …, v^{m}G_{m}).

(b) An m-tuple (G_{1}, G_{2}, …, G_{m}) satisfies the Abhyankar condition, if and only if, (vG_{1}, v^{2}G_{2}, …, v^{m}G_{m}) satisfies the Abhyankar condition.

(c) If the equivalent conditions in (b) are satisfied, then the characteristic numbers of (G_{1}, G_{2}, …, G_{m}) and the characteristic numbers of (vG_{1}, v^{2}G_{2}, …, v^{m}G_{m}) coincide.

Proof. 1. See Lemma 6.5.1.

2. Refer to Definition 6.4.1.

Lemma 9.5. Let (B, W, P) be a framed space germ over k, and (h, g, t) be a Weierstrass representation over (B, W, P). Assume that (h, g, t) satisfies the Abhyankar condition, and (h, g, t) is reductive. We denote P = {x_{1}, x_{2}, …, x_{n}}, Z = k[[P]] = k[[x_{1}, x_{2}, …, x_{n}]], m = deg(g, t). Let G_{1}, G_{2}, …, G_{m} ∈ Z be the coefficient function sequence of (g, t), and b_{1}, b_{2}, …, b_{n} be the characteristic numbers of (g, t). Note that m ≥ 1 by the Abhyankar condition. Let m = lq be the p-decomposition of m.

The following claims 1–9 hold:

1. w(g, t) = ∑_{i=1}^{n} b_{i}.

2. At least one of the characteristic numbers b_{1}, b_{2}, …, b_{n} is not zero.

3. There exists a unique element u_{j} satisfying

G_{j} = u_{j}(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}, u_{j} ∈ Z

for every 1 ≤ j ≤ m.

4. There exists a unique element a ∈ k satisfying

in(g, t) = (t + ax_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{m}, a ≠ 0.

5. u_{j}(0) = \binom{m}{j}a^{j} for every 1 ≤ j ≤ m, where we denote u_{j}(0) = u_{j} mod

M(Z) ∈ Z/M(Z) ≅ k. Besides, u_{q} ∈ Z^{∗}, and u_{m} ∈ Z^{∗}.

6. The following three conditions are equivalent for H ∈ Z:

(a) H is a reducing element of (g, t). In other words, w(g, t) < w(g, t + H).

(b) in(G_{q}) = l in(H)^{q}.

(c) in(H) = ax_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}.

7. There exists a reducing element H ∈ Z of (g, t) satisfying

\frac{H}{x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}} ∈ Z^{∗}.

8. Let H ∈ Z be a reducing element of (g, t) satisfying the condition in 7. Then, there exist m numbered elements F_{1}, F_{2}, …, F_{m} ∈ M(Z) satisfying

g = (t + H)^{m} + ∑_{i=1}^{m}(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}F_{j}(t + H)^{m−j}.

9. Assume that non-negative integers c_{1}, c_{2}, …, c_{n} and elements F′_{1}, F′_{2}, …, F′_{m} ∈

M(Z) satisfy

G_{j} = (x_{1}^{c_{1}}x_{2}^{c_{2}} ⋯ x_{n}^{c_{n}})^{j}F′_{j}

for every 1 ≤ j ≤ m. Then, (F′_{1}, F′_{2}, …, F′_{m}) satisfies the Abhyankar condition. Let d_{1}, d_{2}, …, d_{n} be the characteristic numbers of (F′_{1}, F′_{2}, …, F′_{m}). The following three claims hold:

(a) b_{i} = c_{i} + d_{i} for every 1 ≤ i ≤ n.

(b) w(F′_{1}, F′_{2}, …, F′_{m}) = ∑_{i=1}^{n} d_{i}.

(c) At least one of d_{1}, d_{2}, …, d_{n} is not zero.

Proof. 1. By Lemma 6.12.5 w(g, t) is an integer. By the Abhyankar condition for (g, t) we have

0 ≤ w(g, t) − ∑_{i=1}^{n} b_{i} = min\{\frac{1}{j} ord(\frac{G_{j}}{(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}})\} < 1.

2. It follows from 1 and Lemma 2.20.4.

3. It follows from that (G_{1}, G_{2}, …, G_{m}) satisfies the Abhyankar condition. 4 and 5. By 1 min{ord(u_{j})/j | 1 ≤ j ≤ m} = min{ord(G_{j}/(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j})/j | 1 ≤ j ≤ m} = 0. Thus we have in(g, t) = t^{m} +∑_{i=1}^{m} u_{j}(0)(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}t^{m−j}. On the other hand, by Lemma 6.12.3 there is a unique element K ∈ Z with in(g, t) = (t +

K)^{m}, and K ≠ 0. Thus u_{j}(0)(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j} = \binom{m}{j}K^{j} for every 1 ≤ j ≤ m. By

Lemma 2.4.3 \binom{m}{q} = \binom{l}{1} = l ≠ 0. Thus we have u_{q}(0)(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{q} = l K^{q} ≠ 0.

Let a ∈ k be an element satisfying u_{q}(0) = la^{q}. We have a ≠ 0, since u_{q}(0) ≠ 0. We have K = ax_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}, since l(K −ax_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{q} = lK^{q} −l(ax_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{q} = 0. The uniqueness of a follows from the uniqueness of K. Also we obtain the former half of 5. The latter half of 5 is easy.

6. It follows from Lemma 6.12.6 and above 4.

7. By 6 one knows that H = ax_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}} satisfies the condition.

8. Assume that H ∈ Z, h = H/(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}) ∈ Z^{∗} and w(g, t) < w(g, t + H). We have h(0)x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}} = in(H) = ax_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}} by 6. Thus h(0) = a.

Now, we can write

g = t^{m} + ∑_{j=1}^{m} u_{j}(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}t^{m−j} = (t + H)^{m} + ∑_{j=1}^{m} Φ_{j}(t + H)^{m−j}

with Φ_{1}, Φ_{2}, …, Φ_{m} ∈ Z. Let u_{0} = 1. By Lemma 6.10.2 we have

\begin{aligned}Φ_{j} & = ∑_{i=0}^{j} \binom{m − i}{m − j}(−H)^{j−i}u_{i}(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{i} \\ & = (x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j} ∑_{i=0}^{j} \binom{m − i}{m − j}(−h)^{j−i}u_{i} \end{aligned}

for every 1 ≤ j ≤ m. Putting F_{j} = ∑_{i=0}^{j} \binom{m − i}{m − j}(−h)^{j−i}u_{i} ∈ Z, we have

Φ_{j} = (x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{j}F_{j} for 1 ≤ j ≤ m. For the constant term F_{j}(0) of F_{j} we have

F_{j}(0) = ∑_{i=0}^{j} \binom{m − i}{m − j}(−h(0))^{j−i}u_{i}(0) = ∑_{i=0}^{j} \binom{m − i}{m − j}(−a)^{j−i}\binom{m}{i}a^{i}

= a^{j} ∑_{i=0}^{j} \binom{m − i}{j − i}\binom{m}{i}(−1)^{j−i} = a^{j} ∑_{i=0}^{j} \binom{j}{i}\binom{m}{j}(−1)^{j−i}

= a^{j}\binom{m}{j}\{1 + (−1)\}^{j} = 0.

We used 5 and Lemma 2.3.7. Thus F_{j} ∈ M(Z).

9. Easy.

Definition 9.6. Let (B, W, P) be a framed space germ over k, and δ be a nonnegative integer. Let (Z, U, P) = \underline{(B, W, P)}, 𝔭 be a prime ideal of the valuation ring U, and q = p^{δ}. If an element u ∈ Z has an element h ∈ Z satisfying u − h^{q} ∈ 𝔭, then we say that u is the q-th power modulo 𝔭.

Lemma 9.7. Let (B, W, P) be a framed space germ over k, and δ be a non-negative integer. Let (Z, U, P) = \underline{(B, W, P)}, and u ∈ Z. Let q = p^{δ} and P = \{𝔭 ∈ Spec(U) | u is the q-th power modulo 𝔭}. Then P ≠ ∅ and P has the minimum element.

Proof. Since Z/M(U) ∩ Z ≅ Z/M(Z) ≅ k, and since k is algebraically closed, one has M(U) ∈ P ≠ ∅. One knows that P is a finite set, since P ⊂ Spec(U) and since ♯ Spec(U) − 1 = rank U ≤ dim Z by Corollary 4.15.1 and Corollary 4.22.1. By Corollary 4.15.2 P has the minimum element.

Definition 9.8. Let (B, W, P) be a framed space germ over k, and (h, g, t) be a Weierstrass representation over (B, W, P). We denote P = {x_{1}, x_{2}, …, x_{n}}, (Z, U, P) = \underline{(B, W, P)}, m = deg(g, t). Let G_{1}, G_{2}, …, G_{m} ∈ Z be the coefficient function sequence of (g, t).

1. Assume that (h, g, t) satisfies the Abhyankar condition, and (h, g, t) is reductive. Let b_{1}, b_{2}, …, b_{n} be the characteristic numbers of (g, t), and m = lq be the p-decomposition of m. Note that u_{q} = G_{q}/(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{q} ∈ Z^{∗} by Lemma 9.5.5.

A reducing element H ∈ Z of (g, t) satisfying H/(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}) ∈ Z^{∗} is called an effective reducing element of (g, t).

Let P = {𝔭 ∈ Spec(U) | u_{q} ∈ Z^{∗} is the q-th power modulo 𝔭}, and 𝔭_{0} ∈ P be the minimum element of P.

An effective reducing element H ∈ Z of (g, t) such that

u_{q} − l(\frac{H}{x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}})^{q} ∈ 𝔭_{0}

is called a very effective reducing element of (g, t).

2. Consider a reduction sequence S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν < ω + 1} of (h, g, t). Let (B_{0}, W_{0}, P_{0}) = (B, W, P) and (h_{0}, g_{0}, t_{0}) = (h, g, t). Let {(h_{ν}, g_{ν}, t_{ν}) | 0 ≤ ν < ω + 1} be the transform Weierstrass representation sequence of (h, g, t). For each ν (h_{ν}, g_{ν}, t_{ν}) is a Weierstrass representation over (B_{ν}, W_{ν}, P_{ν}). Let (h′_{ν−1}, g′_{ν−1}, t′_{ν−1}) be the pull-back of (h_{ν−1}, g_{ν−1}, t_{ν−1}) by the associated morphism (B_{ν−1}, W_{ν−1}, P_{ν−1}) → (B_{ν}, W_{ν}, P_{ν}).

We say that the reduction sequence S is effective, if for every integer ν with 1 ≤ ν < ω the element H_{ν} ∈ k[[P_{ν}]] is an effective reducing element of (g′_{ν−1}, t′_{ν−1}).

We say that the reduction sequence S is very effective, if for every integer ν with 1 ≤ ν < ω the element H_{ν} ∈ k[[P_{ν}]] is a very effective reducing element of (g′_{ν−1}, t′_{ν−1}).

Remark . According to the above condition in 2, in the case that ω is a positive integer, H_{ω} is not necessarily an effective reducing element or a very effective reducing element of (g′_{ω−1}, t′_{ω−1}).

If H_{ω} = 0, then H_{ω} is never an effective reducing element of (g′_{ω−1}, t′_{ω−1}).

Lemma 9.9. Let (B, W, P) be a framed space germ over k, and (h, g, t) be a Weierstrass representation over (B, W, P). Assume that (h, g, t) satisfies the Abhyankar condition, and (h, g, t) is reductive. Then, there exists a very effective reducing element H ∈ k[[P]] of the semi-Weierstrass pair (g, t).

Proof. We use notations in Definition 9.8. In particular, Z = k[[P]]. By definition of 𝔭_{0} there is an element h_{1} ∈ Z with u_{q} − h_{1}^{q} ∈ 𝔭_{0}. There is an element s ∈ k such that s ≠ 0 and l = s^{q}. Let h = h_{1}/s ∈ Z. We have u_{q} − lh^{q} = u_{q} − h_{1}^{q} ∈ 𝔭_{0}.

Let H = hx_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}} ∈ Z. We will show that H is a very effective reducing element of (g, t).

We have u_{q}−lh^{q} ∈ M(U)∩Z = M(Z), since u_{q}−lh^{q} ∈ 𝔭_{0} ⊂ M(U). Thus u_{q}(0) = lh(0)^{q}. By Lemma 9.5.5 we have u_{q}(0) ≠ 0. Thus h(0) ≠ 0, and we have in(G_{q}) = u_{q}(0)(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{q} = l(h(0)x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}})^{q} = l in(H)^{q}. By Lemma 9.5.6 one knows that H is a reducing element of (g, t).

We have H/(x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}) = h ∈ Z^{∗}, since h(0) ≠ 0. Thus H is an effective reducing element of (g, t).

Since

u_{q} − l(\frac{H}{x_{1}^{b_{1}}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}})^{q} = u_{q} − lh^{q} ∈ 𝔭_{0},

H is a very effective reducing element of (g, t).

Lemma 9.10. Let C be an admissible class of framed space germs over k, and r be a positive integer.

Assume that RW (B, W, P) is true for every framed space germ (B, W, P) such that (B, W, P) ∈ C and ♯ P ≤ r.

Let (B, W, P) be a framed space germ over k such that (B, W, P) ∈ C and ♯ P ≤ r, and (h, g, t) be a Weierstrass representation over (B, W, P).

Then, the following four conditions are equivalent:

1. w(g, t) < ∞.

2. There exists a reduction sequence of (h, g, t).

3. There exists an effective reduction sequence of (h, g, t).

4. There exists a very effective reduction sequence of (h, g, t).

Proof. The implications 4⇒3⇒2⇒1 are trivial. We have only to show 1⇒4.

Now, we have shown 1⇒2 in Lemma 9.2. Reading this proof again, we notice that if we define the concept of “very effective” also for semi-reduction sequences, and if we apply Lemma 9.9 when we choose a reducing element H_{λ+1} ∈ k[[P_{λ+1}]], then the same proof shows 1⇒4.

In the remaining part of this section we consider properties of effective reduction sequences.

Proposition 9.11. Let (B, W, P) be a framed space germ over k, (h, g, t) be a Weierstrass representation over (B, W, P), and S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν < ∞} be an effective reduction sequence of (h, g, t) with infinite length. Let (B_{0}, W_{0}, P_{0}) = (B, W, P) and (h_{0}, g_{0}, t_{0}) = (h, g, t). Let {(h_{ν}, g_{ν}, t_{ν}) | 0 ≤ ν < ω + 1} be the transform Weierstrass representation sequence of (h, g, t). We denote m = deg(g, t), P_{ν} = {x_{ν,1}, x_{ν,2}, …, x_{ν,n}}, Z_{ν} = k[[P_{ν}]] = k[[x_{ν,1}, x_{ν,2}, …, x_{ν,n}]], and by F_{ν} the quotient field of B_{ν} for every 0 ≤ ν < ∞. By the associated homomorphism F_{ν} → F_{ν+1} we define an inductive system \{F_{ν} | ν ≥ 0\}, and denote F = \underrightarrow{lim}_{ν}F_{ν}. For simplicity, we identify F_{ν} and the image of the canonical homomorphism F_{ν} → F. Under this identification, we have h = h_{ν}, g = g_{ν} and t_{ν} = t + ∑_{μ=1}^{ν} H_{μ} for every non-negative integer ν. We understand ∑_{μ=1}^{0} H_{μ} = 0 here as a convention. Also we denote Z = \underrightarrow{lim}_{ν}Z_{ν} = ∪_{ν=0}^{∞}Z_{ν}. Note that w(g, t) < ∞ and m ≥ 1, since S exists.

Then, there exist an infinite sequence

{(F_{ν,1}, F_{ν,2}, …, F_{ν,m}) | 0 ≤ ν < ∞}

of m-tuples (F_{ν,1}, F_{ν,2}, …, F_{ν,m}) of elements F_{ν,j} ∈ Z, an infinite sequence

{(b_{ν,1}, b_{ν,2}, …, b_{ν,n}) | 1 ≤ ν < ∞}

of n-tuples (b_{ν,1}, b_{ν,2}, …, b_{ν,n}) of non-negative integers b_{ν,i}, and an infinite sequence

{h_{ν} | 1 ≤ ν < ∞}

of elements h_{ν} ∈ Z satisfying the following conditions (1.0) and (1.ν), (2.ν), (3.ν) (4.ν) for every positive integer ν. For simplicity, we write

⟨ν⟩ = x_{ν,1}^{b_{ν,1}}x_{ν,2}^{b_{ν,2}} ⋯ x_{ν,n}^{b_{ν,n}} = ∏_{i=1}^{n} x_{ν,i}^{b_{ν,i}}.

(1.ν) F_{ν,j} ∈ M(Z_{ν}) for every 1 ≤ j ≤ m.

g = t_{ν}^{m} + ∑_{j=1}^{m} (∏_{μ=1}^{ν}⟨μ⟩)^{j} F_{ν,j}t_{ν}^{m−j}.

We understand ∏_{μ=1}^{0}⟨μ⟩ = 1 here as a convention. (2.ν) Note that F_{ν−1,j} ∈ Z_{ν−1} ⊂ Z_{ν} for every 1 ≤ j ≤ m. The m-tuple (F_{ν−1,1},

F_{ν−1,2}, …, F_{ν−1,m}) of elements in Z_{ν} satisfy the Abhyankar condition over the framed space germ (B_{ν}, W_{ν}, P_{ν}). The characteristic numbers of (F_{ν−1,1}, F_{ν−1,2}, …, F_{ν−1,m}) are equal to (b_{ν,1}, b_{ν,2}, …, b_{ν,n}). w(F_{ν−1,1}, F_{ν−1,2}, …,

F_{ν−1,m}) = ∑_{i=1}^{n} b_{ν,i}.

(3.ν) At least one of b_{ν,1}, b_{ν,2}, …, b_{ν,n} is not zero. ⟨ν⟩ ∈ M(Z_{ν}). (4.ν)

H_{ν} = h_{ν} ∏_{μ=1}^{ν}⟨μ⟩, h_{ν} ∈ Z_{ν}^{∗}.

Proof. By induction on ν we define F_{ν,j} (1 ≤ j ≤ m), b_{ν,i} (1 ≤ i ≤ n), and h_{ν}.

First, let F_{0,1}, F_{0,2}, …, F_{0,m} ∈ M(Z_{0}) = M(Z) be the coefficient function sequence of (g_{0}, t_{0}) = (g, t). The condition (1.0) holds.

Let ν be a positive integer. Assume that in addition to F_{0,j} (1 ≤ j ≤ m) which we have just defined, we have already defined F_{μ,j}, b_{μ,i}, h_{μ} for every integers μ, j, i satisfying 1 ≤ μ < ν, 1 ≤ j ≤ m, 1 ≤ i ≤ n, and they satisfy (1.0) and (1.μ), (2.μ), (3.μ) (4.μ) for every positive integer μ with 1 ≤ μ < ν.

We will define b_{ν,1}, b_{ν,2}, …, b_{ν,n}, h_{ν}, and (F_{ν,1}, F_{ν,2}, …, F_{ν,m}) so that they will satisfy (2.ν), (3.ν) (4.ν) and (1.ν).

By definition of the reduction sequence, the Weierstrass representation (h, g, t_{ν−1}) over (B_{ν}, W_{ν}, P_{ν}) satisfies the Abhyankar condition and is reductive. By (1.ν − 1) the coefficient function sequence of (g, t_{ν−1}) is equal to (∏_{μ=1}^{ν−1}⟨μ⟩)^{j}F_{ν−1,j} 1 ≤ j ≤ m. Since the associated morphism (B_{μ}, W_{μ}, P_{μ}) → (B_{ν}, W_{ν}, P_{ν}) for any 0 ≤ μ ≤ ν preserves normal crossings, we have non-negative integers c_{1}, c_{2}, …, c_{n} and an element v ∈ Z_{ν}^{∗} with

∏_{μ=1}^{ν−1}⟨μ⟩ = vx_{ν,1}^{c_{1}}x_{ν,2}^{c_{2}} ⋯ x_{ν,n}^{c_{n}}.

By Lemma 9.5.9 the m-tuple v^{j}F_{ν−1,j} 1 ≤ j ≤ m satisfies the Abhyankar condition, and the weight of it is equal to the sum of the characteristic numbers of it.

By Lemma 9.4.3 also the m-tuple F_{ν−1,j} 1 ≤ j ≤ m satisfies the Abhyankar condition. Besides, the m-tuple v^{j}F_{ν−1,j} 1 ≤ j ≤ m and the m-tuple F_{ν−1,j} 1 ≤ j ≤ m have common weight and common characteristic numbers.

Let b_{ν,1}, b_{ν,2}, …, b_{ν,n} be the characteristic numbers of the m-tuple F_{ν−1,j} 1 ≤ j ≤ m. By Lemma 9.5.9 one knows that they satisfy (2.ν) and (3.ν).

By Lemma 9.5.9 the characteristic numbers of (g, t_{ν−1}) are equal to c_{i} + b_{ν,i} 1 ≤ i ≤ n. By definition of ⟨ν⟩ we have

x_{ν,1}^{c_{1}+b_{ν,1}}x_{ν,2}^{c_{2}+b_{ν,2}} ⋯ x_{ν,n}^{c_{n}+b_{ν,n}} = \frac{1}{v} ∏_{μ=1}^{ν}⟨μ⟩.

Since H_{ν} is an effective reducing element of (g, t_{ν−1}), we have

h = \frac{H_{ν}}{x_{ν,1}^{c_{1}+b_{ν,1}}x_{ν,2}^{c_{2}+b_{ν,2}} ⋯ x_{ν,n}^{c_{n}+b_{ν,n}}} = \frac{vH_{ν}}{∏_{μ=1}^{ν}⟨μ⟩} ∈ Z_{ν}^{∗},

and w(g, t_{ν−1}) < w(g, t_{ν−1} + H_{ν}). Thus h_{ν} = h/v satisfies (4.ν).

Since t_{ν} = t_{ν−1} + H_{ν}, by Lemma 9.5.8 there exists an m-tuple F_{j} ∈ M(Z_{ν}) 1 ≤ j ≤ m satisfying

g = t_{ν}^{m} + ∑_{j=1}^{m}(x_{ν,1}^{c_{1}+b_{ν,1}}x_{ν,2}^{c_{2}+b_{ν,2}} ⋯ x_{ν,n}^{c_{n}+b_{ν,n}})^{j}F_{j}t_{ν}^{m−j}.

Let F_{ν,j} = F_{j}/v^{j} for 1 ≤ j ≤ m. The m-tuple F_{ν,j} 1 ≤ j ≤ m satisfies (1.ν).

Remark . By the equality in (1.ν) it is intuitively quite plausible that

g = lim_{ν→∞} {t_{ν}^{m} + ∑_{j=1}^{m} (∏_{μ=1}^{ν}⟨μ⟩)^{j} F_{ν,j}t_{ν}^{m−j}} = ( lim_{ν→∞} t_{ν})^{m} = (t + ∑_{ν=1}^{∞} H_{ν})^{m}.

If this is true, then we can conclude that g = (t + H)^{m} for some H ∈ Z_{0} by Remark after Proposition 2.18, and we obtain a stronger version of our main theorem Theorem 1.1 without any assumption on valuation rings. (Note that g = (t + H)^{m} ∈ B has a normal crossing.) However, it is very hard to give a meaning to the above limit, since the ring Z is not necessarily noetherian.

Below we apply the valuation theory to overcome this difficulty.

Corollary 9.12. 1. For every positive integer ν also the following (5.ν) holds: (5.ν) For every 1 ≤ j ≤ m

u_{ν,j} = \frac{F_{ν−1,j}}{⟨ν⟩^{j}} ∈ Z_{ν}, u_{ν,j}(0) = \binom{m}{j}h_{ν}(0)^{j}, and

g = t_{ν−1}^{m} + ∑_{j=1}^{m} (∏_{μ=1}^{ν}⟨μ⟩)^{j} u_{ν,j}t_{ν−1}^{m−j}.

2. For every positive integer ν and for every 1 ≤ j ≤ m

⟨ν⟩^{j}F_{ν,j} = \binom{m}{m − j}(−h_{ν}⟨ν⟩)^{j} + ∑_{i=1}^{j} \binom{m − i}{m − j}(−h_{ν}⟨ν⟩)^{j−i}F_{ν−1,i}.

Proof. 1. By (2.ν) we have u_{ν,j} ∈ Z_{ν}. Substituting F_{ν−1,j} = ⟨ν⟩^{j}u_{ν,j} into (1.ν −1), we obtain the equality for g.

For simplicity, we write X = x_{ν,1}^{c_{1}+b_{ν,1}}x_{ν,2}^{c_{2}+b_{ν,2}} ⋯ x_{ν,n}^{c_{n}+b_{ν,n}}. By the equality we just obtained, one knows

in(g, t_{ν−1}) = t_{ν−1}^{m} + ∑_{j=1}^{m}(v(0)X)^{j}u_{ν,j}(0)t_{ν−1}^{m−j}.

By (4.ν) in(H_{ν}) = h_{ν}(0)v(0)X. By Lemma 6.12.2 in(g, t_{ν−1}) = (t_{ν−1} + in(H_{ν}))^{m}. Thus for every 1 ≤ j ≤ m

(v(0)X)^{j}u_{ν,j}(0) = \binom{m}{j}(v(0)X)^{j}h_{ν}(0)^{j}.

Since v(0)X ≠ 0, we obtain the desired equality.

2. It follows from Lemma 6.10.2.

Corollary 9.13. Let m = lq be the p-decomposition of m, ν_{0} be a non-negative integer, and 𝔦 be an ideal of the ring Z. The image of an element Φ ∈ Z by the canonical surjective homomorphism Z → Z/𝔦 is denoted by \overline{Φ} ∈ Z/𝔦.

1. We assume the following two conditions for an integer j_{0}:

(a) 1 ≤ j_{0} < m, j_{0} ≢ 0 mod q.

(b) If 1 ≤ j < j_{0}, j ≢ 0 mod q, then \overline{F}_{ν_{0},j} = 0.

For every integer ν with ν_{0} ≤ ν and for every integer j satisfying 1 ≤ j < j_{0} and j ≢ 0 mod q we have

( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{j} \overline{F}_{ν,j} = 0, and \overline{F}_{ν_{0},j_{0}} = ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{j_{0}} \overline{F}_{ν,j_{0}}.

We understand ∏_{μ=ν_{0}+1}^{ν_{0}} \overline{⟨μ⟩} = 1 here as a convention.

2. Assume that \overline{F}_{ν_{0},j} = 0 for every integer j with 1 ≤ j < q. For every integer ν with ν_{0} ≤ ν

\overline{F}_{ν_{0},q} = l ( ∑_{μ=ν_{0}+1}^{ν} \overline{h}_{μ} ∏_{λ=ν_{0}+1}^{μ} \overline{⟨λ⟩})^{q} + ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{q} \overline{F}_{ν,q}.

We understand ∑_{μ=ν_{0}+1}^{ν_{0}} \overline{h}_{μ} ∏_{λ=ν_{0}+1}^{μ} \overline{⟨λ⟩} = 0 here as a convention.

3. We assume the following three conditions for an integer i_{0}:

(a) If 1 ≤ j ≤ m, j ≢ 0 mod q, then \overline{F}_{ν_{0},j} = 0.

(b) 2 ≤ i_{0} ≤ l.

(c) If 1 ≤ i < i_{0}, then

\overline{F}_{ν_{0},iq} − \binom{l}{i} (\frac{\overline{F}_{ν_{0},q}}{l} )^{i} = 0.

For every integer ν with ν_{0} ≤ ν and for every integer i with 1 ≤ i < i_{0} we have

( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{iq} \{\overline{F}_{ν,iq} − \binom{l}{i} (\frac{\overline{F}_{ν,q}}{l} )^{i}\} = 0, and

\overline{F}_{ν_{0},i_{0}q} − \binom{l}{i_{0}} (\frac{\overline{F}_{ν_{0},q}}{l} )^{i_{0}} = ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{i_{0}q} \{\overline{F}_{ν,i_{0}q} − \binom{l}{i_{0}} (\frac{\overline{F}_{ν,q}}{l} )^{i_{0}}\} .

Proof. 1. If q = 1, then there is no integer j_{0} satisfying (a), and thus the claim is true. We assume q > 1 below.

First, note that by Lemma 2.4.3, \binom{m − i}{m − j} = 0 for any pair of integers i, j

satisfying 0 ≤ i ≤ j ≤ m, i ≡ 0 mod q, and j ≢ 0 mod q.

Let j_{1} be an arbitrary integer satisfying 1 ≤ j_{1} ≤ j_{0} and j_{1} ≢ 0 mod q. By induction on j_{1} we will show the following claim [j_{1}].

[j_{1}] For every integer ν with ν_{0} ≤ ν and for every integer j satisfying 1 ≤ j < j_{1}

and j ≢ 0 mod q, (∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{j} \overline{F}_{ν,j} = 0.

Claim [1] is true by the trivial reason. Assume j_{1} > 1. Let j_{2} be the maximum element of the set of integers j satisfying 1 ≤ j < j_{1} and j ≢ 0 mod q. We assume that [j_{2}] is true. Under these assumptions [j_{1}] is true, if we can show

(∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{j_{2}} \overline{F}_{ν,j_{2}} = 0.

By Corollary 9.12.2 for every integer ν with ν_{0} ≤ ν we have

( ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{j_{2}} \overline{F}_{ν+1,j_{2}}

= ∑_{\begin{matrix}0≤i≤j_{2}, \\ i≢0modq \end{matrix}} \binom{m − i}{m − j_{2}}(−\overline{h}_{ν+1}\overline{⟨ν + 1⟩})^{j_{2}−i} ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{j_{2}} \overline{F}_{ν,i}

= ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{j_{2}} \overline{F}_{ν,j_{2}}.

Thus, for every integer ν with ν_{0} ≤ ν we have (∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{j_{2}} \overline{F}_{ν,j_{2}} = \overline{F}_{ν_{0},j_{2}} = 0 by assumption (b), since 1 ≤ j_{2} < j_{1} ≤ j_{0}. It follows that [j_{1}] is true.

In particular, [j_{0}] is true. We can repeat the above argument until the middle, after replacing j_{2} by j_{0}. We obtain the last claim of 1.

2. By 1 for every integer ν with ν_{0} ≤ ν and for every integer j with 1 ≤ j < q

(∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{j} \overline{F}_{ν,j} = 0.

By Corollary 9.12.2 for every integer ν with ν_{0} ≤ ν we have

( ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{q} \overline{F}_{ν+1,q}

=\binom{m}{m − q} (−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{q}

+ ∑_{j=1}^{q} \binom{m − j}{m − q}(−\overline{h}_{ν+1}\overline{⟨ν + 1⟩})^{q−j} ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{q} \overline{F}_{ν,j}

= l (−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{q} + ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{q} \overline{F}_{ν,q}.

Claim 2 follows from the above equality and induction on ν.

3. Let i_{1} be an arbitrary integer with 2 ≤ i_{1} ≤ i_{0}. By induction on i_{1} we will show the following claim [i_{1}].

[i_{1}] For every integer ν with ν_{0} ≤ ν and for every integer i with 1 ≤ i < i_{1}

( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{iq} \{\overline{F}_{ν,iq} − \binom{l}{i} (\frac{\overline{F}_{ν,q}}{l} )^{i}\} = 0.

Claim [2] is true, since \binom{l}{1} = l. Assume i_{1} > 2 and [i_{1} − 1] is true. Let

i_{2} = i_{1} − 1.

By Corollary 9.12.2 for every integer ν with ν_{0} ≤ ν we have

( ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{i_{2}q} \overline{F}_{ν+1,i_{2}q}

=\binom{lq}{lq − i_{2}q} (−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{i_{2}q}

+ ∑_{r=1}^{i_{2}q} \binom{lq − r}{lq − i_{2}q}(−\overline{h}_{ν+1}\overline{⟨ν + 1⟩})^{i_{2}q−r} ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{i_{2}q} \overline{F}_{ν,r}

=\binom{l}{i_{2}} (−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{i_{2}q}

+ ∑_{i=1}^{i_{2}} \binom{l − i}{l − i_{2}} (−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{(i_{2}−i)q} ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{iq} \overline{F}_{ν,iq}.

We used assumption (a) and Lemma 2.4.3 above.

On the other hand, by the equality in the proof of 2 and by the binomial theorem, we have

− ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{i_{2}q} \binom{l}{i_{2}} (\frac{\overline{F}_{ν+1,q}}{l} )^{i_{2}}

= − \binom{l}{i_{2}} \{(−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{q} + ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{q} (\frac{\overline{F}_{ν,q}}{l} )\}^{i_{2}}

= − \binom{l}{i_{2}} (−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{i_{2}q}

− ∑_{i=1}^{i_{2}} \binom{l}{i_{2}}\binom{i_{2}}{i} (−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{(i_{2}−i)q} ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{iq} (\frac{\overline{F}_{ν,q}}{l} )^{i}

Note here that by Lemma 2.3.3 and 7 \binom{l}{i_{2}}\binom{i_{2}}{i} = \binom{l}{i}\binom{l − i}{i_{2} − i} = \binom{l}{i}\binom{l − i}{l − i_{2}}.

Thus we have

( ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{i_{2}q} \{\overline{F}_{ν+1,iq} − \binom{l}{i_{2}} (\frac{\overline{F}_{ν+1,q}}{l} )^{i_{2}}\}

= ∑_{i=1}^{i_{2}} \binom{l − i}{l − i_{2}} (−\overline{h}_{ν+1} ∏_{μ=ν_{0}+1}^{ν+1} \overline{⟨μ⟩})^{(i_{2}−i)q} ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{iq} \{\overline{F}_{ν,iq} − \binom{l}{i} (\frac{\overline{F}_{ν,q}}{l} )^{i}\}

= ( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{i_{2}q} \{\overline{F}_{ν,i_{2}q} − \binom{l}{i_{2}} (\frac{\overline{F}_{ν,q}}{l} )^{i_{2}}\}

Thus, for every integer ν with ν_{0} ≤ ν we have

( ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩})^{i_{2}q} \{\overline{F}_{ν,i_{2}q} − \binom{l}{i_{2}} (\frac{\overline{F}_{ν,q}}{l} )^{i_{2}}\} = \overline{F}_{ν_{0},i_{2}q} − \binom{l}{i_{2}} (\frac{\overline{F}_{ν_{0},q}}{l} )^{i_{2}} = 0

by assumption (c), since 1 ≤ i_{2} = i_{1} − 1 < i_{1} ≤ i_{0}. It follows that [i_{1}] is true.

In particular, [i_{0}] is true. We can repeat the above argument until the middle, after replacing i_{2} by i_{0}. We obtain the last claim of 3.

10. Ring theory, Part two

In this section we show several technical results in the ring theory. The theory of excellent rings (Theorem 2.2) and Theorem 4.31.3 play the essential role. Theorem 10.8.4 seems to be a key of this paper. It will be applied to avoid purely inseparable extensions of the function field.

Lemma 10.1. Let (Z, U) be a space germ over k. Assume that finite elements a_{1}, a_{2}, …, a_{r} ∈ U are given. Denote n = dim Z.

1. The subdomain Z[a_{1}, a_{2}, …, a_{r}] of the quotient field of Z is an excellent domain contained in U. The intersection 𝔪 = M(U) ∩ Z[a_{1}, a_{2}, …, a_{r}] is a prime ideal of Z[a_{1}, a_{2}, …, a_{r}]. dim Z[a_{1}, a_{2}, …, a_{r}] = n.

2. Let R = Z[a_{1}, a_{2}, …, a_{r}]_{𝔪} be the localization of Z[a_{1}, a_{2}, …, a_{r}] with respect to the prime ideal 𝔪. Ring R is an excellent local domain dominated by U. dim R = n.

Proof. 1. By definition of space germs we have Z ⊂ U. Since U is a ring, and since a_{1}, a_{2}, …, a_{r} ∈ U, we have Z[a_{1}, a_{2}, …, a_{r}] ⊂ U. By Theorem 2.2.3 Z is an excellent ring. By Theorem 2.2.4 also Z[a_{1}, a_{2}, …, a_{r}] is an excellent ring. By Theorem 2.2.6 Z is universally catenary. Obviously Z[a_{1}, a_{2}, …, a_{r}] is a domain, and 𝔪 is a prime ideal of Z[a_{1}, a_{2}, …, a_{r}].

Since k ⊂ Z/𝔪 ∩ Z ⊂ Z[a_{1}, a_{2}, …, a_{r}]/𝔪 ⊂ U/M(U) ≅ k, we have Z/𝔪 ∩ Z ≅ Z[a_{1}, a_{2}, …, a_{r}]/𝔪 ≅ k and 𝔪 ∩ Z = M(Z).

By Theorem 2.1 we have ht(𝔪) = ht(𝔪 ∩ Z) + tr.deg_{Z}Z[a_{1}, a_{2}, …, a_{r}] −tr.deg_{(Z/𝔪∩Z)}(Z[a_{1}, a_{2}, …, a_{r}]/𝔪) = ht(𝔪 ∩ Z) = ht(M(Z)) = dim Z = n. Note here that tr.deg_{Z}Z[a_{1}, a_{2}, …, a_{r}] = 0, since Z and Z[a_{1}, a_{2}, …, a_{r}] have the same quotient field.

Let 𝔭 be an arbitrary prime ideal of Z[a_{1}, a_{2}, …, a_{r}]. By Theorem 2.1 we have ht(𝔭) ≤ ht(𝔭 ∩ Z) ≤ dim Z = n. We conclude dim Z[a_{1}, a_{2}, …, a_{r}] = n.

2. By definition of the localization, every element x ∈ R has an element z ∈ Z[a_{1}, a_{2}, …, a_{r}] ⊂ V and an element y ∈ Z[a_{1}, a_{2}, …, a_{r}] − 𝔪 ⊂ U − M(U) = U^{∗} with x = z/y. Thus z ∈ U and R ⊂ U. Obviously R is a local domain. M(R) = 𝔪R ⊂ M(U)U = M(U). Thus U dominates R. dim R = ht(𝔪) = n.

Lemma 10.2. Let V be a regular local ring of dimension one containing the field k, E be the quotient field of V, E′ = V/M(V) be the residue field of V, and δ be a positive integer. Let q = p^{δ} and let F_{δ} : E → E, F_{δ} : E′ → E′ and F_{δ} : V → V be the Frobenius homomorphisms of degree δ. We denote their images by E^{q}, (E′)^{q} and V^{q} respectively. Assume that the relative degree [E : E^{q}] is finite.

1. q[E′ : (E′)^{q}] ≤ [E : E^{q}]. In particular, the relative degree [E′ : (E′)^{q}] is finite.

2. If q[E′ : (E′)^{q}] = [E : E^{q}], then V is a finitely generated V^{q}-module.

Proof. 1. Ring V^{q} is isomorphic to V, and is a regular local ring of dimension one. By Proposition 4.25 V^{q} is a valuation ring of the quotient field E^{q}. The residue field of V^{q} is (E′)^{q} by Lemma 2.6.2.

By the same reason V is a valuation ring of E. We can check that V dominates V^{q}. Thus the valuation ring V ∩E^{q} of the field E^{q} dominates V^{q}. By Corollary 4.19 we have V ∩ E^{q} = V^{q}.

Let π be a generator of M(V). Obviously π^{q} is a generator of M(V^{q}). By Lemma 4.26 one knows that the ramification index of V over V^{q} is equal to q.

By Theorem 4.31.1 we obtain 1.

2. Since x^{q} ∈ V^{q} for every x ∈ V, V is integral over V^{q}. By Proposition 4.25 V is normal. Thus V coincides with the integral closure of V^{q} in E. By Theorem 4.31.3 we obtain 2.

Lemma 10.3. Let (Z, U) be a space germ over k of iterated divisor type. Denote n = dim Z = rank U. Let R be an excellent ring with Z ⊂ R ⊂ U. By

{0} = 𝔭_{0} ⊂ 𝔭_{1} ⊂ ⋯ ⊂ 𝔭_{i−1} ⊂ 𝔭_{i} ⊂ ⋯ ⊂ 𝔭_{n} = M(U)

we denote the sequence of prime ideals of the valuation ring U. We assume 𝔭_{i−1} ≠ 𝔭_{i} for every 1 ≤ i ≤ n. For each integer i with 1 ≤ i ≤ n we choose an element a_{i} with a_{i} ∈ 𝔭_{i} − 𝔭_{i−1}, and we consider the subring

S = R[a_{1}, a_{2}, …, a_{n}]

of the quotient field of Z.

1. S ⊂ U. S is an excellent domain.

2. Let 𝔮_{i} = 𝔭_{i} ∩ S for every 0 ≤ i ≤ n. Each 𝔮_{i} is a prime ideal of S, and they

satisfy

{0} = 𝔮_{0} ⊂ 𝔮_{1} ⊂ ⋯ ⊂ 𝔮_{i−1} ⊂ 𝔮_{i} ⊂ ⋯ ⊂ 𝔮_{n} ⊂ S.

For every 0 ≤ i ≤ n, ht(𝔮_{i}) = i, and ht(𝔮_{n}/𝔮_{i}) = n − i.

3. Let U_{i} = U/𝔭_{i}, D_{i} denote the quotient field of U_{i}, and \overline{S}_{i} denote the integral closure of the domain S/𝔮_{i} in the field D_{i} for every 0 ≤ i ≤ n. For every 0 ≤ i ≤ n, S/𝔮_{i} is an excellent ring, and \overline{S}_{i} ⊂ U_{i}.

4. For every 0 ≤ i ≤ n, D_{i} coincides with the quotient field of \overline{S}_{i}, and the field D_{i} is a finite algebraic extension of the quotient field of S/𝔮_{i}.

5. For every 0 ≤ i ≤ n, \overline{S}_{i} is a finitely generated S/𝔮_{i}-module, and \overline{S}_{i} is an excellent ring.

6. For every 0 ≤ i ≤ n − 1, by \overline{𝔭}_{i+1} we denote the prime ideal 𝔭_{i+1}/𝔭_{i} of U_{i}, and \overline{𝔮}_{i+1} = \overline{𝔭}_{i+1} ∩ \overline{S}_{i}. We have

(\overline{S}_{i})_{\overline{𝔮}_{i+1}} = (U_{i})_{\overline{𝔭}_{i+1}}.

7. For every 0 ≤ i ≤ n − 1, D_{i+1} coincides with the quotient field of \overline{S}_{i}/\overline{𝔮}_{i+1}, and the integral closure of \overline{S}_{i}/\overline{𝔮}_{i+1} in the field D_{i+1} coincides with \overline{S}_{i+1}.

8. For every 0 ≤ i ≤ n − 1, U_{𝔭_{i+1}}/𝔭_{i}U_{𝔭_{i+1}} is an excellent regular local ring of dimension one.

9. The value group Γ of the valuation defined by U is isomorphic to ℤ^{n} with the lexicographic order as totally ordered abelian groups.

10. For every 0 ≤ i ≤ n and for every positive integer δ

[D_{i} : D_{i}^{p^{δ}}] = p^{(n−i)δ}.

By D_{i}^{p^{δ}} we denoted here the image of the Frobenius homomorphism F_{δ} : D_{i} → D_{i} of degree δ.

Proof. 1. It follows from Theorem 2.2.4.

2. Obviously 𝔮_{i} is a prime ideal of S, and 𝔮_{i−1} ⊂ 𝔮_{i}. a_{i} ∈ 𝔭_{i} ∩ S = 𝔮_{i}. On the other hand, a_{i} ∉ 𝔮_{i−1}, since 𝔮_{i−1} ⊂ 𝔭_{i−1} and a_{i} ∉ 𝔭_{i−1}. Thus 𝔮_{i−1} ≠ 𝔮_{i}.

One concludes ht(𝔮_{i}) ≥ i, and ht(𝔮_{n}/𝔮_{i}) ≥ n − i for every 0 ≤ i ≤ n.

By Theorem 2.1 we have ht(𝔮_{n}) ≤ ht(𝔮_{n} ∩R)+tr.deg_{R}S = ht(𝔮_{n} ∩R) ≤ dim R = n. Note here that tr.deg_{R}S = 0, since R and S have the same quotient field.

We have n ≤ ht(𝔮_{i}) + ht(𝔮_{n}/𝔮_{i}) ≤ ht(𝔮_{n}) ≤ n. We obtain 2.

3. We fix an integer i with 0 ≤ i ≤ n.

By Theorem 2.2.4 S/𝔮_{i} is an excellent ring.

By Lemma 4.16.1 U_{i} = U/𝔭_{i} is a valuation ring of the field D_{i}. By Corollary 4.10.2 U_{i} is normal.

Since S/𝔮_{i} ⊂ U_{i}, we have \overline{S}_{i} ⊂ U_{i} by definition of \overline{S}_{i}. 4, 5, 6 and 7. We will show 4, 5, 6 and 7 together by induction on i. Fixing the integer i, by 4.i, 5.i, 6.i and 7.i we denote the corresponding claims of 4, 5, 6 and 7 with the value i fixed respectively.

First, note that 4.0 is true. By definition of space germs the quotient field of S = S/𝔮_{0} and the quotient field D_{0} of U = U_{0} = U/𝔭_{0} coincide. By definition \overline{S}_{0} is the integral closure of S in the common quotient field D_{0}. Thus three fields, the quotient field of S/𝔮_{0}, the quotient field of \overline{S}_{0} and D_{0} coincide, and 4.0 is true.

Second, note that the implication 4.i ⇒5.i holds. By 4.i the quotient field D_{i} of U_{i} is a finite algebraic extension of the quotient field of S/𝔮_{i}. By definition \overline{S}_{i} is the integral closure of S/𝔮_{i} in the field of D_{i}. By 3 S/𝔮_{i} is an excellent domain. Thus, by Theorem 2.2.7 \overline{S}_{i} is a finitely generated S/𝔮_{i}-module. In particular, \overline{S}_{i} is a finitely generated S/𝔮_{i}-algebra, and by Theorem 2.2.4 \overline{S}_{i} is an excellent ring.

Third, we will show 4.i and 5.i ⇒ 6.i.

By definition \overline{𝔮}_{i+1} ∩ (S/𝔮_{i}) = \overline{𝔭}_{i+1} ∩ (S/𝔮_{i}) = 𝔮_{i+1}/𝔮_{i}. By 2 ht(𝔮_{i+1}/𝔮_{i}) = 1. By 5.i \overline{S}_{i} is integral over S/𝔮_{i}. Thus by Matsumura [7], p.33–p.34, (5.E), Theorem 5, ii), one knows ht(\overline{𝔮}_{i+1}) = 1. Thus dim(\overline{S}_{i})_{\overline{𝔮}_{i+1}} = 1. On the other hand, by 5.i (\overline{S}_{i})_{\overline{𝔮}_{i+1}} is a noetherian normal local ring of dimension one. By Proposition 4.25 (\overline{S}_{i})_{\overline{𝔮}_{i+1}} is a valuation ring of the quotient field of \overline{S}_{i}. By 4.i the quotient field of (\overline{S}_{i})_{\overline{𝔮}_{i+1}} coincides with D_{i}. By definition (U_{i})_{\overline{𝔭}_{i+1}} dominates (\overline{S}_{i})_{\overline{𝔮}_{i+1}}. By Corollary 4.19 we obtain 6.i.

Fourth, we will show 5.i and 6.i ⇒ 7.i. Considering the residue fields of local rings at the both-hand sides of the equality in 6.i, we have

(\overline{S}_{i})_{\overline{𝔮}_{i+1}}/\overline{𝔮}_{i+1}(\overline{S}_{i})_{\overline{𝔮}_{i+1}} ≅ (U_{i})_{\overline{𝔭}_{i+1}}/\overline{𝔭}_{i+1}(U_{i})_{\overline{𝔭}_{i+1}}. (*)

The left-hand side of (∗) coincides with the quotient field of \overline{S}_{i}/\overline{𝔮}_{i+1}, and the righthand side of (∗) coincides with the quotient field D_{i+1} of U_{i+1} = U_{i}/\overline{𝔭}_{i+1}. Thus we obtain the former half of 7.i.

Note that \overline{𝔮}_{i+1} ∩ (S/𝔮_{i}) = 𝔮_{i+1}/𝔮_{i}, (S/𝔮_{i})/(\overline{𝔮}_{i+1} ∩ (S/𝔮_{i})) ≅ S/𝔮_{i+1}. Thus, by 5.i \overline{S}_{i}/\overline{𝔮}_{i+1} is a finitely generated S/𝔮_{i+1}-module. In particular, \overline{S}_{i}/\overline{𝔮}_{i+1} is integral over S/𝔮_{i+1}. By the former half of 7.i \overline{S}_{i}/\overline{𝔮}_{i+1} and U_{i+1} have the same quotient field D_{i+1}. By definition \overline{S}_{i+1} is the integral closure of S/𝔮_{i+1} in the field D_{i+1}. Thus S/𝔮_{i} ⊂ \overline{S}_{i}/\overline{𝔮}_{i+1} ⊂ \overline{S}_{i+1}, and we obtain the latter half of 7.i.

Fifth, we show 5.i and 7.i ⇒ 4.i + 1.

By 7.i the three fields, D_{i+1}, the quotient field of \overline{S}_{i+1} and the quotient field of \overline{S}_{i}/\overline{𝔮}_{i+1}, coincide. Thus, in particular, the former half of 4.i + 1 is true.

By 5.i \overline{S}_{i}/\overline{𝔮}_{i+1} is a finitely generated S/𝔮_{i+1}-module. Let E denote the quotient field of S/𝔮_{i+1}. Let D′ be the set of elements x of the quotient field D_{i+1} of \overline{S}_{i}/\overline{𝔮}_{i+1} such that there are elements y ∈ \overline{S}_{i}/\overline{q}_{i+1} and z ∈ S/𝔮_{i+1} satisfying z ≠ 0 and x = y/z. D′ is a subdomain of D_{i+1}. D′ is a finitely generated E-module. D′ contains \overline{S}_{i}/\overline{𝔮}_{i+1}. Thus D′ is a field, and D′ = D_{i+1}. D_{i+1} is a finite algebraic extension of E.

8. U_{𝔭_{i+1}}/𝔭_{i}U_{𝔭_{i+1}} ≅ (U_{i})_{\overline{𝔭}_{i+1}}. Thus by 6, the claim of 8 is equivalent to that (\overline{S}_{i})_{\overline{𝔮}_{i+1}} is an excellent regular local ring of dimension one. We have already seen that (\overline{S}_{i})_{\overline{𝔮}_{i+1}} is a regular local ring of dimension one in the proof of 4.i and 5.i ⇒ 6.i. By 5 and by Theorem 2.2.5 (\overline{S}_{i})_{\overline{𝔮}_{i+1}} is excellent.

9. It follows from Corollary 4.22.2.

10. We write q = p^{δ}, and let P = {x_{1}, x_{2}, …, x_{n}} be a parameter system of Z. The field D_{0} is the quotient field of the ring Z = k[[x_{1}, x_{2}, …, x_{n}]]. Since k is an algebraically closed field, the Frobenius homomorphism F_{δ} : k → k of degree δ over k is surjective. Thus D_{0}^{q} is the quotient field of Z^{q} = k[[x_{1}^{q}, x_{2}^{q}, …, x_{n}^{q}]]. Obviously Z is a free Z^{q}-module, and q^{n} elements x_{1}^{α_{1}}x_{2}^{α_{2}} ⋯ x_{n}^{α_{n}} 0 ≤ α_{1}, α_{2}, …, α_{n} ≤ q − 1 are a basis of Z over Z^{q}.

Let D′ be the set of elements x of the field D_{0} such that there are elements y ∈ Z and z ∈ Z^{q} satisfying z ≠ 0 and x = y/z. D′ is a subdomain of D_{0}. D′ is a finitely generated D_{0}^{q}-module. D′ contains Z. Thus D′ is a field, and D′ = D_{0}. There exists a surjective homomorphism Z ⊗_{Z^{q}} D_{0}^{q} → D′ = D_{0} of D_{0}^{q}-modules, and one knows [D_{0} : D_{0}^{q}] ≤ q^{n}.

By 8 U_{𝔭_{i+1}}/𝔭_{i}U_{𝔭_{i+1}} is a regular local ring of dimension one, the quotient field of it is D_{i}, and the residue field of it is D_{i+1}. By Lemma 10.2.1 we have [D_{i+1} : D_{i+1}^{q}] ≤ [D_{i} : D_{i}^{q}]/q, if [D_{i} : D_{i}^{q}] is finite. We have above shown that [D_{0} : D_{0}^{q}] is finite. Thus, by induction, one knows that [D_{i} : D_{i}^{q}] is finite for every 0 ≤ i ≤ n. Besides, the following (∗∗) holds:

\begin{aligned}[D_{n} : D_{n}^{q}] ≤ & \frac{[D_{n−1} : D_{n−1}^{q}]}{q} ≤ ⋯ ≤ \frac{[D_{i+1} : D_{i+1}^{q}]}{q^{n−i−1}} ≤ \frac{[D_{i} : D_{i}^{q}]}{q^{n−i}} ≤ \\ & ⋯ ≤ \frac{[D_{0} : D_{0}^{q}]}{q^{n}} ≤ \frac{q^{n}}{q^{n}} = 1 \end{aligned} (**)

By definition [D_{n} : D_{n}^{q}] is a positive integer. Thus by (∗∗) one knows [D_{i} : D_{i}^{q}] = q^{n−i} for every 0 ≤ i ≤ n.

Proposition 10.4. Let (Z, U) be a space germ over k of iterated divisor type, and (Z′, U′) be an elementary monoidal transform of (Z, U). Denote n = dim Z = rank U.

1. A space germ (Z′, U′) is of iterated divisor type. Let

{0} = 𝔭_{0} ⊂ 𝔭_{1} ⊂ ⋯ ⊂ 𝔭_{i−1} ⊂ 𝔭_{i} ⊂ ⋯ ⊂ 𝔭_{n} = M(U),
{0} = 𝔭′_{0} ⊂ 𝔭′_{1} ⊂ ⋯ ⊂ 𝔭′_{i−1} ⊂ 𝔭′_{i} ⊂ ⋯ ⊂ 𝔭′_{n} = M(U′)

be the sequences of prime ideals of U and U′ respectively. For every 0 ≤ i ≤ n 𝔭_{i} = 𝔭′_{i} ∩ U.

2. Let i be an arbitrary integer with 0 ≤ i ≤ n. By D_{i} and D′_{i} denote the quotient field of U/𝔭_{i} and U′/𝔭′_{i} respectively. Let δ be an arbitrary positive integer. The field D′_{i} is generated by D_{i} and (D′_{i})^{p^{δ}}, where the image of the Frobenius homomorphism F_{δ} : D′_{i} → D′_{i} of degree δ is denoted by (D′_{i})^{p^{δ}}.

3. For every 0 ≤ i ≤ n−1 the homomorphism of regular local rings of dimension

one

U_{𝔭_{i+1}}/𝔭_{i}U_{𝔭_{i+1}} → U′_{𝔭′_{i+1}}/𝔭′_{i}U′_{𝔭′_{i+1}}

induced by the associated homomorphism U → U′ sends a generator of the maximal ideal to a generator of the maximal ideal.

4. Let Γ and Γ′ denote the value group of the valuation ring U and U′ respectively. The induced homomorphism Γ → Γ′ by U → U′ is an isomorphism.

Proof. 1. It follows from Corollary 4.22.1 and Corollary 4.15.

2. Let U_{i} = U/𝔭_{i} and U′_{i} = U′/𝔭′_{i}. The injective homomorphism U_{i} = U/𝔭_{i} = U/𝔭′_{i} ∩ U → U′_{i} = U′/𝔭′_{i} induces an injective homomorphism D_{i} → D′_{i}. We regard D_{i} as a subfield of D′_{i} by this homomorphism.

By definition of an elementary monoidal transform, there exists a partial parameter system x_{0}, x_{1}, …, x_{m} of Z with the following properties:

(a) m ≥ 1.

(b) b_{i} = x_{i}/x_{0} ∈ M(U) for every 1 ≤ i ≤ m.

(c) Let 𝔪 = M(U) ∩ Z[b_{1}, b_{2}, …, b_{m}] and R = Z[b_{1}, b_{2}, …, b_{m}]_{𝔪}. The valuation ring U dominates R.

(d) The completion R̂ of the local ring R coincides with Z′.

By Lemma 10.1.2 R is an excellent local domain and n = dim R = dim R̂. By Theorem 2.2.3 also R̂ = Z′ is excellent.

Choosing an element a_{i} ∈ 𝔭_{i} −𝔭_{i−1} for each integer i with 1 ≤ i ≤ n, we consider a subdomain

S = R[a_{1}, a_{2}, ⋯, a_{n}]

of the quotient field D_{0} of Z. Besides, regarding a_{i} ∈ D_{0} ⊂ D′_{0}, we consider the subdomain

S′ = R̂[a_{1}, a_{2}, ⋯, a_{n}]

of the quotient field D′_{0} of Z′. Note that a_{i} ∈ 𝔭′_{i} − 𝔭′_{i−1} for every 1 ≤ i ≤ n by 1 above. Note also that R̂ = Z′ ⊂ U′ by definition of an elementary monoidal transform.

Lemma 10.3.1–10 hold for both pairs (S, U) and (S′, U′).

Choosing an arbitrary integer i with 0 ≤ i ≤ n, and we fix i below. Let \overline{S}_{i} and \overline{S}′_{i} denote the integral closure of the domain S/𝔭_{i} ∩S in the field D_{i} and the integral closure of the domain S′/𝔭′_{i} ∩ S′ in the field D′_{i} respectively. By Lemma 10.3.4 the quotient field of \overline{S}_{i} coincides with D_{i}, and the quotient field of \overline{S}′_{i} coincides with D′_{i}.

By definition, \overline{S}_{i} is integral over S/𝔭_{i} ∩ S. Since S/𝔭_{i} ∩ S ⊂ S′/𝔭′_{i} ∩ S′, \overline{S}_{i} is integral over S′/𝔭′_{i} ∩ S′. Thus we have \overline{S}_{i} ⊂ \overline{S}′_{i}.

\overline{S}′_{i} contains R̂/𝔭′_{i} ∩ R̂ as a subring. The subring of \overline{S}′_{i} generated by \overline{S}_{i} and R̂/𝔭′_{i} ∩ R̂ is denoted by \overline{S}_{i}[R̂/𝔭′_{i} ∩ R̂] or \overline{S}_{i}[R̂] in short. \overline{S}_{i}[R̂] ⊂ \overline{S}′_{i} ⊂ U′_{i}. Let 𝔫_{i} = M(U′_{i}) ∩ \overline{S}_{i}[R̂].

We consider the ring \overline{S}_{i} ⊗_{R} R̂ and the maximal ideal 𝔪_{i} = (M(U_{i}) ∩ \overline{S}_{i}) ⊗ R̂ + \overline{S}_{i} ⊗ M(R̂) here. We need a lemma.

Lemma 10.5. 1. The localization (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} of the ring \overline{S}_{i} ⊗_{R} R̂ with respect

to 𝔪_{i} is a normal local domain.

2. The canonical homomorphism \overline{S}_{i} ⊗_{R} R̂ → \overline{S}_{i}[R̂] is surjective, and it induces an isomorphism (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} → \overline{S}_{i}[R̂]_{𝔫_{i}}.

3. L_{i} = \overline{S}_{i}[R̂] − M(U′_{i}) ∩ \overline{S}_{i}[R̂] is a multiplicative subset of \overline{S}_{i}[R̂]. \overline{S}_{i}[R̂]_{𝔫_{i}} = L_{i}^{−1}\overline{S}_{i}[R̂].

4. L_{i}^{−1}\overline{S}_{i}[R̂] = L_{i}^{−1}\overline{S}′_{i}. The quotient field of \overline{S}_{i}[R̂] coincides with D′_{i}.

5. L_{i}^{−1}\overline{S}′_{i} = (\overline{S}′_{i})_{M(U′_{i})∩\overline{S}′_{i}}.

Proof. 1. For every positive integer ν we have 𝔪_{i}^{ν} ⊃ ((M(U_{i}) ∩ \overline{S}_{i}) ⊗ R̂)^{ν} + (\overline{S}_{i} ⊗ M(R̂))^{ν} ⊃ 𝔪_{i}^{2ν−1}, and \underleftarrow{lim}_{ν}(\overline{S}_{i} ⊗_{R} R̂)/𝔪_{i}^{ν} ≅ \underleftarrow{lim}_{ν}(\overline{S}_{i}/(M(U_{i}) ∩ \overline{S}_{i})^{ν}) ⊗_{R} (R̂/M(R̂)^{ν}) ≅ \underleftarrow{lim}_{ν}(\overline{S}_{i}/(M(U_{i}) ∩ \overline{S}_{i})^{ν}). Thus the completion of the local ring

(\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} coincides with the completion ˆ \overline{S}_{i} of the local ring (\overline{S}_{i})_{M(U_{i})∩\overline{S}_{i}}.

Since (\overline{S}_{i})_{M(U_{i})∩\overline{S}_{i}} is an excellent normal local domain, ˆ \overline{S}_{i} is a noetherian com-

plete normal local domain by Theorem 2.2.9. Since the completion ˆ \overline{S}_{i} of (\overline{S}_{i}⊗_{R}R̂)_{𝔪_{i}} is faithfully flat over (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}}, (Matsumura [7], p. 172, (24.A), Theorem 56.) (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} is a domain. By Matsumura [7], p. 176, (24.E), Proposition ii), (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} is normal.

2. Obviously the homomorphism \overline{S}_{i} ⊗_{R} R̂ → \overline{S}_{i}[R̂] is surjective, and the inverse image of the prime ideal 𝔫_{i} coincides with 𝔪_{i}. Thus we have an induced surjective homomorphism (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} → \overline{S}_{i}[R̂]_{𝔫_{i}}. In particular, dim(\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} ≥ dim \overline{S}_{i}[R̂]_{𝔫_{i}}.

The valuation ring U′_{i} has a sequence

{0} = 𝔭′_{i}/𝔭′_{i} ⊂ 𝔭′_{i+1}/𝔭′_{i} ⊂ ⋯ ⊂ 𝔭′_{n}/𝔭′_{i} = M(U′_{i})

of prime ideals. For every j with i + 1 ≤ j ≤ n we have

(𝔭′_{j−1}/𝔭′_{i}) ∩ \overline{S}_{i}[R̂] ⊂ (𝔭′_{j}/𝔭′_{i}) ∩ \overline{S}_{i}[R̂]. (*)

S′/𝔭′_{i} ∩ S′ ⊂ \overline{S}_{i}[R̂] and by Lemma 10.3.2 for every j with i ≤ j ≤ n we have (𝔭′_{j}/𝔭′_{i}) ∩ (S′/𝔭′_{i} ∩ S′) = (𝔭′_{j} ∩ S′)/(𝔭′_{i} ∩ S′), ht((𝔭′_{j} ∩ S′)/(𝔭′_{i} ∩ S′)) = j − i. Thus, the both-hand sides of (∗) never coincide, and one knows dim \overline{S}_{i}[R̂]_{𝔫_{i}} ≥ n − i.

We consider the homomorphism \overline{S}_{i} → \overline{S}_{i} ⊗_{R} R̂. The inverse image of 𝔪_{i} by it coincides with M(U_{i}) ∩ \overline{S}_{i}. By Matsumura [7], p. 79, (13.B), Theorem 19 (1), we have dim(\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} ≤ dim(\overline{S}_{i})_{M(U_{i})∩\overline{S}_{i}} + dim(\overline{S}_{i}/M(U_{i}) ∩ \overline{S}_{i}) ⊗_{R} R̂ = dim(\overline{S}_{i})_{M(U_{i})∩\overline{S}_{i}}. We used (\overline{S}_{i}/M(U_{i}) ∩ \overline{S}_{i}) ⊗_{R} R̂ ≅ k ⊗_{R} R̂ ≅ k here. Since \overline{S}_{i} is integral over S/𝔭_{i} ∩ S, by Matsumura [7], p. 81, (13.C), Theorem 20 (2) we have dim(\overline{S}_{i})_{M(U_{i})∩\overline{S}_{i}} ≤ dim(S/𝔭_{i} ∩ S)_{M(U_{i})∩(S/𝔭_{i}∩S)} = ht((𝔭_{n} ∩ S)/(𝔭_{i} ∩ S)) = n − i. We used Lemma 10.3.2 here.

By the above we have n − i ≥ dim(\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} ≥ dim \overline{S}_{i}[R̂]_{𝔫_{i}} ≥ n − i, and in particular dim(\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} = dim \overline{S}_{i}[R̂]_{𝔫_{i}}. The surjective ring homomorphism (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} → \overline{S}_{i}[R̂]_{𝔫_{i}} between two local domains of the same dimension is an isomorphism.

3. Trivial.

4. Since S′/𝔭′_{i} ∩S′ ⊂ \overline{S}_{i}[R̂] ⊂ \overline{S}′_{i}, and since \overline{S}′_{i} is the integral closure of S′/𝔭′_{i} ∩S′ in D′_{i}, \overline{S}′_{i} coincides with the integral closure of \overline{S}_{i}[R̂] in D′_{i}. Thus L_{i}^{−1}\overline{S}′_{i} coincides with the integral closure of L_{i}^{−1}\overline{S}_{i}[R̂] in D′_{i}. Since L_{i}^{−1}\overline{S}_{i}[R̂] ≅ (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} is normal by 1, 2 and 3 above, the quotient field of L_{i}^{−1}\overline{S}_{i}[R̂] coincides with D′_{i}, if and only if, L_{i}^{−1}\overline{S}_{i}[R̂] = L_{i}^{−1}\overline{S}′_{i}.

By induction on i we will show that these equivalent conditions are satisfied for every 0 ≤ i ≤ n.

In the case where i = 0, we have D′_{0} ⊃ the quotient field of L_{0}^{−1}\overline{S}_{0}[R̂] ⊃ the quotient field of R̂ = D′_{0}.

Assume that 1 ≤ i ≤ n and L_{i−1}^{−1}\overline{S}_{i−1}[R̂] = L_{i−1}^{−1}\overline{S}′_{i−1}. We have

the quotient field of L_{i}^{−1}\overline{S}_{i}[R̂] = the quotient field of \overline{S}_{i}[R̂]

= the field generated by the quotient field of \overline{S}_{i}
and the quotient field of R̂/𝔭′_{i} ∩ R̂
= the field generated by the quotient field of (\overline{S}_{i−1}/(𝔭_{i}/𝔭_{i−1}) ∩ \overline{S}_{i−1})
and the quotient field of R̂/𝔭′_{i} ∩ R̂
= the quotient field of (\overline{S}_{i−1}[R̂]/(𝔭′_{i}/𝔭′_{i−1}) ∩ \overline{S}_{i−1}[R̂])
= the quotient field of (L_{i−1}^{−1}\overline{S}_{i−1}[R̂]/(𝔭′_{i}/𝔭′_{i−1}) ∩ L_{i−1}^{−1}\overline{S}_{i−1}[R̂])
= the quotient field of (L_{i−1}^{−1}\overline{S}′_{i−1}/(𝔭′_{i}/𝔭′_{i−1}) ∩ L_{i−1}^{−1}\overline{S}′_{i−1})
= the quotient field of (\overline{S}′_{i−1}/(𝔭′_{i}/𝔭′_{i−1}) ∩ \overline{S}′_{i−1}) = the quotient field of \overline{S}′_{i} = D′_{i}.

We used Lemma 10.3.7 and 4.

5. Since \overline{S}_{i}[R̂] ⊂ \overline{S}′_{i}, we have L_{i} ⊂ \overline{S}′_{i} − M(U′_{i}) ∩ \overline{S}′_{i} and L_{i}^{−1}\overline{S}′_{i} ⊂ (\overline{S}′_{i})_{M(U′_{i})∩\overline{S}′_{i}}.

On the other hand, by 3 and 4 we have L_{i}^{−1}\overline{S}′_{i} = \overline{S}_{i}[R̂]_{𝔫_{i}} and \overline{S}_{i}[R̂]_{𝔫_{i}} is a local ring dominated by U′_{i}. Thus one has 5.

Continuation of the proof of Proposition 10.4. By Lemma 10.5.4 the field D′_{i} coincides with the quotient field of \overline{S}_{i}[R̂/𝔭′_{i} ∩ R̂]. Thus D′_{i} coincides with the field generated by D_{i} and R̂/𝔭′_{i} ∩ R̂. Since R̂/𝔭′_{i} ∩ R̂ is a homomorphic image of the completion of the local ring R/𝔭_{i} ∩ R, R̂/𝔭′_{i} ∩ R̂ is generated by R/𝔭_{i} ∩ R and (R̂/𝔭′_{i} ∩ R̂)^{p^{δ}}. Since R/𝔭_{i} ∩ R ⊂ \overline{S}_{i} ⊂ D_{i}, and since (R̂/𝔭′_{i} ∩ R̂)^{p^{δ}} ⊂ (D′_{i})^{p^{δ}}, D′_{i} coincides with the field generated by D_{i} and (D′_{i})^{p^{δ}}.

3. By Lemma 10.3.6 the homomorphism U_{𝔭_{i+1}}/𝔭_{i}U_{𝔭_{i+1}} → U′_{𝔭′_{i+1}}/𝔭′_{i}U′_{𝔭′_{i+1}} can be

identified with the homomorphism (\overline{S}_{i})_{\overline{𝔮}_{i+1}} → (\overline{S}′_{i})\overline{𝔮}′_{i+1}, where \overline{𝔮}_{i+1} = (𝔭_{i+1}/𝔭_{i})∩\overline{S}_{i}

and \overline{𝔮}′_{i+1} = (𝔭′_{i+1}/𝔭′_{i}) ∩ \overline{S}′_{i}.

On the other hand, by Lemma 10.5.1–5 there exists an isomorphism (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}} ≅ (\overline{S}′_{i})_{M(U′_{i})∩\overline{S}′_{i}}, and the homomorphism (\overline{S}_{i})_{M(U_{i})∩\overline{S}_{i}} → (\overline{S}′_{i})_{M(U′_{i})∩\overline{S}′_{i}} can be

identified with the homomorphism (\overline{S}_{i})_{M(U_{i})∩\overline{S}_{i}} → (\overline{S}_{i} ⊗_{R} R̂)_{𝔪_{i}}. Let \overline{𝔮}′′_{i+1} be the

inverse image of the prime ideal \overline{𝔮}′_{i+1}(\overline{S}′_{i})_{M(U′_{i})∩\overline{S}′_{i}} by the homomorphism \overline{S}_{i}⊗_{R} R̂ →

(\overline{S}_{i}⊗_{R}R̂)_{𝔪_{i}} ≅ (\overline{S}′_{i})_{M(U′_{i})∩\overline{S}′_{i}}. \overline{𝔮}′′_{i+1} is a prime ideal of the ring \overline{S}_{i}⊗_{R}R̂, \overline{𝔮}′′_{i+1} ⊂ 𝔪_{i}, and

the inverse image of \overline{𝔮}′′_{i+1} by the homomorphism \overline{S}_{i} → \overline{S}_{i} ⊗_{R} R̂ coincides with \overline{𝔮}_{i+1}. The homomorphism (\overline{S}_{i})_{\overline{𝔮}_{i+1}} → (\overline{S}′_{i})\overline{𝔮}′_{i+1} can be identified with the homomorphism (\overline{S}_{i})_{\overline{𝔮}_{i+1}} → (\overline{S}_{i} ⊗_{R} R̂)\overline{𝔮}′′_{i+1}.

We would like to apply Theorem 2.2.10 for the excellent ring R. Note that \overline{S}_{i}/\overline{𝔮}_{i+1} is a finitely generated R-algebra by Lemma 10.3.5. Let L be the quotient field of \overline{S}_{i}/\overline{𝔮}_{i+1} and 𝔔 = \overline{𝔮}′′_{i+1}(L ⊗_{R} R̂). The field L contains the quotient field of R/\overline{𝔮}_{i+1} ∩ R, and L is finitely generated over the quotient field of R/\overline{𝔮}_{i+1} ∩ R. 𝔔 is a prime ideal of L ⊗_{R} R̂. By Theorem 2.2.10 the ring (L ⊗_{R} R̂)_{𝔔} is a regular local ring, and has no nilpotent elements. Since (L ⊗_{R} R̂)_{𝔔} ≅ ((\overline{S}_{i}/\overline{𝔮}_{i+1}) ⊗_{R} R̂)\overline{𝔮}′′_{i+1} ≅ (\overline{S}_{i} ⊗_{R} R̂)\overline{𝔮}′′_{i+1}/\overline{𝔮}_{i+1}(\overline{S}_{i} ⊗_{R} R̂)\overline{𝔮}′′_{i+1}, the ring (\overline{S}_{i} ⊗_{R} R̂)\overline{𝔮}′′_{i+1}/\overline{𝔮}_{i+1}(\overline{S}_{i} ⊗_{R} R̂)\overline{𝔮}′′_{i+1} has no nilpotent elements, which is equivalent to our claim 3.

4. Let v : D_{0} → Γ ∪ {∞} and v′ : D′_{0} → Γ′ ∪ {∞} denote the valuation defined by valuation rings U and U′ respectively. Let Δ_{i} = v((U_{𝔭_{i}})^{∗}) and Δ′_{i} = v′((U′_{𝔭′_{i}})^{∗}) for every 0 ≤ i ≤ n. By Lemma 4.16.2 each Δ_{i} is an isolated subgroup of Γ, and each Δ′_{i} is an isolated subgroup of Γ′. We have

Γ ⊃ Δ_{0} ⊃ Δ_{1} ⊃ ⋯ ⊃ Δ_{i} ⊃ Δ_{i+1} ⊃ ⋯ ⊃ Δ_{n} = {0},
Γ′ ⊃ Δ′_{0} ⊃ Δ′_{1} ⊃ ⋯ ⊃ Δ′_{i} ⊃ Δ′_{i+1} ⊃ ⋯ ⊃ Δ′_{n} = {0}.

Since (Z′, U′) is an elementary monoidal transform of (Z, U) we have U = U′ ∩ D_{0}. By Lemma 4.13.2 we can regard Γ as a subgroup of Γ′ and v′ | D_{0} = v. By Corollary 4.15.3 Δ_{i} = Δ′_{i} ∩ Γ for every 0 ≤ i ≤ n.

We will show the following claim 4.i by induction on i.

4.i The homomorphism Γ/Δ_{i} → Γ′/Δ′_{i} induced by the inclusion homomorphism

Γ → Γ′ is an isomorphism. Obviously 4.0 is true. Assume that 0 ≤ i ≤ n − 1 and 4.i is true, we have the following commutative diagram:

0 ⟶ Δ_{i}/Δ_{i+1} ⟶ Γ/Δ_{i+1} ⟶ Γ/Δ_{i} ⟶ 0
↓ ↓ ↓
0 ⟶ Δ′_{i}/Δ′_{i+1} ⟶ Γ′/Δ′_{i+1} ⟶ Γ′/Δ′_{i} ⟶ 0

Two rows of the above diagram are exact, and by 4.i the right vertical arrow is an isomorphism. By Lemma 4.16 Δ_{i}/Δ_{i+1} coincides with the value group of the valuation ring U_{𝔭_{i+1}}/𝔭_{i}U_{𝔭_{i+1}}, and Δ′_{i}/Δ′_{i+1} coincides with the value group of the valuation ring U′_{𝔭′_{i+1}}/𝔭′_{i}U′_{𝔭′_{i+1}}. By Lemma 10.3.8 Δ_{i}/Δ_{i+1} ≅ ℤ and Δ′_{i}/Δ′_{i+1} ≅ ℤ. By 3 above the image of the generator of Δ_{i}/Δ_{i+1} is a generator of Δ′_{i}/Δ′_{i+1}. Thus, the left vertical arrow is also an isomorphism. The middle vertical arrow is an isomorphism, and one knows that 4.i + 1 is true.

In particular, 4.n is true. We obtain 4.

Lemma 10.6. Let (Z, U) be a space germ over k of iterated divisor type, and (Z′, U′) be an IAMT of (Z, U). Denote n = dim Z = rank U. The same claims 1–4 as in Proposition 10.4 hold under this assumption.

Proof. By definition there is a finite sequence {(Z_{ν}, U_{ν}) | 0 ≤ ν ≤ ω} of space germs satisfying the following two conditions:

1. (Z, U) = (Z_{0}, U_{0}), (Z′, U′) = (Z_{ω}, U_{ω}).

2. For every 1 ≤ ν ≤ ω (Z_{ν}, U_{ν}) is an elementary monoidal transform of

(Z_{ν−1}, U_{ν−1}). If ω = 0, then obviously our claims hold. If ω = 1, then by Proposition 10.4 our claims holds. Thus by induction on ω we know that our claims hold.

We write only the induction step corresponding to claim 2 of Proposition 10.4, since other parts are trivial.

Assume ω ≥ 2. Let

{0} = 𝔭′′_{0} ⊂ 𝔭′′_{1} ⊂ ⋯ ⊂ 𝔭′′_{i−1} ⊂ 𝔭′′_{i} ⊂ ⋯ ⊂ 𝔭′′_{n} = M(U_{ω−1})

be the sequence of prime ideals of U_{ω−1}. Let i be an integer with 0 ≤ i ≤ n, and by D′′_{i} we denote the quotient field of U_{ω−1}/𝔭′′_{i}.

By induction hypothesis D′′_{i} is generated by D_{i} and (D′′_{i})^{p^{δ}}. By the case ω = 1 D′_{i} is generated by D′′_{i} and (D′_{i})^{p^{δ}}. Thus, D′_{i} is generated by D_{i}, (D′′_{i})^{p^{δ}} and (D′_{i})^{p^{δ}}. Since D′′_{i} ⊂ D′_{i}, we have (D′′_{i})^{p^{δ}} ⊂ (D′_{i})^{p^{δ}}. Thus D′_{i} is generated by D_{i} and (D′_{i})^{p^{δ}}.

Lemma 10.7. Let D be a field containing k, and δ be a positive integer. We consider the Frobenius homomorphism F_{δ} : D → D of degree δ. We write q = p^{δ} and D^{q} = F_{δ}(D). Assume that there exists an infinite sequence of subfields D_{ν} of D numbered by non-negative integers ν satisfying the following three conditions:

1. D_{ν} ⊂ D_{ν+1} for every non-negative integer ν. D = ∪_{ν=0}^{∞}D_{ν}.

2. There exists a non-negative integer ϵ such that for every non-negative integer ν the relative degree [D_{ν} : D_{ν}^{q}] is finite, and it is equal to p^{ϵ}, where D_{ν}^{q} = F_{δ}(D_{ν}).

3. D_{ν} is generated by subfields D_{0} and (D_{ν})^{q} for every non-negative integer ν. Then, the relative degree [D : D^{q}] is finite, and it is equal to p^{ϵ}

Proof. Easy.

Theorem 10.8. Let {(Z_{ν}, U_{ν}) | ν ≥ 0} be an infinite sequence of space germs over k numbered by non-negative integers ν. Assume the following two conditions:

(a) dim Z_{0} = rank U_{0} = n.

(b) (Z_{ν+1}, U_{ν+1}) is an IAMT of (Z_{ν}, U_{ν}) for every non-negative integer ν. We denote the quotient field of Z_{ν} by D_{ν} for every non-negative integer ν. By the associated homomorphism U_{ν} → U_{ν+1}, D_{ν} → D_{ν+1} we define two inductive systems \{U_{ν} | ν ≥ 0\} and \{D_{ν} | ν ≥ 0\}. Let U = \underrightarrow{lim}_{ν}U_{ν} and D = \underrightarrow{lim}_{ν}D_{ν}.

1. U is a valuation ring of the field D with rank U = n. Let ν be an arbitrary non-negative integer. rank U_{ν} = n. Let

{0} = 𝔭_{ν,0} ⊂ 𝔭_{ν,1} ⊂ ⋯ ⊂ 𝔭_{ν,i−1} ⊂ 𝔭_{ν,i} ⊂ ⋯ ⊂ 𝔭_{ν,n} = M(U_{ν}),
{0} = 𝔭̃_{0} ⊂ 𝔭̃_{1} ⊂ ⋯ ⊂ 𝔭̃_{i−1} ⊂ 𝔭̃_{i} ⊂ ⋯ ⊂ 𝔭̃_{n} = M(U)

be the sequences of prime ideals of U_{ν} and U respectively. For every 0 ≤ i ≤ n we have 𝔭_{ν,i} = 𝔭̃_{i} ∩ U_{ν}.

2. Let ν be an arbitrary non-negative integer. For every 0 ≤ i ≤ n − 1

V_{ν,i} = (U_{ν})_{𝔭_{ν,i+1}}/𝔭_{ν,i}(U_{ν})_{𝔭_{ν,i+1}} and
V_{i} = U𝔭̃_{i+1}/𝔭̃_{i}U𝔭̃_{i+1}

are regular local rings of dimension one. The induced homomorphism V_{ν,i} → V_{i} sends a generator of the maximal ideal to a generator of the maximal ideal. The induced homomorphism from the value group of the valuation ring U_{ν} to the value group of the valuation ring U is an isomorphism. In particular, the value group of U is isomorphic to ℤ^{n} with the lexicographic order as totally ordered abelian groups.

3. Let δ be an arbitrary positive integer. We write q = p^{δ}. Let V_{i}^{q} denote the image of the Frobenius homomorphism F_{δ} : V_{i} → V_{i} of degree δ. For every 0 ≤ i ≤ n − 1 V_{i} is a finitely generated V_{i}^{q}-module.

4. Let i be an arbitrary integer with 0 ≤ i ≤ n − 1. Let

V̂_{i} = \underleftarrow{lim}_{ν}V_{i}/M(V_{i})^{ν}

be the completion of the local ring V_{i}. By the canonical homomorphism V_{i} → V̂_{i} we regard V_{i} as a subring of V̂_{i}. If elements \overline{F} ∈ V_{i} and \overline{K} ∈ V̂_{i} satisfy \overline{F} = \overline{K}^{q}, then \overline{K} ∈ V_{i}.

Proof. 1. It is easy to see that U is a valuation ring of the field D. By Proposition 10.4.1 one has rank U_{ν} = n. For every integer μ with ν ≤ μ and for every 0 ≤ i ≤ n we have 𝔭_{ν,i} = 𝔭_{μ,i} ∩ U_{ν}.

Let 𝔭̃ be an arbitrary prime ideal of U. There exists a unique integer i with 0 ≤ i ≤ n and 𝔭_{0,i} = 𝔭̃ ∩ U_{0}. By the above we have 𝔭_{μ,i} = 𝔭̃ ∩ U_{μ} for every non-negative integer μ. We have 𝔭̃ = ∪_{μ=0}^{∞}(𝔭̃ ∩ U_{μ}) = ∪_{μ=0}^{∞}𝔭_{μ,i}. Thus, 𝔭̃ is uniquely determined by the number i. The morphism Spec(U) → Spec(U_{0}) induced by the inclusion homomorphism U_{0} → U is injective. By Corollary 4.15.3 Spec(U) → Spec(U_{0}) is surjective. Thus Spec(U) → Spec(U_{0}) is bijective. One knows that Spec(U) is a finite set with n + 1 elements. By Corollary 4.15.1 we have rank U = n. Since the morphism Spec(U) → Spec(U_{0}) preserves inclusion relations, we obtain the last claim.

2. By Lemma 10.3.8 V_{ν,i} is a regular local ring of dimension one. Let Γ_{ν,i} denote the value group of the valuation ring V_{ν,i}, and D_{ν,i} denote the quotient field of V_{ν,i}. Γ_{ν,i} ≅ D_{ν,i}^{∗}/V_{ν,i}^{∗} ≅ ℤ.

By 1 we have V_{i} = \underrightarrow{lim}_{ν}V_{ν,i}. Let Γ̃_{i} denote the value group of V_{i}, and D_{i} denote the quotient field of V_{i}. Γ̃_{i} ≅ D_{i}^{∗}/V_{i}^{∗} ≅ \underrightarrow{lim}_{ν}D_{ν,i}^{∗}/ \underrightarrow{lim}_{ν}V_{ν,i}^{∗} ≅ \underrightarrow{lim}_{ν}(D_{ν,i}^{∗}/V_{ν,i}^{∗}) ≅ \underrightarrow{lim}_{ν}Γ_{ν,i}. By Proposition 10.4.3 the homomorphism Γ_{ν,i} → Γ_{ν+1,i} is an isomorphism for every non-negative integer ν. Thus the homomorphism Γ_{ν,i} → Γ̃_{i} is an isomorphism for every non-negative integer ν, and we have Γ̃_{i} ≅ ℤ.

By Proposition 4.25 V_{i} is a regular local ring of dimension one. The remaining claims follow from that the homomorphism Γ_{ν,i} → Γ̃_{i} is an isomorphism.

3. We check that the field D and the sequence D_{ν} of subfields satisfy the three assumptions of Lemma 10.7. The first assumption is obviously satisfied. By Lemma 10.3.10 [D_{ν} : D_{ν}^{q}] = q^{n} for every ν. Thus the second assumption is satisfied. By Proposition 10.4.2 and Lemma 10.6 also the third assumption is satisfied. By Lemma 10.7 we have [D : D^{q}] = q^{n}.

Let D_{i} denote the quotient field of U/𝔭̃_{i} for every 0 ≤ i ≤ n. D_{0} = D. For every 0 ≤ i ≤ n − 1 the quotient field of the local ring V_{i} is D_{i}, and the residue field of V_{i} is D_{i+1}.

By applying Lemma 10.2.1 repeatedly as in the proof of Lemma 10.3.10, one knows that [D_{i} : D_{i}^{q}] is finite for every 0 ≤ i ≤ n. Besides,

\begin{aligned}[D_{n} : D_{n}^{q}] ≤ & \frac{[D_{n−1} : D_{n−1}^{q}]}{q} ≤ ⋯ ≤ \frac{[D_{i+1} : D_{i+1}^{q}]}{q^{n−i−1}} ≤ \frac{[D_{i} : D_{i}^{q}]}{q^{n−i}} ≤ \\ & ⋯ ≤ \frac{[D_{0} : D_{0}^{q}]}{q^{n}} ≤ \frac{q^{n}}{q^{n}} = 1 \end{aligned}

Since [D_{n} : D_{n}^{q}] ≥ 1, we have

[D_{i+1} : D_{i+1}^{q}] = \frac{[D_{i} : D_{i}^{q}]}{q}

for every 0 ≤ i ≤ n − 1. By Lemma 10.2.2 one knows that V_{i} is a finitely generated V_{i}^{q}-module.

4. We fix a number i. We write 𝔪 = M(V_{i}^{q}) and L = V_{i}/V_{i}^{q}. Note that V_{i}^{q} (≅ V_{i}) is a noetherian local ring with the maximal ideal 𝔪. We have an exact sequence

0 ⟶ V_{i}^{q} ⟶ V_{i} ⟶ L ⟶ 0

of finitely generated V_{i}^{q}-modules. By Matsumura [7], p.168, (23.I), Theorem 54, the sequence

0 ⟶ \underleftarrow{lim}_{ν}V_{i}^{q}/𝔪^{ν}V_{i}^{q} ⟶ \underleftarrow{lim}_{ν}V_{i}/𝔪^{ν}V_{i} ⟶ \underleftarrow{lim}_{ν}L/𝔪^{ν}L ⟶ 0

is exact. By V̂_{i}^{q} we denote the image of the Frobenius homomorphism F_{δ} : V̂_{i} → V̂_{i}. We have \underleftarrow{lim}_{ν}V_{i}^{q}/𝔪^{ν}V_{i}^{q} ≅ V̂_{i}^{q}. Since 𝔪^{ν}V_{i} = M(V_{i})^{νq}, we have \underleftarrow{lim}_{ν}V_{i}/𝔪^{ν}V_{i} ≅ V̂_{i}. Denoting L̂ = \underleftarrow{lim}_{ν}L/𝔪^{ν}L, we obtain the following commutative diagram:

0 ⟶ V_{i}^{q} ⟶ V_{i} ⟶ L ⟶ 0
↓ ↓ ↓ φ
0 ⟶ V̂_{i}^{q} ⟶ V̂_{i} ⟶ L̂ ⟶ 0

Two rows are exact. Thus our claim 4 is equivalent to that φ : L → L̂ is injective. By Matsumura [7], p.172, (24.A), Theorem 56 (2)⇔(3), φ is injective.

11. Main results

The theorem below is the most essential result of this paper.

Theorem 11.1. Let (B, W, P) be a framed space germ over k, and (h, g, t) be a Weierstrass representation over (B, W, P). Let (Z, U, P) = \underline{(B, W, P)}. Assume that the basic valuation ring U is of normal crossing type. Then, any very effective reduction sequence of (h, g, t) has finite length.

Proof. Assume that there exists a very effective reduction sequence S = {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν < ∞} of (h, g, t) with infinite length. We will deduce a contradiction.

Let (B_{0}, W_{0}, P_{0}) = (B, W, P), and (h_{0}, g_{0}, t_{0}) = (h, g, t). Let {(h_{ν}, g_{ν}, t_{ν}) | 0 ≤ ν < ∞} denote the transform Weierstrass representation sequence of (h, g, t) along S. For every non-negative integer ν, we denote (Z_{ν}, U_{ν}, P_{ν}) = \underline{(B_{ν}, W_{ν}, P_{ν})}, by F_{ν} we denote the quotient field of B_{ν}, by D_{ν} the quotient field of Z_{ν}, and P_{ν} = \{x_{ν,1}, x_{ν,2}, …, x_{ν,n}\}. Let Z = \underrightarrow{lim}_{ν}Z_{ν}, U = \underrightarrow{lim}_{ν}U_{ν}, D = \underrightarrow{lim}_{ν}D_{ν}, and F = \underrightarrow{lim}_{ν}F_{ν}. For simplicity, we identify F_{ν} and the image of the canonical homomorphism F_{ν} → F. Note that under this identification we have h = h_{ν}, g = g_{ν} and t_{ν} = t + ∑_{μ=1}^{ν} H_{μ} for every non-negative integer ν.

By Lemma 8.7.3 for every ν, (B_{ν}, W_{ν}, P_{ν}) has the basic valuation ring of normal crossing type, and in particular, n = dim Z_{ν} = rank U_{ν}. By Theorem 10.8.1 U is a valuation ring of the field D, and rank U = n. By Corollary 4.19 U ∩ D_{ν} = U_{ν} for every ν. Ring Z is a local ring, and U dominates Z. Let

v : D → Γ ∪ {∞}

denote the valuation defined by U. Let

{0} = 𝔭_{0} ⊂ 𝔭_{1} ⊂ ⋯ ⊂ 𝔭_{i} ⊂ 𝔭_{i+1} ⊂ ⋯ ⊂ 𝔭_{n} = M(U)

be the sequence of prime ideals of U.

By Theorem 10.8.2 Γ is isomorphic to ℤ^{n} with the lexicographic order as totally ordered abelian groups. Fixing an appropriate isomorphism Γ ≅ ℤ^{n}, we identify Γ and ℤ^{n}. For every integer i with 0 ≤ i ≤ n, Δ_{i} = v(U_{𝔭_{i}}^{∗}) is an isolated subgroup of Γ, U_{𝔭_{i}}^{∗} = v^{−1}(Δ_{i}), and Δ_{i} = {(α_{1}, α_{2}, …, α_{n}) ∈ ℤ^{n} | α_{j} = 0 for every integer j with 1 ≤ j ≤ i.} under the above identification.

We write

v(Φ) = (v_{1}(Φ), v_{2}(Φ), …, v_{n}(Φ)) ∈ ℤ^{n}

for Φ ∈ D^{∗}. Each v_{i}(Φ) is an integer.

For every 0 ≤ i ≤ n − 1 we denote

V_{i} = U_{𝔭_{i+1}}/𝔭_{i}U_{𝔭_{i+1}}.

By Theorem 10.8.2 V_{i} is a regular local ring of dimension one. Regarding V_{i} as a valuation ring of the quotient field, by \overline{v}_{i+1} we denote the valuation defined by V_{i}. The value group of \overline{v}_{i+1} is ℤ. Fixing an integer i with 0 ≤ i ≤ n − 1, by \overline{Φ} ∈ V_{i} we denote the image of an element Φ ∈ U by the canonical homomorphism U → V_{i}. If \overline{Φ} ≠ 0, then we have v_{i+1}(Φ) = \overline{v}_{i+1}(\overline{Φ}) by Lemma 4.16.

Let m = deg(g, t). Since S exists, we have m ≥ 1. Let m = lq denote the p-decomposition of m.

Since S is an effective reduction sequence with infinite length, we have three infinite sequences {(F_{ν,1}, F_{ν,2}, …, F_{ν,m}) | 0 ≤ ν < ∞}, {(b_{ν,1}, b_{ν,2}, …, b_{ν,n}) | 1 ≤ ν < ∞}, and {h_{ν} | 1 ≤ ν < ∞} satisfying the conditions in Proposition 9.11.

For simplicity, we write

⟨ν⟩ = x_{ν,1}^{b_{ν,1}}x_{ν,2}^{b_{ν,2}} ⋯ x_{ν,n}^{b_{ν,n}} = ∏_{i=1}^{n} x_{ν,i}^{b_{ν,i}} ∈ Z_{ν}

for every positive integer ν. Besides, for every pair ν, μ of integers with 0 ≤ ν ≤ μ we write

⟨⟨ν, μ⟩⟩ = ∏_{λ=ν+1}^{μ}⟨λ⟩ ∈ Z_{μ}.

We understand ⟨⟨ν, ν⟩⟩ = 1 here. For 0 ≤ ν ≤ μ ≤ λ we have ⟨⟨ν, λ⟩⟩ = ⟨⟨ν, μ⟩⟩⟨⟨μ, λ⟩⟩.

By Proposition 9.11 (3.ν) we have v(⟨ν⟩) > 0 for every positive integer ν. Thus v(⟨⟨0, ν⟩⟩) < v(⟨⟨0, ν + 1⟩⟩) = v(⟨⟨0, ν⟩⟩) + v(⟨ν + 1⟩) for every non-negative integer ν.

Since ℤ is discrete, by definition of the lexicographic order, we have two cases. ⟨A⟩ v_{1}(⟨⟨0, ν⟩⟩) ≤ v_{1}(⟨⟨0, ν + 1⟩⟩) for every non-negative integer ν, and

lim_{ν→+∞} v_{1}(⟨⟨0, ν⟩⟩) = +∞. ⟨B⟩ For some integer i with 1 ≤ i ≤ n − 1 and for some non-negative integer ν_{0}

the following two conditions are satisfied: ⟨B.1⟩ v_{j}(⟨⟨0, ν_{0}⟩⟩) = v_{j}(⟨⟨0, ν⟩⟩) for every integer ν with ν ≥ ν_{0} and for every

integer j with 1 ≤ j ≤ i. ⟨B.2⟩ v_{i+1}(⟨⟨0, ν⟩⟩) ≤ v_{i+1}(⟨⟨0, ν + 1⟩⟩) for every non-negative integer ν with

ν ≥ ν_{0}, and lim_{ν→+∞} v_{i+1}(⟨⟨0, ν⟩⟩) = +∞.

In case ⟨A⟩, we put i = 0 and ν_{0} = 0. In case ⟨B⟩, we choose integers i and ν_{0} satisfying the conditions, and we fix them.

Consider the canonical homomorphism Φ ∈ U ↦ \overline{Φ} ∈ V_{i}.

We consider the case ⟨B⟩. Let j be an arbitrary integer with 1 ≤ j ≤ i. If ν ≥ ν_{0}, then by ⟨B.1⟩ we have v_{j}(⟨⟨0, ν⟩⟩) = v_{j}(⟨⟨0, ν + 1⟩⟩) = v_{j}(⟨⟨0, ν⟩⟩) + v_{j}(⟨ν + 1⟩), and thus v_{j}(⟨ν + 1⟩) = 0. One knows that v_{1}(⟨ν⟩) = v_{2}(⟨ν⟩) = ⋯ = v_{i}(⟨ν⟩) = 0 if ν > ν_{0}. In particular, ⟨ν⟩ ∈ U_{𝔭_{i}}^{∗} and \overline{⟨ν⟩} ≠ 0 if ν > ν_{0}.

In the case ⟨A⟩, the homomorphism U → V_{i} is injective, and Φ = \overline{Φ}. Thus \overline{⟨ν⟩} ≠ 0 for ν > ν_{0} also in this case.

In both cases

\overline{⟨⟨ν_{0}, ν⟩⟩} = ∏_{μ=ν_{0}+1}^{ν} \overline{⟨μ⟩} ≠ 0

for every integer ν with ν ≥ ν_{0}.

Below we treat both cases together for a while.

We claim that \overline{F}_{ν_{0},j} = 0 for every integer j satisfying 1 ≤ j ≤ m and j ≢ 0 mod q.

Assume otherwise. There exists an integer j_{0} satisfying the following three conditions:

(a) 1 ≤ j_{0} < m, j_{0} ≢ 0 mod q.

(b) If 1 ≤ j < j_{0}, j ≢ 0 mod q, then \overline{F}_{ν_{0},j} = 0.

(c) \overline{F}_{ν_{0},j_{0}} ≠ 0.

By Corollary 9.13.1

\overline{F}_{ν_{0},j_{0}} = \overline{⟨⟨ν_{0}, ν⟩⟩}^{j_{0}} \overline{F}_{ν,j_{0}}

for every integer ν with ν ≥ ν_{0}. Since F_{ν,j_{0}} ∈ Z_{ν} ⊂ Z ⊂ U, \overline{F}_{ν,j_{0}} ∈ V_{i} and \overline{v}_{i+1}(\overline{F}_{ν,j_{0}}) ≥ 0. Thus

\overline{v}_{i+1}(\overline{F}_{ν_{0},j_{0}}) ≥ j_{0}\overline{v}_{i+1}(\overline{⟨⟨ν_{0}, ν⟩⟩})

for every integer ν with ν ≥ ν_{0}. By (c) above the left-hand side of this equality is an integer independent of the integer ν. Consider the right-hand side. Since \overline{⟨⟨ν_{0}, ν⟩⟩} ≠ 0, we have

\overline{v}_{i+1}(\overline{⟨⟨ν_{0}, ν⟩⟩}) = v_{i+1}(⟨⟨ν_{0}, ν⟩⟩) = v_{i+1}(⟨⟨0, ν⟩⟩) − v_{i+1}(⟨⟨0, ν_{0}⟩⟩).

As ν → +∞, \overline{v}_{i+1}(\overline{⟨⟨ν_{0}, ν⟩⟩}) → +∞, since v_{i+1}(⟨⟨0, ν⟩⟩) → +∞. Thus, there is an integer ν satisfying ν ≥ ν_{0} and

\overline{v}_{i+1}(\overline{F}_{ν_{0},j_{0}}) < j_{0}\overline{v}_{i+1}(\overline{⟨⟨ν_{0}, ν⟩⟩}),

which is a contradiction.

Applying Corollary 9.13.1 again, one has

\overline{⟨⟨ν_{0}, ν⟩⟩}^{j}\overline{F}_{ν,j} = 0

for every integer j satisfying 1 ≤ j ≤ m and j ≢ 0 mod q and for every integer ν with ν ≥ ν_{0}. Since V_{i} is a domain, and since \overline{⟨⟨ν_{0}, ν⟩⟩} ≠ 0, we have \overline{F}_{ν,j} = 0.

One knows that \overline{F}_{ν,j} = 0 for every integer j satisfying 1 ≤ j ≤ m and j ≢ 0 mod q and for every integer ν with ν ≥ ν_{0}.

By the same argument applying Corollary 9.13.3 instead of Corollary 9.13.1, one knows

\overline{F}_{ν,jq} = \binom{l}{j} (\frac{\overline{F}_{ν,q}}{l} )^{j}

for every integer j with 2 ≤ j ≤ l and for every integer ν with ν ≥ ν_{0}. Thus for ν ≥ ν_{0} we can write

T^{m} + ∑_{j=1}^{m} \overline{F}_{ν,j}T^{m−j} = (T^{q} + \frac{\overline{F}_{ν,q}}{l} )^{l} ,

where T is an algebraically independent element. The problem has been reduced to considering only \overline{F}_{ν,q} for ν ≥ ν_{0}.

By Corollary 9.13.2

\overline{F}_{ν_{0},q} = l ( ∑_{μ=ν_{0}+1}^{ν} \overline{h}_{μ}\overline{⟨⟨ν_{0}, μ⟩⟩})^{q} + \overline{⟨⟨ν_{0}, ν⟩⟩}^{q}\overline{F}_{ν,q}

for every integer ν with ν ≥ ν_{0}.

We consider the completion V̂_{i} = \underleftarrow{lim}_{α}V_{i}/M(V_{i})^{α} of a regular local ring V_{i} of dimension one. Note that V_{i} is noetherian, and the canonical homomorphism V_{i} → V̂_{i} is injective.

As ν → +∞, \overline{v}_{i+1}(\overline{⟨⟨ν_{0}, ν⟩⟩}) → +∞. \overline{F}_{ν,q} ∈ V̂_{i} for every integer ν with ν ≥ ν_{0}. Thus, as ν → +∞, \overline{⟨⟨ν_{0}, ν⟩⟩}^{q}\overline{F}_{ν,q} → 0 in the ring V̂_{i}. Besides, an element

\overline{K}_{ν_{0}} = ∑_{μ=ν_{0}+1}^{∞} \overline{h}_{μ}\overline{⟨⟨ν_{0}, μ⟩⟩} ∈ V̂_{i}

is defined, and it satisfies

\overline{F}_{ν_{0},q} = s^{q}\overline{K}_{ν_{0}}^{q},

where s ∈ k is an element with s ≠ 0 and s^{q} = l.

By Theorem 10.8.4

\overline{K}_{ν_{0}} ∈ V_{i}.

By Theorem 10.8.1 Spec(U_{ν}) = {𝔭_{α} ∩ U_{ν} | 0 ≤ α ≤ n} for every non-negative integer ν. By Lemma 8.7.3 for every ν, (B_{ν}, W_{ν}, P_{ν}) has the basic valuation ring of normal crossing type. By Lemma 8.7.2

(Z_{ν})_{(𝔭_{i+1}∩Z_{ν})}/(𝔭_{i} ∩ Z_{ν})(Z_{ν})_{(𝔭_{i+1}∩Z_{ν})} ≅ (U_{ν})_{(𝔭_{i+1}∩U_{ν})}/(𝔭_{i} ∩ U_{ν})(U_{ν})_{(𝔭_{i+1}∩U_{ν})}

for every non-negative integer ν. Thus, one knows

\begin{aligned}V_{i} & ≅ \underrightarrow{lim}_{ν} [(U_{ν})_{(𝔭_{i+1}∩U_{ν})}/(𝔭_{i} ∩ U_{ν})(U_{ν})_{(𝔭_{i+1}∩U_{ν})}] \\ & ≅ \underrightarrow{lim}_{ν} [(Z_{ν})_{(𝔭_{i+1}∩Z_{ν})}/(𝔭_{i} ∩ Z_{ν})(Z_{ν})_{(𝔭_{i+1}∩Z_{ν})}] . \end{aligned}

By the above there exists an integer ν_{1} satisfying ν_{1} ≥ ν_{0} and

\overline{K}_{ν_{0}} ∈ (Z_{ν_{1}})_{(𝔭_{i+1}∩Z_{ν_{1}} )}/(𝔭_{i} ∩ Z_{ν_{1}})(Z_{ν_{1}})_{(𝔭_{i+1}∩Z_{ν_{1}} )}.

Similarly, defining an element

\overline{K}_{ν_{1}} = ∑_{μ=ν_{1}+1}^{∞} \overline{h}_{μ}\overline{⟨⟨ν_{1}, μ⟩⟩} ∈ V̂_{i},

one knows \overline{K}_{ν_{1}} ∈ V_{i} and \overline{F}_{ν_{1},q} = s^{q}\overline{K}_{ν_{1}}^{q}.

\overline{K}_{ν_{1}} = \frac{\overline{K}_{ν_{0}} − ∑_{μ=ν_{0}+1}^{ν_{1}} \overline{h}_{μ}\overline{⟨⟨ν_{0}, μ⟩⟩}}{\overline{⟨⟨ν_{0}, ν_{1}⟩⟩}}.

By the right-hand side, \overline{K}_{ν_{1}} belongs to the quotient field of Z_{ν_{1}}/𝔭_{i} ∩ Z_{ν_{1}}.

Let ν_{2} = ν_{1} + 1. ν_{2} > ν_{0}. \overline{K}_{ν_{1}} belongs to the quotient field of Z_{ν_{2}}/𝔭_{i} ∩ Z_{ν_{2}}. Besides, there exist an element w_{ν_{2}} ∈ Z_{ν_{2}}^{∗} and non-negative integers c_{ν_{2},1}, c_{ν_{2},2}, …, c_{ν_{2},n} satisfying

⟨⟨0, ν_{2} − 1⟩⟩ = ∏_{μ=1}^{ν_{2}−1}⟨μ⟩ = w_{ν_{2}}x_{ν_{2},1}^{c_{ν_{2},1}}x_{ν_{2},2}^{c_{ν_{2},2}} ⋯ x_{ν_{2},n}^{c_{ν_{2},n}}.

The Weierstrass representation (h, g, t_{ν_{2}−1}) over (B_{ν_{2}}, W_{ν_{2}}, P_{ν_{2}}) satisfies the Abhyankar condition, (h, g, t_{ν_{2}−1}) is reductive, and we can write

g = t_{ν_{2}−1}^{m} + ∑_{j=1}^{m}⟨⟨0, ν_{2} − 1⟩⟩^{j}F_{ν_{2}−1,j}t_{ν_{2}−1}^{m−j}.

Let d_{ν_{2},1}, d_{ν_{2},2}, …, d_{ν_{2},n} denote the characteristic numbers of (h, g, t_{ν_{2}−1}). By Lemma 9.5.9 d_{ν_{2},α} = c_{ν_{2},α} + b_{ν_{2},α} for every 1 ≤ α ≤ n. By definition of characteristic numbers

Z_{ν_{2}} ∋ \frac{⟨⟨0, ν_{2} − 1⟩⟩^{q}F_{ν_{2}−1,q}}{(x_{ν_{2},1}^{d_{ν_{2},1}}x_{ν_{2},2}^{d_{ν_{2},2}} ⋯ x_{ν_{2},n}^{d_{ν_{2},n}})^{q}} = \frac{w_{ν_{2}}^{q}F_{ν_{2}−1,q}}{⟨ν_{2}⟩^{q}} .

Thus, we have

Z_{ν_{2}}/𝔭_{i} ∩ Z_{ν_{2}} ∋ \overline{(\frac{w_{ν_{2}}^{q}F_{ν_{2}−1,q}}{⟨ν_{2}⟩^{q}} )} = (\frac{\overline{w}_{ν_{2}}s\overline{K}_{ν_{1}}}{\overline{⟨ν_{2}⟩}} )^{q} .

\overline{⟨ν_{2}⟩} ≠ 0, \overline{⟨ν_{2}⟩} ∈ Z_{ν_{2}}/𝔭_{i} ∩ Z_{ν_{2}}, and \overline{w}_{ν_{2}} ∈ Z_{ν_{2}}/𝔭_{i} ∩ Z_{ν_{2}}. Thus, the element \overline{w}_{ν_{2}}s\overline{K}_{ν_{1}}/\overline{⟨ν_{2}⟩} belongs to the quotient field of Z_{ν_{2}}/𝔭_{i} ∩ Z_{ν_{2}}, and it is integral over Z_{ν_{2}}/𝔭_{i} ∩ Z_{ν_{2}}. On the other hand, by Lemma 8.7.1 Z_{ν_{2}}/𝔭_{i} ∩ Z_{ν_{2}} is a normal ring. We conclude that the element \overline{w}_{ν_{2}}s\overline{K}_{ν_{1}}/\overline{⟨ν_{2}⟩} belongs to Z_{ν_{2}}/𝔭_{i} ∩ Z_{ν_{2}}. One knows that the element \overline{(w_{ν_{2}}^{q}F_{ν_{2}−1,q}/⟨ν_{2}⟩^{q})} is the q-th power modulo 𝔭_{i}.

By Proposition 9.11 (4.ν_{2}) we have

\frac{H_{ν_{2}}}{x_{ν_{2},1}^{d_{ν_{2},1}}x_{ν_{2},2}^{d_{ν_{2},2}} ⋯ x_{ν_{2},n}^{d_{ν_{2},n}}} = w_{ν_{2}}h_{ν_{2}}.

Since H_{ν_{2}} is a very effective reducing element, we have

\overline{(\frac{w_{ν_{2}}^{q}F_{ν_{2}−1,q}}{⟨ν_{2}⟩^{q}} )} = l(\overline{w_{ν_{2}}h_{ν_{2}}})^{q},

and

(\frac{\overline{w}_{ν_{2}} \overline{K}_{ν_{1}}}{\overline{⟨ν_{2}⟩}} )^{q} = (\overline{w_{ν_{2}}h_{ν_{2}}})^{q}.

Thus, we obtain \overline{K}_{ν_{1}} = \overline{⟨ν_{2}⟩h_{ν_{2}}} and \overline{F}_{ν_{1},q} = l(\overline{⟨ν_{2}⟩h_{ν_{2}}})^{q}.

On the other hand, by Corollary 9.13.2

\overline{F}_{ν_{1},q} = l(\overline{⟨ν_{2}⟩h_{ν_{2}}})^{q} + \overline{⟨ν_{2}⟩}^{q}\overline{F}_{ν_{2},q}.

We obtain \overline{⟨ν_{2}⟩}^{q}\overline{F}_{ν_{2},q} = 0 and \overline{F}_{ν_{2},q} = 0. By the above results we conclude that \overline{F}_{ν_{2},j} = 0 for every 1 ≤ j ≤ m.

We consider the case ⟨A⟩. In this case F_{ν_{2},j} = 0 for every 1 ≤ j ≤ m. Thus g = t_{ν_{2}}^{m} and w(g, t_{ν_{2}}) = ∞. On the other hand, since S is a reduction sequence with infinite length, we have w(g, t_{ν_{2}}) < ∞, which is a contradiction.

We proceed to case ⟨B⟩. By definition of characteristic numbers,

\frac{F_{ν_{2},j}}{⟨ν_{2} + 1⟩^{j}} ∈ Z_{ν_{2}+1}

for every 1 ≤ j ≤ m. By Proposition 9.11(2.ν_{2} + 1) w(F_{ν_{2},1}, F_{ν_{2},2}, …, F_{ν_{2},m}) = ∑_{α=1}^{n} b_{ν_{2}+1,α}. By definition of the weight, there is an integer j satisfying 1 ≤ j ≤ m and ord(F_{ν_{2},j})/j = ∑_{α=1}^{n} b_{ν_{2}+1,α}. For this number j, we have ord(F_{ν_{2},j}/⟨ν_{2}+1⟩^{j}) = ord(F_{ν_{2},j}) − j ∑_{α=1}^{n} b_{ν_{2}+1,α} = 0. Thus, F_{ν_{2},j}/⟨ν_{2} + 1⟩^{j} ∈ Z_{ν_{2}+1}^{∗}, and v_{α}(F_{ν_{2},j}) = jv_{α}(⟨ν_{2} + 1⟩) for every 1 ≤ α ≤ n.

On the other hand, \overline{F}_{ν_{2},j} = 0, and F_{ν_{2},j} ∈ 𝔭_{i} ∩Z_{ν_{2}+1}. Thus, there is an integer α satisfying 1 ≤ α ≤ i and v_{α}(F_{ν_{2},j}) ≠ 0. For this number α we have v_{α}(⟨ν_{2} +1⟩) ≠ 0. Note that ν_{2} + 1 = ν_{1} + 2 > ν_{0}. We obtain a contradiction, since for every integer ν with ν > ν_{0} we have v_{1}(⟨ν⟩) = v_{2}(⟨ν⟩) = ⋯ = v_{i}(⟨ν⟩) = 0.

The lemma below solves Problem(C_{0}) we gave at the last part of Section 7 for the admissible class C_{0}. Recall the statement RW (B, W, P) we gave before Lemma 5.12 and the statement AN(B, W, P) we gave before Theorem 7.4.

Lemma 11.2. Let C_{0} be the collection of framed space germs (B, W, P) such that the basic framed space germ (A, V, P) = \underline{(B, W, P)} satisfies dim A = rank V. Let r be a positive integer. Assume that RW (B, W, P) is true for every framed space germ (B, W, P) such that (B, W, P) ∈ C_{0} and ♯ P ≤ r. Then, AN(B, W, P) is true for every framed space germ (B, W, P) such that (B, W, P) ∈ C_{0} and ♯ P ≤ r.

Proof. By Corollary 8.3 C_{0} is an admissible class.

Let (B, W, P) be a framed space germ over k. We assume that (B, W, P) ∈ C_{0} and ♯ P ≤ r. Let (h, g, t) be a Weierstrass representation over (B, W, P).

By Corollary 8.6 there exists an IAMT (B′, W′, P′) with a normal crossing of (B, W, P) such that (B′, W′, P′) has the basic valuation ring of normal crossing type. Let (h′, g′, t′) denote the pull-back of (h, g, t) by the associated morphism (B, W, P) → (B′, W′, P′). If w(g′, t′) = ∞, then putting H′ = 0 ∈ k[[P′]] one knows that AN(B, W, P) is true. We assume w(g′, t′) < ∞ below. Since C_{0} is an admissible class, we have (B′, W′, P′) ∈ C_{0}. We have ♯ P′ = ♯ P ≤ r.

By Lemma 9.10 there exists a very effective reduction sequence of (h′, g′, t′). By Theorem 11.1 any very effective reduction sequence of (h′, g′, t′) has finite length. Thus, there exists a positive integer ω and a very effective reduction sequence {(B_{ν}, W_{ν}, P_{ν}, H_{ν}) | 1 ≤ ν ≤ ω} with length ω. Let (B′′, W′′, P′′) = (B_{ω}, W_{ω}, P_{ω}) and (h′′, g′′, t′′) denote the pull-back of (h′, g′, t′) by the associated morphism (B′, W′, P′) → (B′′, W′′, P′′). Obviously (B′′, W′′, P′′) is an IAMT with a normal crossing of (B, W, P), and (h′′, g′′, t′′) is the pull-back of (h, g, t) by the associated morphism (B, W, P) → (B′′, W′′, P′′). Let H′′ = H_{1} + H_{2} + ⋯ + H_{ω} ∈ k[[P′′]]. By definition of reduction sequences (Definition 9.1.), if w(g′′, t′′ + H′′) < ∞, then (h′′, g′′, t′′ + H′′) satisfies the Abhyankar condition, and moreover (h′′, g′′, t′′ + H′′) is not reductive.

Thus AN(B, W, P) is true.

Corollary 11.3. The statement RW (B, W, Q) is true for every framed space germ (B, W, Q) such that (B, W, Q) ∈ C_{0}.

Proof. Let r be an arbitrary positive integer. By induction on r we will show that RW (B, W, Q) is true for every framed space germ (B, W, Q) such that (B, W, Q) ∈ C_{0} and ♯ Q ≤ r.

If r = 1, then by Lemma 5.14 the claim is true. Assume that the claim is true for r. By Lemma 11.2 and by Theorem 7.4 the

claim is true also for r + 1.

Our main result below has already cited in the introduction as Theorem 1.1.

Theorem 11.4. Let (B, W) be a space germ over k. Assume that the valuation ring W is of iterated divisor type. Let f_{1}, f_{2}, …, f_{ℓ} be a finite number of arbitrary non-zero elements in the ring B. Then, there exists an iterated analytic monoidal transform (B′, W′) of (B, W) such that the product f′_{1}f′_{2} ⋯ f′_{ℓ} ∈ B′ has a normal crossing, and the ideal in B′ generated by any non-empty subset of {f′_{1}, f′_{2}, …, f′_{ℓ}} is a principal ideal, where for every 1 ≤ i ≤ ℓ, f′_{i} ∈ B′ denotes the image of f_{i} ∈ B by the associated homomorphism B → B′.

Proof. By definition, W is of iterated divisor type, if and only if, dim B = rank W.

Let Q be a parameter system of B. Our theorem follows from the statement RS(B, W, Q) we gave after Lemma 5.11. By Corollary 5.13 RS(B, W, Q) and RW (B, W, Q) are equivalent. By Corollary 11.3 RW (B, W, Q) is true. Thus RS(B, W, Q) is true, and our theorem follows.

Problem: Remove the assumption that W is of iterated divisor type from Theorem 11.4.

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Department of Mathematical Sciences, Ibaraki University, Mito, Ibaraki, 310-8512, Japan

Current address: Max-Planck-Institut für Mathematik, Vivatsgasse 1, 53111 Bonn, Germany

  E-mail address: urabe@comp.metro-u.ac.jp, urabe@mito.ipc.ibaraki.ac.jp