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The undersigned, appointed by the Dean of the Graduate School, have examinedthe dissertation entitled

ALGEBRAIC RESOLUTION OF FORMAL IDEALS ALONG A VALUATION

presented by Samar El Hitti

a candidate for the degree of Doctor of Philosophy

and hereby certify that in their opinion it is worthy of acceptance.

ALGEBRAIC RESOLUTION OF FORMAL IDEALSALONG A VALUATION

A Dissertationpresented to

the Faculty of the Graduate SchoolUniversity of Missouri

In Partial Fulfillmentof the Requirements for the Degree

Doctor of Philosophy

bySAMAR EL HITTI

Professor Steven Dale Cutkosky, Dissertation Supervisor

May 2008

ACKNOWLEDGMENTS

As I am about to launch my career, I would like to thank everyone who in one wayor another has contributed to my being here.

I realize that this task is impossible, so I chose to acknowledge some of the peoplein my life who stand out along this journey.

Foremost, I would like to express my gratitude to my adviser, Professor Steven DaleCutkosky, for his endless patience, encouragement and guidance in helping me gainthe knowledge and prepare the thesis.

Special thanks to my committee members, Professor Nakhle Asmar, Professor SashiSatpathy, Professor Hema Srinivasan, and Professor Qi Zhang, for their patienceand support, and to the staff of the University of Missouri Mathematics departmentfor ensuring a very cooperative, positive and helpful atmoshpere.

Finally, and most importantly, non of this would have been possible without myfamily. Words fall short to express the gratitude and respect that I feel toward allfour of them for their ongoing support and belief in my choices in life. I also thankMaurice, and all my friends and their families who, throughout the years becamemy family away from home. I cannot even begin to imagine the possibility of beingwho I am today without any of them being in my life.

ii

RESOLUTION OF FORMAL IDEALSALONG A VALUATION

Samar El HittiProfessor Steven Dale Cutkosky, Dissertation Supervisor ABSTRACT

Let X be a possibly singular complete algebraic variety, defined over a field k

of characteristic zero. X is nonsingular at p ∈ X if OX,p is a regular local ring.

The problem of resolution of singularities is to show that there exists a nonsingular

complete variety X, which birationally dominates X. Resolution of singularities

(in characteristic zero) was proved by Hironaka in 1964. The valuation theoretic

analogue to resolution of singularities is local uniformization.

Let ν be a valuation of the function field of X, ν dominates a unique point

p, on any complete variety Y , which birationally dominates X. The problem of

local uniformization is to show that, given a valuation ν of the function field of

X, there exists a complete variety Y , which birationally dominates X such that

the center of ν on Y , is a regular local ring. Zariski proved local uniformization

(in characteristic zero) in 1944. His proof gives a very detailed analysis of rank 1

valuations, and produces a resolution which reflects invariants of the valuation.

We extend Zariski’s methods to higher rank to give a proof of local uniformiza-

tion which reflects important properties of the valuation. We simultaneously re-

solve the centers of all the composite valuations, and resolve certain formal ideals

associated to the valuation.

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Chapter

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

History of the problem.

Statement of the main result.

2. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

Notations and Definitions

3. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

4. Resolution in highest height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Perron Transforms.

Etale Perron Transforms.

Structure Theorems.

5. Resolution in all height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82

Perron Transforms.

Extension of results to all height.

6. Local Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Proof of the main result.

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103

iv

Chapter 1

Introduction

1.1 History of the problem

One of the fundamental problems in Algebraic Geometry is the problem of Resolu-

tion of Singularities. If Y is a singular algebraic variety, a resolution of singularities

of Y is a proper mapping φ : X → Y which is an isomorphism on a dense open

subset of Y , such that X is nonsingular. Hironaka [29] proved in 1964 that there

exists a resolution of Y if Y is defined over a field of characteristic zero. The proof

uses the existence of a hypersurface of maximal contact to reduce to an induction

on the dimension of Y . There have been significant simplifications of this theo-

rem in recent years, including Abramovich and de Jong [5], Bierstone and Milman

[7], Bogomolov and Pantev [8], Bravo, Encinas and Villamayor [9], Encinas and

Hauser [21], Encinas and Villamayor [22], Hauser [26], Kollar [35], Villamayor [44],

and Wlodarczyk [45].

Since Zariski introduced general valuation theory into algebraic geometry, val-

uations have been important in addressing resolution problems.

1

Suppose that K is an algebraic function field over a base field k, and V is a

valuation ring of K. V determines a unique center on a proper variety X whose

function field is K. The valuation gives a way of reducing a global problem on X,

such as resolution, to a local problem, studying the local rings of centers of V on

different varieties X whose function field is K.

The valuation theoretic analogue of resolution of singularities is local uni-

formization. A variety X is nonsingular at a point p if and only if the local ring

OX,p is a regular local ring.

The problem of local uniformization is to find a regular local ring R essentially of

finite type over k with quotient field K such that the valuation ring V dominates R.

That is, R ⊂ V and mV ∩R = mR. In 1944, Zariski [48] proved local uniformization

over fields of characteristic zero. To be precise, he proved:

Theorem 1.1.1. (Zariski) Suppose that f ∈ R. Then there exists a birational

extension of regular local rings R→ R1 such that R1 is dominated by ν, and

ordR1f ≤ 1 where f is the strict transform of f in R1. If ν has rank 1, then there

exists R1 such that f is a unit in R1.

Zariski first proved local uniformization for two-dimensional function fields over

an algebraically closed field of characteristic zero in [47]. He later proved local uni-

formization for algebraic function fields of characteristic 0 in [48].

Abhyankar has proven local uniformization in positive characteristic for two di-

2

mensional function fields, surfaces and three dimensional varieties [1], [2].

One of the most important techniques in studying resolution problems is to

pass to the completion of a local ring (the germ of a singularity). This allows us

to reduce local questions on singularities to problems on power series.

The first question which arises on completion is if the following generalization

of local uniformization is true:

Question 1.1.2. Given f ∈ R, does there exist a birational extension R → R1

where R1 is a regular local ring dominated by V such that ordR1f ≤ 1, where f is

the strict transform of f in R1 ?

The answer is surprisingly NO! We give a simple counter example to our ques-

tion in Chapter 3, which comes from a discrete valuation.

The example can be understood in terms of an extension of our valuation ring V

to a valuation ring dominating R. It is a fact that the rank (page 7) of a valuation

V dominating R often increases when extending the valuation to a valuation ring

V dominating R. Some papers where this is studied are Spivakovsky [41], Heinzer

and Sally [28], and Cutkosky and Ghezzi [19].

The first measure of this increase of rank is the prime ideal

3

QR = f ∈ R | ν(f) ≥ n for all n ∈ N.

This prime has been previously defined and studied by Teissier [43], Cutkosky

[16] and Spivakovsky.

If the rank V = 1, then there is a unique extension of V to the quotient field of

R/QR dominating R/QR. Thus the rank of the extension does not increase, and

the prime ideal QR led to this obstruction.

In spite of the fact that we cannot resolve the singularity of f = 0 by a birational

extension of R, we can resolve the formal singularity, whose local ring is R/QR, by

a birational extension of R. This is proven for valuations of rank 1 by Cutkosky

and Ghezzi [19].

Theorem 1.1.3. (Cutkosky, Ghezzi) Suppose that rank V = 1. Then there exists

a birational extension R→ R1 where R1 is a regular local ring dominated by ν such

that QR1is a regular prime

Zariski proves local uniformization by constructing special birational extensions

R → R1 dominated by rank 1 valuation rings which he calls Perron transforms.

Cutkosky and Ghezzi make essential use of Perron transforms and Zariski’s reso-

lution algorithm in their proof of Theorem 1.1.3.

We extend Perron transforms to arbitrary rank in Chapter 5, and prove a strong

form of local uniformization, which generalizes both Theorem 1.1.1 and Theorem

1.1.3.

4

1.2 Statement of the main result

First we introduce some notation which is necessary for the statement of out the-

orem. Let

(0) = P tV ⊂ · · · ⊂ P 1

V ⊂ P 0V

be the chain of prime ideals in V . Let

(0) = P tR ⊂ · · · ⊂ P 1

R ⊂ P 0R

be the induced chain of prime ideals in R, where P iR = P i

V ∩R. Let νi be a valuation

whose valuation ring is Vi = VP iV .

Consider the following condition (1.1) on a Cauchy sequence f = fn in RP iR.

For all l ∈ N, there exists nl ∈ N such that νi(fn) ≥ lν(mRPiR

) if n ≥ nl. (1.1)

Define the prime ideal QRPiR

⊂ RP iRfor the valuation ring VP iV by

QRPiR

=f ∈ RP iR

| f satisfies (1.1).

Theorem 1.2.1. Suppose that R is a local domain which is essentially of finite

type over a field k of characteristic zero, and V is a valuation ring of the quotient

field of R which dominates R. Let

5

(0) = P tV ⊂ · · · ⊂ P 0

V

be the chain of prime ideals of V . Then there exists a birational extension R→ R1

such that R1 is a regular local ring and V dominates R1. Further, P iR1

= P iV ∩ R1

are regular primes for all i, and

QR1Pi

R1

⊂ R1P iR1

are regular primes for all i.

6

Chapter 2

Preliminaries

In this section, we introduce some notations and assumptions that will hold in

chapters 4 and 5.

Suppose that T is a regular local ring of dimension q, which is essentially of finite

type over a field k of characteristic zero, with maximal ideal mT , such that T/mT

is an algebraic extension of k. Suppose that ν is a valuation of the quotient field

of T which dominates T . Let QF (T ) be the quotient field of T .

By definition ν is a homomorphism ν : QF (T )∗ → Γ from the multiplicative group

of QF (T ) onto an ordered abelian group Γ such that:

1. ν(ab) = ν(a) + ν(b) for a, b ∈ QF (T )∗,2. ν(a+ b) ≥ min ν(a), ν(b) for a, b ∈ QF (T )∗,3. ν(c) = 0 for 0 6= c ∈ k.

We extend ν to QF (T ) by setting ν(0) =∞.

Let V = a ∈ QF (T )|ν(a) ≥ 0. Then V is a ring and is called the valuation ring

of ν.

Let mV = a ∈ QF (T )|ν(a) > 0. Then mV is the unique maximal ideal of V .

The rank of a valuation ring V is the number of proper prime ideals in V , which

are necessarily ordered by inclusion.

7

The rational rank of ν is the maximal number of rationally independent ele-

ments in Γ, which is bounded above by the dimension of T .

Let

(0) ⊂ · · · ⊂ PV ⊂ mV

be the chain of prime ideals in V .

Let Γ1 ⊂ ΓV be the rank 1 isolated subgroup of the valuation ring V/PV . We

have an embedding Γ1 ⊂ R of ordered groups, such that Z ⊂ Γ1. In this way we

identify the integers with a subgroup of Γ1. We will sometimes say that an element

γ ∈ ΓV is ∞ if γ 6∈ Γ1, and γ < ∞ if γ ∈ Γ1. If γ ∈ Γ1, then there exists n ∈ N

such that γ ≤ n.

The maximal ideal of T is mT = mV ∩ T .

In chapters 4 and 5, we assume that

trdegT/mTV/mV = 0. (2.1)

Let PT = PV ∩ T .

Suppose that V/PV has rational rank s.

In chapters 4 and 5, we assume that

trdegTPT /PTTPTVPV /PV VPV = trdegQF (T/PT )QF (V/PV ) = 0. (2.2)

8

Chapter 3

Example

Example 3.0.2. There exists a discrete valuation ring V dominating a regular

local ring R of dimension 3 such that for all r ≥ 2, there exists f ∈ R such that for

all birational extensions R → R1 of regular local rings dominated by V , the strict

transform of f has order ≥ r in R1.

Proof. Let t, φ(t), ψ(t) ∈ k[[t]] be algebraically independent elements of positive

order.

We have the inclusion k(x, y, z) → k((t)) defined by x = t, y = φ(t) and z = ψ(t).

The order valuation on k((t)) (with valuation ring k[[t]]) restricts to a discrete rank

1 valuation on k(x, y, z), dominating R = k[x, y, z](x,y,z).

QR = (y − φ(x), z − ψ(x)) ⊂ R is a regular prime of height 2, and it defines a

curve γ ⊂ Spec (R).

QR ∩R = (0).

9

Suppose that r ∈ N(r ≥ 2).

Let f = (y − φ(x))r + (z − ψ(x))r+1 ∈ R.

ordRf = r and ordγf = ord(y−φ(x),z−ψ(x))f = r.

Suppose that R → R1 is a birational extension where R1 is a regular local ring

dominated by ν.

In R1, write y − φ(x) = h1g1 and z − ψ(x) = h2g2 where g1 is the strict transform

of y−φ(x) in R1, h1 ∈ R1 and g2 is the strict transform of z−ψ(x) in R1, h2 ∈ R1.

∞ = ν(y − φ(x)) = ν(h1) + ν(g1) and ν(h1) <∞ thus ν(g1) =∞.

Similarly, ν(g2) =∞.

Let γ be the strict transform of γ in Spec(R1), Iγ = (g1, g2) = QR1⊂ mR1

ordR1f ≥ ordγf = ordγf = r

In the previous example, we have f ∈ QR, and has value larger than any element

in the value group.

This means that the rank of the valuation ring V must increase when passing to

the completion R.

10

Chapter 4

Resolution in highest height

4.1 Perron Transforms

Throughout this chapter we assume that the assumptions of chapter 2 hold. We

will define two types of Peron Transforms.

Perron Transforms of type (1, 0):

Suppose that

x1, . . . , xs, . . . , xp, . . . , xq

is a regular system of parameters in T , such that s ≤ p, x1, . . . , xp 6∈ PT ,

xp+1, . . . , xq ∈ PT , and ν(x1), . . . , ν(xs) are rationally independent.

Let τi = ν(xi) for 1 ≤ i ≤ s.

We define two types of transforms of type (1, 0):

• Transforms of type I.

Set τi(0) = τi for 1 ≤ i ≤ s. For each positive integer h define s positive,

rationally independent real numbers τ1(h), . . . , τs(h) by the ”Algorithm of

11

Perron” [48].τ1(h− 1) = τs(h)τ2(h− 1) = τ1(h) + a2(h− 1)τs(h)...τs(h− 1) = τs−1(h) + as(h− 1)τs(h)

Where

aj(h− 1) =

[τj(h− 1)

τ1(h− 1)

], 2 ≤ j ≤ s

the greatest integer inτj(h)

τ1(h). There are nonnegative integers Ai(h) such that

τi = Ai(h)τ1(h) + Ai(h+ 1)τ2(h) + · · ·+ Ai(h+ s− 1)τs(h)

for 1 ≤ i ≤ s.

Then Det(Ai(h+ j)) = (−1)h(s−1) (See [48] page 385).

These numbers have the important property that

limh→∞Ai(h)

A1(h)=ν(xi)

ν(x1)(4.1)

we refer to [48] page 385.

Let Ai(h+ j) = aij+1, and define:

x1 = x1(1)a11 . . . xs(1)a1s

...xs = x1(1)as1 . . . xs(1)ass

xs+1 = xs+1(1)...xq = xq(1).

(4.2)

Then Det(aij) = ±1 and ν(x1(1)) = τ1(h), . . . , ν(xs(1)) = τs(h) are rationally

independent. We necessarily have that x1(1), . . . , xs(1) 6∈ PT1 .

Define a transformation T → T1 of type I along ν by

T1 = T [x1(1), . . . , xs(1)]T [x1(1),...,xs(1)]∩mV .

12

T1 is a regular local ring, QF (T ) = QF (T1) and ν dominates T1.

• Transforms of type IIr.

Now we define T → T1 of type IIr along ν, we refer to [48] (with the restric-

tion that s+ 1 ≤ r ≤ p), as follows :

Set ν(xr) = τr. τr is rationally dependent on τ1, . . . , τs since the ratio-

nal rank of ν is s. There are thus integers λ, λ1, . . . , λs such that λ >

0, (λ, λ1, . . . , λs) = 1 and

λτr = λ1τ1 + · · ·+ λsτs. (4.3)

We first perform a transform T → T (1) of type I along ν where T (1) has

regular parameters x1(1), . . . , xs(1) defined as in (4.2). Then ν(xi(1)) = τi(h)

for 1 ≤ i ≤ s, v(xr(1)) = τr. Set

λi(h) = λ1A1(h+ i− 1) + λ2A2(h+ i− 1) + · · ·+ λsAs(h+ i− 1)

for 1 ≤ i ≤ s. Then

λτr = λ1(h)τ1(h) + · · ·+ λs(h)τs(h).

Take h sufficiently large that all λi(h) > 0. This is possible by (4.1), since

λ1τ1 + · · ·+λsτs > 0. We still have (λ, λ1(h), . . . , λs(h)) = 1 since Det(Ai(h+

j − 1) = ±1. After re-indexing the xi(1), we may suppose that λ1(h) is

not divisible by λ. Let λ1(h) = λµ + λ′, with 0 < λ′ < λ. Now perform

the following transform T (1) → T (2) along ν defined by x1(1) = xr(2),

13

xr(1) = x1(2)xr(2)µ, and xi(1) = xi(2) otherwise. Set τ ′i = ν(xi(2)) for all i.

τ ′1, . . . , τ′s, τ′r are positive and

λ′τ ′r = λ′1τ′1 + · · ·+ λ′sτ

′s

where λ′1 = λ, λ′i = −λi(h) for 2 ≤ i ≤ s. Thus we have achieved a reduction

on λ, and by repeating the above procedure, we reduce to the case

τr = λ1τ1 + · · ·+ λsτs

In this case we define

x1 = N1...xs = Ns

xr = Nλ11 . . . Nλs

s Nr.

Thus in the general case, there exists aij ∈ N, 1 ≤ i, j ≤ s+ 1 such that

x1 = Na111 . . . Na1s

s Na1s+1r

...xs = Nas1

1 . . . Nasss Nass+1

r

xr = Nas+11

1 . . . Nas+1ss Nas+1s+1

r .

where Det(aij) = ±1 and ν(N1), . . . , ν(Ns) are positive and rationally inde-

pendent, and ν(Nr) = 0.

Define a transformation T → T1 of type IIr along ν by

T1 = T [N1, . . . , Ns, Nr]T [N1,...,Ns,Nr]∩mV .

We necessarily have that N1, . . . , Ns, Nr 6∈ PT1 , QF (T ) = QF (T1) and ν dom-

inates T1.

14

Perron Transforms of type (2, 0).

Suppose that

x1, . . . , xs, . . . , xp, . . . , xq

is a regular system of parameters in T such that ν(x1), . . . , ν(xs) are rationally

independent.

Suppose that d1, . . . , ds ∈ N and that xj ∈ PT

Define a transformation T → T1 of type (2, 0) along ν by

T1 = T [xj

xd11 . . . xdss]T [

xj

xd11 ...x

dss

]∩mV .

In all cases, by (2.1), we have that dim(T1) = dim(T ) and T1/mT1 is a finite

extension of T/mT .

For an ideal I ⊂ T , let

ν(I) = minν(f) | f ∈ I.

Consider the following condition (4.4) on a Cauchy sequence fn in T .

For all l ∈ N, there exists nl ∈ N such that ν(fn) ≥ lν(m) if n ≥ nl. (4.4)

Lemma 4.1.1. 1. Suppose that fn and gn are two Cauchy sequences in T

converging to f ∈ T and fn satisfies (4.4). Then gn satisfies (4.4).

2. Let

QT =

f ∈ T | A Cauchy sequence fn in T which

converges to f satisfies (4.4)

.

Then QT is a prime ideal in T .

3. QT ∩ T = PT .

15

Suppose that f ∈ T is not in QT . Let fn be a Cauchy sequence in T which

converges to f . There exists an l ∈ N such that there are arbitrarily large n with

ν(fn) < lν(m). We can thus choose n0 such that fn − fn0 ∈ ml if n ≥ n0 and

ν(fn0) < lν(m). For n ≥ n0, we have fn = fn0 + h with h ∈ ml. ν(h) ≥ lν(m) >

ν(fn0) implies ν(fn) = ν(fn0) for n ≥ n0.

A similar calculation shows that the above value ν(fn0) is independent of choice

of n0 satisfying the above conditions, and is independent of choice of Cauchy se-

quence converging to f .

We may thus define ν(f) = ν(fn) for sufficiently large n, if f ∈ T − QT . We

will sometimes write ν(f) = ∞ if f ∈ QT . Observe that an extension of ν to

a valuation of QF (T ) which dominates T is uniquely determined on elements of

T −QT . We now identify ν with an extension of ν to QF(T ) which dominates T .

Let σ(T ) = dim(T /QT ) and τ(T ) = dim(T/PT ).

We have σ(T ) ≤ τ(T ) with equality if QT = PT T .

Let

ω(T ) = dim(T )− dimT/mT (PT/m2T ∩ PT ).

We have

dimT/mT (PT/m2T ∩ PT ) ≤ height PT

with equality if and only if T/PT is a regular local ring.

Thus w(T ) ≥ dim(T ) − height PT = dim(T/PT ) and w(T ) = dim(T/PT ) if and

only if T/PT is a regular local ring.

16

4.2 Etale Perron Transforms

Suppose that x1, . . . , xs, . . . , xp, . . . , xq is a system of regular parameters in T , such

that x1, . . . , xs, . . . , xp /∈ PT , xp+1, . . . , xq ∈ PT and ν(x1), . . . , ν(xs) are rationally

independent.

Let β0 ∈ T be a primitive element of T/mT over k. Let

U = T [β0]mT∩T[β0]⊂ T .

We thus have that the subfield k[β0] ⊂ U is a coefficient field of U . We will

identify k[β0] with U/mU .

ν extends to a valuation of QF(U) which dominates U by restricting our exten-

sion of ν to QF(T ) to QF(U). ν is uniquely determined on elements of U which

are not in QU = QT .

For the moment, let us call our extension of ν to QF (T ) ν and let V be the

valuation ring of ν. Let ν ′ be the restriction of ν to QF (U) with valuation ring

V ′ = V ∩QF (U).

Since QF (U) is finite over QF (T ), ν and ν ′ both have the same rank and rational

rank.

Thus the chain of prime ideals in V ′ is

(0) ⊂ · · · ⊂ pV ′ ⊂ mV ′

where pV ′ ∩QF (T ) = pV , and the rank 1 valuation rings V ′/pV ′ and V/pV have

the same rational rank s.

Let pU = pV ′ ∩ U .

17

We have τ(U) = τ(T ) by (2.2) and w(U) ≤ w(T ) with equality if PU = PTU .

Let p = ω(T ) and p0 = ω(U), we have p0 ≤ p.

We also define two types of etale Perron transforms:

• Etale Perron transform of type I:

We have that x1, . . . , xs /∈ PU .

Let

U → U1 = U [x1(1), . . . , xs(1)]mV ∩U [x1(1),...,xs(1)]

be a transformation of type I along ν. Define x1(1), . . . , xs(1) by:

x1 = x1(1)a11 . . . xs(1)a1s

...xs = x1(1)as1 . . . xs(1)ass

xs+1 = xs+1(1)...xq = xq(1)

Where Det(aij) = ±1.

We have that the field k[β0] ⊂ U1 has the property that k[β0] = U/mU =

U1/mU1 . We further have that U → U1 is a birational extension.

We identify ν with an extension of ν to the Quotient field of U1 which domi-

nates U1. Let p1 = ω(U1). We have that

s ≤ p1 = ω(U1) ≤ ω(U) = p0 ≤ w(T ) = p.

We have that

x1(1), . . . , xs(1), xs+1, . . . , xq

18

a regular system of parameters in U1, such that ν(x1(1)), . . . , ν(xs(1)) are

rationally independent.

T → U → U1

(with the regular parameters x1, . . . , xs, . . . , xq and

x1(1), . . . , xs(1), xs+1, . . . , xq) is called an etale Perron transform of type I

along ν.

• Etale Perron transform of type IIr : (s+ 1 ≤ r ≤ p)

Let λ(t1, . . . , tr−1) be a polynomial in the polynomial ring

U/mU [t1, . . . , tr−1], in the variables t1, . . . , tr−1.

We have x1, . . . , xs 6∈ PU and we assume that

x′r = xr − λ(x1, . . . , xr−1) 6∈ QU .

Let

U → U ′ = U [N1, . . . , Ns, Nr]mV ∩U [N1,...,Ns,Nr]

be a transformation of type IIr along ν. Define N1, . . . , Ns, Nr by

x1 = Na111 . . . Na1s

s Na1s+1r

...xs = Nas1

1 . . . Nasss Nass+1

r

x′r = Nas+11

1 . . . Nas+1ss Nas+1s+1

r

19

where Det(aij) = ±1, ν(N1), . . . , ν(Ns) are rationally independent and

ν(Nr) = 0. Define x1(1) = N1, . . . , xs(1) = Ns, xr(1) = Nr.

Let β1 ∈ U ′ be a primitive element of U ′/mU ′ over k. Let

U1 = U ′[β1]mU′∩U ′[β1].

We identify ν with an extension of ν to the Quotient field of U1 which domi-

nates U1.

Let α be the residue of xr(1) in U1/mU1 . Let xi(1) = xi if s < i ≤ p, i 6= r

and xr(1) = xr(1)− α.

x1(1), . . . , xs(1), xs+1(1), . . . , xr(1), . . . , xp(1) , xp+1, . . . , xq are regular param-

eters in U1. We necessarily have that ν(x1(1)), . . . , ν(xs(1)) are rationally

independent and x1(1), . . . , xs(1) 6∈ PU1 .

Let p1 = ω(U1). We have that

s ≤ p1 = ω(U1) ≤ ω(U) = p0 ≤ ω(T ) = p.

We choose a regular system of parameters

x1(1), . . . , xp1(1), . . . , xp(1), xp+1, . . . , xq (4.5)

in U1 such that

x1(1), . . . , xp1(1) /∈ PU1 , xp1+1(1), . . . , xp(1), xp+1, . . . , xq ∈ PU1 ,

and there exists a 1-1 map σ : s + 1, . . . , p1 → s + 1, . . . , p such that

xi(1) = xσ(i)(1) for s+ 1 ≤ i ≤ p1.

20

T → U → U1

(with the regular parameters x1, . . . , xs, . . . , xp, . . . , xq and

x1(1), . . . , xs(1), . . . , xp(1), xp+1, . . . , xq) is called an etale Perron transform

of type IIr along ν.

We say that T → U → U1 is an etale Perron transform along ν, if it is an etale

Perron transform of type I or IIr.

If T → U → U1 is an etale Perron transform along ν, we have

dim(T ) = dim(U) = dim(U1) and U1/mU1 is finite over T/mT by (2.1),

σ(T ) = σ(U) ≥ σ(U1), τ(T ) = τ(U) = τ(U1), by (2.2) and ω(T ) ≥ ω(U) ≥ ω(U1).

Suppose that

T → U → U1 → U2 → · · · → Un

is a sequence of etale Perron transforms along ν. Further suppose that σ(Un) =

σ(T ) and τ(Un) = τ(T ). Let g = 0 be a local equation of the exceptional locus

of spec(Un) → spec(T ). Suppose that J ⊂ T is an ideal. We define the strict

transform J of J in Un by

J = ∪∞j=1(JUn : gjUn).

For an ideal I ⊂ T , we define the strict transform I of I in Un by

I = ∪∞j=1(IUn : gjUn).

21

Definition 4.2.1. We will say that T has property (A) if whenever T0 is an alge-

braic regular local ring of K, such that V dominates T , T dominates T0, and

T0 → U → U1 → · · · → Un (4.6)

is a sequence of etale Perron transforms along ν, then

σ(Un) = σ(T ).

Given a sequence (4.6), we have that τ(Un) = τ(T ) by (2.2).

From now on in this section, we assume that T satisfies property (A). Then

for a sequence (4.6), we have that PUn is an irreducible component of the strict

transform of PT in Un and QUnis an irreducible component of the strict transform

of QT in Un.

4.3 Structure theorems

Theorem 4.3.1. Let p = ω(T ). Suppose that x1, . . . , xp, xp+1, . . . , xq are regular

parameters in T such that x1, . . . , xp /∈ PT , xp+1, . . . , xq ∈ PT and ν(x1), . . . , ν(xs)

are rationally independent. Suppose that

T → U → U1 → · · · → Un

is a sequence of etale Perron transforms along ν (where we have identified

ν with an extension of ν to QF(Un) which dominates Un). Let kn ∼= Un/mUn

be an coefficient field of Un. Suppose that the prescribed regular parameters of

Un are x1(n), . . . , xs(n), . . . , xpn(n), . . . , xp(n), xp+1, . . . , xq, where pn = ω(Un), and

ν(x1(n)), . . . , ν(xs(n)) are rationally independent which satisfy

xi = φni (x1(n), . . . , xp(n), xp+1, . . . , xq)

22

for 1 ≤ i ≤ p, where φni ∈ kn[[x1(n), . . . , xp(n), xp+1, . . . , xq]].

Then there exists l0 ∈ N such that for all l ≥ l0, there exists a sequence of

transforms T → Tn,l along ν of type (1, 0) with the following properties:

1. Tn,l/mTn,l∼= kn.

2. Tn,l has regular parameters x1,l(n), . . . , xpn,l(n), . . . , xp,l(n), xp+1, . . . , xq such

that there exists hi ∈ mlTn,l

satisfying

xi = φni (x1,l(n), . . . , xp,l(n), xp+1, . . . , xq) + hi

for 1 ≤ i ≤ p.

3. For 1 ≤ i ≤ pn, ν(xi,l(n)) = ν(xi(n)) < l0

4. ν(xi,l(n)) > l if pn l for 1 ≤ i ≤ p.

Proof. We prove the theorem by induction on the length of a factorization of

U → Un into a sequence of etale Perron transforms along ν.

We thus assume that the conclusions of the theorem are true for a sequence

of etale Perron transforms U → U1 along ν (which could be more than one),

and U1 → U2 is a single etale Perron transform along ν. Let k1 = U1/mU1 and

k2 = U2/mU2 .

We have prescribed regular parameters x1, . . . , xp, . . . , xq in U . Let p1 = ω(U1).

Assume that x1(1), . . . , xp1(1), . . . , xp(1), xp+1, . . . , xq are the prescribed regular pa-

rameters in U1. There exist

φ1i ∈ U1/m1[[x1(1), . . . , xp(1), xp+1, . . . , xq]]

23

such that

xi = φ1i (x1(1), . . . , xp(1), xp+1, . . . , xq)

for 1 ≤ i ≤ p.

By induction, there exists a positive integer l0(1) such that for all l(1) ≥ l0(1),

there exist T1,l(1) such that (1) - (5) of the conclusions of the theorem hold for

U → U1 and T → T1,l(1).

(3) implies that l0(1) > maxν(x1(1)), . . . , ν(xp1(1)).

It also follows that ν(xi,l(1)(1)) = ν(xi(1)) for 1 ≤ i ≤ p1 and ν(mU1) = ν(mT1,l(1)).

We assume that U1 → U2 is of type IIr. The case when U1 → U2 is of type I

is similar. U1 → U2 is then defined as follows. There exists j with s + 1 ≤ j ≤ p1

and

λ(x1(1), . . . ,xj−1(1), xj+1(1), . . . , xp(1))

∈ U1/mU1 [x1(1), . . . , xj−1(1), xj+1(1), . . . , xp(1)]

such that

xj(1)′ = xj(1)− λ(x1(1), . . . , xj−1(1), xj+1(1), . . . , xp(1))

and we have regular parameters

x1(2), . . . , xs(2), xs+1(1), . . . , xj−1(1), xj(2), xj+1(1), . . . , xp1(1), . . . , xp(1),

xp+1, . . . , xq in U2 where x1(2), . . . , xs(2) and xj(2) are defined by

x1(1) = x1(2)a11 . . . xs(2)a1s(xj(2) + α2)a1s+1

...xs(1) = x1(2)as1 . . . xs(2)ass(xj(2) + α2)

ass+1

xj(1)′ = x1(2)as+11 . . . xs(2)as+1s(xj(2) + α2)as+1s+1

(4.7)

24

for appropriate 0 6= α2 ∈ U2/m2 where Det(aij) = ±1.

Let (bij) = (aij)−1.

Let d = ν(x′j(1)) <∞. There exists n such that nν(mU1) > d.

Let l0(2) be chosen so that

l0(2) > max ν(xp1+1(1)), . . . , ν(xs(1)), ν(xj(2)), l0(1), l0(1)s∑

k=1

bik; i = 1, . . . , s.

Suppose l ≥ l0(2).

Let ξ1 = maxl − [∑s

i=1,i 6=k bs+1iν(xi(1)) + (bs+1k − 1)ν(xk(1))], k = 1, . . . , s.

Let ξ2 = maxl − [bts+1ν(xj(1)′) +∑s

i=1,i 6=k btixi(1) + (btk − 1)ν(xk(1))]

where the maximum is over k = 1, . . . , s and t = 1, . . . , s.

Let ξ3 = maxl − [(bks+1 − 1)ν(xj(1)′) +∑s

i=1(bki)ν(xi(1))], k = 1, . . . , s.

Choose l(1) such that l(1) ≥ maxl0(2), d, l0(1), ξ1, ξ2, ξ3.

Choose n0 ∈ N such that n0ν(mU1) > l(1), and n0 > l.

Choose n1 ∈ N such that n1ν(mU2) > l.

There exists λl ∈ T1,l(1) such that

λl = λ(x1,l(1)(1), . . . , xj−1,l(1)(1), xj+1,l(1)(1), . . . , xp1,l(1)(1), . . . . . . , xp,l(1)(1)) + h′

with h′ ∈ mn0

T1,l(1). Let xj,l(1)(1)′ = xj,l(1)(1) − λl. There exist gi,l(1) ∈ QF (U1)

such that xi,l(1)(1) = xi(1) + gi,l(1) for 1 ≤ i ≤ p, and ν(gi,l(1)) > l(1).

xj,l(1)(1)′ = xj,l(1)(1)− λl= xj,l(1)(1)− λ(x1,l(1)(1), . . . , xj−1,l(1)(1), xj+1,l(1)(1), . . . , xp,l(1)(1))− h′= xj(1) + gj,l(1) − λ(x1(1) + g1,l(1), . . . , xp(1) + gp,l(1))− h′= xj(1)− λ(x1(1), . . . , xp(1)) +

∑pi=1 gi,l(1)Λi − h′

= xj(1)′ +∑p

i=1 gi,l(1)Λi − h′,25

where Λi ∈ k1[g1,l(1), . . . , gp,l(1), x1(1), . . . , xp(1)]. Thus ν(Λi) ≥ 0 for 1 ≤ i ≤ p.

We have

ν(

p∑i=1

gi,l(1)Λi) > mingi,l(1) > l(1) > d,

and

ν(h′) > n0ν(mT1,l(1)) = n0ν(mU1) > l(1) > d.

Thus ν(xj,l(1)(1)′) = d = ν(xj(1)′).

Let g′j,l(1) = xj,l(1)(1)′ − xj(1)′.

xj,l(1)(1)′ − xj(1)′ =

p∑i=1

gi,l(1)Λi − h′.

Thus ν(g′j,l(1)) > l(1).

With aij as defined in (4.7), set

x1,l(1)(1) = x1,l(2)a11 . . . xs,l(2)a1sxj,l(2)a1s+1

...xs,l(1)(1) = x1,l(2)as1 . . . xs,l(2)assxj,l(2)ass+1

xj,l(1)(1)′ = x1,l(2)as+11 . . . xs,l(2)as+1sxj,l(2)as+1s+1 .

Let (bij) = (aij)−1.

We have

x1,l(2) = x1,l(1)(1)b11 . . . xs,l(1)(1)b1s [xj,l(1)(1)′]b1s+1

...xs,l(2) = x1,l(1)(1)bs1 . . . xs,l(1)(1)bss [xj,l(1)(1)′]bss+1

xj,l(2) = x1,l(1)(1)bs+11 . . . xs,l(1)(1)bs+1s [xj,l(1)(1)′]bs+1s+1 .

ν(xj,l(1)(1)′) = ν(xj(1)′) and ν(xi,l(1)(1)) = ν(xi(1)) for i = 1, . . . , s.

Thus ν(xi,l(2)) = ν(xi(2)) > 0 for i = 1, . . . , s and ν(xj,l(2)) = 0.

Thus T1,l(1)[x1,l(2), . . . , xs,l(2), xj,l(2)] ⊂ V .

Let

T2,l = T1,l(1)[x1,l(2), . . . , xs,l(2), xj,l(2)]T1,l(1)[x1,l(2),...,xs,l(2),xj,l(2)]∩mV .

26

Let V be the valuation ring of an extension of ν to QF(U2) which dominates

U2. Let L = T2,l/mT2,l= k1[β] ⊂ V /mV , where β is the residue of xj,l(2) in L.

β =[

[xj,l(1)(1)′]bs+1s+1

x1,l(1)(1)bs+11 ...xs,l(1)(1)bs+1s

]=

[(xj(1)′+g′

j,l(1))bs+1s+1

(x1(1)+g1,l(1))bs+11 ...(xs(1)+gs,l(1))

bs+1s

]=

[(xj(1)′)a1

x1(1)bs+11 ...xs(1)bs+1s

]= α2,

since

ν(gi,l(1)) > l(1) > l0(2) > ν(xi(1)),

for i = 1 . . . , s.

and

ν(g′j,l(1)) > l(1) > d = ν(xj(1)′).

Thus

T2,l/mT2,l= k1[α2] = U2/mU2 = k2 ⊂ V /mV .

And x1,l(2), . . . , xs,l(2), xs+1,l(1)(1), . . . , xj−1,l(1)(1), xj,l(2), xj+1,l(1)(1),

. . . , xp1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq are regular parameters in T2,l, where

xj,l(2) = xj,l(2)− α.

Let p2 = ω(U2). We have a prescribed regular system of parameters

x1(2), . . . , xp2(2), xp2+1(2), . . . , xp1(2), xp1+1(1), . . . , xp(1), xp+1, . . . , xq

of U2, where

xp2+1(2), . . . , xp1(2), xp1+1(1), . . . , xp(1), xp+1, . . . , xq

are a basis of PU2/PU2 ∩m2U2

. There exists a 1-1 map

σ : s+ 1, . . . , p2 → s+ 1, . . . , p127

such that xi(2) = xσ(i)(1) if s+ 1 ≤ i ≤ p2 and σ(i) 6= j, and

xi(2) = xj(2) if σ(i) = j.

Let

ψi(x1(2), . . . , xs(2), xs+1(1), . . . , xj−1(1), xj(2), . . . , xp(1))∈ U2/m2[[x1(2), . . . , xs(2), xs+1(1), . . . , xj(2), . . . , xp(1), xp+1, . . . , xq]]

be defined by

ψi = φ1i (x1(2)a11 . . . xs(2)a1s(xj(2) + α2)

a1s+1 , . . . , x1(2)as1 . . . xs(2)ass

(xj(2) + α2)ass+1 , xs+1(1), . . . , xj−1(1), x1(2)as+11 . . . xs(2)as+1s(xj(2)+

α2)as+1s+1 + λ(x1(2)a11 . . . xs(2)a1s(xj(2) + α2)

a1s+1 , . . . , x1(2)as1 . . . xs(2)ass

(xj(2) + α2)ass+1 , xs+1(1), . . . , xj−1(1), xj+1(1), . . . , xp(1)), xj+1(1), . . . , xp(1),

xp+1, . . . , xq).

There exist series

∆i(x1(2), . . . , xp2(2), xp2+1(2), . . . , xq) ∈ U2/mU2 [[x1(2), . . . , xp(2), xp+1, . . . , xq]]

for 1 ≤ i ≤ p, such that

x1(2) = ∆1, . . . , xs(2) = ∆s,

xs+1(1) = ∆s+1, . . . , xj−1(1) = ∆j−1,

xj(2) = ∆j,

xp+1(1) = ∆p+1, . . . , xp(1) = ∆p.

We have

xi = φ2i (x1(2), x2(2), . . . , xp(2), xp+1, . . . , xq)

28

for 1 ≤ i ≤ p are defined by

φ2i (x1(2), x2(2), . . . , xp(2), xp+1, . . . , xq) = ψi(∆1,∆2, . . . ,∆p, xp+1, . . . , xq)∈ U2/m2[[x1(2), . . . , xp2(2), . . . , xp(2), xp+1, . . . , xq]].

Invert the series ∆i to get series

Ei(x1(2), . . . , xs+1(2), xs+1(1), . . . , xj−1(1), xj(2), xj+1(1), . . . , xp(1), xp+1, . . . , xq)

such that xi(2) = Ei for 1 ≤ i ≤ p.

There exist hi ∈ mn1

T2,lsuch that

Ei(x1,l(2), . . . ,xs,l(2), xs+1,l(1)(1), . . . , xj−1,l(1)(1), xj,l(2),

xj+1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq)− hi ∈ T2,l.

Define regular parameters x1,l(2), . . . , xp,l(2), xp+1, . . . , xq in T2,l by

xi,l(2) = Ei(x1,l(2), . . . , xs,l(2), xs+1,l(1)(1), . . . , xj−1,l(1)(1),

xj,l(2), xj+1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq)− hi for 1 ≤ i ≤ p.

Now we show that for 1 ≤ i ≤ p, xi = φ2i (x1,l(2), . . . , xp,l(2), xp+1, . . . , xq) + gi,

with gi ∈ mlT2,l

, and thus proving (2) of the conclusions of the Theorem.

By induction, there exists Hi ∈ ml(1)

T1,l(1)such that

xi = φ1i (x1,l(1)(1), x2,l(1)(1), . . . , xj,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq) +Hi

for 1 ≤ i ≤ p.

29

Thus

xi = φ1i (x1,l(2)a11 . . . xs,l(2)a1s(xj,l(2) + α2)

a1s+1 , . . . , x1,l(2)as1 . . . xs,l(2)ass

(xj,l(2) + α2)ass+1 , xs+1,l(1)(1), . . . , xj−1,l(1)(1), x1,l(2)as+11 . . . xs,l(2)as+1s

(xj,l(2) + α2)as+1s+1 + λl, xj+1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq) +Hi

= φ1i (x1,l(2)a11 . . . xs,l(2)a1s(xj,l(2) + α2)

a1s+1 , . . . , x1,l(2)as1 . . . xs,l(2)ass


(xj,l(2) + α2)as+1s+1 + λ(x1,l(2)a11 . . . xs,l(2)a1s(xj,l(2) + α2)

a1s+1 ,. . . , x1,l(2)as1 . . . xs,l(2)ass(xj,l(2) + α2)

ass+1 , xs+1,l(1)(1), . . . , xj−1,l(1)(1),xj+1,l(1)(1), . . . , xp,l(1)(1)) + h′, xj+1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq) +Hi.

Where h′ ∈ mn0

T1,l(1)thus h′ ∈ mn0

T2,land thus h′ ∈ ml

T2,lsince n0 > l,

and similarly Hi ∈ mlT2,l

since l(1) > l.

Thus the expression of xi becomes:

xi = φ1i (x1,l(2)a11 . . . xs,l(2)a1s(xj,l(2) + α2)

a1s+1 , . . . , x1,l(2)as1 . . . xs,l(2)ass



a1s+1 , . . . ,x1,l(2)as1 . . . xs,l(2)ass(xj,l(2) + α2)

ass+1 , xs+1,l(1)(1), . . . , xj−1,l(1)(1),xj+1,l(1)(1), . . . , xp,l(1)(1)), xj+1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq) +Hi

+ h′∆1.

For some ∆1 ∈ T2,l.

Let Hi = Hi + h′∆1.

Then Hi ∈ mlT2,l

.

On the other hand:

ψi(x1,l(2), . . . , xs,l(2), xs+1,l(1)(1), . . . , xj−1,l(1)(1), xj,l(1)(2), . . . , xp,l(1)(1), xp+1, . . . ,

xq) = φ1i (x1,l(2)a11 . . . xs,l(2)a1s(xj,l(2) + α2)

a1s+1 , . . . , x1,l(2)as1 . . . xs,l(2)ass(xj,l(2) +

α2)ass+1 , xs+1,l(1)(1), . . . , xj−1,l(1)(1), x1,l(2)as+11 . . . xs,l(2)as+1s(xj,l(2) + α2)

as+1s+1

+ λ(x1,l(2)a11 . . . xs,l(2)a1s(xj,l(2) + α2)a1s+1 , . . . , x1,l(2)as1 . . . xs,l(2)ass

(xj,l(2) + α2)ass+1 , xs+1,l(1)(1), . . . , xj−1,l(1)(1), xj+1,l(1)(1), . . . , xp,l(1)(1)) + h′,

xj+1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq)

= φ1i (x1,l(2)a11 . . . xs,l(2)a1s(xj,l(2) + α2)

a1s+1 , . . . , x1,l(2)as1 . . . xs,l(2)ass


30


a1s+1 , . . . , x1,l(2)as1 . . .

xs,l(2)ass(xj,l(2) + α2)ass+1 , xs+1,l(1)(1), . . . , xj−1,l(1)(1), xj+1,l(1)(1), . . . , xp,l(1)(1)),

xj+1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq) + h′φi

Where φi ∈ T2,l.

This implies

ψi(x1,l(2), . . . , xs,l(2), xs+1,l(1)(1), . . . , xj−1,l(1)(1), xj,l(2), xj,l(1)(1), . . . , xp,l(1)(1),

xp+1, . . . , xq) = xi − Hi + h′φi.

Let gi = −Hi + h′φi thus gi ∈ mlT2,l

and ψi(x1,l(2), . . . , xs,l(2), xs+1,l(1)(1), . . . ,

xj−1,l(1)(1), xj,l(2), xj,l(1)(1) . . . , xp,l(1)(1), xp+1, . . . , xq) = xi + gi. Now, we have that

xi,l(2) = Ei(x1,l(2), . . . , xs,l(2), xs+1,l(1)(1), . . . , xj−1,l(1)(1), xj,l(2),

xj,l(1)(1) . . . , xp,l(1)(1), xp+1, . . . , xq)− hi for 1 ≤ i ≤ p.

Thus xi,l(2) + hi = Ei(x1,l(2), x2,l(1)(1), . . . , xj,l(2), . . . , xp,l(1)(1), xp+1, . . . , xq) for

1 ≤ i ≤ p.

This implies that

x1,l(2) = ∆1(x1,l(2) + h1, x2,l(2) + h2, . . . , xp,l(2) + hp, xp+1, . . . , xq)

...

xs,l(2) = ∆s(x1,l(2) + h1, x2,l(2) + h2, . . . , xp,l(2) + hp, xp+1, . . . , xq)

xs+1,l(1)(1) = ∆s+1(x1,l(2) + h1, x2,l(2) + h2, . . . , xp,l(2) + hp, xp+1, . . . , xq)

...

xj−l,l(1)(1) = ∆j−1(x1,l(2) + h1, x2,l(2) + h2, . . . , xp,l(2) + hp, xp+1, . . . , xq)

xj,l(2) = ∆j(x1,l(2) + h1, x2,l(2) + h2, . . . , xp,l(2) + hp, xp+1, . . . , xq)

xj+1,l(1)(1) = ∆j+1(x1,l(2) + h1, x2,l(2) + h2, . . . , xp,l(2) + hp, xp+1, . . . , xq)

31

...

xp,l(1)(1) = ∆p(x1,l(2) + h1, x2,l(2) + h2, . . . , xp,l(2) + hp, xp+1, . . . , xq)

Thus

x1,l(2) = ∆1(x1,l(2), x2,l(2), . . . , xp,l(2), xp+1, . . . , xq) +H1...xs,l(2) = ∆s(x1,l(2), x2,l(2), . . . , xp,l(2), xp+1, . . . , xq) +Hs

xs+1,l(1)(1) = ∆s+1(x1,l(2), x2,l(2), . . . , xp,l(2), xp+1, . . . , xq) +Hs+1...xj−l,l(1)(1) = ∆j−1(x1,l(2), x2,l(2), . . . , xp,l(2), xp+1, . . . , xq) +Hj−1

xj,l(2) = ∆j(x1,l(2), x2,l(2), . . . , xp,l(2), xp+1, . . . , xq) +Hj

xj+1,l(1)(1) = ∆j+1(x1,l(2), x2,l(2), . . . , xp,l(2), xp+1, . . . , xq) +Hj+1...xp,l(1)(1) = ∆p(x1,l(2), x2,l(2), . . . , xp,l(2), xp+1, . . . , xq) +Hp.

Where H i ∈ mlT2,l

for 1 ≤ i ≤ p.

Thus ψi(x1,l(2), . . . , xs,l(2), xs+1,l(1)(1), . . . , xj,l(2), . . . , xp,l(1)(1), xp+1, . . . , xq)= xi + gi,

becomes ψi(∆1 +H1, . . . ,∆p +Hp, xp+1, . . . , xq)

= xi + gi(∆1 +H1, . . . ,∆p +Hp, xp+1, . . . , xq).

This implies

xi = ψi(∆1, . . . ,∆p, xp+1, . . . , xq) +Gi where Gi ∈ mlT2,l

= φ2i (x1,l(2), . . . , xs,l(2), . . . , xp,l(2), xp+1, . . . , xq) +Gi.

Thus this proves (2) of the conclusions of the theorem.

Now we proceed to prove conclusion (3) of the theorem:

We have S1 : x1,l(2), . . . , xs,l(2)xs+1,l(1)(1), . . . , xj−1,l(1)(1), xj,l(2),

32

xj+1,l(1)(1), . . . , xp,l(1)(1), xp+1, . . . , xq a system of regular parameters of T2,l where

x1,l(2) = x1,l(1)(1)b11 . . . xs,l(1)(1)b1s [xj,l(1)(1)′]b1s+1

...xs,l(2) = x1,l(1)(1)bs1 . . . xs,l(1)(1)bss [xj,l(1)(1)′]bss+1

xj,l(2) = x1,l(1)(1)bs+11 . . . xs,l(1)(1)bs+1s [xj,l(1)(1)′]bs+1s+1

and S1 : x1(2), . . . , xs(2), xs+1(1), . . . , xj−1(1), xj(2), xj+1(1), . . . , xp(1), xp+1,

. . . , xq a system of regular parameters of U2 where

x1(2) = x1(1)b11 . . . xs(1)b1s [xj(1)′]b1s+1

...xs(2) = x1(1)bs1 . . . xs(1)bss [xj(1)′]bss+1

xj(2) = x1(1)bs+11 . . . xs(1)bs+1s [xj(1)′]bs+1s+1

where xj(2) = xj(2) + α2.

By induction we have that ν(xi,l(1)(1)) = ν(xi(1)) < l0(1).

Notice that ν(xi(2)) =∑s

k=1 bkiν(xi(1)) <∑s

k=1 bkil0(1) < l0(2) for i = 1, . . . , s.

Thus ν(xi(2)) = ν(xi,l(2)) < l0(2) for i = 1, . . . , s.

xj,l(2) = xj,l(2)− α2

= x1,l(1)(1)bs+11 . . . xs,l(1)(1)bs+1s [xj,l(1)(1)′]bs+1s+1 − α2

= (x1(1) + g1,l(1))bs+11 . . . (xs(1) + gs,l(1))

bs+1s(xj(1)′ + g′j,l(1))bs+1s+1 − α2

= x1(1)bs+11 [1 +g1,l(1)x1(1)

]bs+11 . . . xs(1)bs+1s [1 +gs,l(1)xs(1)

]bs+1s(xj(1)′ + g′j,l(1))bs+1s+1

− α2

= x1(1)bs+11 . . . xs(1)bs+1s [1 +g1,l(1)x1(1)

]bs+11 . . . [1 +gs,l(1)xs(1)

]bs+1s(xj(1)′+

g′j,l(1))bs+1s+1 − α2

= x1(1)bs+11 . . . xs(1)bs+1s [1 +g1,l(1)x1(1)

Ω1] . . . [1 +gs,l(1)xs(1)

Ωs](xj(1)′ + g′j,l(1))bs+1s+1

− α2

= x1(1)bs+11 . . . xs(1)bs+1s [1 +g1,l(1)x1(1)

Ω′1 + · · ·+ gs,l(1)xs(1)

Ω′s](xj(1)′ + g′j,l(1))bs+1s+1

− α2

= x1(1)bs+11 . . . xs(1)bs+1s(xj(1)′ + g′j,l(1))bs+1s+1 − α2+

x1(1)bs+11 . . . xs(1)bs+1s [g1,l(1)x1(1)

Ω′1 + · · ·+ gs,l(1)xs(1)

Ω′s]

= xj(2) + x1(1)bs+11 . . . xs(1)bs+1s [g1,l(1)x1(1)

Ω′1 + · · ·+ gs,l(1)xs(1)

Ω′s]

33

where Ωi,Ω′i ∈ Z[

g1,l(1)x1(1)

, . . . ,gs,l(1)xs(1)

], for i = 1, . . . , s, are of value greater or equal

to zero, by choice of l(1)(≥ ξ1). Now ν(x1(1)bs+11 . . . xi(1)bs+1i . . . xs(1)bs+1sgi,l(1)xi(1)

Ω′1)

= ν(x1(1)bs+11 . . . xi(1)bs+1i−1 . . . xs(1)bs+1sgi,l(1)Ω′1) > l > ν(xj(2)) by choice of l(1),

for i = 1, . . . , s.

Thus

ν(xj,l(2)) = ν(xj(2)) < l0(2).

[Moreover, we also conclude that ν(xj,l(2)− xj(2)) > l.]

And by induction we have:

• ν(xi(1)) = ν(xi,l(1)(1)) < l0(1) <∞ for s+ 1 ≤ i ≤ p1, σ(i) 6= j.

• ν(xi,l(1)) > l and ν(xi(2)) =∞ for p2 < i ≤ p.

• ν(xi) =∞ for p < i ≤ q.

This implies that ν(mT2,l) = ν(mU2).

This also implies that ν(mT2,l) = ν(mU2).

Now for s+ 1 ≤ i ≤ p2 and σ(i) 6= j, we have xi(2) = Ei(x1(2), . . . ,

xs(2), xs+1(1), . . . , xj(2), . . . , xp(1), xp+1, . . . , xq) = xσ(i)(1)

and we have defined the regular parameters x1,l(2), . . . , xp,l(2), xp+1, . . . , xq in T2,l

by

xi,l(2) =Ei(x1,l(2), . . . , xs,l(2), xs+1,l(1)(1), . . . , xj,l(2)− α2, . . . , xp,l(1)(1),

xp+1, . . . , xq)− hi.

Thus xi,l(2) = xσ(i),l(1)(1) − hi where hi ∈ mn1

T2,land n1v(mT2,l

) = n1v(mU2) >

l0(2) > l0(1) > ν(xσ(i),l(1)(1)) = ν(xσ(i)(1)) since σ(i) ∈ s+ 1, . . . , p1 for

34

i ∈ s+ 1, . . . , p2.

This implies ν(hi) ≥ n1v(mT2,l) > ν(xσ(i),l(1)(1)),

thus ν(xi,l(2)) = ν(xσ(i),l(1)(1) − hi) = ν(xσ(i),l(1)(1)) = ν(xσ(i)(1)) = ν(xi(2)) <

l0(2).

Now if σ(i) = j then

xi(2) = xj(2)

= xj(2)− α2

= x1(1)bs+11 . . . xs(1)bs+1s [xj(1)′]bs+1s+1 − α2

thus

xi,l(2) = x1,l(1)(1)bs+11 . . . xs,l(1)(1)bs+1s [xj,l(1)(1)′]bs+1s+1 − α2 − hi= xj,l(2)− α2 − hi

where ν(hi) > ν(xj(2)) by the choice of n1, thus ν(hi) > ν(xj(2)) = ν(xj,l(2)−α2).

Thus, ν(xi,l(2)) = ν(xj,l(2)− α2 − hi) = ν(xj,l(2)− α2) = ν(xj(2))

= ν(xi(2)) < l0(2) and thus (3) of the conclusions of the

theorem is proved.

To prove (4) and (5) we first compare the values of the regular parameters of

S1 and S1.

We already showed that ν(xj,l(2) − xj(2)) > l and we get by induction that

ν(xi,l(1)(1)− xi(1)) > l(1) > l for 1 ≤ i ≤ p and i 6= j.

35

Now, for 1 ≤ i ≤ s

xi,l(2) = x1,l(1)(1)bi1 . . . xs,l(1)(1)bisx′j,l(1)(1)bis+1

= (x1(1) + g1,l(1))bi1 . . . (xs(1) + gs,l(1))

bis(x′j(1) + g′j,l(1))bis+1

= x1(1)bi1(1 +g1,l(1)x1(1)

)bi1 . . . xs(1)bis(1 +gs,l(1)xs(1)

)bisx′j(1)bis+1(1 +g′j,l(1)

x′j(1))bis+1

= x1(1)bi1 . . . xs(1)bisx′j(1)bis+1(1 +g1,l(1)x1(1)

Ω1i) . . . (1 +gs,l(1)xs(1)

Ωsi)(1 +g′j,l(1)

x′j(1)

Ωs+1i)

= x1(1)bi1 . . . xs(1)bisx′j(1)bis+1(1 +g1,l(1)x1(1)

Ω′1i + · · ·+ gs,l(1)xs(1)

Ω′si +g′j,l(1)

x′j(1)Ω′s+1i)

= x1(1)bi1 . . . xs(1)bisx′j(1)bis+1 + x1(1)bi1−1 . . . xs(1)bisx′j(1)bis+1g1,l(1)Ω′1i

+ · · ·+ x1(1)bi1 . . . xs(1)bisx′j(1)bis+1−1g′j,l(1)Ω′s+1i

= xi(2) + x1(1)bi1−1 . . . xs(1)bisx′j(1)bis+1g1,l(1)Ω′1i

+ · · ·+ x1(1)bi1 . . . xs(1)bisx′j(1)bis+1−1g′j,l(1)Ω′s+1i

where Ωki,Ω′ki ∈ Z [

g1,l(1)x1(1)

, . . . ,gs,l(1)xs(1)

,g′j,l(1)

x′j(1)] of value greater or equal to zero, by

choice of l(1), for 1 ≤ k ≤ s + 1 and thus ν(xi,l(2) − xi(2)) > l, by choice of

l(1)(≥ ξ2 and ξ3).

Let si and si be the i-th regular parameter in S1 and S1 respectively, in the

given order, for 1 ≤ i ≤ p and let di = si − si for 1 l.

For all 1 ≤ i ≤ p we have

xi(2)− xi,l(2) = Ei(s1, s2, . . . , sj, . . . , sp, xp+1, . . . , xq)−Ei(s1, s2, . . . , sj, . . . , sp, xp+1, . . . , xq) + hi.

We write Ei(t1, . . . , tq) ∈ k2[[t1, . . . , tq]] as a sum of it’s monomials:

Ei =∑n

Mn

where Mn = antn11 . . . t

nqq with an ∈ k2 and n1, . . . , nq are non-negative integers.

Let Mn(2) = Mn(s1, s2, . . . , sj, . . . , sp, xp+1, . . . , xq) then

xi(2) =∑n

Mn(2)

36

Let Mn,l(2) = Mn(s1, s2, . . . , sj, . . . , sp, xp+1, . . . , xq) then

xi,l(2) =∑n

Mn,l(2)− hi.

This implies xi(2)− xi,l(2) =∑

n[Mn(2)−Mn,l(2)] + hi, where by the choice of

n1ν(hi) > l.

Now,

Mn(2)−Mn,l(2) = an[sn11 s

n22 . . . s

njj . . . s

npp x

sp+1

p+1 . . . xsqq − sn1

1 sn22 . . . s

njj

. . . snpp x

sp+1

p+1 . . . xsqq ]

= an[(s1 + d1)n1(s2 + d2)

n2 . . . (sj + dj)nj . . . (sp + dp)

np

xsp+1

p+1 . . . xsqq − sn1

1 sn22 . . . s

njj . . . s

npp x

sp+1

p+1 . . . xsqq ]

= an[sn11 s

n22 . . . s

njj . . . s

npp x

sp+1

p+1 . . . xsqq + d1ξ1 + · · ·+ dpξp

− sn11 s

n22 . . . s

njj . . . s

npp x

sp+1

p+1 . . . xsqq ],

where ξi ∈ Z [s1, s2, . . . , sj, . . . , sp, xp+1, . . . , xq, d1, . . . , dp], thus ν(ξi) ≥ 0,

and thus Mn(2)−Mn,l(2) = an[d1ξ1 + · · ·+ dpξp] so ν(Mn(2)−Mn,l(2)) > l

and thus ν(xi,l(2)− xi(2)) > l, for 1 ≤ i ≤ p and this proves (5) of the conclusions

of the theorem.

Moreover, for p2 l and thus

proving (4).

Theorem 4.3.2. Suppose that x1, . . . , xs, . . . , xt, xt+1, . . . , xq is a regular system of

parameters in T such that ν(x1), . . . , ν(xs) are rationally independent of finite value,

t ≥ s and QT ∩ T/mT [[x1, . . . , xt]] = (0). Suppose that 0 6= f ∈ T/mT [[x1, . . . , xt]].

Then there exists a sequence of etale Perron transforms

T → U → U1 → · · · → Un

37

along ν such that Un has a system of regular parameters x1(n) . . . , xt(n), xt+1,

. . . , xq such that ν(x1(n)), . . . , ν(xs(n)) are rationally independent and one of the

following two cases holds:

1. f = x1(n)a1 . . . xs(n)asγ(x1(n), . . . , xs(n), xt(n)),

where γ ∈ Un/mUn [[x1(n), . . . , xs(n), xt(n)]] is a unit series.

or

2. QUn∩ Un/mUn [[x1(n), . . . , xt(n)]] 6= (0).

Proof. We first consider the case where t = s.

f ∈ T/mT [[x1, . . . , xs]] and we write f =∑

i≥1 ai1...isxi11 . . . x

iss where ai1...is ∈

T/mT .

Notice that if ai1...is and aj1...js 6= 0, and ν(ai1...isxbi11 . . . x

biss ) = ν(aj1...jsx

bj11 . . . xb

jss )

then bi1 = bj1, . . . , bis = bjs since ν(x1(1)), . . . , ν(xs(1)) are rationally independent.

Thus we may assume that ai1...isxi11 . . . x

iss is of minimum value (ai1...is 6= 0), and

that ν(ai1...isxi11 . . . x

iss ) > ν(ai1...isx

i11 . . . x

iss ) for i 6= i.

Let I =⟨ai1...isx

i11 . . . x

iss , i ≥ 1

⟩⊆ T/mT [[x1, . . . , xs]].

Then there exists a finite set S ⊂ N such that I =⟨ai1...isx

i11 . . . x

iss , i ∈ S

⟩and we necessarily have that i ∈ S.

Thus f = xi11 . . . xiss γ +

∑i∈S−i x

i11 . . . x

iss f′.

Where f ′ ∈ T/mT [[x1, . . . , xs]] and necessarily γ ∈ T/mT is a unit.

38

moreover we still have that ν(xi11 . . . xiss ) < ν(xi11 . . . x

iss ) for i ∈ S − i.

Let T → U → U1 be a sequence of etale Perron transforms of type I along ν

such that U1 has regular parameters x1(1), . . . , xs(1), xs+1, . . . , xq where

x1 = x1(1)a11 . . . xs(1)a1s

...xs = x1(1)as1 . . . xs(1)ass

and ν(x1(1)), . . . , ν(xs(1)) are rationally independent.

Thus we have f = x1(1)bi1 . . . xs(1)b

isγ+

∑i∈S−i x1(1)b

i1 . . . xs(1)b

isf ′(x1(1) . . . xs(1))

and ν(x1(1)bi1 . . . xs(1)b

is) < ν(x1(1)b

i1 . . . xs(1)b

is).

Then by Lemma 4.2 [16] there exists a finite sequence of etale Perron transforms of

type I along ν U1 → U2 such that U2 has regular parameters x1(2), . . . , xs(2), xs+1,

. . . , xq such that

f = x1(2)a1 . . . xs(2)asγ

where γ = γ +∑

i∈S−i x1(2)di1 . . . xs(2)d

isf ′(x1(2), . . . , xs(2)) is a unit in

U2/mU2 [[x1(2), . . . , xs(2)]] since γ 6= 0 and dil 6= 0 for some 1 ≤ l ≤ s.

Now consider the case t > s.

Assume by induction that the theorem is true for t− 1. We will prove it for t.

Let r = order(f(0, . . . , 0, xt)). We have 0 ≤ r ≤ ∞. If r = 0, then 1 of the

conclusions of the theorem hold in T/mT [[x1, . . . , xt]]. Assume that r ≥ 1. We

39

have an expression

f =m∑i=1

ai(x1, . . . , xt−1)xdit +

∑m<i≤n

ai(x1, . . . , xt−1)xdit +

∑n ρ if i > n, and dn+1 < di if i > n+1 . We necessarily

have that ν(xrt ) ≥ ρ.

Thus

di ≤ r for 1 ≤ i ≤ m. (4.8)

By induction, applied to ai(x1, . . . , xt−1), for 1 ≤ i ≤ n, we either construct a

sequence of etale Perron transforms along ν satisfying 2 of the conclusions of the

theorem, or we have a sequence of etale Perron transforms T → U → U1 along ν

such that U1 has regular parameters x1(1), . . . , xt−1(1), xt, . . . , xq such that there

exist unit series

ai(x1(1), . . . , xt−1(1)) ∈ U1/mU1 [[x1(1), . . . , xt−1(1)]],

and cji ∈ N satisfying

ai(x1, . . . , xt−1) = x1(1)c1i . . . xs(1)c

siai(x1(1), . . . , xt−1(1))

for 1 ≤ i ≤ n, and there exist series

bi(x1(1), . . . , xt−1(1)) ∈ U1/mU1 [[x1(1), . . . , xt−1(1)]],

40

satisfying

ai(x1, . . . , xt−1) = bi(x1(1), . . . , xt−1(1))

for n < i. Thus we have

f =n∑i=1

aix1(1)c1i . . . xs(1)c

sixdit +

∑n<i

bixdit . (4.9)

We further may assume that

QU1∩ U1/mU1 [[x1(1), . . . , xt−1(1), xt]] = (0).

Now we define an etale Perron transform of type IIt U1 → U2 by

x1(1) = x1(2)a11 . . . xs(2)a1sxt(2)a1,s+1

...xs(1) = x1(2)as1 . . . xs(2)assxt(2)as,s+1

xt = x1(2)as+1,1 . . . xs(2)as+1,sxt(2)as+1,s+1

(4.10)

We have that ν(x1(2)), . . . , ν(xs(2)) are rationally independent and ν(xt(2)) =

0. Let α be the residue of xt(2) in U2/mU2 . Let xt(2) = xt(2)− α. Then

x1(2), . . . , xs(2), xs+1(1), . . . , xt−1(1), xt(2), xt+1, . . . , xq

are regular parameters in U2. We may assume that

QU2∩ U2/mU2 [[x1(2), . . . , xs(2), xs+1(1), . . . , xt−1(1), xt(2)]] = (0),

since otherwise we have achieved 2. of the conclusions of the theorem.

We will now show that the following formula holds

f =m∑i=1

aix1(2)bi1 . . . xs(2)b

is(xt(2) + α)b

is+1+ (4.11)

41

n∑i=m+1

aix1(2)bi1 . . . xs(2)b

is(xt(2) + α)b

is+1 + (x1(2)as+1,1 . . . xs(2)as+1,s)dn+1Λ

with Λ ∈ U2/mU2 [[x1(2), . . . , xs(2), xt(2)]],

bij = c1i a1j + · · ·+ csiasj + dias+1,j for 1 ≤ i ≤ m and 1 ≤ j ≤ s+ 1,

bi1 = b11, . . . , bis = b1s, for 1 ≤ i ≤ m,

ν(x1(2)bi1 . . . xs(2)b

is) > ρ for m ρ.

To establish (4.11), we first recall that we have

f =n∑i=1

aix1(1)c1i . . . xs(1)c

sixdit +

∑n dn+1 for i > n+ 1 then:

f =n∑i=1

aix1(1)c1i . . . xs(1)c

sixdit + x

dn+1

t

∑n<i

bixdi−dn+1

t .

Substituting (4.10) into f , we have

f =m∑i=1

aix1(2)bi1 . . . xs(2)b

is(xt(2) + α)b

is+1 +

n∑i=m+1

aix1(2)bi1 . . . xs(2)b

is

(xt(2) + α)bis+1 + (x1(2)as+1,1 . . . xs(2)as+1,s)dn+1

∑n<i

bix1(2)as+1,1(di−dn+1) . . .

xs(2)as+1,s(di−dn+1)(xt(2) + α)as+1,s+1di ,

where bij = c1i a1j + · · ·+ csiasj + dias+1,j for 1 ≤ i ≤ m and 1 ≤ j ≤ s+ 1.

We have that:

ρ = ν(a1x1(2)b11 . . . xs(2)b

1s(xt(2) + α)b

1s+1) = ν(aix1(2)b

i1 . . . xs(2)b

is(xt(2) + α)b

is+1)

for 1 ≤ i ≤ m.

Thus ν(x1(2))b11 + · · ·+ ν(xs(2))b1s = ν(x1(2))bi1 + · · ·+ ν(xs(2))bis.

Hence ν(x1(2))(b11 − bi1) + · · ·+ ν(xs(2))(b1s − bis) = 0,

42

and thus b11 = bi1, . . . , b1s = bis since ν(x1(2)), . . . , ν(xs(2)) are rationally indepen-

dent.

Moreover ν(aix1(2)bi1 . . . xs(2)b

is(xt(2) + α)b

is+1) > ρ for m ρ then ν((x1(2)as+1,1 . . . xs(2)as+1,s)dn+1) > ρ.

Let Λ =∑

n<i bix1(2)as+1,1(di−dn+1) . . . xs(2)as+1,s(di−dn+1)(xt(2) + α)as+1,s+1di , then f

has the desired form (4.11).

Now we will show that there exists an etale Perron transform of type I:

x1(2) = x1(3)a′11 . . . xs(3)a

′1s

...xs(2) = x1(3)a

′s1 . . . xs(3)a

′ss

(4.12)

such that (x1(2)b11 . . . xs(2)b

1s)x1(3) . . . xs(3) divides x1(2)b

i1 . . . xs(2)b

is for m <

i ≤ n and (x1(2)b11 . . . xs(2)b

1s)x1(3) . . . xs(3) divides (x1(2)as+1,1 . . . xs(2)as+1,s)dn+1

in U2/mU2 [[x1(3), . . . , xs(3), xt(2)]].

To establish (4.12), we first observe that ν(x1(2)bi1 . . . xs(2)b

is) =

ν(aix1(2)bi1 . . . xs(2)b

is(xt(2) + α)b

is+1) > ρ = ν(x1(2)b

11 . . . xs(2)b

1s)

for m ρ = ν(x1(2)b11 . . . xs(2)b

1s).

Then by Lemma 4.2 of [16] there exists a finite sequence of etale Perron transform

along ν of type I, U2 → U3 such that U3 has regular parameters x1(3), . . . , xs(3),

xs+1(2), . . . , xt(2), xt+1, . . . , xq defined by

x1(2) = x1(3)a′11 . . . xs(3)a

′1s

...xs(2) = x1(3)a

′s1 . . . xs(3)a

′ss

(4.13)

such that x1(2)b11 . . . xs(2)b

1s divides x1(2)b

i1 . . . xs(2)b

is for m < i ≤ n, and

x1(2)b11 . . . xs(2)b

1s divides (x1(2)as+1,1 . . . xs(2)as+1,s)dn+1 in

43

U2/mU2 [[x1(3), . . . , xs(3), xt(2)]].

We have x1(2)b11 . . . xs(2)b

1s = x1(3)b

′11 . . . xs(3)b

′1s and

x1(2)bi1 . . . xs(2)b

is = x1(3)b

′i1 . . . xs(3)b

′is

where b′ji are natural numbers, and since ρ is the minimal value, then for every

i > m, there exists j, with 1 ≤ j ≤ s such that b′ij − b′1j > 0.

Since the exponents a′ij in an etale Perron transform of type I (4.12) are all

positive integers, after possibly performing a finite etale Perron transform of type

I (4.12), we obtain that b′ij − b′1j > 0 for all j with 1 ≤ j ≤ s and i > m.

Define Aij ∈ N by

a11 ... a1,s+1

... ... ...as+1,1 ... as+1,s+1

a′11 ... a′1s 0... ... ...a′s1 ... a′ss 00 ... 0 1

=

A11 ... A1,s+1

... ... ...As+1,1 ... As+1,s+1

.

After making the substitution (4.10) in (4.12) we get

x1(1) = x1(3)a′11a11+···+a′s1a1s . . . xs(3)a

′1sa11+···+a′ssa1sxt(2)a1,s+1

...xs(1) = x1(3)a

′11as1+···+a′s1ass . . . xs(3)a

′1sas1+···+a′ssassxt(2)as,s+1

xt = x1(3)a′11as+1,1+···+a′s1as+1,s . . . xs(3)a

′1sas+1,1+···+a′ssas+1,sxt(2)as+1,s+1 .

Thus by the definition of Aij:

x1(1) = x1(3)A11 . . . xs(3)A1sxt(2)A1,s+1

...xs(1) = x1(3)As1 . . . xs(3)Assxt(2)As,s+1

xt = x1(3)As+1,1 . . . xs(3)As+1,sxt(2)As+1,s+1 .

(4.14)

Now we show that we have an expression

f =m∑i=1

aix1(3)Bi1 . . . xs(3)B

is(xt(2) + α)B

is+1 + x1(3)B

11+1 . . . xs(3)B

1s+1Ω (4.15)

44

with Ω ∈ U2/mU2 [[x1(3), . . . , xs(3), xt(2)]],

Bij = c1iA1j + · · ·+ csiAsj + diAs+1,j for 1 ≤ i ≤ m and 1 ≤ j ≤ s+ 1,

Bi1 = B1

1 , . . . , Bis = B1

s , for 1 ≤ i ≤ m.

To establish (4.15) let Bij = c1iA1j + · · · + csiAsj + diAs+1,j for 1 ≤ i ≤ m and

1 ≤ j ≤ s+ 1,

then

(Bi1 . . . B

is) = (bi1 . . . b

is)

a′11 ... a′1s... ... ...a′s1 ... a′ss

and making the substitution (4.14) into (4.11), we have:

f =m∑i=1

aix1(3)Bi1 . . . xs(3)B

is(xt(2) + α)B

is+1 +

n∑i=m+1

aix1(3)Bi1 . . . xs(3)B

is

(xt(2) + α)Bis+1 + (x1(3)As+1,1 . . . xs(3)As+1,s+1)dn+1Ω′.

Where Ω′ = Λ(x1(3) . . . xs(3)) ∈ U2/mU2 [[x1(3), . . . , xs(3), xt(2)]],

and

f =m∑i=1

aix1(3)Bi1 . . . xs(3)B

is(xt(2) + α)B

is+1 + x1(3)B

11+1 . . . xs(3)B

1s+1Ω

.

Moreover, since ν(x1(3)), . . . , ν(xs(3)) are rationally independent, then Bij = B1

j ,

for 1 ≤ i ≤ m, and 1 ≤ j ≤ s.

By the above analysis, we see that we may thus replace if necessary (aij) with (Aij),

and we have

f =m∑i=1

aix1(2)bi1 . . . xs(2)b

is(xt(2) + α)b

is+1 + x1(2)b

11+1 . . . xs(2)b

1s+1Ω

45

, where bij = c1i a1j + · · ·+ csiasj + dias+1j for 1 ≤ i ≤ m and 1 ≤ j ≤ s+ 1,

and b11 = bi1, . . . , b1s = bis for 1 ≤ i ≤ m.

Define

f = x1(2)b11 . . . xs(2)b

1s(xt(2) + α)b

1s+1f1,

where

f1 =m∑i=1

ai(xt(2) + α)bis+1−b1s+1 + x1(2) . . . xs(2)Ω

. Now b11 = bi1, . . . , b1s = bis for 1 ≤ i ≤ m which gives the following

a11 ... as+1,1

... ... ...a1s ... as+1,s

a1,s+1 ... as+1,s+1

c1i − c11...

csi − cs1di − d1

=

bi1 − b11......

bis+1 − b1s+1

=

0...0

bis+1 − b1s+1

.

By Cramer’s rule

di − d1 =

Det

a11 ... as1 0... ... ... 0

a1,s+1 ... as,s+1 bis+1 − b1s+1

Det(aij)

=

(bis+1−b1s+1)Det

a11 ... as1... ... ...a1s ... ass

Det(aij)

.

This implies

bis+1 − b1s+1 =Det(aij)(di − d1)

Det

a11 ... as1... ... ...a1s ... ass

=di − d1

afor 1 ≤ i ≤ m,

where a = ±Det

a11 ... as1... ... ...a1s ... ass

∈ Z.

Moreover a|di − d1 for 1 ≤ i ≤ m.

46

From now on, we assume a > 0. The case a < 0 is similar.

Let ei = di−d1a

.

Then

f1 =m∑i=1

ai(xt(2) + α)di−d1a + x1(2) . . . xs(2)Ω.

Let r1 = ordf1(0, . . . , 0, xt(2)) = ord∑m

i=1 ai(xt(2)+α)ei where ai = ai(0, . . . , 0)

∈ U2/mU2 .

Then r1 ≤ dm−d1a≤ r since dm ≤ r by (4.8).

Assume that r1 = r.

Then d1 = 0, dm = r and a = 1, so that di = ei for 1 ≤ i ≤ m.

ord f(0, . . . , 0, xt) = r and dm = r implies am is a unit in T/mT [[x1, . . . , xt−1]] and

thus am = am.

So c1m = · · · = csm = 0.

Define

ξ(t) = f1(0, . . . , 0, t− α) =m∑i=1

aitei ,

ordξ(t+ α) = ordf1(0, . . . , 0, t) = r = degξ(t+ α) since em = r.

Thus ξ(t+ α) = amtr.

Now

amtr =ξ(t+ α) =

m∑i=1

ai(t+ α)ei

=am(t+ α)r +m−1∑i=1

ai(t+ α)ei

=amtr + rαamt

r−1 + am−1tem−1 + terms of order < r − 1.

47

Thus em−1 = r − 1.

And moreover

rαam + am−1 = 0 ∈ U2/mU2 . (4.16)

Thus

em = dm = r and em−1 = dm−1 = r − 1, (4.17)

since d1 = 0 and a = 1.

Now, we compute using (4.16) and (4.17)

amxtam−1

=amxrt

am−1xr−1t

=amx

dmt

am−1xdm−1t

= amx1(2)bm1 ...xs(2)b

ms (xt(2)+α)

bms+1

am−1x1(2)bm−11 ...xs(2)b

m−1s (xt(2)+α)

bm−1s+1

= am(xt(2)+α)bms+1−b

m−1s+1

am−1.

We have

bms+1 − bm−1s+1 = (bms+1 − b1s+1)− (bm−1

s+1 − b1s+1)= em − em−1 = r − (r − 1) = 1.

Then xtam−1

= xt(2)+αam−1

.

Taking the residue λ′

of xtam−1

in U2/mU2 , using (4.16) we have that

λ′=

[xtam−1

]=

α

am−1

= − 1

ram∈ T/mT .

Thus ν(xt − λ′am−1) > ν(xt).

xt − λ′am−1 6∈ QT since xt − λ

′am−1 ∈ T/mT [[x1, . . . , xt]].

Let β = ν(xt − λ′1am−1). There exists γ ∈ N such that ν(mγ

T ) > β and there

exists ϕ1 ∈ T/mT [x1, . . . , xt−1] ⊂ U such that

λ′am−1 − ϕ1 ∈ (x1, . . . , xt−1)

γT/mT [[x1, . . . , xt−1]].

48

Then β = ν(xt − ϕ1).

Let x′t = xt − ϕ1.

Let

f ′(x1, . . . , xt−1, x′t) = f(x1, . . . , xt−1, x

′t + ϕ1) = f(x1, . . . , xt).

ord(f ′(0, . . . , 0, x′t)) = r. We repeat the above construction to either achieve a

reduction r1 < r, or we obtain a new change of variables in U with ν(x′′t ) > ν(x′t)

and ν(xt) < ν(x′t) < ν(x′′t ) ≤ ν(f).

Since we cannot have an infinite sequence of this kind, we eventually either

achieve 2 of the conclusions of the theorem, or find a change of variables in xt in

U , which leads to a reduction r1 < r.

Suppose that r1 < r.

x1(2), . . . , xs(2), xs+1, . . . , xt−1, xt(2), xt+1, . . . , xq

are then a regular system of parameters in U2, and we have an expression

f = x1(2)b11 . . . xs(2)b

1sf1(x1(2), . . . , xs(2), xs+1, . . . , xt−1, xt(2))

in U2, where r1 = ord(f1(0, . . . , 0, xt(2))) < r.

We iterate the above construction to achieve either 2. of the conclusions of the

theorem, or a reduction to r = 0, so that the conclusions of the theorem hold for

t, and by the induction, the conclusion of the theorem hold.

Theorem 4.3.3. Suppose that x1, . . . , xs, . . . , xt, xt+1, . . . , xq is a regular sequence

in T , such that ν(x1), . . . , ν(xs) are rationally independent and

49

QT ∩ T/mT [[x1, . . . , xt+1]] 6= (0) and QT ∩ T/mT [[x1, . . . , xt]] = (0). Then there

exists a sequence of etale Perron transforms

T → U → U1 → · · · → Un

along ν such that Un has regular parameters x1(n), . . . , xt+1(n), xt+2, . . . , xq such

that ν(x1(n)), . . . , ν(xs(n)) are rationally independent and and one of the following

holds:

1. There exists a series φ(x1(n), . . . , xt(n)) ∈ Un/mUn [[x1(n), . . . , xt(n)]] such

that xt+1(n)− φ ∈ QUn, and QUn

∩ Un/mUn [[x1(n), . . . , xt(n)]] = (0).

2. QUn∩ Un/mUn [[x1(n), . . . , xt(n)]] 6= (0).

Proof. With our assumptions, there exists 0 6= f ∈ QT ∩ T/mT [[x1, . . . , xt+1]]. We

have that ord(f(x1, . . . , xt+1)) ≥ 1.

Let r = order(f(0, . . . , 0, xt+1)). If r = 1, then 1. of the conclusions of

the theorem holds in T , by the Weierstrass preparation theorem. We may thus

assume that 2 ≤ r ≤ ∞. Since ν(x1), . . . , ν(xs) are rationally independent,

QT ∩ T/mT [[x1, . . . , xs]] = (0) so t ≥ s.

We have an expression

f =m∑i=1

ai(x1, . . . , xt)xdit+1 +

∑m<i≤n

ai(x1, . . . , xt)xdit+1 +

∑n<i

ai(x1, . . . , xt)xdit+1

where ai ∈ T/mT [[x1, . . . , xt]] and the first sum is over the terms aixdit+1 of

minimal value ρ,

d1 < d2 < · · · < dm

50

and we have that diν(xt+1) > ρ if i > n and dn+1 > di if i > n. By our

assumption, ν(ai) <∞ for all i. We have that ν(xrt+1) ≥ ρ. Thus

di ≤ r for 1 ≤ i ≤ m. (4.18)

Then by Theorem 4.3.2, we have a sequence of etale Perron transforms T →

U → U1 along ν such that U1 has regular parameters x1(1), . . . , xt(1), xt+1, . . . , xq

and such that we have either achieved 2. of the conclusions of the theorem, or

there exist unit series

ai(x1(1), . . . , xt(1)) ∈ U1/mU1 [[x1(1), . . . , xt(1)]],

and ci ∈ N satisfying

ai(x1, . . . , xt) = x1(1)c1i . . . xs(1)c

siai(x1(1), . . . , xt(1))

for 1 ≤ i ≤ n, and there exist series

bi(x1(1), . . . , xt(1)) ∈ U1/mU1 [[x1(1), . . . , xt(1)]],

satisfying

ai(x1, . . . , xt) = bi(x1(1), . . . , xt(1))

for n < i. Thus we have

f =n∑i=1

aix1(1)c1i . . . xs(1)c

sixdit+1 +

∑n<i

bixdit+1 (4.19)

We further assume that

QU1∩ U1/mU1 [[x1(1), . . . , xt(1)]] = (0).

51

Now we define an etale Perron transform of type IIt+1 U1 → U2 by

x1(1) = x1(2)a11 . . . xs(2)a1sxt+1(2)a1,s+1

...xs(1) = x1(2)as1 . . . xs(2)assxt+1(2)as,s+1

xt+1 = x1(2)as+1,1 . . . xs(2)as+1,sxt+1(2)as+1,s+1 .

(4.20)

We have that ν(x1(2)), . . . , ν(xs(2)) are rationally independent and

ν(xt+1(2)) = 0.

Let α be the residue of xt+1(2) in U2/mU2 . Let xt+1(2) = xt+1(2)− α. Then

x1(2), . . . , xs(2), xs+1(1), . . . , xt(1), xt+1(2), xt+2, . . . , xq

are regular parameters in U2, and

U2 = U2/mU2 [[x1(2), . . . , xs(2), xs+1(1), . . . , xt(1), xt+1(2), xt+2, . . . , xq]].

We may assume that

QU2∩ U2/mU2 [[x1(2), . . . , xs(2), xs+1(1), . . . , xt(1)]] = (0)

since otherwise we have achieved 2. of the conclusions of the theorem.

We will now show that the following formula holds

f =m∑i=1

aix1(2)bi1 . . . xs(2)b

is(xt+1(2) + α)b

is+1+ (4.21)

n∑i=m+1

aix1(2)bi1 . . . xs(2)b

is(xt+1(2) + α)b

is+1 + (x1(2)as+1,1 . . . xs(2)as+1,s)dn+1Λ

with Λ ∈ U2/mU2 [[x1(2), . . . , xs(2), xt+1(2)]],

bij = c1i a1j + · · ·+ csiasj + dias+1,j for 1 ≤ i ≤ m and 1 ≤ j ≤ s+ 1,

bi1 = b11, . . . , bis = b1s, for 1 ≤ i ≤ m,

52

ν(x1(2)bi1 . . . xs(2)b

is) > ρ for m ρ.

To establish (4.21), we first recall that we have

f =n∑i=1

aix1(1)c1i . . . xs(1)c

sixdit+1 +

∑n dn+1 for i > n+ 1 then:

f =n∑i=1

aix1(1)c1i . . . xs(1)c

sixdit+1 + x

dn+1

t+1

∑n<i

bixdi−dn+1

t+1 .

Substituting (4.19) into f , we have

f =m∑i=1

aix1(2)bi1 . . . xs(2)b

is(xt+1(2) + α)b

is+1 +

n∑i=m+1

aix1(2)bi1 . . . xs(2)b

is

(xt+1(2) + α)bis+1 + (x1(2)as+1,1 . . . xs(2)as+1,s)dn+1

∑n<i

bix1(2)as+1,1(di−dn+1)

. . . xs(2)as+1,s(di−dn+1)(xt+1(2) + α)as+1,s+1di ,

where bij = c1i a1j + · · ·+ csiasj + dias+1,j for 1 ≤ i ≤ m and 1 ≤ j ≤ s+ 1.

We have that

ρ = ν(a1x1(2)b11 . . . xs(2)b

1s(xt+1(2) + α)b

1s+1)

= ν(aix1(2)bi1 . . . xs(2)b

is(xt+1(2) + α)b

is+1) for 1 ≤ i ≤ m,

thus ν(x1(2))b11 + · · ·+ ν(xs(2))b1s = ν(x1(2))bi1 + · · ·+ ν(xs(2))bis,

thus ν(x1(2))(b11 − bi1) + · · ·+ ν(xs(2))(b1s − bis) = 0,

and hence b11 = bi1, . . . , b1s = bis since ν(x1(2)), . . . , ν(xs(2)) are rationally indepen-

dent.

Moreover ν(aix1(2)bi1 . . . xs(2)b

is(xt+1(2) + α)b

is+1) > ρ for m ρ then ν((x1(2)as+1,1 . . . xs(2)as+1,s)dn+1) > ρ.

53

Let

Λ =∑n<i

bix1(2)as+1,1(di−dn+1) . . . xs(2)as+1,s(di−dn+1)(xt+1(2) + α)as+1,s+1di ,

then f has the desired form (4.21).

Now we will show that there exists an etale Perron transform of type I:

x1(2) = x1(3)a′11 . . . xs(3)a

′1s

...xs(2) = x1(3)a

′s1 . . . xs(3)a

′ss

(4.22)

such that (x1(2)b11 . . . xs(2)b

1s)x1(3) . . . xs(3) divides x1(2)b

i1 . . . xs(2)b

is for m < i ≤ n

and (x1(2)b11 . . . xs(2)b

1s)x1(3) . . . xs(3) divides (x1(2)as+1,1 . . . xs(2)as+1,s)dn+1

in U2/mU2 [[x1(3), . . . , xs(3), xt+1(2)]].

To establish (4.22) we first observe that

ν(x1(2)bi1 . . . xs(2)b

is) = ν(aix1(2)b

i1 . . . xs(2)b

is(xt+1(2) + α)b

is+1) > ρ

for m ν(x1(2)b

i1 . . . xs(2)b

is) for

m ρ = ν(x1(2)b11 . . . xs(2)b

1s).

Then by Lemma 4.2 [16] there exists a finite sequence of etale Perron transform

along ν of type I, U2 → U3 such that U3 has regular parameters

x1(3), . . . , xs(3), xs+1(2), . . . , xt+1(2), xt+2 . . . , xq defined by

x1(2) = x1(3)a′11 . . . xs(3)a

′1s

...xs(2) = x1(3)a

′s1 . . . xs(3)a

′ss

(4.23)

such that x1(2)b11 . . . xs(2)b

1s divides x1(2)b

i1 . . . xs(2)b

is for m < i ≤ n and

x1(2)b11 . . . xs(2)b

1s divides (x1(2)as+1,1 . . . xs(2)as+1,s)dn+1 in

54

U2/mU2 [[x1(3), . . . , xs(3), xt+1(2)]].

We have x1(2)b11 . . . xs(2)b

1s = x1(3)b

′11 . . . xs(3)b

′1s and x1(2)b

i1 . . . xs(2)b

is =

x1(3)b′i1 . . . xs(3)b

′is , where b′ji are natural numbers, and since ρ is the minimal value,

then for all i > m there exists j; 1 ≤ j ≤ s such that b′ij − b′1j > 0.

Since the exponents a′ij in an etale Perron transform of type I (4.22) are all positive

integers, after possibly performing a finite etale Perron transform of type I (4.22),

we obtain that b′ij − b′1j > 0 for all j with 1 ≤ j ≤ s and i > m.

Define Aij ∈ N by

a11 ... a1,s+1

... ... ...as+1,1 ... as+1,s+1

a′11 ... a′1s 0... ... ...a′s1 ... a′ss 00 ... 0 1

=

A11 ... A1,s+1

... ... ...As+1,1 ... As+1,s+1

.

After making the substitution (4.19) in (4.22) we get

x1(1) = x1(3)a′11a11+···+a′s1a1s . . . xs(3)a

′1sa11+···+a′ssa1sxt+1(2)a1,s+1

...xs(1) = x1(3)a

′11as1+···+a′s1ass . . . xs(3)a

′1sas1+···+a′ssassxt+1(2)as,s+1

xt = x1(3)a′11as+1,1+···+a′s1as+1,s . . . xs(3)a

′1sas+1,1+···+a′ssas+1,sxt+1(2)as+1,s+1 .

Thus by the definition of Aij:

x1(1) = x1(3)A11 . . . xs(3)A1sxt+1(2)A1,s+1

...xs(1) = x1(3)As1 . . . xs(3)Assxt+1(2)As,s+1

xt = x1(3)As+1,1 . . . xs(3)As+1,sxt+1(2)As+1,s+1 .

(4.24)

Now we show that we have an expression

f =m∑i=1

aix1(3)Bi1 . . . xs(3)B

is(xt+1(2) +α)B

is+1 +x1(3)B

11+1 . . . xs(3)B

1s+1Ω (4.25)

with Ω ∈ U2/mU2 [[x1(3), . . . , xs(3), xt+2(2)]], Bij = c1iA1j + · · ·+ csiAsj +diAs+1,j for

1 ≤ i ≤ m and 1 ≤ j ≤ s+ 1, and Bi1 = B1

1 , . . . , Bis = B1

s , for 1 ≤ i ≤ m.

55

To establish (4.25) let Bij = c1iA1j + · · · + csiAsj + diAs+1,j for 1 ≤ i ≤ m and

1 ≤ j ≤ s+ 1.

Then

(Bi1 . . . B

is) = (bi1 . . . b

is)

a′11 ... a′1s... ... ...a′s1 ... a′ss

Making the substitution (4.24) into (4.21), we have:

f =m∑i=1

aix1(3)Bi1 . . . xs(3)B

is(xt+1(2) + α)B

is+1+

n∑i=m+1

aix1(3)Bi1 . . . xs(3)B

is(xt+1(2) + α)B

is+1 + (x1(3)As+1,1 . . .

xs(3)As+1,s+1)dn+1Ω′

where Ω′ = Λ(x1(3) . . . xs(3)) ∈ U2/mU2 [[x1(3), . . . , xs(3), xt+1(2)]],

and

f =m∑i=1

aix1(3)Bi1 . . . xs(3)B

is(xt+1(2) + α)B

is+1 + x1(3)B

11+1 . . . xs(3)B

1s+1Ω.

Moreover, since ν(x1(3)), . . . , ν(xs(3)) are rationally independent, then Bij = B1

j

for 1 ≤ i ≤ m, and 1 ≤ j ≤ s.

By the above analysis, we see that we may thus replace if necessary (aij) with (Aij),

and we have

f =m∑i=1

aix1(2)bi1 . . . xs(2)b

is(xt+1(2) + α)b

is+1 + x1(2)b

11+1 . . . xs(2)b

1s+1Ω,

where bij = c1i a1j + · · ·+ csiasj + dias+1j for 1 ≤ i ≤ m and 1 ≤ j ≤ s+ 1,

and b11 = bi1, . . . , b1s = bis for 1 ≤ i ≤ m.

Define

f = x1(2)b11 . . . xs(2)b

1s(xt+1(2) + α)b

1s+1f1

56

where

f1 =m∑i=1

ai(xt+1(2) + α)bis+1−b1s+1 + x1(2) . . . xs(2)Ω.

Now b11 = bi1, . . . , b1s = bis for 1 ≤ i ≤ m which gives the following

a11 ... as+1,1

... ... ...a1s ... as+1,s

a1,s+1 ... as+1,s+1

c1i − c11...

csi − cs1di − d1

=

bi1 − b11......

bis+1 − b1s+1

=

0...0

bis+1 − b1s+1

.

By Cramer’s rule

di − d1 =

Det

a11 ... as1 0... ... ... 0

a1,s+1 ... as,s+1 bis+1 − b1s+1

Det(aij)

=

(bis+1−b1s+1)Det

a11 ... as1... ... ...a1s ... ass

Det(aij)

.

Thus bis+1 − b1s+1 =Det(aij)(di−d1)

Det

a11 ... as1... ... ...a1s ... ass

= di−d1

afor 1 ≤ i ≤ m where

a = ±Det

a11 ... as1... ... ...a1s ... ass

∈ Z.

Moreover a|di − d1 for 1 ≤ i ≤ m.

From now on, we assume a > 0. The case a < 0 is similar.

Let ei = di−d1a

.

This implies

f1 =m∑i=1

ai(xt+1(2) + α)di−d1a + x1(2) . . . xs(2)Ω.

57

Let r1 = ordf1(0, . . . , 0, xt+1(2)) = ord∑m

i=1 ai(xt+1(2) + α)ei where

ai = ai(0, . . . , 0) ∈ U2/mU2 .

Then r1 ≤ dm−d1a≤ r since dm ≤ r by (4.18).

Assume that r1 = r.

Then d1 = 0, dm = r and a = 1, so that di = ei for 1 ≤ i ≤ m.

ord f(0, . . . , 0, xt+1) = r and dm = r imply that am is a unit in T/mT [[x1, . . . , xt]]

thus am = am and c1m = · · · = csm = 0.

Define ξ(t) = f1(0, . . . , 0, t− α) =∑m

i=1 aitei .

ordξ(t+ α) = ordf1(0, . . . , 0, t) = r = degξ(t+ α), since em = r.

Thus ξ(t+ α) = amtr.

Now

amtr =ξ(t+ α) =

m∑i=1

ai(t+ α)ei

=am(t+ α)r +m−1∑i=1

ai(t+ α)ei

=amtr + rαamt

r−1 + am−1tem−1 + terms of order < r − 1.

This implies that em−1 = r − 1

and moreover

rαam + am−1 = 0 ∈ U2/mU2 . (4.26)

Thus

em = dm = r and em−1 = dm−1 = r − 1, (4.27)

58

since d1 = 0 and a = 1.

Now, we compute using (4.26) and (4.27)

amxt+1

am−1=

amxrt+1

am−1xr−1t+1

=amx

dmt+1

am−1xdm−1t+1

= amx1(2)bm1 ...xs(2)b

ms (xt+1(2)+α)

bms+1

am−1x1(2)bm−11 ...xs(2)b

m−1s (xt+1(2)+α)

bm−1s+1

= am(xt+1(2)+α)bms+1−b

m−1s+1

am−1.

We have

bms+1 − bm−1s+1 = (bms+1 − b1s+1)− (bm−1

s+1 − b1s+1)= em − em−1 = r − (r − 1) = 1.

Then xt+1

am−1= xt+1(2)+α

am−1.

Taking the residue λ′

of xt+1

am−1in U2/mU2 , we have that

λ′=

[xt+1

am−1

]=

α

am−1

= − 1

ram∈ T/mT .

Thus ν(xt+1− λ′am−1) > ν(xt+1). If xt+1− λ

′am−1 ∈ QT , then we have reached

the conclusions of 1. of the theorem. Assume that xt+1 − λ′am−1 6∈ QT . Let

β = ν(xt+1 − λ′am−1).

There exists γ ∈ N such that ν(mγT ) > β and there exists ϕ1 ∈ T/mt[x1, . . . , xt] ⊂

U such that λ′am−1 − ϕ1 ∈ (x1, . . . , xt)

γT/mT [[x1, . . . , xt]].

Then β = ν(xt+1 − ϕ1).

Let x′t+1 = xt+1 − ϕ1.

f ′(x1, . . . , xt, x′t+1) = f(x1, . . . , xt, x

′t+1 + ϕ1) = f(x1, . . . , xt+1).

ord(f ′(0, . . . , 0, x′t+1)) = r. We repeat the above construction to either achieve a

reduction r1 < r, or we obtain a new change of variables with

ν(x′′t+1) > ν(x′t+1) > ν(xt+1).

59

Iterating this step, we either construct an infinite sequence

0 < ν(xt+1) < ν(x′t+1) < · · · ν(x(n)t+1) < · · · (4.28)

or we find a change of variables in xt+1 which leads to a reduction r1 < r.

Suppose that (4.28) holds, then by construction, there exist polynomials a(n) ∈

T/mT [x1, . . . , xt], such that x(n+1)t+1 = x

(n)t+1 − a(n), ν(a(n)) = ν(x

(n)t+1) for n ≥ 0 where

x(1)t+1 = x′t+1, x

(0)t+1 = xt+1.

Let τn = ν(x(n)t+1) for n ≥ 0 and τ0 = ν(xt+1).

Then we have an infinite sequence τ0 < τ1 < · · · < τn < . . .

and thus

limn→∞

τn =∞ by lemma 2.3 of [16].

Let A = T/mT [[x1, . . . , xt]].

Let

α(n) =n−1∑j=0

a(j) ∈ A,

then x(n)t+1 = xt+1 − α(n) and ν(α(n)) = ν(x

(n)t+1) = τn.

Let δn = f ∈ A/ν(f) ≥ τn ⊆ A.

This implies that

α(n) − α(n+i) = x(n)t+1 − x

(n+i)t+1 =

n+i−1∑j=n

a(j) ∈ δn.

If f ∈ ∩∞n=0δn then ν(f) =∞ and thus f ∈ QT ∩ A = (0) by assumption,

thus proving that ∩∞n=0δn = (0). By Theorem (13) p.270 of [46] there exists an

integral valued function S(n) such that S(n) → ∞ as n → ∞ and such that

δn ⊆ mS(n)A . This implies that α(n) − α(n+i) ∈ mS(n)

A for all i ≥ 0 and thus α(n) is

60

a Cauchy Sequence in A.

So x(n)t+1 is a Cauchy sequence in T .

Then

limn→∞

x(n)t+1 = xt+1 − lim

n→∞α(n)

where

ν( limn→∞

x(n)t+1) =∞.

Let

φt+1(x1, . . . , xt) = limn→∞

α(n)

then xt+1 − φt+1 ∈ QT .

Thus 1. of the conclusions of the theorem holds.

Suppose that r1 < r.

x1(3), . . . , xs(3), xs+1(1), . . . , xt(1), xt+1(2), xt+2, . . . , xq

are then a regular system of parameters in U3, and we have an expression

f = x1(3)B11 . . . xs(3)B

1sf1

in U3, where r1 = ord(f1(0, . . . , 0, xt+1(2))) < r.

Iteration We iterate the above construction, applied to f1 ∈ U3.

We must eventually reach a local ring Un where either 2. of the conclusions of

the theorem hold, or f = x1(n)a1 . . . xs(n)asfn where rn =

ord(fn(x1(n), . . . , xt+1(n))) ≤ 1. f ∈ QT implies that rn = 1. By the Weierstrass

preparation theorem, there exists a series φt+1(x1(n), . . . , xt(n)) such that

61

xt+1 − φt+1(x1(n), . . . , xt(n)) ∈ QUn, so that 1. of the conclusions of the theorem

holds.

Theorem 4.3.4. Suppose that x1, . . . , xs, . . . , xa, . . . xq is a regular system of pa-

rameters in T such that ν(x1), . . . , ν(xs) are rationally independent, and there exist

φi ∈ T/mT [[x1, . . . , xi−1]] such that xi−φi ∈ QT for a+1 ≤ i ≤ q. Further suppose

that QT ∩ T/mT [[x1, . . . , xa]] 6= (0). Then there exists a sequence of etale Perron

transforms

T → U → U1 → · · · → Un

along ν such that Un has a regular system of parameters x1(n), . . . , xa(n), xa+1, . . . ,

xq, and φa ∈ Un/mUn [[x1(n), . . . , xa−1(n)]] such that ν(x1(n)), . . . , ν(xs(n)) are ra-

tionally independent and such that xa(n)− φa ∈ QUn. We further have

φi ∈ Un/mUn [[x1(n), . . . , xa(n), xa+1, . . . , xi−1]]

for a+ 1 ≤ i ≤ q.

Proof. Set a(T ) = a. Let t = t(T ) be defined from the (ordered) regular sequence

x1, . . . , xa, . . . , xq by the condition that QT ∩ T/mT [[x1, . . . , xt]] = (0) and QT ∩

T/mT [[x1, . . . , xt+1]] 6= (0). We have 1 ≤ t(T ) ≤ a− 1.

By Theorem 4.3.3, we can construct a sequence of etale Perron transforms

T → U → U1 along ν such that U1 has regular parameters

x1(1), . . . , xt(1), xt+1(1), . . . , xa+1, . . . , xq

such that either 1. or 2. of the conclusions of Theorem 4.3.3 hold.

62

Suppose that 1. of the conclusions of Theorem 4.3.3 holds. Then after inter-

changing xt+1(1) and xa (if t + 1 6= a) we have attained the conclusions of the

theorem.

Suppose that 2 of the conclusions of Theorem 4.3.3 holds. Then the ordered reg-

ular sequence x1(1), . . . , xt(1), xt+1(1), . . . , xa+1, . . . , xq in U1 satisfies the assump-

tions of Theorem 4.3.4, with a(U1) < a(T ).

We now apply Theorem 4.3.3 to this regular sequence in U1, with t(U1) ≤

a(U1) − 1. After finitely many iterations of Theorem 4.3.3 we must achieve the

conclusions of Theorem 4.3.4.

Lemma 4.3.5. Suppose that x1, . . . , xt, . . . xq is a regular system of parameters in

T , and there exist φi ∈ T/mT [[x1, . . . , xi−1]] such that xi−φi ∈ QT for t+1 ≤ i ≤ q.

Then there exists φi ∈ T/mT [[x1, . . . , xt]] such that xi − φi ∈ QT for t+ 1 ≤ i ≤ q.

Proof. We prove this by induction:

Set φt+1 = φt+1 ∈ T/mT [[x1, . . . , xt]].

Then

xt+2 − φt+2(x1, . . . , xt+1) = xt+2 − φt+2(x1, . . . , xt, (xt+1 − φt+1) + φt+1)

= xt+2 − φt+2(x1, . . . , xt, φt+1) + Ω(1)t+2[xt+1 − φt+1].

Set φt+2 = φt+2(x1, . . . , xt, φt+1) ∈ U1/mU1 [[x1, . . . , xt]] since

φt+1 ∈ T/mT [[x1, . . . , xt]]. Then xt+2−φt+2 ∈ QT since xt+2−φt+2 and xt+1−φt+1

are in QT .

Assume that it is true up to j − 1.

63

Then

xt+j − φt+j(x1, . . . , xt+j−1) = xt+j − φt+j(x1, . . . , xt, (xt+1 − φt+1)+

φt+1, . . . , (xt+j−1 − φt+j−1) + φt+j−1)

= xt+j − φt+j(x1, . . . , xt, φt+1, . . . , φt+j−1)+

Ω(1)t+j[xt+1 − φt+1] + · · ·+ Ω

(j−1)t+j [xt+j−1 − φt+j−1].

Set φt+j = φt+j(x1, . . . , xt, φt+1, . . . , φt+j−1) ∈ T/mT [[x1, . . . , xt]], since φt+1, . . . ,

φt+j−1 ∈ T/mT [[x1, . . . , xt]].

Then xt+j−φt+j ∈ QT , since xt+j−φt+j, xt+1−φt+1, . . . , xt+j−1−φt+j−1 ∈ QT .

Lemma 4.3.6. Suppose that x1, . . . , xa, . . . xq is a regular system of parameters in

T , and there exist φi ∈ T/mT [[x1, . . . , xa]] such that xi−φi ∈ QT for a+1 ≤ i ≤ q.

Suppose also that QT ∩T/mT [[x1, . . . , xa]] = (0), then QT = (xa+1−φa+1, . . . , xq−

φq).

Proof. Let f ∈ QT ⊆ T and write f =∑

I aIxi11 . . . x

iaa x

ia+1

a+1 · · ·xiqq where aI ∈

T /mT .

Expanding

f =∑I

aIxi11 . . . x

iaa ([xa+1 − φa+1] + φa+1)

ia+1 . . . ([xq − φq] + φq)iq

we obtain

f = (xa+1 − φa+1)Ωa+1 + · · ·+ (xq − φq)Ωq + Ω

where Ωi ∈ T/mT [[x1, . . . , xq]] and Ω ∈ T/mT [[x1, . . . , xa]] since φi ∈

T/mT [[x1, . . . , xa]].

Since f, xi − φi ∈ QT for a + 1 ≤ i ≤ q and Ω ∈ T/mT [[x1, . . . , xa]] where

QT ∩ T/mT [[x1, . . . , xa]] = (0) this implies that Ω = 0. Thus QT = (xa+1 −

φa+1, . . . , xq − φq)

64

Corollary 4.3.7. Suppose that x1, . . . , xs, . . . , xa, . . . xq is a regular system of pa-

rameters in T , such that ν(x1), . . . , ν(xs) are rationally independent, and there

exist φi ∈ T/mT [[x1, . . . , xi−1]] such that xi − φi ∈ QT for a + 1 ≤ i ≤ q. Then

there exists a sequence of etale Perron transforms T → U → U1 along ν such that

U1 has a regular system of parameters x1(1), . . . , xt(1), . . . , xa(1), xa+1, . . . , xq, and

φi ∈ U1/mU1 [[x1(1), . . . , xa(1)]] for t+ 1 ≤ i ≤ q such that

QU1= (xt+1(1)− φt+1, . . . , xa(1)− φa, xa+1 − φa+1, . . . , xq − φq).

Proof. Suppose that QT ∩ T/mT [[x1, . . . , xa]] = (0). Then there exists φi ∈

T/mT [[x1, . . . , xa]] for a+ 1 ≤ i ≤ q such that QT = (xa+1 − φa+1, . . . , xq − φq) by

Lemma 4.3.5 and Lemma 4.3.6.

Suppose that QT ∩ T/mT [[x1, . . . , xa]] 6= (0), then by Theorem 4.3.4, there ex-

ists a sequence of etale Perron transforms T → U → U1, along ν such that U1

has regular parameters x1(1), . . . , xa(1), xa+1, . . . , xq and there exist series φ1i ∈

U1/mU1 [[x1(1), . . . , xi−1(1)]] for a ≤ i ≤ q such that xa(1)−φ1a ∈ QU1

and xi−φ1i ∈

QU1for a < i ≤ q.

If QU1∩ U1/mU1 [[x1(1), . . . , xa(1)]] 6= (0), then apply Theorem 4.3.4 again.

In this way we construct a sequence of etale Perron transforms along ν

T → U → U1 → . . .

The sequence must terminate in Ur with

xa−r+1(r)− φra−r+1, . . . , xa(r)− φra, xa+1 − φra+1, . . . , xq − φrq ∈ QUr,

φri ∈ Ur/mUr [[x1(r), . . . , xi−1(r)]] for a− r + 1 ≤ i ≤ q, and

65

QUr∩ Ur/mUr [[x1(r), . . . , xa(r)]] = (0).

The conclusions of the corollary follow from the first part of the proof.

Theorem 4.3.8. Suppose that x1, . . . , xs, . . . , xp, . . . xq is a regular system of pa-

rameters in T , such that ν(x1), . . . , ν(xs) are rationally independent and xi ∈ PT

for p + 1 ≤ i ≤ q, and l > 0. Then there exists a sequence of transforms of type

(1, 0) and (2, 0) T → T1 along ν such that T1 has a regular system of parameters

y1, . . . , ys, . . . , yt, . . . , yp, . . . , yq,

and φi ∈ T1/mT1 [[y1, . . . , yi−1]], τi ∈ mlT1,l

for t+1 ≤ i ≤ p such that ν(y1), . . . , ν(ys)

are rationally independent, and

QT1= (yt+1 − φt+1 − τt+1, . . . , yp − φp − τp, yp+1, . . . , yq).

Further, there exist d1, . . . , ds ∈ N such that yd11 . . . ydss yi = xi for p+ 1 ≤ i ≤ q

and TPT = (T1)PT1.

Proof. By Corollary 4.3.7 (with φi = 0 for p+ 1 ≤ i ≤ q) and Theorem 4.3.2, there

exists a sequence of etale Perron transforms T → U → U1 such that

QU1= (xt+1(1)− φt+1, . . . , xp(1)− φp, xp+1, . . . , xq)

has the form of the conclusions of Corollary 4.3.7.

In particular

φi ∈ U1/mU1 [[x1(1), . . . , xt(1)]], for every i. (4.29)

66

Let α = max ν(x1(1)ci1 . . . xs(1)cis) for t + 1 ≤ i ≤ p. Where the cij are

defined on page 67. There exists n ∈ N such that

α < nv(x1(1)). (4.30)

Choose m0 ∈ N so that m0 > nv(x2(1)), and m ∈ N such that m >

max αν(mT1

),m0, l0. Where l0 is the constant of the conclusion of Theorem 4.3.1.

Let T → T1,m be the corresponding sequence of transforms of type (1, 0) of the


By property (A), QU1is a component of the strict transform of QT in U1.

Since x1(1) . . . xs(1) = 0 is a local equation of the exceptional locus of spec(U1)→

spec(T ), we see that there exist ft+1, . . . , fp ∈ QT and ci1, . . . , cis ∈ N such that

ft+1

x1(1)ct+1,1 . . . xs(1)ct+1,s, . . . ,

fpx1(1)cp1 . . . xs(1)cps

, xp+1, . . . , xq

is a basis of QU1.

Since ν(xi(1) − φi) = ∞ for t + 1 ≤ i ≤ p, we may replace xi(1) with the

subtraction of the first part of the series φi from xi(1) to obtain

ν(xi(1)) = ν(φi) > m0 (4.31)

for t+ 1 ≤ i ≤ p, and

φi ∈ mmT1,m

for every i. (4.32)

Let

σi =

ft+i

x1(1)ct+i,1 ...xs(1)ct+i,sfor 1 ≤ i ≤ p− t

xi+t for p− t+ 1 ≤ i ≤ q − t.

ψi =

xt+i(1)− φt+i for 1 ≤ i ≤ p− txi+t for p− t+ 1 ≤ i ≤ q − t.

67

There exist bij ∈ U1 such that

σi =

q−t∑j=1

bijψj

for 1 ≤ i ≤ q − t, with

bij = δij

for p − t + 1 ≤ i, such that Det(bij) is a unit in U1. There exist hi ∈ mmT1,m

such

that

ft+i = x1,m(1)ct+i,1 . . . x1,m(s)ct+i,s [

p−t∑j=1

bij(x1,m(1), . . . , x1,m(s), . . . , xp,m(1),

xp+1, . . . , xq)ψj(x1,m(1), . . . , x1,m(s), . . . , xp,m(1), xp+1, . . . , xq)+

q∑j=p−t+1

bij(x1,m(1), . . . , x1,m(s), . . . , xp,m(1), xp+1, . . . , xq)xj] + hi

for 1 ≤ i ≤ p− t.

Perform successive sequences of transforms of type IIr Ti−1,m → Ti,m, defined

for 2 ≤ i ≤ t− s+ 1, by

x1,m(i− 1) = x1,m(i)a(i)11 . . . xs,m(i)a

(i)1sNs+i−1,m(i)

......

xs,m(i− 1) = x1,m(i)a(i)s1 . . . xs,m(i)a

(i)ssNs+i−1,m(i)

xs+i−1,m(i− 1) = x1,m(i)a(i)s+1,1 . . . xs,m(i)a

(i)s+1,sNs+i−1,m(i)

and xk,m(i) = xk,m(i− 1) if k /∈ 1, . . . , s, s+ i− 1.

Let xs+i−1,m(i) ∈ Ti+1,m be such that

x1,m(i), . . . , xs+i−1,m(i), xs+i,m(1), xp,m(1), . . . , xp+1, . . . , xq

are regular parameters in Ti+1,m. In Tt−s+1,m,

x1,m(t− s+ 1), . . . , xs,m(t− s+ 1), . . . , xt,m(t− s+ 1), xt+1,m(1), . . . , xp,m(1),xp+1, . . . , xq

68

are then regular parameters. For all i, we have finite sums

hi =∑j

x1,m(t−s+1)dij1 . . . xs,m(t−s+1)d

ijsΛj+

∑t+1≤k≤p

Ωkxk,m(1)2+∑

p+1≤k≤q

Ψkxk

(4.33)

where ν(x1,m(t− s + 1)dij1 . . . xs,m(t− s + 1)d

ijs) > α for all j, and Λi,Ωk,Ψk ∈

Tt−s+1,m, for all i, k.

Let φi = φi(x1,m(1), . . . , xt,m(1)).

We further have by (4.29) and (4.32) that there are finite sums φi =∑

j eijx1,m(t−

s+ 1)fij1 . . . xs,m(t− s+ 1)f

ijs for 1 ≤ i ≤ p− 1.

with ν(x1,m(t− s+ 1)fij1 . . . xs,m(t− s+ 1)f

ijs) > α, for all i and j.

We have an expression

x1,m(1) = x1,m(t− s+ 1)g1 . . . xs,m(t− s+ 1)gsΩ

where Ω is a unit in Tt−s+1,m.

For t+ 1 ≤ k ≤ p, by (4.31) and Theorem 4.3.1 we have that ν(xk,m(1)) > m0,

so by (4.30) ν(xk,m(1)) > ν((x1,m(t − s + 1)g1 . . . xs,m(t − s + 1)gs)n) > α. Recall

that ν(xi) =∞ for p+ 1 ≤ i ≤ q. We may then define a transformation

Tt−s+1,m → Tt−s+2,m by

xk,m(1) = (x1,m(t−s+1)g1 . . . xs,m(t−s+1)gs)n−xk,m(t−s+1) for t+1 ≤ k ≤ q.

Now substitute into (4.33), and perform a final transform Tt−s+2,m → Tt−s+3,m of

type I, to obtain by Lemma 4.2 [16] regular parameters x1,m(t−s+3), . . . , xt+1,m(t−

s+3) = xt+1,m(t−s+1), . . . , xp,m(t−s+3) = xp,m(t−s+1), xp+1(t−s+3), . . . , xq(t−

s+ 3) in Tt−s+3,m.

69

Let Gt+1, . . . , Gp be the strict transforms of ft+1, . . . , fp in Tt−s+3,m.

We have that Gt+i−∑p−t

j=1 bij(0, . . . , 0)xt+j,m(t−s+3) ∈ m2Tt−s+3,m

for 1 ≤ i ≤ p− t

and

Det

b11(0, . . . , 0) ... b1,p−t(0, . . . , 0)... ... ...

bp−t,1(0, . . . , 0) ... bp−t,p−t(0, . . . , 0)

6= 0.

Thus by property (A), Lemma (4.3.6) and the Weierstrass preparation Theorem,

there exists a regular system of parameters in Tt−s+3,m of the form of the conclusion

of Theorem 4.3.8.

Theorem 4.3.9. Suppose that ν has rank ≥ 2, and x1, . . . , xs, . . . , xp, xp+1, . . . , xq

is a regular sequence in T such that ν(x1), . . . , ν(xs) are rationally independent,

and

(xp+1, . . . , xq) ⊂6= PT .

Then there exists a sequence of etale Perron transforms along ν

T → U → U1 → · · · → Ue

such that Ue has regular parameters x1(e), . . . , xs(e), . . . , xp(e), xp+1, . . . , xq such

that ν(x1(e)), . . . , ν(xs(e)) are rationally independent, and such that

(xp(e), xp+1, . . . , xq) ⊂ PUe .

Proof. If we achieve

ω(Ui) = dim(Ui)− dimUi/mUi(PUi/m

2Ui∩ PUi) < p

70

in some Ui, then we terminate the algorithm, as we can make a change of variables

in Ui so that the conclusions of the theorem hold in Ui. We may thus assume that

ω(Ui) = p (4.34)

throughout the proof.

By Corollary 4.3.7, there exists a sequence of etale Perron transforms U → U1

along ν such that there exists m ≥ s such that there exist regular parameters

x1(1), . . . , xs(1), . . . , xm(1), . . . , xp(1), xp+1, . . . , xq

in U1, series φi ∈ (U1/mU1)[[x1(1), . . . , xm(1)]] for m+ 1 ≤ i ≤ p such that

QU1= (xm+1(1)− φm+1, . . . , xp(1)− φp, xp+1, . . . , xq).

Let xi = xi(1)− φi for m+ 1 ≤ i ≤ p.

By our assumptions, there exists g ∈ PU1 − (xp+1, . . . , xq)U1. In U1 we have an

expansion:

g =∑

im+1+···ip>0

aim+1,...,ip(x1(1), . . . , xm(1))xim+1

m+1 · · · xipp +

q∑i=p+1

xiΩi. (4.35)

Since some aim+1,...,ip(x1(1), . . . , xm(1)) 6= 0, differentiating with respect to the

partials computed from the regular system of parameters

x1(1), . . . , xp(1), xp+1, . . . , xq,

we see that for some i with m+ 1 ≤ i ≤ p,

∂g

∂xi(1)(x1(1), . . . , xp(1), 0 . . . , 0)

71

has smaller order with respect to the ideal

(xm+1, . . . , xp) ⊂ T/m[[x1(1), . . . , xp]] ⊂ T/m[[x1(1), . . . , xp(1)]]

than g(x1(1), . . . , xp(1), 0, . . . , 0) does.

Suppose that ∂g∂xi(1)

∈ PU1 = QU1∩ U1. Then we repeat the above argument

with g replaced with ∂g∂xi(1)

. After a finite number of iterations we find g ∈ PU1 such

that ∂g∂xi(1)

6∈ QU1. Thus ∂g

∂xi(1)6∈ PU1 . After possibly interchanging xi(1) and xp(1),

we may assume that i = p. Thus ∂g∂xp(1)

6∈ PU1 . In particular, a0···01 6= 0.

We have QU1∩ U1/mU1 [[x1(1), . . . , xm(1)]] = (0).

We have ν(xp(1)) = ν(φp) since xp ∈ QU1(xp(1) 6∈ QU1

by (4.34)). There exists

a polynomial

hp(x1(1), . . . , xm(1)) ∈ U1/mU1 [x1(1), . . . , xm(1)]

and a series Ψp ∈ U1/mU1 [[x1(1), . . . , xm(1)]] such that φp = hp + Ψp with ν(Ψp) >

ν(a0···01). We make a change of variables in U1, replacing xp(1) with xp(1) − hp.

We now have xp = xp(1)−Ψp, so we have ν(xp(1)) = ν(Ψp) > ν(a0,...,0,1).

In the same way, we make changes of variables in xm+1(1), . . . , xp−1(1), so we

may assume that

ν(xi(1)) = ν(Ψi) > ν(a0,...,0,1xp(1)) for m+ 1 ≤ i ≤ p− 1, andν(xp(1)) > ν(a0,...,0,1).

(4.36)

By Theorem 4.3.2, and since we are assuming that T satisfies property (A) of

Definition (4.2.1), there exists a sequence of etale Perron transforms U1 → U2 along

ν such that U2 has regular parameters

x1(2), . . . , xs(2), . . . , xm(2), xm+1(1), . . . , xp(1), xp+1, . . . , xq

72

with

a0,...,0,1 = x1(2)c1 . . . xs(2)csγ(x1(2), . . . , xm(2))

where γ ∈ U2 is a unit series, and

Ψi = x1(2)di1 . . . xs(2)disδi(x1(2), . . . , xm(2))

for m+ 1 ≤ i ≤ p, where γ and δi ∈ U2 are unit series.

For 2 ≤ i ≤ p−m+1, now perform etale Perron transforms along ν, Ui → Ui+1,

defined by

x1(i) = x1(i+ 1)a(i+1)11 . . . xs(i+ 1)a

(i+1)1s (xm+i−1(i+ 1) + αi+1)

a(i+1)1,s+1

......

xs(i) = x1(i+ 1)a(i+1)s1 . . . xs(i+ 1)a

(i+1)ss (xm+i−1(i+ 1) + αi+1)

a(i+1)s,s+1

xm+i−1(1) = x1(i+ 1)a(i+1)s+1,1 . . . xs(i+ 1)a

(i+1)s+1,s(xm+i−1(i+ 1) + αi+1)

a(i+1)s+1,s+1

and xk(i+ 1) = xk(i) if k /∈ 1, . . . , s,m+ i− 1.

In Uj−m+1,

x1(j −m+ 1), . . . , xs(j −m+ 1), . . . , xj−1(j −m+ 1), xj(1), . . . , xp(1), xp+1, . . . , xq

are regular parameters, and there exists a unit series

εj(x1(j −m+ 1), . . . , xj−1(j −m+ 1))

∈ Uj−m+1/mUj−m+1[[x1(j −m+ 1), . . . , xj−1(j −m+ 1)]]

and d1j, . . . , dsj ∈ N such that

xj = xj(1)− x1(j −m+ 1)d1j . . . xs(j −m+ 1)dsjεj.

73

In Uj−m+2,

xj = x1(j −m+ 2)a(j−m+2)s+1,1 . . . xs(j −m+ 2)a

(j−m+2)s+1,s (xj(j −m+ 2)+

αj−m+2)a(j−m+2)s+1,s+1 − [x1(j −m+ 2)a

(j−m+2)11 . . . xs(j −m+ 2)a

(j−m+2)1s

(xj(j −m+ 2) + αj−m+2)a(j−m+2)1,s+1 ]d1j . . . x1(j −m+ 2)a

(j−m+2)s1 . . .

[xs(j −m+ 2)a(j−m+2)ss (xj(j −m+ 2) + αj−m+2)

a(j−m+2)s,s+1 ]dsjεj.

Set xi = xi(j−m+ 2) for 1 ≤ i, α = αj−m+2, aij = aij(j−m+ 2). There exists

0 6= β ∈ Uj−m+2/mUj−m+2and Λi ∈ Uj−m+2/mUj−m+2

[[x1, . . . , xj]] such that

εj = β + Λ1x1 + · · ·+ Λj−1xj−1.

Since ν(x1), . . . , ν(xs) are rationally independent we have :

as+1,1 = a11d1j + · · ·+ as1dsj...

as+1,s = a1sd1j + · · ·+ assdsj

(4.37)

and thus

xj = xas+1,1

1 . . . xas+1,ss [(xj + α)as+1,s+1 − β(xj + α)a1,s+1d1j+···+as,s+1dsj + Λ1x1

+ · · ·+ Λj−1xj−1]

for some

Λ1, . . . ,Λj−1 ∈ Uj−m+2/mUj−m+2[[x1, . . . , xj]].

Let dj = a1,s+1d1j + · · ·+ as,s+1dsj.

We expand

(xj + α)as+1,s+1 − β(xj + α)dj = (αas+1,s+1 − βαdj)+

(as+1,s+1αas+1,s+1−1 − βdjαdj−1)xj + x2

jΣ.

αas+1,s+1 − βαdj = 0 since xj ∈ QUj−m+2.

Thus β = αas+1,s+1−dj

as+1,s+1αas+1,s+1−1 − βdjαdj−1 = βαdj−1[as+1,s+1 − dj].

74

By (4.37), we have :a11 ... as+1,1

... ... ...a1s ... as+1,s

a1,s+1 ... as+1,s+1

d1j

...

dsj−1

=

0...0

dj − as+1,s+1

.

By Cramer’s rule

−1 =

Det

a11 ... as1 0... ... ... 0

a1,s+1 ... as,s+1 dj − as+1,s+1

Det(aij)

.

Since Det(aij) = ±1, we have

±1 = Det

a11 ... as1... ...a1s ass

(dj − as+1,s+1), and thus dj − as+1,s+1 6= 0.

By the Weierstrass preparation theorem, there exists series Ωj and ψj such that

Ωj is a unit series in Uj−m+2/mUj−m+2[[x1, . . . , xj]] and ψj is a series in

Uj−m+2/mUj−m+2[[x1, . . . , xj−1]], such that

xj = xas+1,1

1 . . . xas+1,ss Ωj(xj − ψj(x1, . . . , xj−1)).

Thus in Up−m+2, we have an expression

xj = x1(p−m+ 2)c1j . . . xs(p−m+ 2)csjΩj(xj(p−m+ 2)−

σj(x1(p−m+ 2), . . . , xj−1(p−m+ 2), xj+1(p−m+ 2), . . . , xp(p−m+ 2)))

where Ωj ∈ Uj−m+2/mUj−m+2[[x1(p−m+ 2), . . . , xp(p−m+ 2)]] is a unit series and

ν(x1(p−m+ 2)c1j . . . xs(p−m+ 2)csj) = ν(xj(1)). (4.38)

We have

a0,...,0,1 = x1(2)c1 . . . xs(2)csγ(x1(2), . . . , xm(2)) = x1(p−m+2)c1 . . . xs(p−m+2)csγ

75

where γ ∈ Uj−m+2/mUj−m+2[[x1(p−m+ 2), . . . , xp(p−m+ 2)]] is a unit series.

There exists F,Gi ∈ U1/mU1 [[x1(1), . . . , xp(1)]] such that

g = a0,...,0,1xp + Fx2p +

p−1∑i=m+1

Gixi +

q∑j=p+1

xjΩj.

In Up−m+2, we have F ,Gj,Σ ∈ Uj−m+2/mUj−m+2[[x1(p−m+2), . . . , xp(p−m+2)]]

such that Σ is a unit and

g = x1(p−m+ 2)c1+c1p . . . xs(p−m+ 2)cs+csp(xp(p−m+ 2)− σp)Σ+

x1(p−m+ 2)2c1p . . . xs(p−m+ 2)2cspF +

p−1∑j=m+1

x1(p−m+ 2)c1j . . .

xs(p−m+ 2)csjGj +

q∑j=p+1

xiΩi.

By (4.38) and (4.36), and Lemma 4.2 [16], after possibly performing an etale

Perron transform of type I along ν, we have that

x1(p−m+ 2)c1+c1p . . . xs(p−m+ 2)cs+csp properly divides

x1(p−m+ 2)2c1p . . . xs(p−m+ 2)2csp and x1(p−m+ 2)c1j . . . xs(p−m+ 2)csj

m+ 1 ≤ j ≤ p− 1.

Let a = p−m+ 2, ci = ci + cip for 1 ≤ i ≤ s. By the Weierstrass Preparation

Theorem, we have an expression

g = x1(a)c1 . . . xs(a)csγ(xp(a)− Σ(x1(a), . . . , xp−1(a))) +

q∑j=p+1

xjΩj,

where γ ∈ Ua/mUa [[x1(a), . . . , xp(a)]] is a unit series. Extend xp+1, . . . , xq to a

minimal system of generators xp+1, . . . , xq, f1, . . . , ft of PUa . We have

x1(a)c1 . . . xs(a)cs(xp(a)− Σ) ∈ PUaUa.76

Since we are assuming that T satisfies property (A) of Definition (4.2.1), PUa

is a component of the strict transform of PU in Ua.

Since x1(a) . . . xs(a) = 0 is a local equation of the exceptional locus of spec(Ua)→

spec(U), we have that x1(a) . . . xs(a) /∈ PUa .

Thus xp(a)− Σ ∈ PUaUa We have an expression

xp(a)− Σ =

q∑i=p+1

bixi +t∑

j=1

difi

with bi, di ∈ Ua. Thus ∂fi∂xp(a)

(0, . . . , 0) 6= 0 for some 1 ≤ i ≤ t, which implies

that

dimUa/mUa(PUa/m

2Ua ∩ PUa) > q − p

and thus

ω(Ua) = dim(Ua)− dimUa/mUa(PUa/m

2Ua ∩ PUa) < p.

Corollary 4.3.10. Suppose that ν has rank ≥ 2, and x1, . . . , xs, . . . , xp, xp+1, . . . , xq

is a regular sequence in T such that ν(x1), . . . , ν(xs) are rationally independent, and

such that

(xp+1, . . . , xq) ⊂6= PT .

Then there exists a sequence of etale Perron transforms along ν

T → U → U1 → · · · → Ue

such that Ue has regular parameters

x1(e), . . . , xs(e), . . . , xσ(1)(e), xσ(1)+1(e), . . . , xp(e), xp+1, . . . xq

77

such that ν(x1(e)), . . . , ν(xs(e)) are rationally independent and such that

(xσ(1)+1(e), . . . , xp(e), xp+1, . . . , xq) = PUe .

Proof. By Theorem 4.3.9, there exists a sequence of etale Perron transforms along

ν.

T → U → U1

such that U1 has regular parameters

x1(1), . . . , xs(1), . . . , xp(1), xp+1, . . . xq

such that

(xp(1), xp+1, . . . , xq) ⊂ PU1 .

and we have a regular sequence of length q − p+ 1 in PU1 .

If PU1 = (xp(1), xp+1, . . . , xq), then we are done; otherwise, we repeat the algorithm

of Theorem 4.3.9, each time getting a longer regular sequence in PUi . Since the

length of a regular sequence is ≤ q, this process must terminate after finitely many

steps, and thus achieving the conclusion of the Corollary.

Theorem 4.3.11. Suppose that ν has rank ≥ 2, and x1, . . . , xs, . . . , xp, xp+1, . . . , xq

is a regular sequence in T such that ν(x1), . . . , ν(xs) are rationally independent and

such that

(xp+1, . . . , xq) ⊂6= PT .

78

Then there exists a sequence of transforms of types (1, 0) and (2, 0) along ν

T → T1 → · · · → Te

such that Te has regular parameters

x1(e), . . . , xs(e), . . . , xσ(1)(e), xσ(1)+1(e), . . . , xp+1, . . . , xq(e)

such that ν(x1(e)), . . . , ν(xs(e)) are rationally independent and

(xσ(1)+1(e), . . . xp+1(e), . . . , xq(e)) = PTe .

Further, there exist n1, . . . , ns ∈ N such that x1(e)n1 . . . xs(e)

nsxi(e) = xi for

p+ 1 ≤ i ≤ q and TPT = (Te)PTe .

Proof. By Corollary 4.3.10 there exists a sequence of etale Perron transforms T →

U → U1 such that U1 had regular parameters

x1(1), . . . , xt(1), . . . , xp(1), xp+1, . . . , xq,

such that:

PU1 = (xt+1(1), . . . , xp(1), xp+1, . . . , xq).

Let T → T1,m be the corresponding sequence of transforms of type (1, 0) of the


By property (A), PU1 is a component of the strict transform of PT in U1 (and

PT1 is a component of the strict transform of PT in T1). Since x1(1) . . . xs(1) = 0 is

a local equation of the exceptional locus of spec(U1)→ spec(T ), we see that there

exist ft+1, . . . , fp ∈ PT and ci1, . . . , cis ∈ N such that

ft+1

x1(1)ct+1,1 . . . xs(1)ct+1,s, . . . ,

fpx1(1)cp1 . . . xs(1)cps

, xp+1, . . . , xq

79

is a basis of PU1 .

Let

σi =

ft+i

x1(1)ct+i,1 ...xs(1)ct+i,sfor 1 ≤ i ≤ p− t

xi+t for p− t+ 1 ≤ i ≤ q − t.

ψi =

xt+i(1)− φt+i for 1 ≤ i ≤ p− txi+t for p− t+ 1 ≤ i ≤ q − t.

There exist bij ∈ U1 such that

σi =

q−t∑j=1

bijψj

for 1 ≤ i ≤ p− t, with

bij = δij

for p − t + 1 ≤ i, such that Det(bij) is a unit in U1. There exist hi ∈ mmT1,m

such

that

ft+i = x1,m(1)ct+i,1 . . . x1,m(s)ct+i,s [

p−t∑j=1

bij(x1,m(1), . . . , x1,m(s), . . . , xp,m(1),

xp+1, . . . , xq)ψj(x1,m(1), . . . , x1,m(s), . . . , xp,m(1), xp+1, . . . , xq)+

q∑j=p−t+1

bijxj] + hi

for 1 ≤ i ≤ p− t.

As in the proof of Theorem 4.3.8, there exists a sequence of transforms T →

T1,m → T2 along ν satisfying the conclusion of Theorem 4.3.11.

Theorem 4.3.12. Suppose that x1, . . . , xs, . . . , xq are regular parameters in T such

that ν(x1), . . . , ν(xs) are rationally independent. Suppose that f ∈ T .

80

1. If f 6∈ QT , then there exists a sequence of transforms of type (1, 0) and (2, 0),

T → T1 such that

f = x1(1)c1 . . . xs(1)csγ

where γ ∈ T1 is a unit.

2. If f ∈ QT and d > 0, then there exists a sequence of transforms of type (1, 0)

and (2, 0), T → T1 such that

f = x1(1)c1 . . . xs(1)csγ

where γ ∈ T1, and ν(x1(1)c1 . . . xs(1)cs) > d.

Proof. If f ∈ T − QT then there exists l ∈ N such that ν(f) < lv(m), thus

f = g + h where h ∈ ml and ν(f) = ν(g) < lv(m). By the methods presented

in this section, there exists a sequence of transforms of type (1, 0) followed by a

sequence of transforms of type (2, 0), T → T1 such that

f = x1(1)c1 . . . xs(1)csγ

where γ ∈ T1 is a unit.

If f ∈ QT then, ν(f) > lv(m) for all l ∈ N, thus given d > 0 , there exists a

sequence of transforms of type (1, 0) followed by a sequence of transforms of type

(2, 0), T → T1 such that

f = x1(1)c1 . . . xs(1)csγ

where γ ∈ T1 and ν(x1(1)c1 . . . xs(1)cs) > d.

81

Chapter 5

Resolution in all height

Let

(0) = P tV ⊂ · · · ⊂ P 1

V ⊂ P 0V = mV

be the chain of prime ideals in V . Let

(0) = P tT ⊂ · · · ⊂ P 1

T ⊂ P 0T = mT

be the chain of prime ideals in T , where P iT = P i

V ∩ T .

Let νi be a valuation whose valuation ring is Vi = VP iV .

Consider the following condition (5.1) on a Cauchy sequence fn in TP iT .

For all l ∈ N, there exists nl ∈ N such that νi(fn) ≥ lν(mTPiT

) if n ≥ nl. (5.1)

Define QTPiT

⊂ TP iT for the valuation ring VP iV by the following:

QTPiT

= f ∈ TP iT | A Cauchy sequencefn in TP iT which converges to f

satisfies (5.1).

We assume throughout this section that

trdegTPiT/P iTTPi

T

(VP iV /PiV VP iV ) = trdegQF (T/P iT )QF (V/P i

V ) = 0

82

for all i, and TP iT satisfies property (A) for VP iV for all i.

Let

τ(i) = dim(T/P iT ), σ(i) = dim(TP iT /QT

PiT

)

for 0 ≤ i ≤ t.

Let si be the rational rank of (V/P i+1V )P iV for 0 ≤ i ≤ t. We will write s = s0

and ν = ν0.

Lemma 5.0.13. Suppose there exists a regular system of parameters z1, . . . , zq in T

such that ν(zi) <∞ and ν(z1), . . . , ν(zs) are rationally independent. Let R = TP 1T

.

Suppose that P 1T is a regular prime in T and

yτ(1)+1, . . . , yq

are regular parameters in R. Then there exists a sequence of Perron transforms of

types (1, 0) and (2, 0) T → T1 along ν such that

(T1)P 1T1

= TP 1T,

and T1 has a regular system of parameters

x1(1), . . . , xs(1), . . . , xτ(1)+1(1), . . . , xq(1)

such that ν(x1(1)), . . . , ν(xs(1)) are rationally independent and

P 1T1

= (xτ(1)+1(1), . . . , xq(1))

and

yi = x1(1)d1i . . . xs(1)d

sixi(1)

83

for some dji ∈ N and τ(1) + 1 ≤ i ≤ q.

Proof. There are regular parameters

x1, . . . , xτ(1)+1, . . . , xq

in T such that

P 1T = (xτ(1)+1, . . . , xq).

There exist λij ∈ R such that

yτ(1)+i =

q−τ(1)∑j=1

λijxτ(1)+j

for 1 ≤ i ≤ q − τ(1).

For 1 ≤ i ≤ q − τ(1), there exist fij, gij ∈ T with gij 6∈ P 1T such that

λij =fijgij.

By Theorem 4.3.12, there exists a sequence of transforms of types (1, 0) and (2, 0)

T → T1 such that T1 has regular parameters

x1(1), . . . , xσ(1)(1), xσ(1)+1, . . . , xq,

such that

gij = x1(1)d1ij . . . xs(1)d

sijγij

where γij ∈ T1 are units for all i, j. Now we perform a transform of type (2, 0)

T1 → T ′1, defined by

xi = x1(1)d1i . . . xs(1)d

sixi(1)

84

for τ(1) + 1 ≤ i ≤ q, and appropriately large dji , followed by a transform of type I,

so that (by Lemma 4.2 [16]) T ′1 has regular parameters

x1(1), . . . , xq(1)

such that there are expansions

yτ(1)+i =

q−τ(1)∑j=1

λ′ijxτ(1)+j(1) (5.2)

for 1 ≤ i ≤ q − τ(1), with λ′ij ∈ T ′1.

We will now show that we can make a further sequence of transforms T ′1 → T2

of types (1, 0) and (2, 0) such that T2 has regular parameters

x1(2), . . . , xq(2)

with P 1T2

= (xτ(1)+1(2), . . . , xq(2)), and there are expressions

yτ(1)+i = x1(2)d1i . . . xs(2)d

sixτ(1)+i(2) (5.3)

for 1 ≤ i ≤ q − τ(1).

We prove this as follows. Since Det(λ′ij) is a unit in R, there exists a λ′ik such

that λ′ik 6∈ P 1T ′1

. Without loss of generality, i = 1. By Theorem 4.3.12, there exists

a sequence of transforms of types (1, 0) and (2, 0) T ′1 → T ′′1 such that T ′′1 has a

regular system of parameters

x1(1)′′, . . . , xτ(1)(1)′′, xτ(1)+1(1), . . . , xq(1)

such that for 1 ≤ j ≤ q − τ(1) λ′1j = [x1(1)′′]c1j . . . [xs(1)′′]c

sjγj, where γj ∈ T ′′1 is a

unit if λ′1j 6∈ P 1T ′1

, and γj ∈ T ′′1 and ν([x1(1)′′]c1j . . . [xs(1)′′]c

sj ) is arbitrarily large if

85

λ′1j ∈ P 1T ′1

. After permuting xτ(1)+1, . . . , xq, we may assume that

ν([x1(1)′′]c1j . . . [xs(1)′′]c

sj ) has minimal value for 1 ≤ j ≤ q − τ(1).

After performing a transform of type I, we then have an expression

yτ(1)+1 =[x1(1)′′]c11 . . . [xs(1)′′]c

1s [(γ1xτ(1)+1(1)) +

q−τ(1)∑j=2

[x1(1)′′]c1j−c11 . . .

[xs(1)′′]csj−cs1γjxτ(1)+j(1)].

We may make a change of variables in T ′′1 , replacing xτ(1)+1 with

xτ(1)(1)′′ =yτ(1)+1

[x1(1)′′]c11 ...[xs(1)′′]c

1s. We now have an expression similar to (5.2)

yτ(1)+i =

q−τ(1)∑j=1

λ′′ijxτ(1)+j(1) (5.4)

for 1 ≤ i ≤ q − τ(1) in T ′′1 , with yτ(1)+1 = [x1(1)′′]c11 . . . [xs(1)′′]c

1sxτ(1)+1(1)′′. Since

Det(λ′′ij) is a unit in P 1T ′′1

, we have λ′′ij 6∈ P 1T ′′1

for some 2 ≤ i.

We iterate our construction of T ′1 → T ′′1 to construct T ′′1 → T2 such that (5.3)

holds.

5.1 Perron Transforms

Perron Transforms of type (1,m)

Suppose that 1 ≤ m ≤ n (with n ≤ t) and there exists a regular system of

parameters

x1, . . . , xq

in T such that

P iT = (xτ(i)+1, . . . , xq)

86

for 0 ≤ i ≤ n, and vi(xτ(i)+1), . . . , vi(xτ(i)+si) are rationally independent for 0 ≤

i ≤ n.

Analogous to a transform of type (1, 0) in Chapter 4, we define two types of

such transforms along ν T → T1, transforms of type I and type IIr

Type I is defined by a transformation

xτ(m)+1 = xτ(m)+1(1)a11 . . . xτ(m)+sm(1)a1sm

...xτ(m)+sm = xτ(m)+1(1)asm1 . . . xτ(m)+sm(1)asmsm

(5.5)

where Det (aij) = ±1 and vm(xτ(m)+1(1)), . . . , vm(xτ(m)+sm(1)) are rationally

independent.

We define T1 = T [xτ(m)+1(1), . . . , xτ(m)+sm(1)]T [xτ(m)+1(1),...,xτ(m)+sm (1)]∩mV .

Suppose that τ(m) + sm < r and vm(xr) <∞. We will say that

xτ(m)+1, . . . , xτ(m)+sm , xr are permissible of type m if there exists a transformation

xτ(m)+1 = Na11

τ(m)+1 . . . Na1sm

τ(m)+smNa1,sm+1r

...xτ(m)+sm = N

asm1

τ(m)+1 . . . Nasm,smτ(m)+sm

Nasm,sm+1r

xr = Nasm+1,1

τ(m)+1 . . . Nasm+1,sm

τ(m)+smNasm+1,sm+1r

(5.6)

where Det (aij) = ±1, 0 < vm(Ni) < ∞ for τ(m) + 1 ≤ i ≤ τ(m) + sm,

vm(Nτ(m)+1), . . . , vm(Nτ(m)+sm) are rationally independent, vm(Nr) = 0, and

ν(Nr) ≥ 0.

If such a transformation exists, we define

T1 = T [Nτ(m)+1, . . . , Nτ(m)+sm , Nr]T [Nτ(m)+1,...,Nτ(m)+sm ,Nr]∩mV and call T → T1 a

87

transformation of type IIr.

Let (bij) = (aij)−1. We have

Ni = xbi1τ(m)+1 . . . xbi,smτ(m)+sm

xbi,sm+1r (5.7)

for τ(m) + 1 ≤ i ≤ τ(m) + sm and i = r.

Remark 5.1.1. 1. We could have that P jT1

are not regular primes if j ≤ m.

2. We have P iT1

= (xτ(i)+1, . . . , xq) are regular primes for m < i ≤ n.

3. Type (1,m) is a generalization to higher rank of the transform of type (1, 0)

of Chapter4.

Transforms of type (2,m)

Suppose that xj ∈ Pm+1T and d1, . . . , dsm ∈ N.

Define a transformation T → T1 of type (2,m) by

T1 = T [xj

xd1τ(m)+1 . . . xdsmτ(m)+sm

]T [

xj

xd1τ(m)+1

...xdsmτ(m)+sm

]∩mV .

Let xj(1) =xj

xd1τ(m)+1

...xdsmτ(m)+sm

, and xi(1) = xi if i 6= j.

Remark 5.1.2. 1. We have P iT1

= (xτ(i)+1(1), . . . , xq(1)) for 1 ≤ i ≤ n.

2. Type (2,m) is a generalization to higher rank of the transform of type (2, 0)

of Chapter 4.

88

5.2 Extension of results to higher rank


such that ν(zi) <∞ and ν(z1), . . . , ν(zs) are rationally independent. Suppose that

R = TP 1T

has regular parameters

yτ(1)+1, . . . , yτ(2)+1, . . . , yτ(m)+1, . . . , yq

such that vm(yτ(m)+1), . . . , vm(yτ(m)+sm) are rationally independent, and τ(m) +

sm < r is such that vm(yr) < ∞, and yτ(m)+1, . . . , yτ(m)+sm , yr are permissible of

type m. Let R→ R1 be the transformation of type (1,m− 1) and type IIr defined

by (5.6) so that

R1 = R[Nτ(m)+1, . . . , Nr]B

Where B = R[Nτ(m)+1, . . . , Nr] ∩mVP1V

.

Then there exists a sequence of transformations of type (1, 0) and (2, 0) T → T1

such that (T1)P 1T1

= R1 and T1 has regular parameters

x1(1), . . . , xτ(1)(1), xτ(1)+1(1), . . . , xτ(2)+1(1), . . . , xτ(m)+1(1), . . . , xq(1)

such that ν(x1(1)), . . . , ν(xs(1)) are rationally independent, P 1T1

= (xτ(1)+1(1), . . . ,

xq(1)), there exist dji ∈ N such that yi = x1(1)d1i . . . xs(1)d

sixi(1) for τ(1) + 1 ≤ i,

and xτ(m)+1(1), . . . , xτ(m)+sm(1), xr(1) are permissible of type m.

Let bij = (a−1ij ) as in (5.6), and let

N τ(m)+i = xτ(m)+1(1)bi1 . . . xτ(m)+sm(1)bi,smxr(1)bi,sm+1

for 1 ≤ i ≤ sm and N r = xτ(m)+1(1)bsm+1,1 . . . xr(1)bsm+1,sm+1

89

Let T1 → T2 be the resulting transform of type (1,m) where

T2 = T1[N τ(m)+1, . . . , N τ(m)+sm , N r]B1

where B1 = T1[N τ(m)+1, . . . , N τ(m)+sm , N r] ∩mV .

Then (T2)P 1T2

= R1.

Proof. We first apply Theorem 4.3.11 to construct T → T1 such that P 1T1

is a regular

prime and (T1)P 1T1

= R. Then we apply Lemma 5.0.13 to construct T1 → T2 such

that (T2)P 1T2

= (T1)P 1T1

and T2 has a regular system of parameters

x1(2), . . . , xτ(1)+1(2), . . . , xτ(m)+1(2), . . . , xq(2)

satisfying yi = x1(2)d1i . . . xs(2)d

sixi(2) for τ(1) + 1 ≤ i ≤ q and some dji ∈ N. We

have that bsm+1,i < 0 for some i (as in (5.7)). After possibly permuting the xj(1),

we may assume that i 6= 1.

Now make the transformation of type (2,m) T2 → T3 defined by

xτ(m)+i(2) = x1(3)ci1 . . . xs(1)c

isxτ(m)+i(3)

for 1 ≤ i ≤ sm, where cji is chosen so that

ν(xτ(m)+1(3)bsm+1,1 . . . xr(3)bsm+1,sm+1) > 0.

We have that (T3)P 1T3

= (T2)P 1T2

.

We now have that xτ(m)+1(3), . . . , xτ(m)+sm(3), xr(3) are permissible of type m.

Define Mτ(m)+i = xτ(m)+1(3)bi1 . . . xτ(m)+sm(3)bi,smxr(3)bi,sm+1 for 1 ≤ i ≤ sm and

Mr = xτ(m)+1(3)bsm+1,1 . . . xτ(m)+sm(3)bsm+1,smxr(3)bsm+1,sm+1 .

90

We may define the transformation of type (1,m) T3 → T4, where

T4 = T3[Mτ(m)+1, . . . ,Mτ(m)+sm ,Mr]B3

with B3 = T3[Mτ(m)+1, . . . ,Mτ(m)+sm ,Mr] ∩mV .

There exists nji ∈ Z such that Mi = Nix1(3)ni1 . . . xs(3)n

is for all i.

Since P 1T4∩ T3 = P 1

T3, R = (T3)P 1

T3and x1(3), . . . , xs(3) 6∈ P 1

T3, we have that

(T4)P 1T4

= (T3)P 1T3

[Mτ(m)+1, . . . ,Mτ(m)+sm ,Mr]P 1T4

=R[Nτ(m)+1, . . . , Nτ(m)+sm , Nr]R[Nτ(m)+1,...,Nτ(m)+sm ,Nr]∩mVP1V

= R1.

Remark 5.2.2. A similar but simpler statement holds for transformations R→ R1

of type (1,m− 1) and type I.


such that ν(zi) <∞ and ν(z1), . . . , ν(zs) are rationally independent. Suppose that

R = TP 1T

has regular parameters

yτ(1)+1, . . . , yτ(m)+1, . . . , yσ(m)+sm , . . . , yj, . . . , yq

such that ν(yτ(m)+1), . . . , ν(yτ(m)+sm) are rationally independent, and

yj ∈ Pm+1R .

Suppose that R→ R1 is the transformation of type (2,m− 1) defined by

R1 = R[yj

yd1τ(m)+1 . . . ydsmτ(m)+sm

]R[

yj

yd1τ(m)+1

...ydsmτ(m)+sm

]∩mVP1V

for some d1, . . . , dsm ∈ N. Then there exists a sequence of transformations of types

(1, 0) and (2, 0), T → T1 such that (T1)P 1T1

= R and T1 has regular parameters

x1(1), . . . , xτ(1)+1(1), . . . , xτ(m)+1(1), . . . xq

91

such that ν(x1(1)), . . . , ν(xs(1)) are rationally independent and there exists dji ∈ N

with yi = x1(1)d1i . . . xs(1)d

sixi(1) for τ(1) + 1 ≤ i. Let T1 → T2 where

T2 = T1[xj(1)

xτ(m)+1(1)d1 . . . xτ(m)+sm(1)dsm]T1[

xj(1)

xτ(m)+1(1)d1 ...xτ(m)+sm(1)dsm

]∩mV

be the resulting transformation of type (2,m). Then (T2)P 1T2

= R1.

Proof. The proof is similar to the proof of Lemma 5.2.1, but is a little simpler.

Theorem 5.2.4. Suppose that T satisfies the assumptions of this section, and that

T has a regular system of parameters x1, . . . , xq such that xτ(i)+1, . . . , xτ(i)+si ∈

P iT −P i+1

T , for every i and vi(xτ(i)+1), . . . , vi(xτ(i)+si) are rationally independent for

all i. Then there exists a sequence of transformations T → T1 (of types (1, a) and

(2, b) for appropriate a, b ∈ N) such that

(0) = P tT1⊂ · · · ⊂ P 0

T1= mT1

are regular primes.

Proof. The proof is by induction on t. The case t = 0 is trivial. Assume that the

theorem is true for t − 1. Let R = TP 1T. By induction, there exists a sequence of

transformations R → R1 such that the conclusions of the theorem hold for ν1 on

R1, so that R1 has a system of regular parameters

yτ(1)+1, . . . , yτ(2)+1, . . . , yτ(t−1)+1, . . . , yq

satisfying the conclusions of the theorem.

By Lemma 5.2.1 , Remark 5.2.2 and Lemma 5.2.3, there exists a sequence of

transformations T → T1 such that (T1)P 1T1

= R1.

92

By Theorem 4.3.11, there exists a sequence of transformations of type (1, 0)

and (2, 0) T1 → T2 such that P 1T2

is a regular prime and (T2)P 1T2

= R1.

By Lemma 5.0.13, there exists a sequence of transformations of types (1, 0) and

(2, 0) T2 → T3 such that (T3)P 1T3

= R1 and T3 has a system of regular parameters

x1(3), . . . , xτ(1)+1(3), . . . , xq(3)

such that yi = x1(3)d1i . . . xs(3)d

sixi(3) for τ(1) + 1 ≤ i, and some di ∈ N. Thus the

conclusions of the theorem hold for T3.

93

Chapter 6

Local Uniformization

Normal uniformization transformation sequences (NUTS) are defined in Definition

6.1 of [19]. Etale Perron transforms, and any birational extensions of normal local

rings dominated by a valuation are examples of NUTS.

Lemma 6.0.5. Suppose that K is an algebraic function field over a field k of

characteristic zero, V is a valuation ring of K, and R is an algebraic local ring of

K such that V dominates R. Let

(0) = P tV ⊂ · · · ⊂ P 1

V ⊂ P 0V = mV

be the chain of prime ideals in V . Let Vi = VP iV for 0 ≤ i, νi be a valuation

whose valuation ring is Vi. Then there exists a normal algebraic local ring S of K,

such that V dominates S and S dominates R, with the property that for all i, if

SP iV ∩S → S1 is a NUTS along νi, then σ(SP iV ∩S) = σ(S1) and τ(SP iV ∩S) = τ(S1).

Thus SP iV ∩S satisfies property (A) for all i. Further,

trdegQF (S1/mVi∩S1)QF (V/P iV ) = 0

for all i.

94

Proof. We choose λ1, . . . , λn ∈ V such that

trdegQF (R[λ1,...,λn]/P iV ∩R[λ1,...,λn])QF (V/P iV ) = 0

for all i. Let

S = R[λ1, . . . , λn]R[λ1,...,λn]∩mV .

By Theorem 6.3 [19], there exists a normal algebraic local ring S ofK, with quotient

field K such that V dominates S and S dominates S, with the property that for

all i, if SP iV ∩S → S1 is a NUTS along νi then σ(SP iV ∩S) = σ(S1). Since SP iV ∩S

dominates SP iV ∩S, we have that

trdegQF(S1/mVi∩S1)QF(V/P iV ) = trdegQF (S/P iV ∩S)

QF (V/P iV ) = 0,

and τ(S1) = τ(SP iV ∩S) for all i.

The following theorem is stronger than Zariski’s local uniformization theorem.

Theorem 6.0.6. Suppose that R is a local domain which is essentially of finite

type over a field k of characteristic zero, and ω is a valuation of the quotient field

of R which dominates R. Let W be the valuation ring of ω, and let

(0) = P tW ⊂ · · · ⊂ P 0

W = mW

be the chain of prime ideals of W . Then there exists a birational extension R→ R1

such that R1 is a regular local ring and W dominates R1. Further, P iR1

= P iW ∩R1

are regular primes for all i, and QR1Pi

R1

⊂ R1P iR1are regular primes for all i.

Proof. Let S be the extension of R satisfying the conclusions of Lemma 6.0.5. Since

S is essentially of finite type over k, there exists a regular local ring T , which is

95

essentially of finite type over k, and a prime ideal P in T such that T/P = S.

Let νt be the PTP adic valuation of the regular local ring TP , and let ν be the

composite of ω and νt. Let V be the valuation ring of ν.

trdegQF (T/P iT )QF (V/P iV ) = 0

for all i and TP iT satisfies property (A) for all i.

Let

(0) ⊂ P tV ⊂ · · · ⊂ P 0

V = mV

be the chain of prime ideals in V . We have V/P iV∼= W/P i

W for 0 ≤ i ≤ t. Let si be

the rational rank of (V/P i+1V )P iV for all i. Let τ(i) = dim(T/P i

V ∩T ). By adjoining

appropriate elements of k to S, we may further assume that T is a localization of

a polynomial ring, T = k[x1, . . . , xq](x1,...,xq) where xτ(i)+1, . . . , xτ(i)+si ∈ P iT −P i+1

T ,

for every i, and vi(xτ(i)+1), . . . , vi(xτ(i)+si) are rationally independent for all i.

By Theorem 5.2.4, There exists a sequence of transforms T → T1 such that T1

is a regular local ring, P iT1

are regular primes for all i, and TP ∼= TP tT∼= (T1)P tT1

.

Now by applying the methods of Chapter 4 and by Theorem 4.3.8, we achieve

Q (T1)PiT1

⊂ (T1)P iT1are regular primes for all i. Thus R → R1 = T1/P

tT1

is a

birational extension of R with the desired properties.

96

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102

VITA

Samar El Hitti is Lebanese born on September 24 1979, in Constantine, Algeria.

Her family moved back to Lebanon in 1980, where she grew up. There she com-

pleted her elementary and secondary education, and graduated high school with an

emphasis in Mathematics in 1997. She received her ”Maitrise es Sciences” degree in

Statistics and Probability in 2001 from the Lebanese University II, Beirut. In Win-

ter 2002 she began her Ph.D. studies at the University of Missouri - Columbia under

the supervision of Professor Steven Dale Cutkosky. Her graduation is planned on

May 2008.

103

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