# A counterexample to strong local monomialization in a tower of two
independent defect Artin-Schreier extensions

Steven Dale Cutkosky  Steven Dale Cutkosky, Department of Mathematics,
University of Missouri, Columbia, MO 65211, USA<EMAIL_ADDRESS>
###### Abstract.

We give an example of an extension of two dimensional regular local rings in a
tower of two independent defect Artin-Schreier extensions for which strong
local monomialization does not hold.

###### Key words and phrases:

valuation, positive characteristic, defect, strong monomialization

###### 2000 Mathematics Subject Classification:

primary 14B05; secondary 14B25, 13A18

Partially supported by NSF grant DMS-2054394

##  1\. Introduction

In characteristic zero, there is a very nice local form for morphisms, called
local monomialization. This result is a little stronger than what comes
immediately from the assumption that toroidalization is possible. If
$R\rightarrow S$ is an extension of local rings such that the maximal ideal of
$S$ contracts to the maximal ideal of $R$ then we say that $S$ dominates $R$.
If $S$ is dominated by the valuation ring $\mathcal{O}_{\omega}$ of a
valuation $\omega$ we say that $\omega$ dominates $S$.

######  Theorem 1.1.

(local monomialization)([2], [3]) Suppose that $k$ is a field of
characteristic zero and $R\rightarrow S$ is an extension of regular local
rings such that $R$ and $S$ are essentially of finite type over $k$ and
$\omega$ is a valuation of the quotient field of $S$ which dominates $S$ and
$S$ dominates $R$. Then there is a commutative diagram

$\begin{array}[]{ccc}R_{1}&\rightarrow&S_{1}\\\ \uparrow&&\uparrow\\\
R&\rightarrow&S\end{array}$

such that $\omega$ dominates $S_{1}$, $S_{1}$ dominates $R_{1}$ and the
vertical arrows are products of monoidal transforms; that is, these arrows are
factored by the local rings of blowups of prime ideals whose quotients are
regular local rings. In particular, $R_{1}$ and $S_{1}$ are regular local
rings. Further, $R_{1}\rightarrow S_{1}$ has a locally monomial form; that is,
there exist regular parameters $u_{1},\ldots,u_{m}$ in $R_{1}$ and
$x_{1},\ldots,x_{n}$ in $S_{1}$, an $m\times n$ matrix $A=(a_{ij})$ with
integral coefficients such that $\mbox{rank}(A)=m$ and units $\delta_{i}\in
S_{1}$ such that

$u_{i}=\delta_{i}\prod_{j=1}^{n}x_{j}^{a_{ij}}$

for $1\leq i\leq m$.

The difficulty in the proof is to obtain the condition that
$\mbox{rank}(A)=m$. To do this, it is necessary to blow up above both $R$ and
$S$.

In the case when the extension of quotient fields $K\rightarrow L$ of the
extension $R\rightarrow S$ is a finite extension and $k$ has characteristic
zero, it is possible to find a local monomialization such that the structure
of the matrix of coefficients recovers classical invariants of the extension
of valuations in $K\rightarrow L$, and this form holds stably along suitable
sequences of birational morphisms which generate the respective valuation
rings. This form is called strong local uniformization. It is established for
rank 1 valuations in [2] and for general valuations in [8]. The case which has
the simplest form and will be of interest to us in this paper is when the
valuation has rational rank 1. In this case, if $R_{1}\rightarrow S_{1}$ is a
strong local monomialization, then there exist regular parameters
$u_{1},\ldots,u_{m}$ in $R_{1}$ and $v_{1},\ldots,v_{m}$ in $S_{1}$ , a
positive integer $a$ and a unit $\delta\in S_{1}$ such that

(1) $u_{1}=\delta v_{1}^{a},u_{2}=v_{2},\ldots,u_{m}=v_{m}.$

The stable forms of mappings in positive characteristic and dimension $\geq 2$
are much more complicated. For instance, local monomialization does not always
hold. An example is given in [5] where $R\rightarrow S$ are local rings of
points on nonsingular algebraic surfaces over an algebraically closed field
$k$ of positive characteristic $p$ and $k(X)\rightarrow k(Y)$ is finite and
separable.

The obstruction to local monomialization is the defect. The defect
$\delta(\omega/\nu)$, which is a power of the residue characteristic $p$ of
$\mathcal{O}_{\omega}$, is defined and its basic properties developed in [21,
Chapter VI, Section 11], [12], [8, Section 7.1]. The defect is discussed in
Subsection 2.1. We have the following theorem, showing that the defect is the
only obstruction to strong local monomialization for maps of surfaces.

######  Theorem 1.2.

([8, Theorem 7.35]) Suppose that $K\rightarrow L$ is a finite, separable
extension of algebraic function fields over an algebraically closed field $k$
of characteristic $p>0$, $R\rightarrow S$ is an extension of local domains
such that $R$ and $S$ are essentially of finite type over $k$ and the quotient
fields of $R$ and $S$ are $K$ and $L$ respectively such that $S$ dominates
$R$. Suppose that $\omega$ is valuation of $L$ which dominates $S$. Let $\nu$
be the restriction of $\omega$ to $K$. Suppose that the extension is
defectless ($\delta(\omega/\nu)=1$). Then the conclusions of Theorem 1.1 hold.
In particular, $R\rightarrow S$ has a local monomialization (and a strong
local monomialization) along $\omega$.

Suppose that $K\rightarrow L$ is a Galois extension of fields of
characteristic $p>0$ and $\omega$ is a valuation of $L$, $\nu$ is the
restriction of $\omega$ to $K$. Then there is a classical tower of fields
([10, page 171])

$K\rightarrow K^{s}\rightarrow K^{i}\rightarrow K^{v}\rightarrow L.$

where $K^{s}$ is the splitting field, $K^{i}$ is the inertia field, $K^{v}$ is
the ramification field and the extension $K\rightarrow K^{v}$ has no defect.
Thus the essential difficulty comes from the extension from $K^{v}$ to $L$
which could have defect. The extension $K^{v}\rightarrow L$ is a tower of
Artin-Schreier extensions, so the Artin-Schreier extension is of fundamental
importance in this theory.

Kuhlmann has extensively studied defect in Artin-Schreier extensions in [13].
He separated these extensions into dependent and independent defect Artin-
Schreier extensions. This definition is reproduced in Subsection 2.4. Kuhlmann
also defined an invariant called the distance to distinguish the natures of
Artin-Schreier extensions. This definition is given in Subsections 2.3 and
2.4.

We now specialize to the case of a finite separable extension $K\rightarrow L$
of two dimensional algebraic function fields over an algebraically closed
field $k$ of characteristic $p>0$, and suppose that $\omega$ is a valuation of
$L$ which is trivial on $k$ and $\nu$ is the restriction of $\omega$ to $K$.
If $L/K$ has defect then $\omega$ must have rational rank 1 and be
nondiscrete. We will assume that $\omega$ has rational rank 1 and is
nondiscrete for the remainder of the introduction.

With these restrictions, the distance $\delta$ of an Artin-Schreier extension
is $\leq$ $0^{-}$ when the extension has defect. If it is a defect extension
with $\delta=0^{-}$ then it is an independent defect extension. If it is a
defect extension and the distance is less than $0^{-}$ then the extension is a
dependent defect extension.

A quadratic transform along a valuation is the center of the valuation at the
blow up of a maximal ideal of a regular local ring. There is the sequence of
quadratic transforms along $\nu$ and $\omega$

(2) $R\rightarrow R_{1}\rightarrow R_{2}\rightarrow\cdots\mbox{ and
}S\rightarrow S_{1}\rightarrow S_{2}\rightarrow\cdots.$

We have that $\cup_{i=1}^{\infty}R_{i}=\mathcal{O}_{\nu}$, the valuation ring
of $\nu$, and $\cup_{i=1}^{\infty}S_{i}=\mathcal{O}_{\omega}$, the valuation
ring of $\omega$. These sequences can be factored by standard quadratic
transform sequences (defined in Section 3). It is shown in [8] that given
positive integers $r_{0}$ and $s_{0}$, there exists $r\geq r_{0}$ and $s\geq
s_{0}$ such that $R_{r}\rightarrow S_{s}$ has the following form:

(3) $u=\delta x^{a},v=x^{b}(y^{d}\gamma+x\Omega)$

where $u,v$ are regular parameters in $R_{r}$, $x,y$ are regular parameters in
$S_{s}$, $\gamma$ and $\tau$ are units in $S_{s}$, $\Omega\in S_{s}$, $a$ and
$d$ are positive integers and $b$ is a non negative integer. If we choose
$r_{0}$ sufficiently large, then we have that the complexity $ad$ of the
extension $R_{r}\rightarrow S_{s}$ is a constant which depends on the
extension of valuations, which we call the stable complexity of (2). When
$R_{r}\rightarrow S_{s}$ has this stable complexity, we call the forms (3)
stable forms.

The strongly monomial form is the case when $b=0$ and $d=1$; that is, after
making a change of variables in $y$,

$u=\delta x^{a},v=y.$

As we observed earlier (Theorem 1.2) if the extension $K\rightarrow L$ has no
defect, then the stable form is the strongly monomial form. If there is
defect, then it is possible for the $a$ and $d$ in stable forms along a
valuation to vary wildly, even though their product $ad$ is fixed by the
extension, as shown in [6, Theorem 5.4].

An example is constructed in [8], showing failure of strong local
monomialization. It is a tower of two defect Artin-Schreier extensions, each
of the type of [6, Theorem 5.4] referred to above. The first extension is of
type 1 for even integers and of type 2 for odd integers. The second extension
is of type 2 for even integers and of type 1 for odd integers. The composite
gives a sequence of extensions of regular local rings $R_{i}\rightarrow
S_{i}$, where $R_{i}$ has regular parameters $u_{i},v_{i}$ and $S_{i}$ has
regular parameters $x_{i},y_{i}$ such that the stable form is

(4) $u_{i}=\gamma x_{i}^{p},v_{i}=y_{i}^{p}\tau+x_{i}\Omega$

for all $i$. Both of these Artin-Schreier extensions are dependent. This is
calculated in [11] and in [6, Section 6]. In keeping with the philosophy that
independent Artin-Schreier extensions are better behaved than dependent ones,
this leads to the question of if strong monomialization holds in towers of
independent Artin-Schreier extensions. However, this is not true as is shown
in Theorem 4.1 of this paper. In this theorem, we construct an example in a
tower of two independent defect extensions such that strong local
monomialization does not hold.

Suppose that $K\rightarrow L$ is a finite extension of fields of positive
characteristic and $\omega$ is a valuation of $L$ with restriction $\nu$ to
$K$. It is known that there is no defect in the extension if and only if there
is a finite generating sequence in $L$ for the valuation $\omega$ over $K$
([19], [16]). The calculation of generating sequences for extensions of
Noetherian local rings which are dominated by a valuation is extremely
difficult. This has been accomplished for two dimensional regular local rings
in [18] and [9] and for many hypersurface singularities above a regular local
ring of arbitrary dimension in [7].

The nature of a generating sequence in an extension of $S$ over $R$ determines
the nature of the mappings in the stable forms. It is shown in [4, Theorem 1]
that if $R\rightarrow S$ is an extension of two dimensional excellent regular
local rings whose quotient fields give a finite extension $K\rightarrow L$ and
$\omega$ is a valuation of $L$ which dominates $S$ then the extension is
without defect if and only if there exist sequences of quadratic transform
$R\rightarrow R_{1}$ and $S\rightarrow S_{1}$ along $\nu$ such that $\omega$
has a finite generating sequence in $S_{1}$ over $R_{1}$. This shows us that
we can expect good stable forms (as do hold by Theorem 1.2) if there is no
defect, but not otherwise.

##  2\. Preliminaries

###  2.1. Some notation

Let $K$ be a field with a valuation $\nu$. The valuation ring of $\nu$ will be
donoted by $\mathcal{O}_{\nu}$, $\nu K$ will denote the value group of $\nu$
and $K\nu$ will denote the residue field of $\mathcal{O}_{\nu}$.

The maximal ideal of a local ring $A$ will be denoted by $m_{A}$. If
$A\rightarrow B$ is an extension (inclusion) of local rings such that
$m_{B}\cap A=m_{A}$ we will say that $B$ dominates $A$. If a valuation ring
$\mathcal{O}_{\nu}$ dominates $A$ we will say that the valuation $\nu$
dominates $A$.

Suppose that $K$ is an algebraic function field over a field $k$. An algebraic
local ring $A$ of $K$ is a local domain which is a localization of a finite
type $k$-algebra whose quotient field is $K$. A $k$-valuation of $K$ is a
valuation of $K$ which is trivial on $k$.

Suppose that $K\rightarrow L$ is a finite algebraic extension of fields, $\nu$
is a valuation of $K$ and $\omega$ is an extension of $\nu$ to $L$. Then the
reduced ramification index of the extension is $e(\omega/\nu)=[\omega L:\nu
K]$ and the residue degree of the extension is $f(\omega/\nu)=[L\omega:K\nu]$.

The defect $\delta(\omega/\nu)$, which is a power of the residue
characteristic $p$ of $\mathcal{O}_{\omega}$, is defined and its basic
properties developed in [21, Chapter VI, Section 11], [12] and [8, Section
7.1]. In the case that $L$ is Galois over $K$, we have the formula

(5) $[L:K]=e(\omega/\nu)f(\omega/\nu)\delta(\omega/\nu)g$

where $g$ is the number of extensions of $\nu$ to $L$. In fact, we have the
equation (c.f. [13] or Section 7.1 [8])

$|G^{s}(\omega/\nu)|=e(\omega/\nu)f(\omega/\nu)\delta(\omega/\nu),$

where $G^{s}(\omega/\nu)$ is the decomposition group of $L/K$.

If $K\rightarrow L$ is a finite Galois extension, then we will denote the
Galois group of $L/K$ by $\mbox{Gal}(L/K)$.

###  2.2. Initial and final segments and cuts

We review some basic material about cuts in totally ordered sets from [13].
Let $(S,<)$ be a totally ordered set. An initial segment of $S$ is a subset
$\Lambda$ of $S$ such that if $\alpha\in\Lambda$ and $\beta<\alpha$ then
$\beta\in\Lambda$. A final segment of $S$ is a subset $\Lambda$ of $S$ such
that if $\alpha\in\Lambda$ and $\beta>\alpha$ then $\beta\in\Lambda$. A cut in
$S$ is a pair of sets $(\Lambda^{L},\Lambda^{R})$ such that $\Lambda^{L}$ is
an initial segment of $S$ and $\Lambda^{R}$ is a final segment of $S$
satisfying $\Lambda^{L}\cup\Lambda^{R}=S$ and
$\Lambda^{L}\cap\Lambda^{R}=\emptyset$. If $\Lambda_{1}$ and $\Lambda_{2}$ are
two cuts in $S$, write $\Lambda_{1}<\Lambda_{2}$ if
$\Lambda_{1}^{L}\subsetneq\Lambda_{2}^{L}$. Suppose that $S\subset T$ is an
order preserving inclusion of ordered sets and
$\Lambda=(\Lambda^{L},\Lambda^{R})$ is a cut in $S$. Then define the cut
induced by $\Lambda=(\Lambda^{L},\Lambda^{R})$ in $T$ to be the cut
$\Lambda\uparrow T=(\Lambda^{L}\uparrow T,T\setminus(\Lambda^{L}\uparrow T))$
where $\Lambda^{L}\uparrow T$ is the least initial segment of $T$ in which
$\Lambda^{L}$ forms a cofinal subset.

We embed $S$ in the set of all cuts of $S$ by sending $s\in S$ to

$s^{+}=(\\{t\in S\mid t\leq s\\},\\{t\in S\mid t>s\\}).$

we may identify $s$ with the cut $s^{+}$. Define

$s^{-}=(\\{t\in S\mid t<s\\},\\{t\in S\mid t\geq s\\}).$

Given a cut $\Lambda=(\Lambda^{L},\Lambda^{R})$, we define
$-\Lambda=(-\Lambda^{R},-\Lambda^{L})$ where $-\Lambda^{L}=\\{-s\mid
s\in\Lambda^{L}\\}$ and $-\Lambda^{R}=\\{-s\mid s\in\Lambda^{R}\\}$. We have
that if $\Lambda_{1}$ and $\Lambda_{2}$ are cuts, then
$\Lambda_{1}<\Lambda_{2}$ if and only if $-\Lambda_{2}<-\Lambda_{1}$.

Observe that for $s\in S$, $-s=-(s^{+})=(-s)^{-}$ and $-(s^{-})=(-s)^{+}=-s$.

###  2.3. Distances

Let $K\rightarrow L$ be an extension of fields and $\omega$ be a valuation of
$L$ with restriction $\nu$ to $K$. Let $\widetilde{\nu K}$ be the divisible
hull of $\nu K$. Suppose that $z\in L$. Then the distance of $z$ from $K$ is
defined in [13, Section 2.3] to be the cut $\mbox{dist}(z,K)$ of
$\widetilde{\nu K}$ in which the initial segment of $\mbox{dist}(z,K)$ is the
least initial segment of $\widetilde{\nu K}$ in which $\omega(z-K)$ is
cofinal. That is,

$\mbox{dist}(z,K)=(\Lambda^{L}(z,K),\Lambda^{R}(z,K))\uparrow\widetilde{\nu
K}$

where

$\Lambda^{L}(z,K)=\\{\omega(z-c)\mid c\in K\mbox{ and }\omega(z-c)\in\nu
K\\}.$

The following notion of equivalence is defined in [13, Section 2.3]. If
$y,z\in L$, then $z\sim_{K}y$ if $\omega(z-y)>\mbox{dist}(z,K)$.

###  2.4. Artin-Schreier extensions

Let $K\rightarrow L$ be an Artin-Schreier extension of fields of
characteristic $p>0$ and $\omega$ be a valuation of $L$ with restriction $\nu$
to $K$. The field $L$ is Galois over $K$ with Galois group
$G\cong{\mathbb{Z}}_{p}$, where $p$ is the characteristic of $K$.

Let $\Theta\in L$ be an Artin-Schreier generator of $K$; that is, there is an
expression

$\Theta^{p}-\Theta=a$

for some $a\in K$. We have that

$\mbox{Gal}(L/K)\cong{\mathbb{Z}}_{p}=\\{{\rm
id},\sigma_{1},\ldots,\sigma_{p-1}\\},$

where $\sigma_{i}(\Theta)=\Theta+i$.

Since $L$ is Galois over $K$, we have that
$ge(\omega/\nu)f(\omega/\nu)\delta(\omega/\nu)=p$ where $g$ is the number of
extensions of $\nu$ to $L$. So we either have that $g=1$ or $g=p$. If $g=1$,
then $\omega$ is the unique extension of $\nu$ to $L$ and either
$e(\omega/\nu)=p$ and $\delta(\omega/\nu)=1$ or $e(\omega/\nu)=1$ and
$\delta(\omega/\nu)=p$. In particular, the extension is defect if and only if
is an immediate extension ($e=f=1$) and $\omega$ is the unique extension of
$\nu$ to $L$.

From now on in this subsection, suppose that $L$ is a defect extension of $K$.
By [13, Lemma 4.1], the distance $\delta=\mbox{dist}(\Theta,K)$ does not
depend on the choice of Artin-Schreier generator $\Theta$, so $\delta$ can be
called the distance of the Artin-Schreier extension. Since $L/K$ is an
immediate extension, the set $\omega(\Theta-K)$ is an initial segment in $\nu
K$ which has no maximal element by [13, Theorem 2.19].

We have, since the extension is defect, that

(6) $\delta={\rm dist}(\Theta,K)\leq 0^{-}$

by [13, Corollary 2.30].

A defect Artin-Schreier extension $L$ is defined in [13, Section 4] to be a
dependent defect Artin-Schreier extension if there exists an immediate purely
inseparable extension $K(\eta)$ of $K$ of degree $p$ such that
$\eta\sim_{K}\Theta$. Otherwise, $L/K$ is defined to be an independent defect
Artin-Schreier defect extension. We have by [13, Proposition 4.2] that for a
defect Artin- Schreier extension,

(7) $L/K$ is independent if and only if the distance $\delta={\rm
dist}(\Theta,K)$ satisfies $\delta=p\delta$.

###  2.5. Extensions of rank 1 valuations in an Artin-Schreier extension

In this subsection, we suppose that $L$ is an Artin-Schreier extension of a
field $K$ of characteristic $p$, $\omega$ is a rank 1 valuation of $L$ and
$\nu$ is the restriction of $\omega$ to $K$. We suppose that $L$ is a defect
extension of $K$. To simplify notation, we suppose that we have an embedding
of $\nu L$ in ${\mathbb{R}}$. Since $L$ has defect over $K$ and $L$ is
separable over $K$, $\nu L$ is nondiscrete by the corollary on page 287 of
[20], so that $\nu L$ is dense in ${\mathbb{R}}$.

We define a cut in ${\mathbb{R}}$ by extending the cut $\mbox{dist}(\Theta,K)$
in $\widetilde{\nu K}$ to a cut of ${\mathbb{R}}$ by taking the initial
segment of the extended cut to be the least initial segment of ${\mathbb{R}}$
in which the cut $\mbox{dist}(\Theta,K)$ is confinal. This cut is then
$\mbox{dist}(\Theta,K)\uparrow{\mathbb{R}}$. This cut is either $s$ or $s^{-}$
for some $s\in{\mathbb{R}}$. If $L$ is a defect extension of $K$ then
$\mbox{dist}(\Theta,K)\uparrow{\mathbb{R}}=s^{-}$ where $s$ is a non positive
real number by [13, Theorem 2.19] and [13, Corollary 2.30]. We will set
$\mbox{dist}(\omega/\nu)$ to be this real number $s$, so that

$\mbox{dist}(\Theta,K)\uparrow{\mathbb{R}}=s^{-}=(\mbox{dist}(\omega/\nu))^{-}.$

The real number $\mbox{dist}(\omega/\nu)$ is well defined since it is
independent of choice of Artin-Schreier generator of $L/K$ by Lemma 4.1 [13].

With the assumptions of this subsection, by (6) and (7), the distance
$\delta=\mbox{dist}(\Theta,K)$ of an Artin-Schreier extension is $\leq$
$0^{-}$ when the extension has defect. If it is a defect extension with
distance equal to $0^{-}$ then it is an independent defect extension. If it is
a defect extensions and the distance is less than $0^{-}$ then the extension
is a dependent defect extension. Thus if $L/K$ is a defect extension, we have
that $\mbox{dist}(\omega/\nu)\leq 0$ and the defect extension $L/K$ is
independent if and only if $\mbox{dist}(\omega/\nu)=0$.

##  3\. Calculations in two dimensional Artin-Schreier Extensions

Suppose that $M$ is a two dimensional algebraic function field over an
algebraically closed field $k$ of characteristic $p>0$ and $\mu$ is a
nondiscrete rational rank 1 valuation of $M$. Suppose that $A$ is an algebraic
regular local ring of $M$ such that $\mu$ dominates $A$. A quadratic transform
of $A$ is an extension $A\rightarrow A_{1}$ where $A_{1}$ is a local ring of
the blowup of the maximal ideal of $A$ such that $A_{1}$ dominates $A$ and
$A_{1}$ has dimension two. A quadratic transform $A\rightarrow A_{1}$ is said
to be along the valuation $\mu$ if $\mu$ dominates $A_{1}$.

Let

$A=A_{0}\rightarrow A_{1}\rightarrow A_{2}\rightarrow\cdots$

be the sequence of quadratic transforms along $\mu$. Then the valuation ring
$\mathcal{O}_{\mu}=\cup A_{i}$ (by [1, Lemma 12]).

Suppose that $K\rightarrow L$ is a finite extension of two dimensional
algebraic function fields, $R$ is an algebraic regular local ring of $K$ which
is dominated by a regular algebraic local ring $S$ of $L$ such that $\dim
R=\dim S=2$. Let $x,y$ be regular parameters in $S$ and $u,v$ be regular
parameters in $R$. Then we can form the Jacobian ideal

$J(S/R)=(\frac{\partial u}{\partial x}\frac{\partial v}{\partial
y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}).$

This ideal is independent of choice of regular parameters.

The following proposition is proven in [17].

######  Proposition 3.1.

Suppose that $K\rightarrow L$ is an Artin-Schreier extension of two
dimensional algebraic function fields over an algebraically closed field $k$
of characteristic $p>0$, $\omega$ is a rational rank 1 nondiscrete valuation
of $L$ with restriction $\nu=\omega|K$. Further suppose that $A$ is an
algebraic local ring of $K$ and $B$ is an algebraic local ring of $L$ which is
dominated by $\omega$ such that $B$ dominates $A$. Then there exists a
commutative diagram of homomorphisms

$\begin{array}[]{ccc}R&\rightarrow&S\\\ \uparrow&&\uparrow\\\
A&\rightarrow&B\end{array}$

such that $R$ is a regular algebraic local ring of $K$ with regular parameters
$u,v$, $S$ is a regular algebraic local ring of $L$ with regular parameters
$x,y$ such that $S$ is dominated by $\omega$, $S$ dominates $R$, $R\rightarrow
S$ is quasi finite, $J(S/R)=(x^{\overline{c}})$ for some non negative integer
$\overline{c}$ and one of the following three cases holds:

  1. 0)

$u=x$, $v=y$ ($R\rightarrow S$ is unramified).

  2. 1)

$u=x$, $v=y^{p}\gamma+x\Sigma$ where $\gamma$ is a unit in $S$ and $\Sigma\in
S$.

  3. 2)

$u=\gamma x^{p}$, $v=y$ where $\gamma$ is a unit in $S$.

Let $K\rightarrow L$ be an Artin-Schreier extension of two dimensional
algebraic function fields over an algebraically closed field $k$ of
characteristic $p>0$. Let $R\rightarrow S$ be an extension from a regular
algebraic local ring of $K$ to a regular algebraic local ring of $L$ such that
$S$ dominates $R$.

Let $u,v$ be regular parameters in $R$ and $x,y$ be regular parameters in $S$.
We will say that $R\rightarrow S$ is of type 0 with respect to these
parameters if

$\mbox{Type 0:}\,\,\,\,\,u=\gamma x,v=y\tau+x\Omega$

where $\gamma,\tau$ are units in $S$ and $\Omega\in S$, so that $R\rightarrow
S$ is unramified. We will say that $R\rightarrow S$ is of type 1 with respect
to these parameters if

$\mbox{Type 1:}\,\,\,\,\,u=\gamma x,v=y^{p}\tau+x\Omega$

where $\gamma,\tau$ are units in $S$ and $\Omega\in S$. We will say that
$R\rightarrow S$ is of type 2 with respect to these parameters if

$\mbox{Type 2:}\,\,\,\,\,u=\gamma x^{p},v=y\tau+x\Omega$

where $\gamma,\tau$ are units in $S$ and $\Omega\in S$.

These definitions are such that if one these types hold, and
$\overline{u},\overline{v}$ are regular parameters in $R$,
$\overline{x},\overline{y}$ are regular parameters in $S$ such that
$\overline{u}$ is a unit in $R$ times $u$ and $\overline{x}$ is a unit in $S$
times $x$ then $R\rightarrow S$ is of the same type for the new parameters
$\overline{u},\overline{v}$ and $\overline{x},\overline{y}$.

In the construction of our example (Theorem 4.1), we will make use of some
results from [6].

######  Theorem 3.2.

([6, Theorem 4.1]) Suppose that $R\rightarrow S$ is of type 1 with respect to
regular parameters $x,y$ in $S$ and $u,v$ in $R$ and that
$J(S/R)=(x^{\overline{c}})$. Let $\overline{x}=u$,
$\overline{y}=y-g(\overline{x})$ where $g(\overline{x})\in k[\overline{x}]$ is
a polynomial with zero constant term, so that $\overline{x},\overline{y}$ are
regular parameters in $S$. Computing the Jacobian determinate $J(S/R)$, we see
that

$u=\overline{x},v=\overline{y}^{p}\gamma+\overline{x}^{\overline{c}}\overline{y}\tau+f(\overline{x})$

where $\gamma,\tau$ are unit series in $\hat{S}$ and $f(\overline{x})=\sum
e_{i}\overline{x}^{i}\in k[[\overline{x}]]$. Make the change of variables
$\overline{v}=v-\sum e_{i}u^{i}$ where the sum is over $i$ such that
$i\leq\frac{pq}{m}$ so that $u,\overline{v}$ are regular parameters in $R$.

Suppose that $m,q$ are positive integers with $m>1$ and $\mbox{gcd}(m,q)=1$.
Let $\alpha$ be a nonzero element of $k$. Consider the sequence of quadratic
transforms $S\rightarrow S_{1}$ so that $S_{1}$ has regular parameters
$x_{1},y_{1}$ defined by

$\overline{x}=x_{1}^{m}(y_{1}+\alpha)^{a^{\prime}},\overline{y}=x_{1}^{q}(y_{1}+\alpha)^{b^{\prime}}$

where $a^{\prime},b^{\prime}\in{\mathbb{N}}$ are such that
$mb^{\prime}-qa^{\prime}=1$.

We have that $R\rightarrow S$ is of type 1 with respect to the regular
parameters $\overline{x},\overline{y}$ and $u,v$. Let
$\sigma=\mbox{gcd}(m,pq)$ which is 1 or $p$.

There exists a unique sequence of quadratic transforms $R\rightarrow R_{1}$
such that $R_{1}$ has regular parameters $u_{1},v_{1}$ defined by

$u=u_{1}^{\overline{m}}(v_{1}+\beta)^{c^{\prime}},\overline{v}=u_{1}^{\overline{q}}(v_{1}+\beta)^{d^{\prime}}$

with $0\neq\beta\in k$ giving a commutative diagram of homomorphisms

$\begin{array}[]{lll}R_{1}&\rightarrow&S_{1}\\\ \uparrow&&\uparrow\\\
R&\rightarrow&S\end{array}$

such that $R_{1}\rightarrow S_{1}$ is quasi finite. We have that
$J(S_{1}/R_{1})=(x_{1}^{c_{1}})$ for some positive integer $c_{1}$ and
$R_{1}\rightarrow S_{1}$ is quasi finite. Further:

  1. 0)

If $\frac{q}{m}\geq\frac{\overline{c}}{p-1}$ then $R_{1}\rightarrow S_{1}$ is
of type 0.

  2. 1)

If $\frac{q}{m}<\frac{\overline{c}}{p-1}$ and $\sigma=1$ then
$R_{1}\rightarrow S_{1}$ is of type 1 and

$\left(\frac{c_{1}}{p-1}\right)=\left(\frac{\overline{c}}{p-1}\right)m-q.$

  3. 2)

If $\frac{q}{m}<\frac{\overline{c}}{p-1}$ and $\sigma=p$ then
$R_{1}\rightarrow S_{1}$ is of type 2 and

$\left(\frac{c_{1}}{p-1}\right)=\left(\frac{\overline{c}}{p-1}\right)m-q+1.$

In cases 1) and 2), $m=\sigma\overline{m}$, $pq=\sigma\overline{q}$ and
$\overline{m}c^{\prime}-\overline{q}d^{\prime}=1$.

######  Theorem 3.3.

([6, Theorem 4.3]) Suppose that $R\rightarrow S$ is of type 2 with respect to
regular parameters $x,y$ in $S$ and $u,v$ in $R$ and that
$J(S/R)=(x^{\overline{c}})$. Let $g(u)\in k[u]$ be a polynomial with no
constant term. Make the change of variables, letting $\overline{v}=v-g(u)$ and
$\overline{y}=\overline{v}$, so that $x,\overline{y}$ are regular parameters
in $S$ and $u,\overline{v}$ are regular parameters in $R$.

Suppose that $m,q$ are positive integers with $\mbox{gcd}(m,q)=1$. Let
$\alpha$ be a nonzero element of $k$. Consider the sequence of quadratic
transforms $S\rightarrow S_{1}$ so that $S_{1}$ has regular parameters
$x_{1},y_{1}$ defined by

$x=x_{1}^{m}(y_{1}+\alpha)^{a^{\prime}},\overline{y}=x_{1}^{q}(y_{1}+\alpha)^{b^{\prime}}$

where $a^{\prime},b^{\prime}\in{\mathbb{N}}$ are such that
$mb^{\prime}-qa^{\prime}=1$.

Let $\sigma=\mbox{gcd}(pm,q)$ which is 1 or $p$. There exists a unique
sequence of quadratic transforms $R\rightarrow R_{1}$ such that $R_{1}$ has
regular parameters $u_{1},v_{1}$ defined by

$u=u_{1}^{\overline{m}}(v_{1}+\beta)^{c^{\prime}},\overline{v}=u_{1}^{\overline{q}}(v_{1}+\beta)^{d^{\prime}}$

where $pm=\sigma\overline{m}$, $q=\sigma\overline{q}$,
$\overline{m}d^{\prime}-c^{\prime}\overline{q}=1$ and $0\neq\beta\in k$,
giving a commutative diagram of homomorphisms

$\begin{array}[]{lll}R_{1}&\rightarrow&S_{1}\\\ \uparrow&&\uparrow\\\
R&\rightarrow&S\end{array}$

such that $R_{1}\rightarrow S_{1}$ is quasi finite. We have that
$J(S_{1}/R_{1})=(x_{1}^{c_{1}})$ for some positive integer $c_{1}$. Further:

  1. 1)

If $\sigma=1$ then $R_{1}\rightarrow S_{1}$ is of type 1 and

$\left(\frac{c_{1}}{p-1}\right)=\left(\frac{\overline{c}}{p-1}\right)m-m.$

  2. 2)

If $\sigma=p$ then $R_{1}\rightarrow S_{1}$ is of type 2 and

$\left(\frac{c_{1}}{p-1}\right)=\left(\frac{\overline{c}}{p-1}\right)m-m+1.$

A proof of the following proposition is given in [6, Proposition 7.9]. More
general results are proven in [15].

######  Proposition 3.4.

(Kuhlmann and Piltant, [14]) Suppose that $K$ and $L$ are two dimensional
algebraic function fields over an algebraically closed field $k$ of
characteristic $p>0$ and $K\rightarrow L$ is an Artin-Schreier extension. Let
$\omega$ be a rational rank one nondiscrete valuation of $L$ and let $\nu$ be
the restriction of $\omega$ to $K$. Suppose that $L$ is a defect extension of
$K$.

Suppose that $R$ is a regular algebraic local ring of $K$ and $S$ is a regular
algebraic local ring of $L$ such that $\omega$ dominates $S$, $S$ dominates
$R$ and $R\rightarrow S$ is of type 1 or 2. Inductively applying Theorems 3.2
and 3.3, we construct a diagram where the horizontal sequences are birational
extensions of regular local rings

(8)
$\begin{array}[]{ccccccc}S=S_{0}&\rightarrow&S_{1}&\rightarrow&S_{2}&\rightarrow&\cdots\\\
\uparrow&&\uparrow&&\uparrow&&\\\
R=R_{0}&\rightarrow&R_{1}&\rightarrow&R_{2}&\rightarrow&\cdots\end{array}$

with $\cup_{i=1}^{\infty}S_{i}=\mathcal{O}_{\omega}$. Further assume that for
each map $R_{i}\rightarrow S_{i}$, there are regular parameters $u,v$ in
$R_{i}$ and $x,y$ in $S_{i}$ such that one of the following forms hold:

(9) $u=x,v=f$

where $\dim_{k}S_{i}/(x,f)=p$, or

(10) $u=\delta x^{p},v=y$

where $\delta$ is a unit in $S_{i}$ and in both cases that $x=0$ is a local
equation of the critical locus of
$\mbox{Spec}(S_{i})\rightarrow\mbox{Spec}(R_{i})$. Let

$J_{i}=J(S_{i}/R_{i})=(\frac{\partial u}{\partial x}\frac{\partial v}{\partial
y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x})$

be the Jacobian ideal of the map $R_{i}\rightarrow S_{i}$.

Then the distance $\mbox{dist}(\omega/\nu)$ is computed by the formula

$-{\rm dist}(\omega/\nu)=\frac{1}{p-1}\inf_{i}\\{\omega(J(S_{i}/R_{i}))\\}$

where the infimum is over the $R_{i}\rightarrow S_{i}$ in the sequence (8).

##  4\. An example of a tower of independent defect extensions in which strong
local monomialization doesn’t hold

######  Theorem 4.1.

There exists a tower $(K,\nu)\rightarrow(L,\omega)\rightarrow(M,\mu)$ of
independent defect Artin-Schreier extensions of valued two dimensional
algebraic function fields over an algebraically closed field $k$ of
characteristic $p>0$ such that there exist algebraic regular local rings $A$
of $K$ and $C$ of $M$ such that $\mu$ dominates $C$ and $C$ dominates $A$ but
strong local monomialization along $\mu$ does not hold above $A\rightarrow C$.

######  Remark 4.2.

Let $\delta\in{\mathbb{R}}_{\geq 0}$ be a fixed ratio. Suppose that
$R\rightarrow S$ is of type 1. By taking $m$ and $q$ sufficiently large in
Theorem 3.2 such that $R_{1}\rightarrow S_{1}$ is of type 2, we can achieve
that $v_{1}=\lambda y_{1}+g(x_{1})$ where $\lambda$ is a unit in $S_{1}$ and
the order of $g(x_{1})$ is arbitrarily large. Suppose that $R\rightarrow S$ is
of type 2. By taking $m$ and $q$ sufficiently large in Theorem 3.3 such that
$R_{1}\rightarrow S_{1}$ is of type 1 we can achieve that
$v_{1}=y_{1}^{p}\gamma+x_{1}^{c_{1}}y_{1}\tau+f(x_{1})$ where $\gamma$ and
$\tau$ are unit series in $S_{1}$ and the order of $f(x_{1})$ is arbitrarily
large. In both cases, we can choose $m$ and $q$ so that $\frac{q}{m}$ is
arbitrarily close to $\delta$.

######  Remark 4.3.

In Theorem 3.3, we have an expression $\overline{v}=\tau y+f(x)$ where $\tau$
is a unit in $S$. Suppose that $m$ and $q$ are positive integers with
$\mbox{gcd}(m,q)=1$ and such that $\mbox{ord }f(x)>\frac{q}{m}$. Then the
proof of Theorem 3.3 extends to show that the conclusions of Theorem 3.3 hold
with $\overline{y}$ replaced with $y$.

We now give the proof of Theorem 4.1.

###### Proof.

Let $K$ be a two dimensional algebraic function field over an algebraically
closed field, and let $R_{-2}$ be a two dimensional algebraic regular local
ring of $K$. Let $u_{-2},v_{-2}$ be regular parameters in $R_{-2}$.

Let $e$ be a positive integer. Let $c_{-2}=(p-1)e$. Let $\Theta$ be a root of
the Artin-Schreier polynomial $X^{p}-X-v_{-2}u_{-2}^{-pe}$. Let $L=K(\Theta)$.
Set $x_{-2}=u_{-2}$, $y_{-2}=u_{-2}^{e}\Theta$. Let
$S_{-2}=R_{-2}[y_{-2}]_{(x_{-2},y_{-2})}$, which is an algebraic regular local
ring of $L$ which dominates $R_{-2}$. The regular parameters $x_{-2},y_{-2}$
in $S_{-2}$ satisfy $u_{-2}=x_{-2},v_{-2}=y_{-2}^{p}-x_{-2}^{e(p-1)}y_{-2}$,
so that the extension $R_{-2}\rightarrow S_{-2}$ is of type 1. We have that
$J(S_{-2}/R_{-2})=(x_{-2}^{c_{-2}})$, with $\frac{c_{-2}}{p-1}>0$.

We first construct a commutative diagram

$\begin{array}[]{lll}S_{-2}&\rightarrow&S_{-1}\\\ \uparrow&&\uparrow\\\
R_{-2}&\rightarrow&R_{-1}\end{array}$

using Theorem 3.2 so that $R_{-1}\rightarrow S_{-1}$ is of type 2. Let
$\Sigma$ be a root of the Artin-Schreier polynomial
$X^{p}-X-y_{-1}x_{-1}^{-pe}$. Let $M=L(\Sigma)$. Set $z_{-1}=x_{-1}$,
$w_{-1}=x_{-1}^{e}\Sigma$. Let $T_{-1}=S_{-1}[w_{-1}]_{(z_{-1},w_{-1})}$,
which is an algebraic regular local ring of $M$ which dominates $S_{-1}$. The
regular parameters $z_{-1},w_{-1}$ in $T_{-1}$ satisfy
$x_{-1}=z_{-1},y_{-1}=w_{-1}^{p}-z_{-1}^{e(p-1)}w_{-1}$, so that the extension
$S_{-1}\rightarrow T_{-1}$ is of type 1. We have that
$J(T_{-1}/S_{-1})=(z_{-1}^{c_{-1}^{\prime}})$, with
$\frac{c_{-1}^{\prime}}{p-1}>0$.

From Theorems 3.2 and 3.3, we construct

$\begin{array}[]{ccc}T_{-1}&\rightarrow&T_{0}\\\ \uparrow&&\uparrow\\\
S_{-1}&\rightarrow&S_{0}\\\ \uparrow&&\uparrow\\\ R_{-1}&\rightarrow&R_{0}\\\
\end{array}$

such that $R_{0}\rightarrow S_{0}$ is of type 1 and $S_{0}\rightarrow T_{0}$
is of type 2. Explicitely, $R_{-1},R_{0},S_{-1},S_{0},T_{-1},T_{0}$ have
respective regular parameters $(u_{-1},v_{-1})$, $(u_{0},v_{0})$,
$(x_{-1},y_{-1})$, $(x_{0},y_{0})$ and $(z_{-1},w_{-1})$, $(z_{0},w_{0})$
which are related by equations

$\begin{array}[]{l}u_{-1}=u_{0}^{pm_{0}}(v_{0}+\beta_{0})^{d_{0}^{\prime}},v_{-1}=u_{0}^{q_{0}}(v_{0}+\beta_{0})^{e_{0}^{\prime}}\\\
x_{-1}=x_{0}^{m_{0}}(y_{0}+\alpha_{0})^{a_{0}^{\prime}},y_{-1}=x_{0}^{q_{0}}(w_{0}+\alpha_{0})^{g_{0}^{\prime}}\\\
z_{-1}=z_{0}^{pm_{0}}(v_{0}+\gamma_{0})^{f_{0}^{\prime}},w_{-1}=z_{0}^{q_{0}}(w_{0}+\gamma_{0})^{g_{0}^{\prime}}\end{array}$

where $p\not|q_{0}$ and $\frac{q_{0}}{pm_{0}}<\frac{c_{-1}^{\prime}}{p-1}$
where $J(T_{-1}/S_{-1})=(z_{-1}^{c_{-1}^{\prime}})$.

By Remarks 4.2 and 4.3, we can construct $R_{0}\rightarrow S_{0}\rightarrow
T_{0}$ so that we have expressions $y_{0}=\lambda_{0}w_{0}+g_{0}(z_{0})$ where
$\lambda_{0}$ is a unit in $T_{0}$ and $\mbox{ord }g_{0}(z_{0})$ is
arbitrarily large and
$v_{0}=\sigma_{0}y_{0}^{p}+\tau_{0}x_{0}^{c_{0}}y_{0}+f_{0}(x_{0})$ where
$\sigma_{0},\tau_{0}$ are units in $S_{0}$ and $\mbox{ord }f_{0}(x_{0})$ is
arbitrarily large.

We will inductively construct a commutative diagram within $K\rightarrow
L\rightarrow M$ of two dimensional regular algebraic local rings

(11)
$\begin{array}[]{ccccccc}T_{0}&\rightarrow&T_{1}&\rightarrow&T_{2}&\rightarrow&\cdots\\\
\uparrow&&\uparrow&&\uparrow&&\\\
S_{0}&\rightarrow&S_{1}&\rightarrow&S_{2}&\rightarrow&\cdots\\\
\uparrow&&\uparrow&&\uparrow&&\\\
R_{0}&\rightarrow&R_{1}&\rightarrow&R_{2}&\rightarrow&\cdots\\\ \end{array}$

such that $R_{i}\rightarrow S_{i}$ is of type 1 if $i$ is even and is of type
2 if $i$ is odd, $S_{i}\rightarrow T_{i}$ is of type 2 if $i$ is even and is
of type 1 if $i$ is odd. Further, valuations $\nu$, $\omega$ and $\mu$ of the
respective function fields $K$, $L$ and $M$ determined by these sequences are
such that $K\rightarrow L$ and $L\rightarrow M$ are independent defect
extensions. We will have that $R_{i}$ has regular parameters $(u_{i},v_{i})$,
$S_{i}$ has regular parameters $(x_{i},y_{i})$ and $T_{i}$ has regular
parameters $(z_{i},w_{i})$ such that

$u_{i}=u_{i+1}^{\overline{m}_{i+1}}(v_{i+1}+\beta_{i+1})^{d_{i+1}^{\prime}},v_{i}=u_{i+1}^{\overline{q}_{i+1}}(v_{i+1}+\beta_{i+1})^{e_{i+1}^{\prime}},$
$x_{i}=x_{i+1}^{m_{i+1}}(y_{i+1}+\alpha_{i+1})^{a_{i+1}^{\prime}},y_{i}=x_{i+1}^{q_{i+1}}(y_{i+1}+\alpha_{i+1})^{b_{i+1}^{\prime}},$
$z_{i}=z_{i+1}^{m_{i+1}^{\prime}}(w_{i+1}+\gamma_{i+1})^{f_{i+1}^{\prime}},w_{i}=z_{i+1}^{q_{i+1}^{\prime}}(w_{i}+\gamma_{i+1})^{g_{i+1}^{\prime}}$

with $\overline{m}_{i},m_{i}$ and $m_{i}^{\prime}$ larger than 1 for all $i$.

Let $J(S_{i}/R_{i})=(x_{i}^{c_{i}})$ and
$J(T_{i}/S_{i})=(z_{i}^{c_{i}^{\prime}})$.

If $i$ is even, then
$m_{i+1}=p\overline{m}_{i+1},m_{i+1}^{\prime}=\overline{m}_{i+1},q_{i+1}=\overline{q}_{i+1},q_{i+1}^{\prime}=q_{i+1}$
and

$\frac{q_{i+1}}{m_{i+1}}<\frac{c_{i}}{p-1}.$

If $i$ is odd, then
$\overline{m}_{i+1}=pm_{i+1},m_{i+1}^{\prime}=\overline{m}_{i+1},q_{i+1}=\overline{q}_{i+1},q_{i+1}^{\prime}=q_{i+1}$
and

$\frac{q_{i+1}^{\prime}}{m_{i+1}^{\prime}}<\frac{c_{i}^{\prime}}{p-1}.$

In our construction, if $r$ is even, we will have that

(12) $y_{r}=\lambda_{r}w_{r}+g_{r}(z_{r})$

where $\lambda_{r}$ is a unit in $T_{r}$ and $\mbox{ord }g_{r}(z_{r})$ is
arbitrarily large and

(13) $v_{r}=\sigma_{r}y_{r}^{p}+\tau_{r}x_{r}^{c_{r}}y_{r}+f_{r}(x_{r})$

where $\sigma_{r},\tau_{r}$ are units in $S_{r}$ and $\mbox{ord }f_{r}(x_{r})$
is arbitrarily large. If $r$ is even, we will have

(14) $y_{r}=\sigma_{r}w_{r}^{p}+\tau_{r}z_{r}^{c_{r}^{\prime}}w_{r}+f(z_{r})$

where $\sigma_{r},\tau_{r}$ are units in $T_{r}$ and $\mbox{ord }f(z_{r})$ is
arbitrarily large and

(15) $v_{r}=\lambda_{r}y_{r}+g_{r}(x_{r})$

where $\lambda_{r}$ is a unit in $S_{r}$ and $\mbox{ord }g_{r}(x_{r})$ is
arbitrarily large.

Suppose that $r$ is even, and we have constructed $R_{r}\rightarrow
S_{r}\rightarrow T_{r}$. We will construct

$\begin{array}[]{ccccc}T_{r}&\rightarrow&T_{r+1}&\rightarrow&T_{r+2}\\\
\uparrow&&\uparrow&&\uparrow\\\
S_{r}&\rightarrow&S_{r+1}&\rightarrow&S_{r+2}\\\
\uparrow&&\uparrow&&\uparrow\\\
R_{r}&\rightarrow&R_{r+1}&\rightarrow&R_{r+2}.\end{array}$

There exists an integer $\lambda(r+1)>1$ and $q_{r+1}\in{\mathbb{Z}}_{+}$ such
that $\mbox{gcd}(q_{r+1},p)=1$ and

(16)
$\frac{c_{r}}{p-1}>\frac{q_{r+1}}{p^{\lambda(r+1)}}>\frac{c_{r}}{p-1}-\frac{1}{2^{r+1}}m_{1}\cdots
m_{r}.$

In fact, we can find $\lambda(r+1)$ arbitrarily large satisfying the
inequality. Set $m_{r+1}=p^{\lambda(r+1)}$. We have that
$\frac{q_{r+1}}{m_{r+1}}<\frac{c_{r}}{p-1}$ with
$\mbox{gcd}(m_{r+1},pq_{r+1})=p$. This choice of $m_{r+1}$ and $q_{r+1}$
(along with a choice of $0\neq\alpha_{r+1}\in k$) determines $S_{r}\rightarrow
S_{r+1}$. We have an expression
$v_{r}=\sigma_{r}y_{r}^{p}+\tau_{r}x_{r}^{c_{r}}y_{r}+f_{r}(x_{r})$ where
$\mbox{ord }f_{r}(x_{r})$ is arbitrarily large. In particular, we can assume
that $\mbox{ord }f_{r}(x_{r})>\frac{pq_{r+1}}{m_{r+1}}$. Then
$R_{r}\rightarrow R_{r+1}$ is defined as desired by Theorem 3.2. By Remark
4.2, since we can take $\lambda(r+1)$ to be arbitrarily large, we can assume
that $v_{r+1}=\lambda_{r+1}y_{r+1}+g_{r+1}(x_{r+1})$ where $\mbox{ord
}g_{r+1}(x_{r+1})$ is arbitrarily large.

By Remark 4.3 and Theorem 3.3, $T_{r}\rightarrow T_{r+1}$ is defined as
desired, with $m_{r+1}^{\prime}=\frac{m_{r+1}}{p}$,
$q_{r+1}^{\prime}=q_{r+1}$. Since we can take $\lambda(r+1)$ to be arbitrarily
large, we can assume that
$y_{r+1}=\sigma_{r+1}w_{r+1}^{p}+\tau_{r+1}z_{r+1}^{c_{r+1}^{\prime}}w_{r}+f_{r+1}(z_{r+1})$
where $\mbox{ord }f_{r+1}(z_{r+1})$ is arbitrarily large.

We have defined a commutative diagram

(17) $\begin{array}[]{ccc}T_{r}&\rightarrow&T_{r+1}\\\ \uparrow&&\uparrow\\\
S_{r}&\rightarrow&S_{r+1}\\\ \uparrow&&\uparrow\\\
R_{r}&\rightarrow&R_{r+1}\end{array}$

with the desired properties; in particular, $R_{r+1}\rightarrow S_{r+1}$ is of
type 2 with

$\frac{c_{r+1}}{p-1}=\left(\frac{c_{r}}{p-1}\right)m_{r+1}-q_{r+1}+1$

and $S_{r+1}\rightarrow T_{r+1}$ is of type 1, with

$\frac{c_{r+1}^{\prime}}{p-1}=\frac{c_{r}^{\prime}}{p-1}m_{r+1}^{\prime}-m_{r+1}^{\prime}.$

Now choose $q_{r+2}^{\prime}$, $m_{r+2}^{\prime}=p^{\lambda(r+2)}$ such that
$p\not|q_{r+2}^{\prime}$ and

(18)
$\frac{c_{r+1}^{\prime}}{p-1}>\frac{q_{r+2}^{\prime}}{m_{r+2}^{\prime}}>\frac{c_{r+1}^{\prime}}{p-1}-\frac{1}{2^{r+2}}m_{1}^{\prime}\cdots
m_{r+1}^{\prime}.$

We can take $\lambda(r+2)$ arbitrarily large. Set
$m_{r+2}=\frac{m_{r+2}^{\prime}}{p}=p^{\lambda(r+2)-1}$,
$q_{r+2}=q_{r+2}^{\prime}$. By (18),
$\frac{q_{r+2}^{\prime}}{m_{r+2}^{\prime}}<\frac{c_{r+1}^{\prime}}{p-1}$.

Now construct, as in the construction of (17), using Theorems 3.2 and 3.3 and
Remark 4.3 and these values of $m_{r+2}$ and $q_{r+2}$,

$\begin{array}[]{ccc}T_{r+1}&\rightarrow&T_{r+2}\\\ \uparrow&&\uparrow\\\
S_{r+1}&\rightarrow&S_{r+2}\\\ \uparrow&&\uparrow\\\
R_{r+1}&\rightarrow&R_{r+2},\end{array}$

so that $R_{r+2}\rightarrow S_{r+2}$ is of type 1 and $S_{r+2}\rightarrow
T_{r+2}$ is of type 2. By Remark 4.2, we obtain expressions (12) and (13) for
$r+2$.

By induction, we construct the diagram (11).

Let $A=R_{0}$ and $C=T_{0}$. We will show that strong local monomialization
doesn’t hold above $A\rightarrow C$ along $\mu$. Suppose that
$R^{\prime}\rightarrow T^{\prime}$ has a strongly monomial form above
$A\rightarrow C$. Then $R^{\prime}$ has regular parameters
$u^{\prime},v^{\prime}$ and $T^{\prime}$ has regular parameters
$z^{\prime},w^{\prime}$ such that $u^{\prime}=\lambda(z^{\prime})^{m}$ and
$v^{\prime}=w^{\prime}$ where $m\in{\mathbb{Z}}_{>0}$ and $\lambda$ is a unit
in $T^{\prime}$. We will show that this cannot occur. There exists a
commutative diagram

$\begin{array}[]{ccccc}T_{s}&\rightarrow&T^{\prime}&\rightarrow&T_{s+1}\\\
\uparrow&&\uparrow&&\uparrow\\\
R_{s}&\rightarrow&R^{\prime}&\rightarrow&R_{s+1}\end{array}$

for some $s$. The ring $T^{\prime}$ has regular parameters
$\overline{z},\overline{w}$ such that

(19)
$z_{s}=\overline{z}^{a}\overline{w}^{b},w_{s}=\overline{z}^{c}\overline{w}^{d}$

for some $a,b,c,d\in{\mathbb{N}}$ with $ad-bc=\pm 1$, and $R^{\prime}$ has
regular parameters $\overline{u},\overline{v}$ such that
$u_{s}=\overline{u}^{\overline{a}}\overline{v}^{\overline{b}}$,
$v_{s}=\overline{u}^{\overline{c}}\overline{v}^{\overline{d}}$, where
$\overline{a}\overline{d}-\overline{b}\overline{c}=\pm 1$. We have an
expression

(20) $u_{s}=\alpha z_{s}^{p},v_{s}=\beta w_{s}^{p}+\Omega$

where $\alpha,\beta$ are units in $T_{s}$ and where

(21) $\Omega=\varepsilon z_{s}^{pc_{s}}w_{s}+M$

or

(22) $\Omega=\varepsilon z_{s}^{c_{s}^{\prime}}w_{s}+M$

where $\varepsilon\in T_{s}$ is a unit and $M$ is a sum of monomials in
$z_{s},w_{s}$ of high order in $z_{s}$. Further,
$\mu(w_{s}^{p})<\mu(z_{s}^{pc_{s}}w_{s})$ in (21) and
$\mu(w_{s}^{p})<\mu(z_{s}^{c_{s}^{\prime}}w_{s})$ in (22).

In particular, $R_{s}\rightarrow T_{s}$ is not a strongly monomial form.

Substituting (19) into $u_{s}$ and $v_{s}$ in (20), we have

(23)
$u_{s}=\alpha\overline{z}^{ap}\overline{w}^{bp},v_{s}=\beta\overline{z}^{cp}\overline{w}^{dp}+\Omega.$

We necessarily have that $u_{s}|v_{s}$ or $v_{s}|u_{s}$ in $T^{\prime}$.

First suppose that $c\geq a$ and $d\geq b$. Then we have that

$u_{s}=\alpha\overline{z}^{ap}\overline{w}^{bp},\frac{v_{s}}{u_{s}}=\beta\overline{z}^{(c-a)p}\overline{w}^{(d-b)p}+\frac{\Omega}{\alpha\overline{z}^{ap}\overline{w}^{bp}}$

giving an expression of the form (23). We will show that this is not a
strongly monomial form. If it is, then we must have that $a=0$ or $b=0$ so
that either

(24) $z_{s}=\overline{w},w_{s}=\overline{z}\overline{w}^{d}$

or

(25) $z_{s}=\overline{z},w_{s}=\overline{z}^{c}\overline{w}$

and we must have that $\frac{\Omega}{u_{s}}$ is part of a regular system of
parameters in $T^{\prime}$. Substituting into (21) or (22), we see that this
cannot occur except possibly in the case that (22) holds and
$\frac{z_{s}^{c_{s}^{\prime}}w_{s}}{u_{s}}$ is part of a regular system of
parameters in $T^{\prime}$.

Suppose that (22) and (24) hold with

$\frac{z_{s}^{c_{s}^{\prime}}w_{s}}{u_{s}}=\frac{\overline{w}^{c_{s}^{\prime}+d}\overline{z}}{\alpha\overline{w}^{p}}$

being part of a regular system of parameters in $T^{\prime}$. Now in this
case, $\mu(w_{s})>\mu(z_{s})$ and
$\mu(w_{s}^{p})<\mu(z_{s}^{c_{s}^{\prime}}w_{s})$ so $p\leq c_{s}^{\prime}$.
Thus
$\frac{\overline{w}^{c_{s}^{\prime}+d}\overline{z}}{\alpha\overline{w}^{p}}$
cannot be part of a regular system of parameters in $T^{\prime}$. A similar
argument shows that we do not obtain a strongly monomial form when (22) and
(25) hold.

Suppose that $c<a$ and $d<b$. Then we have expressions

$v_{s}=\gamma\overline{z}^{cp}\overline{w}^{dp},\frac{u_{s}}{v_{s}}=\alpha\gamma^{-1}\overline{z}^{(a-c)p}\overline{w}^{(b-d)p}$

where $\gamma\in T^{\prime}$ is a unit, giving an expression of the form of
(23), which is not strongly monomial. Thus we reduce to the case where
$(c-a)(d-b)<0$. We then have that $u_{s}\not|v_{s}$ since
$u_{s}\not|\overline{z}^{cp}\overline{w}^{dp}$. Suppose that $v_{s}|u_{s}$.
Then $v_{s}=\lambda\overline{z}^{cp}\overline{w}^{dp}$ where $\lambda$ is a
unit in $T^{\prime}$. But this is impossible since $(c-a)(d-b)<0$. Thus
$R^{\prime}\rightarrow T^{\prime}$ has a form (23) with $a,b,c,d>0$ and so
cannot be a strongly monomial form. We have established that strong local
monomialization along $\mu$ does not hold above $A\rightarrow C$.

From Theorem 3.2, we have that

(26) $\left(\frac{c_{r+1}}{p-1}\right)\frac{1}{m_{1}\cdots
m_{r+1}}=\left(\frac{c_{r}}{p-1}\right)\frac{1}{m_{1}\cdots
m_{r}}-\frac{q_{r+1}}{m_{r+1}}\left(\frac{1}{m_{1}\cdots
m_{r}}\right)+\frac{1}{m_{1}\cdots m_{r+1}}.$

Then from Theorem 3.3, we have that

$\frac{c_{r+2}}{p-1}=\left(\frac{c_{r+1}}{p-1}\right)m_{r+2}-m_{r+2},$

and so

(27) $\left(\frac{c_{r+2}}{p-1}\right)\frac{1}{m_{1}\cdots
m_{r+2}}=\left(\frac{c_{r}}{p-1}\right)\frac{1}{m_{1}\cdots
m_{r}}-\frac{q_{r+1}}{m_{1}\cdots m_{r+1}}.$

By equation (16) we have

(28) $\frac{1}{2^{r+1}}>\left(\frac{c_{r}}{p-1}\right)\frac{1}{m_{1}\cdots
m_{r}}-\left(\frac{q_{r+1}}{m_{r+1}}\right)\frac{1}{m_{1}\cdots m_{r}}>0.$

By Theorem 3.2,

$\left(\frac{c_{r+2}^{\prime}}{p-1}\right)\frac{1}{m_{1}^{\prime}\cdots
m_{r+2}^{\prime}}=\left(\frac{c_{r+1}^{\prime}}{p-1}\right)\frac{1}{m_{1}^{\prime}\cdots
m_{r+1}^{\prime}}-\frac{q_{r+2}^{\prime}}{m_{1}^{\prime}\cdots
m_{r+2}^{\prime}}+\frac{1}{m_{1}^{\prime}\cdots m_{r+2}^{\prime}}$

and by Theorem 3.3,

$\frac{c_{r+3}^{\prime}}{p-1}=\left(\frac{c_{r+2}^{\prime}}{p-1}\right)m_{r+3}^{\prime}-m_{r+3}^{\prime}.$

We thus have that

(29) $\left(\frac{c_{r+3}^{\prime}}{p-1}\right)\frac{1}{m_{1}^{\prime}\cdots
m_{r+3}^{\prime}}=\left(\frac{c_{r+1}^{\prime}}{p-1}\right)\frac{1}{m_{1}^{\prime}\cdots
m_{r+1}^{\prime}}-\frac{q_{r+2}^{\prime}}{m_{1}^{\prime}\cdots
m_{r+2}^{\prime}}.$

Equation (18) implies

(30)
$\frac{1}{2^{r+2}}>\left(\frac{c_{r+1}^{\prime}}{p-1}\right)\frac{1}{m_{1}^{\prime}\cdots
m_{r+1}^{\prime}}-\frac{q_{r+2}^{\prime}}{m_{1}^{\prime}\cdots
m_{r+2}^{\prime}}>0.$

Now $J(S_{i}/R_{i})=(x_{i}^{c_{i}})$ and $x_{0}=x_{i}^{m_{1}\cdots m_{i}}$ so
$\omega(J(S_{i}/R_{i}))=\frac{c_{i}}{m_{1}\cdots m_{i}}\omega(x_{0})$ and thus
by Proposition 3.4, (27) and (28), we have that

$-{\rm dist}(\omega/\nu)=\frac{1}{p-1}\inf_{i}\\{\omega(J(S_{i}/R_{i}))\\}=0.$

We have that $J(T_{i}/S_{i})=(z_{i}^{c_{i}^{\prime}})$ and
$z_{0}=z_{i}^{m_{1}^{\prime}\cdots m_{i}^{\prime}}$ so
$\omega(J(T_{i}/S_{i}))=\frac{c_{i}^{\prime}}{m_{1}^{\prime}\cdots
m_{i}^{\prime}}\omega(z_{0})$ and thus by Proposition 3.4, (29) and (30), we
have that

$-{\rm dist}(\mu/\omega)=\frac{1}{p-1}\inf_{i}\\{\omega(J(T_{i}/S_{i}))\\}=0.$

∎

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