# On the structure of certain valued fields

Junguk Lee and Wan Lee  Department of Mathematics, Yonsei University, 134
Sinchon-Dong, Seodaemun-Gu, Seoul 120-749, South Korea
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For any two complete discrete valued fields $K_{1}$ and $K_{2}$ of mixed
characteristic with perfect residue fields, we show that if each pair of
$n$-th residue rings is isomorphic for each $n\geq 1$, then $K_{1}$ and
$K_{2}$ are isometric and isomorphic. More generally, for $n_{1},n_{2}\geq 1$,
if $n_{2}$ is large enough, then any homomorphism from the $n_{1}$-th residue
ring of $K_{1}$ to the $n_{2}$-th residue ring of $K_{2}$ can be lifted to a
homomorphism between the valuation rings. We can find a lower bound for
$n_{2}$ depending only on $K_{2}$. Moreover, we get a functor from a category
of certain principal Artinian local rings of length $n$ to a category of
certain complete discrete valuation rings of mixed characteristic with perfect
residue fields, which naturally generalizes the functorial property of
unramified complete discrete valuation rings. The result improves Basarab’s
generalization of the AKE-principle for finitely ramified henselian valued
fields, which solves a question posed by Basarab, in the case of perfect
residue fields.

###### Key words and phrases:

finitely ramified valued fields, functorial property of the ring of Witt
vectors, Krasner’s lemma, lifting number, Ax-Kochen-Ershov prinicple, Keisler-
Shelah isomorphism theorem, coarsening of valuation

##  1\. Introduction

In this paper, we consider some problems on valued fields arising from the
interaction of number-theoretic approaches and model-theoretic approaches. In
number theory, it is well-known that the following are equivalent.

  * •

There is an isometric isomorphism between two complete unramified discrete
valued fields $K_{1}$ and $K_{2}$ of mixed characteristic $(0,p)$ with perfect
residue fields.

  * •

There is an isomorphism between residue fields of $K_{1}$ and $K_{2}$.

In model theory, as a counterpart, there is a principle called Ax-Kochen-
Ershov-principle(briefly, AKE-principle) which states the following are
equivalent.

  * •

Two absolutely unramified henselian valued fields $K_{1}$ and $K_{2}$ of the
same type of mixed characteristic $(0,p)$ with value groups as
${\mathbb{Z}}$-groups are elementarily equivalent.

  * •

Residue fields of $K_{1}$ and $K_{2}$ are elementarily equivalent.

We introduce more elementary classes of valued fields satisfying the AKE-
principle:
a) Algebraically closed valued fields by Robinson in [29].
b) Henselian fields with residue fields of characteristic $0$ by Ax and Kochen
in [4] and independently by Ershov in [14].
c) $p$-adically closed fields by Ax and Kochen in [4] and independently by
Ershov in [14].
d) Algebraically maximal Kaplansky fields by Ershov in [15] and independently
by Ziegler in [32].
e) Tame fields of equal characteristic by Kuhlmann in [22].
f) Separably tame fields of equal characteristic by Kuhlmann and Pal in [23].
Some elementary classes of valued fields with additional structures are also
known to satisfy the AKE-principle.
g) The ring of Witt vectors after adding a predicate for a unique
multiplicative set of representatives for the residue field by van den Dries
in [12].
h) Some valued difference fields by Bélair, Macintyre, and Scanlon in [9], by
Azgin and van den Dries in [1], by Pal in [26], and by Rideau in [28].

Most of all, in this paper, we are interested in finitely ramified valued
fields. Prestel and Roquette in [27] considered the class of $\wp$-closed
fields which are finite extensions of $p$-closed fields so that the residue
fields are finite. They showed that the theory of $\wp$-closed fields of a
fixed $p$-rank is model complete. Basarab in [6] extended this result and
generalized the AKE-principle for the case of finitely ramified valued fields.
Actually, he showed that for any two finitely ramified henselian valued fields
of mixed characteristic, they are elementarily equivalent if and only if their
value groups are elementarily equivalent, and their $n$-th residue rings are
elementarily equivalent for each $n\geq 1$, where the $n$-th residue ring is
the quotient of the valuation ring by the $n$-th power of the maximal ideal.
And the theory of a finitely ramified henselian valued field is model complete
if and only if each theory of its $n$-th residue ring and its value group are
model complete. Motivated from the relation between number theory and model
theory for the unramified case, we ask whether there is a number-theoretic
part which corresponds to Basarab’s result on the AKE-principle:

######  Question 1.1.

Are two finitely ramified complete discrete valued fields $K_{1}$ and $K_{2}$
of mixed characteristic with perfect residue fields isomorphic if the $n$-th
residue rings of $K_{1}$ and $K_{2}$ are isomorphic for each $n\geq 1$ ?

We report some known necessary and sufficient conditions for certain valued
fields to be isomorphic. For a $p$-valued field $(K,\nu)$ and any two
$\wp$-closed fields $(L_{1},\nu)$ and $(L_{2},\nu)$ of the same $p$-rank as
$(K,\nu)$, Prestel and Roquette in [27] showed that $L_{1}$ and $L_{2}$ are
$K$-isomorphic as valued fields if and only if the $n$-th powers of $L_{1}$
and $L_{2}$ contained in $K$ are the same for each $n$. Basarab and Kuhlmann
in [7] introduced a notion of the $\delta$-ring for each $\delta\geq 0$ in the
value group of a valued field, which generalizes the notion of $n$-th residue
rings. For $\delta\geq 0$ in a value group, the $\delta$-ideal is the set of
elements whose valuations are strictly larger than $\delta$. The $\delta$-ring
is the quotient of the valuation ring by the $\delta$-ideal. They showed that
for a base field $K$ and separable extensions $L_{1}$ and $L_{2}$ of $K$, if
each $\delta$-rings of $L_{1}$ and $L_{2}$ are isomorphic over the
$\delta$-ring of $K$ for each $\delta$ in the value group of $K$, then $L_{1}$
and $L_{2}$ are isomorphic over $K$. Also if $L_{1}$ and $L_{2}$ are finite
extensions of $K$, it is enough to check whether $\delta$-rings are isomorphic
for a large enough $\delta$. These two results required the existence of base
field $K$. To generalize a functorial property of Witt rings(Question 1.3) and
to answer Basarab’s question(Question 1.4), we should avoid this requirement.
In Section 2, we use Witt rings to avoid this problem and answer Question 1.1
positively. In Section 4, combining with properties of different(Lemma 2.15),
we get an explicit upper bound of Basarab’s invariant(Definition 4.11).

We return to the unramified case. The previous equivalence for the unramified
case in number theory is a corollary of the following well-known
theorem([30]).

  * •

For any perfect field $k$ of characteristic $p$, there exists a unique
unramified complete discrete valuation ring $R$, called the ring of Witt
vectors of $k$, of characteristic $0$ which has $k$ as its residue field.

  * •

For any two unramified complete discrete valuation rings $R_{1}$ and $R_{2}$
of mixed characteristic with perfect residue fields $k_{1}$ and $k_{2}$
respectively, suppose that there is a homomorphism $\phi:k_{1}\longrightarrow
k_{2}$. Then there is a unique lifting homomorphism $g:R_{1}\longrightarrow
R_{2}$ such that $g$ induces $\phi$.

In categorical setting, the theorem above is equivalent to the following
statement.

  * •

Let ${\mathcal{C}}_{p}$ be the category of complete unramified discrete
valuation rings of mixed characteristic $(0,p)$ with perfect residue fields
and ${\mathcal{R}}_{p}$ the category of perfect fields of characteristic $p$.
Then ${\mathcal{C}}_{p}$ is equivalent to ${\mathcal{R}}_{p}$. More precisely,
there is a functor $\operatorname{L}^{\prime}:\
{\mathcal{R}}_{p}\rightarrow{\mathcal{C}}_{p}$ which satisfies:

    * –

$\Pr\circ\operatorname{L}^{\prime}$ is equivalent to the identity functor
$\operatorname{Id}_{{\mathcal{R}}_{p}}$ where
$\Pr:{\mathcal{C}}_{p}\longrightarrow{\mathcal{R}}_{p}$ is the natural
projection functor.

    * –

$\operatorname{L}^{\prime}\circ\Pr$ is equivalent to
$\operatorname{Id}_{{\mathcal{C}}_{p}}$.

Based on Question 1.1 and the statements above, we raise generalized
questions.

######  Question 1.2.

  1. (1)

For a principal Artinian local ring $\overline{R}$ of length $n$ with certain
conditions, is there a unique complete discrete valuation ring $R$ which has
$\overline{R}$ as its residue ring ?

  2. (2)

For any two finitely ramified complete discrete valuation rings $R_{1}$ and
$R_{2}$ of mixed characteristic with perfect residue fields, let $R_{1,n_{1}}$
and $R_{2,n_{2}}$ be the $n_{1}$-th residue ring of $R_{1}$ and the $n_{2}$-th
residue ring of $R_{2}$ respectively. Under certain conditions on $n_{1}$ and
$n_{2}$, given a homomorphism $\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$,
is there a unique lifting homomorphism $g:R_{1}\longrightarrow R_{2}$ such
that $g$ induces $\phi$ ?

######  Question 1.3.

Let ${\mathcal{C}}_{p,e}$ be the category of complete discrete valuation rings
of mixed characteristic $(0,p)$ with perfect residue fields and absolute
ramification index $e$. Let ${\mathcal{R}}_{p,e}^{n}$ be a category of
principal Artinian local rings of length $n$ with certain conditions. Let
$\Pr_{n}:{\mathcal{C}}_{p,e}\longrightarrow{\mathcal{R}}_{p,e}^{n}$ be the
natural projection functor. Is there a functor
$\operatorname{L}:{\mathcal{R}}_{p,e}^{n}\longrightarrow{\mathcal{C}}_{p,e}$
which satisfies:

  * •

$\Pr_{n}\circ\operatorname{L}$ is equivalent to
$\operatorname{Id}_{{\mathcal{R}}_{p,e}^{n}}$.

  * •

$\operatorname{L}\circ\Pr_{n}$ is equivalent to
$\operatorname{Id}_{{\mathcal{C}}_{p,e}}$.

Question 1.2.(2) is not true in general, that is, there is a homomorphism
$\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$ such that any homomorphism from
$R_{1}$ into $R_{2}$ does not induce $\phi$. In this paper, the main result
shows that for sufficiently large $n_{2}$, if there is a given homomorphism
$\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$, then there is a homomorphism
$g:R_{1}\longrightarrow R_{2}$ rather naturally related with $\phi$. In the
beginning of Section 2, we show that Question 1.1 is true in a special case of
local fields. The main ingredient in the proof is to use the compactness of
the valuation rings of local fields. In order to extend the result to the case
of infinite perfect residue fields, we need the Witt subring. Since valuation
rings are not compact in general, we use Krasner’s lemma instead. More
precisely, the main result shows that for sufficiently large $n_{2}$, if there
is a given homomorphism $\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$, then
there is a homomorphism $\operatorname{L}(\phi):R_{1}\longrightarrow R_{2}$
satisfying a lifting property similar to that of the unramified case. Even
though the construction of $\operatorname{L}(\phi)$ depends on the choice of
uniformizer, it turns out that $\operatorname{L}(\phi)$ does not depend on the
choice of uniformizer. Moreover, when $\phi$ is an isomorphism, so is
$\operatorname{L}(\phi)$. This provides an answer for Question 1.1. We define
$\operatorname{L}(\phi)$ as the lifting of $\phi$ even though
$\operatorname{L}(\phi)$ does not induce $\phi$. The lifting map
$\operatorname{L}$ provides an answer for Question 1.2.(2) and Question
1.2.(1) where the latter follows from $\operatorname{L}$ and the Cohen
structure theorem for complete local ring([19]).

In Section 3, we concentrate on Question 1.3. By using the fact that the
definition of the lifting map $\operatorname{L}$ is independent of the choice
of uniformizer, we can show that $\operatorname{L}$ is compatible with
composition of homomorphisms between residue rings. More precisely,
$\operatorname{L}(\phi_{2}\circ\phi_{1})=\operatorname{L}(\phi_{2})\circ\operatorname{L}(\phi_{1})$
for any $\phi_{1}:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$ and
$\phi_{2}:R_{2,n_{2}}\longrightarrow R_{3,n_{3}}$. This defines a functor
$\operatorname{L}:{\mathcal{R}}_{p,e}^{n}\longrightarrow{\mathcal{C}}_{p,e}$
for sufficiently large $n$. We prove that a lower bound for $n$ depends only
on the ramification index $e$ and the prime number $p$. Even though
$\operatorname{L}$ does not give an equivalence between
${\mathcal{R}}_{p,e}^{n}$ and ${\mathcal{C}}_{p,e}$, it turns out that
$\operatorname{L}$ satisfies a similar functorial property to
$\operatorname{L}^{\prime}:\ {\mathcal{R}}_{p}\rightarrow{\mathcal{C}}_{p}$.
This provides an answer for Question 1.3.

We define the lifting number depending only on $p$ and $e$ as the least number
$n$ such that there is a lifting functor
$\operatorname{L}:{\mathcal{R}}_{p,e}^{n}\longrightarrow{\mathcal{C}}_{p,e}$,
which turns out to be greater than or equal to Basarab’s invariant
$\lambda(T)$ of the theory of henselian valued fields. For the tamely ramified
case, we prove that the lifting number for ${\mathcal{C}}_{p,e}$ is $e+1$ when
$e\geq 2$. For the wildly ramified case, we have that the lifting number for
${\mathcal{C}}_{p,e}$ is at least $e+1$. Finally we conclude that the lifting
number for ${\mathcal{C}}_{p,e}$ is either $=1$ or $\geq 3$ for any case. We
note that the lifting number for ${\mathcal{C}}_{p,e}$ is $1$ if and only if
$e=1$.

In [6], Basarab posed the following question:

######  Question 1.4.

Given a finitely ramified henselian valued field $K$ of ramification index
$e\geq 2$, is there a finite integer $N^{\prime}\geq 1$ depending on $K$ such
that any other finitely ramified henselian valued field of the same
ramification index $e$ is elementarily equivalent to $K$ if and only if their
$N^{\prime}$-th residue rings are elementarily equivalent and their value
groups are elementarily equivalent?

In Section 4, for given valued fields, each of whose value groups has the
least positive element, we reduce the problem determining elementary
equivalence between them to the problem determining whether certain complete
discrete valued fields related with them are isomorphic. Using results in
Section 2, we improve Basarab’s result on the AKE-principle which gives a
positive answer for Question 1.4 when the residue fields are perfect.

Given a finitely ramified henselian valued field $K$, Basarab([6]) denoted the
minimal number $N^{\prime}$ which satisfies the equivalence in Question 1.4 by
$\lambda(T)$ for a complete theory $T$ of $K$. He showed $\lambda(T)$ for a
local field $K$ is finite but did not give any explicit value of $\lambda(T)$.
Under certain conditions, we calculate $\lambda(T)$ explicitly for the tame
case and get a lower bound of $\lambda(T)$ for the wild case. Surprisingly we
show that $\lambda(T)$ can be $1$ even when $K$ is not unramified. As a
special case, we conclude that $\lambda(T)$ is $1$ or $e+1$ if $p\not|e$, and
$\lambda(T)\geq e+1$ if $p|e$ when $K$ is a finitely ramified henselian
subfield of ${\mathbb{C}}_{p}$ with ramification index $e$.

We introduce basic notations and terminologies which will be used in this
paper. We denote a valued field by a tuple $(K,R,\mathfrak{m},\nu,k,\Gamma)$
consisting of the following data : $K$ is the underlying field, $R$ is the
valuation ring, $\mathfrak{m}$ is the maximal ideal of $R$, $\nu$ is the
valuation map, $k$ is the residue field, and $\Gamma$ is the value group.
Hereafter, the full tuple $(K,R,\mathfrak{m},\nu,k,\Gamma)$ will be
abbreviated in accordance with the situational need for the components.

######  Definition 1.5.

Let $(K,\nu,k,\Gamma)$ be a valued field of characteristic zero. We say
$(K,\nu)$ is absolutely unramified if $\mathop{char}(k)=0$, or
$\mathop{char}(k)=p$ and $\nu(p)$ is the minimal positive element in $\Gamma$
for $p>0$. We say $(K,\nu)$ is absolutely ramified if it is not absolutely
unramified.

######  Definition 1.6.

Let $(K,R,\nu,k,\Gamma)$ be a valued field whose residue field has prime
characteristic $p$.

  1. (1)

We say $(K,R,\nu,k,\Gamma)$ is absolutely finitely ramified if the set
$\\{\gamma\in\Gamma|\ 0<\gamma\leq\nu(p)\\}$ is finite. The cardinality of
$\\{\gamma\in\Gamma|\ 0<\gamma\leq\nu(p)\\}$ is called the absolute
ramification index of $(K,\nu)$, denoted by $e(K,\nu)$ or $e(R)$. If $K$ or
$\nu$ is clear from context, we write $e(K)$ or $e$ for $e(K,\nu)$. For $x\in
R$, we write $e_{\nu}(x):=|\\{\gamma\in\Gamma|\ 0<\gamma\leq\nu(x)\\}|$. If
there is no confusion, we write $e(x)$ for $e_{\nu}(x)$

  2. (2)

Let $(K,R,\nu,k,\Gamma)$ be finitely ramified. If $p$ does not divide
$e_{\nu}(p)$, we say $(K,\nu)$ is _absolutely tamely ramified_. Otherwise, we
say $(K,\nu)$ is _absolutely wildly ramified_.

Note that if a valued field of mixed characteristic has the absolute finite
ramification index, then its value group has the minimum positive element.

######  Definition 1.7.

Let $(K_{1},\nu_{1})$ and $(K_{2},\nu_{2})$ be valued fields. Let $R_{1}$ and
$R_{2}$ be subrings of $K_{1}$ and $K_{2}$ respectively. Let
$f:R_{1}\rightarrow R_{2}$ be a injective ring homomorphism. We say $f$ is an
isometry if for $a,b\in R_{1}$,

$\nu_{1}(a)>\nu_{1}(b)\Leftrightarrow\nu_{2}(f(a))>\nu_{2}(f(b)).$

######  Definition 1.8.

For a local ring $R$ with maximal ideal $\mathfrak{m}$, we denote
$R/\mathfrak{m}^{n}$ by $R_{n}$, and we call $R_{n}$ the _$n$ -th residue
ring_ of $R$. In particular, $R_{1}$ is the residue field of $R$.

For each $m>n$, let $\operatorname{pr}_{n}:\ R\rightarrow R_{n}$ and
$\operatorname{pr}^{m}_{n}:\ R_{m}\rightarrow R_{n}$ be the canonical
projection maps respectively. For $R$-algebras $S_{1}$ and $S_{2}$, we denote
the set of $R$-algebra homomorphisms from $S_{1}$ to $S_{2}$ by
$\mathrm{Hom}_{R}(S_{1},S_{2})$, and we briefly write
$\mathrm{Hom}(S_{1},S_{2})$ for $\mathrm{Hom}_{\mathbb{Z}}(S_{1},S_{2})$. We
denote the set of $R$-algebra isomorphisms by $\mathrm{Iso}_{R}(S_{1},S_{2})$,
and we write $\mathrm{Iso}(S_{1},S_{2})$ for
$\mathrm{Iso}_{\mathbb{Z}}(S_{1},S_{2})$. We write $\mathrm{Iso}(R)$ for
$\mathrm{Iso}(R,R)$. We denote a primitive $n$-th root of unity by
$\zeta_{n}$.

##  2\. Lifting homomorphisms

We start from the following proposition.

######  Proposition 2.1.

Let $K_{1}$ and $K_{2}$ be finite extensions of ${\mathbb{Q}}_{p}$ for some
prime $p$. Let $R_{1,n}$ and $R_{2,n}$ be the $n$-th residue rings of $K_{1}$
and $K_{2}$ respectively. Suppose that there is an isomorphism
$\iota_{n}:R_{1,n}\rightarrow R_{2,n}$ for each $n>0$. Then there is an
isomorphism $\iota:K_{1}\rightarrow K_{2}$ over ${\mathbb{Q}}_{p}$.

###### Proof.

(1) First method: Let $\mathrm{Iso}(R_{n})$ be the set of isomorphisms from
$R_{1,n}$ onto $R_{2,n}$, and $\xi_{n+1,n}$ be the natural reduction map from
$\mathrm{Iso}(R_{n+1})$ to $\mathrm{Iso}(R_{n})$. Then
$\\{\mathrm{Iso}(R_{n}),\xi_{n+1,n}\\}$ forms an inverse system. Since each
residue ring $R_{i,n}$ is finite, $\mathrm{Iso}(R_{n})$ is finite,in
particular compact for each $n$. By the theory of topological algebra,
$\varprojlim\mathrm{Iso}(R_{n})$ is not empty([25]). This shows there exists
an isomorphism $\iota:K_{1}\longrightarrow K_{2}$. $\iota$ is defined over
${\mathbb{Q}}_{p}$ since all elements of $\mathrm{Iso}(R_{n})$ are continuous.

(2) Second method: Let $R_{1}$ and $R_{2}$ be valuation rings of $K_{1}$ and
$K_{2}$ respectively. Take an element $a$ in $R_{1}$ satisfying
$K_{1}=\mathbb{Q}_{p}(a)$. Let $f$ be the monic irreducible polynomial of $a$
over ${\mathbb{Z}}_{p}$. Consider a sequence $(a^{\prime}_{n}\in R_{2})_{n\geq
1}$ such that
$\operatorname{pr}_{2,n}(a^{\prime}_{n})=\iota_{n}(\operatorname{pr}_{1,n}(a))$
where $\operatorname{pr}_{i,n}$ denotes a $n$-th natural projection from
$R_{i}$ to $R_{i,n}$. We note that each $\iota_{n}$ is an
${\mathbb{Z}}_{p}$-algebra isomorphism since $\iota_{n}$ is continuous. Since
$f(a)=0$,
$f(\iota_{n}(\operatorname{pr}_{1,n}(a)))=\iota_{n}(f(\operatorname{pr}_{1,n}(a)))=\iota_{n}(\operatorname{pr}_{1,n}(f(a)))=0$
in $R_{2,n}$. First equality follows from that fact that $\iota_{n}$ is an
${\mathbb{Z}}_{p}$-algebra homomorphism. Hence,
$f(\operatorname{pr}_{2,n}(a^{\prime}_{n}))=\operatorname{pr}_{2,n}(f(a^{\prime}_{n}))=0$
in $R_{2,n}$, that is, $f(a^{\prime}_{n})\in\mathfrak{m}_{2}^{n}$ where
$\mathfrak{m}_{2}$ is the maximal ideal of $R_{2}$. Since $R_{2}$ is compact,
there is a subsequence $(a^{\prime}_{n_{i}})$ which converges to
$a^{\prime}\in R_{2}$, and since $f$ is continuous,
$f(a^{\prime})=\lim_{n_{i}\rightarrow\infty}f(a^{\prime}_{n_{i}})=0$ in
$R_{2}$. Thus $K_{2}$ contains a zero $a^{\prime}$ of $f$. Therefore, there is
an injection $\iota:K_{1}\rightarrow K_{2},\ a\mapsto a^{\prime}$ over
${\mathbb{Q}}_{p}$, and hence, we obtain an inequality
$[K_{1}:{\mathbb{Q}}_{p}]\leq[K_{2}:{\mathbb{Q}}_{p}]$ between the field
extension degrees. Similarly, one can show
$[K_{1}:{\mathbb{Q}}_{p}]\geq[K_{2}:{\mathbb{Q}}_{p}]$. Hence, $\iota$ is an
isomorphism over ${\mathbb{Q}}_{p}$.

∎

Since the proof of the fact that the inverse limit of $\mathrm{Iso}(R_{n})$ is
not empty uses Zorn’s lemma, we can only prove the existence of an isomorphism
in the first method. The second method does not use Zorn’s lemma and the given
isomorphism is more easier to construct. But both methods use the fact that
the homomorphisms are defined over $\mathbb{Q}_{p}$ or $\mathbb{Z}_{p}$
crucially. For the case of infinite perfect residue fields, we need the
absolutely unramified discrete valuation rings called the ring of Witt
vectors. By Krasner’s lemma, it suffices to consider a single $n$-th residue
ring for sufficiently large $n$.

The following theorem is well-known.

######  Theorem 2.2.

  1. (1)

Let $k$ be a perfect field of characteristic $p$. Then there exists a complete
discrete valuation ring of characteristic $0$ which is absolutely unramified
and has $k$ as its residue field. Such a ring is unique up to isomorphism.
This unique ring is called the ring of Witt vectors of $k$, denoted by $W(k)$.

  2. (2)

Let $R_{1}$ and $R_{2}$ be complete discrete valuation rings of mixed
characteristic with perfect residue fields $k_{1}$ and $k_{2}$ respectively.
Suppose $R_{1}$ is absolutely unramified. Then for every homomorphism
$\phi:k_{1}\longrightarrow k_{2}$, there exists a unique homomorphism
$g:R_{1}\longrightarrow R_{2}$ making the following diagram commutative:

$\begin{CD}R_{1}@>{{g}}>{}>R_{2}\\\
@V{{\operatorname{pr}_{1,1}}}V{}V@V{{\operatorname{pr}_{2,1}}}V{}V\\\
k_{1}@>{\phi}>{}>k_{2}\end{CD}$

###### Proof.

Chapter 2, section 5 of [30].

∎

Before stating main theorems, we need some lemmas.

######  Lemma 2.3.

Let $R$ be a complete discrete valuation ring of characteristic $0$ with
perfect residue field $k$ of characteristic $p$ and corresponding valuation
$\nu$. Then $W(k)$ can be embedded as a subring of $R$ and $R$ is a free
$W(k)$-module of rank $\nu(p)$. Moreover, $R=W(k)[\pi]$ where $\pi$ is a
uniformizer of $R$.

###### Proof.

Chapter 2, Section 5 of [30] ∎

######  Lemma 2.4.

Let $A$ be a ring that is Hausdorff and complete for a topology defined by a
decreasing sequence $\mathfrak{a}_{1}\supset\mathfrak{a}_{2}\supset...$ of
ideals such that
$\mathfrak{a}_{n}\cdot\mathfrak{a}_{m}\subset\mathfrak{a}_{n+m}$. Assume that
the residue ring $A_{1}=A/\mathfrak{a}_{1}$ is a perfect field of
characteristic $p$. Then:

  1. (1)

There exists one and only one system of representatives
$h:A_{1}\longrightarrow A$ which commutes with $p$-th powers:
$h(\lambda^{p})=h(\lambda)^{p}$. This system of representatives is called the
set of Teichmüller representatives.

  2. (2)

In order that $a\in A$ belong to $S=h(A_{1})$, it is necessary and sufficient
that $a$ be a $p^{n}$-th power for all $n\geq 0$.

  3. (3)

This system of representatives is multiplicative which means

$h(\lambda\mu)=h(\lambda)h(\mu)$

for all $\lambda,\mu\in A_{1}$.

  4. (4)

$S$ contains $0$ and $1$.

  5. (5)

$S\setminus\\{0\\}$ is a subgroup of the unit group of $A$.

###### Proof.

(1)(2)(3): Chapter 2, Section 4 of [30]

(4): $0$ and $1$ satisfy (2).

(5): (3) and (4) show that $S\setminus\\{0\\}$ is a subgroup of the unit group
of $A$. ∎

######  Lemma 2.5.

Let $R_{1}$ and $R_{2}$ be discrete valuation rings of characteristic $0$ with
residue characteristic $p$. Let $\mathfrak{m}_{i}$ be the maximal ideal of
$R_{i}$ generated by $\pi_{i}$ and $\nu_{i}$ corresponding discrete valuation
of $R_{i}$ for $i=1,2$. Suppose there is a homomorphism
$\iota:R_{1}\longrightarrow R_{2,n}$. If $n>a\nu_{2}(p)$ for some real number
$a\geq 1$, kernel of $\iota$ is equal to $\mathfrak{m}_{1}^{m}$ for some
$m>a\nu_{1}(p)$.

###### Proof.

Let $\overline{\mathfrak{m}}_{2}=\mathfrak{m}_{2}/\mathfrak{m}_{2}^{n}$ be the
maximal of $R_{2,n}$. If we write
$\iota(\pi_{1})R_{2,n}=\overline{\mathfrak{m}}_{2}^{x}$,

$\displaystyle\overline{\mathfrak{m}}_{2}^{\nu_{2}(p)}=pR_{2,n}$
$\displaystyle=\iota(p)R_{2,n}$
$\displaystyle=\iota\left(\pi_{1}^{\nu_{1}(p)}\right)R_{2,n}$
$\displaystyle=\overline{\mathfrak{m}}_{2}^{x\nu_{1}(p)}.$

In particular $x=\nu_{2}(p)/\nu_{1}(p)$ and $\nu_{2}(p)/\nu_{1}(p)$ is a
positive integer. Suppose

$\iota(\pi_{1}^{m})R_{2,n}=\overline{\mathfrak{m}}_{2}^{\frac{m\nu_{2}(p)}{\nu_{1}(p)}}=0$

in $R_{2,n}$ for some $m$. Then we obtain

$\frac{m\nu_{2}(p)}{\nu_{1}(p)}\geq n>a\nu_{2}(p),$

and hence $m>a\nu_{1}(p)$. ∎

For any field $L$, $L^{alg}$ denotes a fixed algebraic closure of $L$. Let
$(L,\nu)$ be a valued field whose value group is contained in $\mathbb{R}$. If
$L$ is of characteristic $0$ and of residue characteristic $p$, we define a
normalized valuation $\overline{\nu}$ on $L$ of $\nu$ by the property
$\overline{\nu}(p)=1$, that is, $\nu(p)\overline{\nu}=\nu$. We denote an
extended valuation of $\overline{\nu}$ on $L^{alg}$ by $\widetilde{\nu}$. When
$L$ is henselian, $\widetilde{\nu}$ is unique.

######  Lemma 2.6.

Let $(K_{1},\nu_{1})$ and $(K_{2},\nu_{2})$ be valued fields whose value
groups are contained in ${\mathbb{R}}$. Let $f:K_{1}\longrightarrow K_{2}$ be
an isometric homomorphism. Suppose $K_{1}$ is henselian. Let
$\widetilde{f}:K_{1}^{alg}\longrightarrow K_{2}^{alg}$ be an extended
homomorphism of $f$. Then $\widetilde{f}$ is an isometry.

###### Proof.

There are two valuations on $\widetilde{f}(K_{1}^{alg})$,
$\widetilde{\nu_{1}}\circ\widetilde{f}^{-1}$ and
$\widetilde{\nu_{2}}|_{\widetilde{f}(K_{1}^{alg})}$ where
$\widetilde{\nu_{2}}|_{\widetilde{f}(K_{1}^{alg})}$ is the restriction of
$\widetilde{\nu_{2}}$ to $\widetilde{f}(K_{1}^{alg})$. Since $f$ is an
isometry, the restrictions of $\widetilde{\nu_{1}}\circ\widetilde{f}^{-1}$ and
$\widetilde{\nu_{2}}|_{\widetilde{f}(K_{1}^{alg})}$ to $f(K_{1})$ are
equivalent, in fact, they are equal since
$(\widetilde{\nu_{1}}\circ\widetilde{f}^{-1})(p)=\widetilde{\nu_{2}}|_{\widetilde{f}(K_{1}^{alg})}(p)=1$.
Since $K_{1}$ is henselian, $f(K_{1})$ is Henselian. Hence,
$\widetilde{\nu_{1}}\circ\widetilde{f}^{-1}$ is equal to
$\widetilde{\nu_{2}}|_{\widetilde{f}(K_{1}^{alg})}$ by the henselian property.
This shows that $\widetilde{f}$ is an isometry. ∎

Let $R$ be a complete discrete valuation ring of mixed characteristic with
perfect residue field. Let $\pi$ be a uniformizer of $R$ and $\nu$
corresponding valuation of $R$. Let $L$ and $K$ be the fraction fields of $R$
and $W(k)$ respectively. We denote the maximal number

$\max\left\\{\widetilde{\nu}\big{(}\pi-\sigma(\pi)\big{)}:\sigma\in\mathrm{Hom}_{K}(L,L^{alg}),\sigma(\pi)\neq\pi\right\\}$

by $M(R)_{\pi}$ or $M(L)_{\pi}$.

######  Definition 2.7.

Let $R_{1}$ and $R_{2}$ be complete discrete valuation rings of characteristic
$0$ with perfect residue fields $k_{1}$ and $k_{2}$ of characteristic $p$
respectively. Let $\mathfrak{m}_{i}$ be the maximal ideal of $R_{i}$ generated
by $\pi_{i}$ and $\nu_{i}$ corresponding valuation of $R_{i}$ for $i=1,2$. Let
$L_{i}$ and $K_{i}$ be the fraction fields of $R_{i}$ and $W(k_{i})$ for
$i=1,2$ respectively. For any homomorphism $\phi:R_{1,n_{1}}\longrightarrow
R_{2,n_{2}}$, we say that a homomorphism $g:R_{1}\longrightarrow R_{2}$ is a
_$(n_{1},n_{2})$ -lifting_ of $\phi$ at $\pi_{1}$ if $g$ satisfies the
following:

  * •

There exists a representaive $\beta$ of
$\phi(\pi_{1}+\mathfrak{m}_{1}^{n_{1}})$ which satisfies

$\widetilde{\nu_{2}}\big{(}g(\pi_{1})-\beta\big{)}>\max_{\sigma}\limits\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-\beta\Big{)}:\sigma\big{(}g(\pi_{1})\big{)}\neq
g(\pi_{1})\right\\}$

where $\sigma$ runs through all of $\mathrm{Hom}_{K_{2}}(L_{2},L_{2}^{alg})$.

  * •

$\phi_{red,1}\circ\operatorname{pr}_{1,1}=\operatorname{pr}_{2,1}\circ g$
where $\phi_{red,1}:k_{1}\longrightarrow k_{2}$ denotes the natural reduction
map of $\phi$.

When such $g$ is unique, we denote $g$ by
$\operatorname{L}_{\pi_{1},n_{1},n_{2}}(\phi)$. When
$\operatorname{L}_{\pi_{1},n_{1},n_{2}}(\phi)$ exists for all
$\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$, we write
$\operatorname{L}_{\pi_{1},n_{1},n_{2}}:\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})\longrightarrow\mathrm{Hom}(R_{1},R_{2})$.
When $n_{1}=n_{2}=n$, we briefly write
$\operatorname{L}_{\pi_{1},n_{1},n_{2}}=\operatorname{L}_{\pi_{1},n}$ and say
that $\operatorname{L}_{\pi_{1},n}$ is the _$n$ -lifting at $\pi_{1}$_.

If $n_{2}\nu_{1}(p)/\nu_{2}(p)\leq n_{1}$, there is a natural projection map
$\operatorname{pr}^{n_{1},n_{2}}:\mathrm{Hom}(R_{1},R_{2})\longrightarrow\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})$
such that for any $g$ in $\mathrm{Hom}(R_{1},R_{2})$,
$\operatorname{pr}_{2,n_{2}}\circ
g=\operatorname{pr}^{n_{1},n_{2}}(g)\circ\operatorname{pr}_{1,n_{1}}.$ In
particular, $g$ is a $(n_{1},n_{2})$-lifting of
$\operatorname{pr}^{n_{1},n_{2}}(g)$ at $\pi_{1}$. Note that when
$n_{2}>\nu_{2}(p)$ and $\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})$ is not empty,
$n_{2}\nu_{1}(p)/\nu_{2}(p)\leq n_{1}$ by Lemma 2.5.

The definition of liftings does not depend on the choice of uniformizer. In
order to prove this, we need the following lemmas.

######  Lemma 2.8.

Let $R_{i}$ be a complete discrete valuation ring of characteristic $0$ with
perfect residue field $k_{i}$ of characteristic $p$ for $i=1,2$. Let
$\mathfrak{m}_{i}$ be the maximal ideal of $R_{i}$ generated by $\pi_{i}$ and
$S_{i}$ the set of Teichmüller representatives of $R_{i}$ for $i=1,2$.

  1. (1)

For any homomorphism $\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$,
$\phi(S_{1}+\mathfrak{m}_{1}^{n_{1}})$ is contained in
$S_{2}+\mathfrak{m}_{2}^{n_{2}}$. Similarly, for any homomorphism
$g:R_{1}\longrightarrow R_{2}$, $g(S_{1})$ is contained in $S_{2}$.

  2. (2)

For any homomorphism $\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$,
$\phi\left((W(k_{1})+\mathfrak{m}_{1}^{n_{1}})/\mathfrak{m}_{1}^{n_{1}}\right)$
is contained in
$(W(k_{2})+\mathfrak{m}_{2}^{n_{2}})/\mathfrak{m}_{2}^{n_{2}}$. Similarly, for
any homomorphism $g:R_{1}\longrightarrow R_{2}$, $g(W(k_{1}))$ is contained in
$W(k_{2})$.

###### Proof.

(1) Since $W(k_{i})/pW(k_{i})\cong R_{i}/\mathfrak{m}_{i}\cong k_{i}$, $S_{i}$
is contained in $W(k_{i})$ by Lemma 2.4. For any $\lambda\in S_{1}$, let
$\eta_{s}$ be any representative of
$\phi(\lambda^{1/{p^{s}}}+\mathfrak{m}_{1}^{n_{1}})$. We note that
$\lambda^{1/{p^{s}}}$ exists in $S_{1}$ by Lemma 2.4 and
$\eta_{s}^{p^{s}}+\mathfrak{m}_{2}^{n_{2}}=\phi(\lambda+\mathfrak{m}_{1}^{n_{1}})$.
If $\theta_{s}$ is any other representative of
$\phi(\lambda^{1/{p^{s}}}+\mathfrak{m}_{1}^{n_{1}})$, then
$\eta_{s}-\theta_{s}\in\mathfrak{m}_{2}^{n_{2}}$. Hence, if we write
$\eta_{s}=\theta_{s}+\pi_{2}^{n_{2}}a$ for some $a$ in $R_{2}$, the following
binomial expansion

$\displaystyle\eta_{s}^{p^{s}}$
$\displaystyle=(\theta_{s}+\pi_{2}^{n_{2}}a)^{p^{s}}$
$\displaystyle=\theta_{s}^{p^{s}}+p^{s}\theta_{s}^{p^{s}-1}\pi_{2}^{n}a+...+(\pi_{2}^{n}a)^{p^{s}}$

shows $\eta_{s}^{p^{s}}-\theta_{s}^{p^{s}}\in\mathfrak{m}_{2}^{s}$. Since
$\eta_{s+1}^{p}$ is a representative of
$\phi(\lambda^{1/{p^{s}}}+\mathfrak{m}_{1}^{n_{1}})$, the calculation above
shows that $\\{\eta_{s}^{p^{s}}\\}$ is a Cauchy sequence and
$\lim_{s\rightarrow\infty}\eta_{s}^{p^{s}}$ is well-defined in $R_{2}$. Since
$\eta_{s}^{p^{s}}+\mathfrak{m}_{2}^{n_{2}}=\phi(\lambda+\mathfrak{m}_{1}^{n_{1}})$
and $\mathfrak{m}_{2}^{n_{2}}$ is topologically closed in $R_{2}$,

$\phi\left(\lambda+\mathfrak{m}_{1}^{n_{1}}\right)=\left(\lim_{s\rightarrow\infty}\eta_{s}^{p^{s}}\right)+\mathfrak{m}_{2}^{n_{2}}.$

Similarlly, we have

$\phi\left(\lambda^{1/{p}}+\mathfrak{m}_{1}^{n_{1}}\right)=\left(\lim_{s\rightarrow\infty}\eta_{s}^{p^{s-1}}\right)+\mathfrak{m}_{2}^{n_{2}}.$

Since

$\lim_{s\rightarrow\infty}\eta_{s}^{p^{s}}=\left(\lim_{s\rightarrow\infty}\eta_{s}^{p^{s-1}}\right)^{p},$

we obtain

$\lim_{s\rightarrow\infty}\eta_{s}^{p^{s}}\in S_{2}$

by Lemma 2.4.

Since $g(\lambda)^{1/p}=g(\lambda^{1/p})$, $g(S_{1})$ is contained in $S_{2}$
by Lemma 2.4.

(2) For any element $a$ in $W(k_{1})$, we can write
$a=\sum_{r=0}^{\infty}\lambda_{r}p^{r}$ uniquely where $\lambda_{r}$ is in
$S_{1}$ by Lemma 2.4. Then by Lemma 2.8.(1),
$g(a)=\sum_{r=0}^{\infty}g(\lambda_{r}){p^{r}}$ is in $W(k_{2})$.

Let
$\phi_{res}:(W(k_{1})+\mathfrak{m}_{1}^{n_{1}})/\mathfrak{m}_{1}^{n_{1}}\longrightarrow
R_{2,n_{2}}$ be the restrition map of $\phi$ to the domain
$W(k_{1})/(W(k_{1})\bigcap\mathfrak{m}_{1}^{n_{1}})\cong(W(k_{1})+\mathfrak{m}_{1}^{n_{1}})/\mathfrak{m}_{1}^{n_{1}}$.
By Theorem 2.2, we define $g_{res}$ to be the $(1,1)$-lifting of
$\phi_{res,red,1}$ where $\phi_{res,red,1}:k_{1}\longrightarrow k_{2}$ denotes
the natural reduction map of $\phi_{res}$. We claim that $g_{res}$ induces
$\phi_{res}$. For any $\lambda$ in $S_{1}$, $g_{res}(\lambda)=\tau$ where
$\tau$ is a unique representative of $\phi(\lambda+\mathfrak{m}_{1}^{n_{1}})$
contained in $S_{2}$ by Lemma 2.4. Since $g_{res}$ is a ring homomorphism,
$g_{res}(a)=\sum_{r=0}^{\infty}\tau_{r}p^{r}$ where $\tau_{r}$ is a unique
representative of $\phi_{res}(\lambda_{r}+\mathfrak{m}_{1}^{n_{1}})$ which is
contained in $S_{2}$. This shows

$\displaystyle g_{res}(a)+\mathfrak{m}^{n_{2}}$
$\displaystyle=\left(\sum_{r=0}^{\infty}\tau_{r}p^{r}\right)+\mathfrak{m}_{2}^{n_{2}}$
$\displaystyle=\sum_{r=0}^{\infty}p^{r}\phi_{res}(\lambda_{r}+\mathfrak{m}_{1}^{n_{1}})$
$\displaystyle=\phi_{res}(a+\mathfrak{m}_{1}^{n_{1}}),$

and hence, $g_{res}$ induces $\phi_{res}$. Since the image of $g_{res}$ is
contained in $W(k_{2})$, the image of $\phi_{res}$ is contained in
$(W(k_{2})+\mathfrak{m}_{2}^{n_{2}})/\mathfrak{m}_{2}^{n_{2}}$.

∎

######  Lemma 2.9.

Let $R_{1}$ and $R_{2}$ be complete discrete valuation rings of characteristic
$0$ with perfect residue fields $k_{1}$ and $k_{2}$ of characteristic $p$
respectively. Let $\mathfrak{m}_{i}$ be the maximal ideal of $R_{i}$ generated
by $\pi_{i}$ and $\nu_{i}$ corresponding valuation of $R_{i}$ for $i=1,2$. Let
$L_{i}$ and $K_{i}$ be the fraction fields of $R_{i}$ and $W(k_{i})$ for
$i=1,2$ respectively.

  1. (1)

Let $\alpha$ be a uniformizer of $R_{1}$ other than $\pi_{1}$. Then
$M(R_{1})_{\pi_{1}}=M(R_{1})_{\alpha}$. We briefly write
$M(R_{1})_{\pi_{1}}=M(R_{1})$.

  2. (2)

Suppose $[L_{1}:K_{1}]=[L_{2}:K_{2}]=e$, that is, $\nu_{1}(p)=\nu_{2}(p)=e$.
Suppose there is an isometry $g:L_{1}\longrightarrow L_{2}$. Then
$M(R_{1})=M(R_{2})$.

###### Proof.

(1) By Lemma 2.4, we can write
$\alpha=\sum_{r=1}^{\infty}\lambda_{r}\pi_{1}^{r}$ where $\lambda_{r}$ is a
Teichmüller representative of $R_{1}$ for each $r$ and $\lambda_{1}\neq 0$.
Since $R_{1}/\mathfrak{m}_{1}=k_{1}$, $\lambda_{r}$ is in $W(k_{1})$ for each
$r$ by Lemma 2.4. For any $\sigma$ in
$\mathrm{Hom}_{K_{1}}(L_{1},K_{1}^{alg})$,

$\displaystyle\alpha-\sigma(\alpha)$
$\displaystyle=\sum_{r=1}^{\infty}\lambda_{r}\pi_{1}^{r}-\sigma\left(\sum_{r=1}^{\infty}\lambda_{r}\pi_{1}^{r}\right)$
$\displaystyle=\sum_{r=1}^{\infty}\lambda_{r}\Big{(}\pi_{1}^{r}-\sigma(\pi_{1}^{r})\Big{)}$
$\displaystyle=\Big{(}\pi_{1}-\sigma(\pi_{1})\Big{)}\sum_{r=1}^{\infty}\lambda_{r}\left(\sum_{j=0}^{r-1}\pi_{1}^{r-1-j}\sigma(\pi_{1}^{j})\right)$

shows
$\widetilde{\nu_{1}}(\alpha-\sigma(\alpha))=\widetilde{\nu_{1}}(\pi_{1}-\sigma(\pi_{1}))$
since

$\widetilde{\nu_{1}}\left(\sum_{r=1}^{\infty}\lambda_{r}\left(\sum_{j=0}^{r-1}\pi_{1}^{r-1-j}\sigma(\pi_{1}^{j})\right)\right)=0.$

We have $M(R_{1})_{\pi_{1}}=M(R_{1})_{\alpha}$.

(2) By Lemma 2.8.(2), $g(K_{1})$ is contained in $K_{2}$. Let $f_{1}$ be the
monic irreducible polynomial of $\pi_{1}$ over $W(k_{1})$. Since $g$ is an
isometry, we have
$\overline{\nu_{2}}(g(\pi_{1}))=\overline{\nu_{1}}(\pi_{1})=1/e$, and hence,
$g(\pi_{1})$ is a uniformizer of $L_{2}$. Let
$\widetilde{g}:L_{1}^{alg}\longrightarrow L_{2}^{alg}$ be an extended
homomorphism of $g$. Let $g(f_{1})$ be the monic irreducible polynomial of
$g(\pi_{1})$ over $K_{2}$. If we write $f_{1}=x^{e}+...a_{1}x+a_{0}$, we have

$\displaystyle g(f_{1})$ $\displaystyle=x^{e}+...g(a_{1})x+g(a_{0})$

since $g(K_{1})$ is contained in $K_{2}$. Then by Lemma 2.9.(1) and Lemma 2.6,

$\displaystyle M(R_{2})$
$\displaystyle=\max\left\\{\widetilde{\nu_{2}}\big{(}g(\pi_{1})-\eta\big{)}:g(f_{1})(\eta)=0,\eta\neq
g(\pi_{1})\right\\}$
$\displaystyle=\max\left\\{\widetilde{\nu_{2}}\big{(}g(\pi_{1})-\widetilde{g}(\pi_{1}^{\prime})\big{)}:f_{1}(\pi_{1}^{\prime})=0,\pi_{1}^{\prime}\neq\pi_{1}\right\\}$
$\displaystyle=\max\left\\{\widetilde{\nu_{1}}(\pi_{1}-\pi_{1}^{\prime}):f_{1}(\pi_{1}^{\prime})=0,\pi_{1}^{\prime}\neq\pi_{1}\right\\}$
$\displaystyle=M(R_{1})$

∎

######  Proposition 2.10.

Let $R_{1}$ and $R_{2}$ be complete discrete valuation rings of characteristic
$0$ with perfect residue fields of characteristic $p$. Let $\mathfrak{m}_{i}$
be the maximal ideal of $R_{i}$ generated by $\pi_{i}$ and $\nu_{i}$
corresponding valuation of $R_{i}$ for $i=1,2$. Let $L_{i}$ and $K_{i}$ be the
fraction fields of $R_{i}$ and $W(k_{i})$ for $i=1,2$ respectively.

  1. (1)

Let $g:R_{1}\longrightarrow R_{2}$ be a $(n_{1},n_{2})$-lifting of
$\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$ at $\pi_{1}$ which satisfies

$\widetilde{\nu_{2}}\big{(}g(\pi_{1})-\beta\big{)}>\max_{\sigma}\limits\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-\beta\Big{)}:\sigma\big{(}g(\pi_{1})\big{)}\neq
g(\pi_{1})\right\\}$

where $\sigma$ runs through all of $\mathrm{Hom}_{K_{2}}(L_{2},L_{2}^{alg})$
and $\beta$ is a representative of $\phi(\pi_{1}+\mathfrak{m}_{1}^{n_{1}})$.
Then

$\max_{\sigma}\limits\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-\beta\Big{)}:\sigma\big{(}g(\pi_{1})\big{)}\neq
g(\pi_{1})\right\\}=M(R_{1}).$

  2. (2)

The definition of liftings is independent of the choice of uniformizer of
$R_{1}$. More precisely, saying that $g:R_{1}\longrightarrow R_{2}$ is a
$(n_{1},n_{2})$-lifting of $\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$ at
$\pi_{1}$ is equivalent to the following:

     * •

For any $x$ in $R_{1}$, there exists a representative $\beta_{x}$ of
$\phi(x+\mathfrak{m}_{1}^{n_{1}})$ which satisfies

$\widetilde{\nu_{2}}\big{(}g(x)-\beta_{x}\big{)}>M(R_{1})$

     * •

$\phi_{red,1}\circ\operatorname{pr}_{1,1}=\operatorname{pr}_{2,1}\circ g$

We write
$\operatorname{L}_{\pi_{1},n_{1},n_{2}}=\operatorname{L}_{n_{1},n_{2}}$ and
say that $\operatorname{L}_{n_{1},n_{2}}$ is the _$(n_{1},n_{2})$ -lifting_.

###### Proof.

(1) For $\sigma\in\mathrm{Hom}_{K_{2}}(L_{2},L_{2}^{alg})$ with
$\sigma(g(\pi_{1}))\neq g(\pi_{1})$,

$\displaystyle\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-\beta\Big{)}$
$\displaystyle=\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-g(\pi_{1})+g(\pi_{1})-\beta\Big{)}$
$\displaystyle=\min\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-g(\pi_{1})\Big{)},\widetilde{\nu_{2}}\big{(}g(\pi_{1})-\beta\big{)}\right\\}$
$\displaystyle=\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-g(\pi_{1})\Big{)}$

Since
$\widetilde{\nu_{2}}(g(\pi_{1})-\beta)>\widetilde{\nu_{2}}(\sigma(g(\pi_{1}))-\beta)$.
This shows

$\displaystyle M(R_{1})$
$\displaystyle=\max_{\sigma}\left\\{\widetilde{\nu_{1}}\big{(}\pi_{1}-\sigma(\pi_{1})\big{)}:\sigma(\pi_{1})\neq\pi_{1}\right\\}$
$\displaystyle=\max_{\sigma}\left\\{\widetilde{\nu_{2}}\Big{(}g(\pi_{1})-\sigma\big{(}g(\pi_{1})\big{)}\Big{)}:\sigma\big{(}g(\pi_{1})\big{)}\neq
g(\pi_{1})\right\\}$
$\displaystyle=\max_{\sigma}\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-\beta\Big{)}:\sigma\big{(}g(\pi_{1})\big{)}\neq
g(\pi_{1})\right\\}$

where the second equality follows from Lemma 2.9.(2) since
$[K_{2}(g(\pi_{1})):K_{2}]$ is equal to $[L_{1}:K_{1}]$ and $g(\pi_{1})$ is a
uniformizer of $K_{2}(g(\pi_{1}))$.

(2) Let $g:R_{1}\longrightarrow R_{2}$ be a $(n_{1},n_{2})$-lifting of
$\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$ at $\pi_{1}$ which satisfies

$\widetilde{\nu_{2}}\big{(}g(\pi_{1})-\beta\big{)}>\max_{\sigma}\limits\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-\beta\Big{)}:\sigma\big{(}g(\pi_{1})\big{)}\neq
g(\pi_{1})\right\\}$

where $\sigma$ runs through all of $\mathrm{Hom}_{K_{2}}(L_{2},L_{2}^{alg})$
and $\beta$ is a representative of $\phi(\pi_{1}+\mathfrak{m}_{1}^{n_{1}})$.
For any $x$ in $R_{1}$, we can write
$x=\sum_{r=0}^{\infty}\lambda_{r}\pi_{1}^{r}$ where $\lambda_{r}$ is in the
set $S_{1}$ of Teichmüller representatives for each $r$. Then

$\phi(x+\mathfrak{m}_{1}^{n_{1}})=\phi\left(\left(\sum_{r=0}^{\infty}\lambda_{r}\pi_{1}^{r}\right)+\mathfrak{m}_{1}^{n_{1}}\right)\\\
=\left(\sum_{r=0}^{\infty}\tau_{r}\beta^{r}\right)+\mathfrak{m}_{2}^{n_{2}}$

where $\tau_{r}$ is a representative of
$\phi(\lambda_{r}+\mathfrak{m}_{1}^{n_{1}})$ contained in $S_{2}$ guaranteed
by Lemma 2.8.(1). In particular $\sum_{r=0}^{\infty}\tau_{r}\beta^{r}$ is a
representative of $\phi(x+\mathfrak{m}_{1}^{n_{1}})$, say $\beta_{x}$. By the
second condition of the definition of liftings and Lemma 2.4, we have
$g(\lambda_{r})=\tau_{r}$, and hence,

$g(x)=g\left(\sum_{r=0}^{\infty}\lambda_{r}\pi_{1}^{r}\right)=\sum_{r=0}^{\infty}\tau_{r}g(\pi_{1})^{r}.$

We obtain

$\displaystyle\widetilde{\nu_{2}}(g(x)-\beta_{x})$
$\displaystyle=\widetilde{\nu_{2}}\left(\sum_{r=0}^{\infty}\tau_{r}g(\pi_{1})^{r}-\sum_{r=0}^{\infty}\tau_{r}\beta^{r}\right)$
$\displaystyle=\widetilde{\nu_{2}}\left(\Big{(}g(\pi_{1})-\beta\Big{)}\sum_{r=1}^{\infty}\tau_{r}\left(\sum_{j=0}^{r-1}g(\pi_{1})^{r-1-j}\beta^{j}\right)\right)$
$\displaystyle>M(R_{1})$

since

$\displaystyle\widetilde{\nu_{2}}\big{(}g(\pi_{1})-\beta\big{)}$
$\displaystyle>\max_{\sigma}\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-\beta\Big{)}:\sigma\big{(}g(\pi_{1})\big{)}\neq
g(\pi_{1})\right\\}$ $\displaystyle=M(R_{1}).$

∎

######  Lemma 2.11 (Krasner’s lemma).

Let $(K,\nu)$ be henseilan valued field whose value group is contained in
${\mathbb{R}}$ and let $a,b\in K^{alg}$. Suppose $a$ is separable over $K(b)$.
Suppose that for all embeddings $\sigma(\neq id)$ of $K(a)$ over $K$, we have

$\widetilde{\nu}(b-a)>\widetilde{\nu}\big{(}\sigma(a)-a\big{)}.$

Then $K(a)\subset K(b)$.

###### Proof.

Chapter 2 of [24]. ∎

The following theorem shows that there is a unique lifting if we enlarge the
lengths of residue rings.

######  Theorem 2.12.

Let $R_{1}$ and $R_{2}$ be complete discrete valuation rings of characteristic
$0$ with perfect residue fields $k_{1}$ and $k_{2}$ of characteristic $p$
respectively. Let $\mathfrak{m}_{i}$ be the maximal ideal of $R_{i}$ generated
by $\pi_{i}$ and $\nu_{i}$ corresponding valuation of $R_{i}$ for $i=1,2$. Let
$L_{i}$ and $K_{i}$ be the fraction fields of $R_{i}$ and $W(k_{i})$ for
$i=1,2$ respectively. Suppose $n_{2}>M(R_{1})\nu_{1}(p)\nu_{2}(p)$ and
$\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})$ is not empty. Then there exists a
unique $(n_{1},n_{2})$-lifting
$\operatorname{L}_{n_{1},n_{2}}:\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})\longrightarrow\mathrm{Hom}(R_{1},R_{2})$.
$\operatorname{L}_{n_{1},n_{2}}(\phi)$ is also an isomorphism when
$\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$ is an isomorphism.

###### Proof.

Let $S_{i}$ be the set of Teichmüller representatives of $R_{i}$. By Lemma
2.8.(2), let
$\phi_{res}:(W(k_{1})+\mathfrak{m}_{1}^{n_{1}})/\mathfrak{m}_{1}^{n_{1}}\longrightarrow(W(k_{2})+\mathfrak{m}_{2}^{n_{2}})/\mathfrak{m}_{2}^{n_{2}}$
be the restrition map of $\phi$. For an element
$a=\sum_{r=0}^{\infty}\lambda_{r}p^{r}$ in $W(k_{1})$, as in the proof of
Lemma 2.8.(2), we define $g_{res}:W(k_{1})\longrightarrow W(k_{2})$ by
$g_{res}(a)=\sum_{r=0}^{\infty}\tau_{r}p^{r}$ where $\tau_{r}$ is a unique
representative of $\phi_{res}(\lambda_{r}+\mathfrak{m}_{1}^{n_{1}})$ which is
contained in $S_{2}$. Then $g_{res}$ induces $\phi_{res}$. By Lemma 2.3,
$L_{1}=K_{1}(\alpha)$ is totally ramified of degree $\nu_{1}(p)$ over $K_{1}$
where $\alpha=\pi_{1}$. Let $f$ be the monic irreducible polynomial of
$\alpha$ over $K_{1}$. The ring homomorphism $g_{res}$ induces the field
homomorphism from $K_{1}$ into $K_{2}$. We still denote the fraction field
homomorphism by $g_{res}$ if there is no confusion. Then
$g_{res}:K_{1}\longrightarrow K_{2}$ is an isometry. Let
$\widetilde{g_{res}}:K_{1}^{alg}\longrightarrow K_{2}^{alg}$ be an extended
field homomorphism of $g_{res}$. Then $\widetilde{g_{res}}$ is an isometry by
Lemma 2.6. Let $g_{res}(f)$ be the monic irreducible polynomial of
$\widetilde{g_{res}}(\alpha)$ over $K_{2}$. If we write

$\displaystyle f=$ $\displaystyle x^{\nu_{1}(p)}+...+a_{1}x+a_{0}$
$\displaystyle=$ $\displaystyle(x-\alpha_{1})...(x-\alpha_{\nu_{1}(p)}),$

where $\alpha=\alpha_{1}$, then

$\displaystyle g_{res}(f)=$ $\displaystyle
x^{\nu_{1}(p)}+...+g_{res}(a_{1})x+g_{res}(a_{0})$ $\displaystyle=$
$\displaystyle\big{(}x-\widetilde{g_{res}}(\alpha_{1})\big{)}...\big{(}x-\widetilde{g_{res}}(\alpha_{\nu_{1}(p)})\big{)}$

since $[K_{2}(\widetilde{g_{res}}(\alpha)):K_{2}]\leq\nu_{1}(p)$ and
$\widetilde{\nu_{2}}(\widetilde{g_{res}}(\alpha))=1/\nu_{1}(p)$. Let $\beta$
be any representative of $\phi(\alpha+\mathfrak{m}_{1}^{n_{1}})$. Since
$g_{res}$ induces $\phi_{res}$, we can write

$\displaystyle 0+\mathfrak{m}_{2}^{n_{2}}$
$\displaystyle=\phi(f(\alpha)+\mathfrak{m}_{1}^{n_{1}})$
$\displaystyle=\phi(\alpha+\mathfrak{m}_{1}^{n_{1}})^{\nu_{1}(p)}+...+\phi(a_{1}+\mathfrak{m}_{1}^{n_{1}})\phi(\alpha+\mathfrak{m}_{1}^{n_{1}})+\phi(a_{0}+\mathfrak{m}_{1}^{n_{1}})$
$\displaystyle=g_{res}(f)(\beta)+\mathfrak{m}_{2}^{n_{2}}.$

This shows that $g_{res}(f)(\beta)$ is in $\mathfrak{m}_{2}^{n_{2}}$ and

$\nu_{2}\big{(}g_{res}(f)(\beta)\big{)}\geq
n_{2}>M(R_{1})\nu_{1}(p)\nu_{2}(p).$

We claim that there exists an index $i_{0}$ satisfying
$\widetilde{\nu_{2}}(\beta-\widetilde{g_{res}}(\alpha_{i_{0}}))>M(R_{1})$. If
$\widetilde{\nu_{2}}(\beta-\widetilde{g_{res}}(\alpha_{i}))\leq M(R_{1})$ for
all $i$, then

$\widetilde{\nu_{2}}\big{(}g_{res}(f)(\beta)\big{)}=\widetilde{\nu_{2}}\left(\prod_{i}\Big{(}\beta-\widetilde{g_{res}}(\alpha_{i})\Big{)}\right)\leq
M(R_{1})\nu_{1}(p).$

This shows

$\nu_{2}\big{(}g_{res}(f)(\beta)\big{)}=\nu_{2}(p)\widetilde{\nu_{2}}\big{(}g_{res}(f)(\beta)\big{)}\leq
M(R_{1})\nu_{1}(p)\nu_{2}(p).$

Thus there is an index $i_{0}$ satisfying

$\widetilde{\nu_{2}}\big{(}\beta-\widetilde{g_{res}}(\alpha_{i_{0}})\big{)}>M(R_{1})=\max\left\\{\widetilde{\nu_{2}}\big{(}\widetilde{g_{res}}(\alpha_{1})-\widetilde{g_{res}}(\alpha_{j})\big{)}:j=2,...,\nu_{1}(p)\right\\}$

where the equality follows from the fact that $\widetilde{g_{res}}$ is an
isometry. Hence, Krasner’s lemma 2.11 shows
$K_{2}(\widetilde{g_{res}}(\alpha_{i_{0}}))\subset K_{2}(\beta)\subset L_{2}$.
We define an extended homomorphism $g:L_{1}\longrightarrow L_{2}$ of
$g_{res}:K_{1}\longrightarrow K_{2}$ by the rule $\pi_{1}\mapsto
g(\pi_{1})=\widetilde{g_{res}}(\alpha_{i_{0}})$. $g$ induces the restricted
homomorphism from $R_{1}$ to $R_{2}$ which is still denoted by $g$. Since
$g_{res}$ induces $\phi_{res}$ and

$M(R_{1})=\max_{\sigma}\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g(\pi_{1})\big{)}-\beta\Big{)}:\sigma\big{(}g(\pi_{1})\big{)}\neq
g(\pi_{1})\right\\}$

by Lemma 2.10.(1), $g$ is a $(n_{1},n_{2})$-lifting of $\phi$.

Suppose that $g_{1}:R_{1}\longrightarrow R_{2}$ is a $(n_{1},n_{2})$-lifting
of $\phi$ other than $g$. Then we have

$\widetilde{\nu_{2}}\big{(}g_{1}(\pi_{1})-\beta\big{)}>\max_{\sigma}\left\\{\widetilde{\nu_{2}}\Big{(}\sigma\big{(}g_{1}(\pi_{1})\big{)}-\beta\Big{)}:\sigma\big{(}g\left(\pi_{1}\right)\big{)}\neq
g(\pi_{1})\right\\}$

by the first condition of the definition of liftings. By the second condition
of the definition of liftings and by Theorem 2.2, we obtain the restriction
$g_{1}|_{W(k_{1})}$ of $g_{1}$ to $W(k_{1})$ is equal to $g|_{W(k_{1})}$. This
shows that the monic irreducible polynomial $g_{1}(f)$ of $g_{1}(\pi_{1})$ is
equal to the monic irreducible polynomial $g(f)$ of $g(\pi_{1})$ and

$\left\\{\sigma\big{(}g_{1}(\pi_{1})\big{)}:\sigma\in\mathrm{Hom}_{K_{2}}(L_{2},L_{2}^{alg})\right\\}=\left\\{\sigma\big{(}g(\pi_{1})\big{)}:\sigma\in\mathrm{Hom}_{K_{2}}(L_{2},L_{2}^{alg})\right\\}.$

In particular $g_{1}(\pi_{1})=\sigma(g(\pi_{1}))$ for some
$\sigma\in\mathrm{Hom}_{K_{2}}(L_{2},L_{2}^{alg})$. But since
$g_{1}(\pi_{1})\neq g(\pi_{1})$, we have the inequalities
$\widetilde{\nu_{2}}(g_{1}(\pi_{1})-\beta)>\widetilde{\nu_{2}}(g(\pi_{1})-\beta)$
and
$\widetilde{\nu_{2}}(g_{1}(\pi_{1})-\beta)<\widetilde{\nu_{2}}(g(\pi_{1})-\beta)$
simultaneously. This gives a contradiction. Hence we obtain the uniqueness of
the lifting.

When $\phi$ is an isomorphism, so are $\phi_{res}$ and $g_{res}$. We obtain
$[L_{2}:K_{2}]=[L_{1}:K_{1}]$, and hence,
$\operatorname{L}_{n_{1},n_{2}}(\phi)$ is also an isomorphism.

∎

We note that the proof of Theorem 2.12 works for any representative $\beta$ of
$\phi(\pi_{1}+\mathfrak{m}_{1}^{n_{1}})$.

######  Example 2.13.

  1. (1)

Let $R_{1}={\mathbb{Z}}_{3}[\sqrt{3}]$ and
$R_{2}={\mathbb{Z}}_{3}[\sqrt{-3}]$. There is no homomorphism between $R_{1}$
and $R_{2}$ by Kummer theory. But there is an isomorphism

$\phi:R_{1,2}=\frac{{\mathbb{Z}}_{3}[\sqrt{3}]}{3{\mathbb{Z}}_{3}[\sqrt{3}]}\longrightarrow
R_{2,2}=\frac{{\mathbb{Z}}_{3}[\sqrt{-3}]}{3{\mathbb{Z}}_{3}[\sqrt{-3}]}$

given by the rule $a+b\sqrt{3}\mapsto a+b\sqrt{-3}$. Since
$\nu_{1}(3)=\nu_{2}(3)=2$ and
$M(R_{1})=\widetilde{\nu_{1}}(\sqrt{3}-(-\sqrt{3}))=1/2$, we obtain
$M(R_{1})\nu_{1}(3)\nu_{2}(3)=2$. Hence the lower bound for $n_{2}$ in
_Theorem 2.12_ is the best possible in this case. This phenomenon will be
generalized in _Proposition 3.11_.

  2. (2)

If we take $R_{1}=R_{2}=\mathbb{Z}_{3}[\sqrt{3}]$ and $n_{1}=n_{2}=2n$, then
$R_{1,2n}=R_{2,2n}\cong(\mathbb{Z}_{3}/3^{n}\mathbb{Z}_{3})[x]/(x^{2}-3)$.
Then $\phi:a+bx\mapsto a+(1+3^{n-1})bx=\phi(a+bx)$ defines an isomorphism
between $R_{1}$ and $R_{2}$. But when $n>1$, there is no homomorphism
$g:R_{1}\longrightarrow R_{2}$ which induces $\phi$ since Galois conjugates of
$\sqrt{3}$ are $\pm\sqrt{3}$. This shows that in _Theorem 2.12_, we can not
guarantee that the following diagram is commutative:

$\begin{CD}R_{1}@>{\operatorname{L}_{n_{1},n_{2}}(\phi)}>{}>R_{2}\\\
@V{}V{}V@V{}V{}V\\\ R_{1,n_{1}}@>{\phi}>{}>R_{2,n_{2}}\end{CD}$

######  Remark 2.14.

We regard _Theorem 2.12_ as a generalization of _Theorem 2.2.(2)_. We can
restate _Theorem 2.2.(2)_ as follows. For $\phi:k_{1}\longrightarrow k_{2}$,
there exists a unique homomorphism $g:W(k_{1})\longrightarrow W(k_{2})$ which
is characterized by the following property:

  * •

For any $x$ in $W(k_{1})$, there exists a representative $\beta_{x}$ of
$\phi(x+pW(k_{1}))$ which satisfies $\nu_{2}(g(x)-\beta_{x})=\infty$.

In general, the property above does not hold as is seen in _Example 2.13.(2)_.
But we can restate _Theorem 2.12_ as follows. For
$\phi:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$, there exists a unique
homomorphism $g:R_{1}\longrightarrow R_{2}$ which is characterized by the
following property:

  * •

There exists $N$ depending on $R_{1}$ only such that for any $x$ in $R_{1}$,
there exists a representative $\beta_{x}$ of
$\phi(x+\mathfrak{m}_{1}^{n_{1}})$ which satisfies
$\nu_{2}(g(x)-\beta_{x})>N$.

This follows from _Proposition 2.10.(2)_.

Note that $M(R)e(R)$ can vary when $R$ changes arbitrarily. The following
lemma will play an important role for bounding $M(R)e(R)$ when $e(R)$ is
fixed.

######  Lemma 2.15.

Let $R\subset S$ be discrete valuation rings and $S$ a finitely generated
$R$-module. Suppose $S=R[\alpha]$ for some $\alpha$ in $S$. Let $f(x)$ in
$R[x]$ be the monic irreducible polynomial of $\alpha$ over $R$.

  1. (1)

The different $\mathfrak{D}_{S/R}$ of $S/R$ is a principal ideal generated by
$f^{\prime}(\alpha)$

  2. (2)

Let $\mathfrak{B}$ be the maximal ideal of $S$. Let $e$ be the ramification
index of $S$ over $R$ and $\nu_{S}$ the valuation corresponding to $S$. Let
$s$ be the power which satisfies $\mathfrak{B}^{s}=\mathfrak{D}_{S/R}$. Then
one has

$\displaystyle s=e-1$ $\displaystyle\textrm{ if}\ S\ \textrm{is tamely
ramified,}$ $\displaystyle e\leq s\leq e-1+\nu_{S}(e)$
$\displaystyle\textrm{if}\ S\textrm{ is wildly ramified.}$

###### Proof.

Chapter 3, Section 2 of [25]. ∎

The following theorem can be regarded as a generalized version of Theorem
2.2.(1) for the ramified case.

######  Theorem 2.16.

Let $\overline{R}$ be a principal Artinian local ring of length $n$ with
perfect residue field $k$ of characteristic $p$ and maximal ideal
$\overline{\mathfrak{m}}$. Here length $n$ means
$\overline{\mathfrak{m}}^{n}=0$ and $\overline{\mathfrak{m}}^{n-1}\neq 0$
which is denoted by $l(\overline{R})=n$. Suppose that $\overline{R}$ has no
finite subfield as a subring. For any positive integer $a$, if $a$ generates
an ideal $\overline{\mathfrak{m}}^{k}$, we denote $k$ by $\nu(a)$. Suppose

$l(\overline{R})=n>\nu(p)+\nu(p)\nu\big{(}\nu(p)\big{)}.$

Then there exists a complete discrete valuation ring of characteristic $0$
which has $\overline{R}$ as its $n$-th residue ring. Such a ring is unique up
to isomorphism.

###### Proof.

Any principal Artinian local ring is a homomorphic image of a discrete
valuation ring. This can be proved by Cohen structure theorem for complete
local rings([19]) or, more directly, by the property of CPU-rings([17]). Since
the completion of a discrete valuation ring $R$ has the same $n$-th residue
ring as that of $R$, we may assume that there are complete discrete valuation
rings $R_{1}$ and $R_{2}$ which have $\overline{R}$ as isomorphic copies of
$R_{1,n}$ and $R_{2,n}$ respectively. We note that $R_{i}$ is of
characteristic $0$ for $i=1,2$ since $\overline{R}$ has no finite subfield as
a subring. Let $L_{i}$ and $K_{i}$ be the fraction fields of $R_{i}$ and
$W(k_{i})$ for $i=1,2$ respectively. Then by Lemma 2.3, $L_{1}=K_{1}(\alpha)$
where $\alpha=\pi_{1}$ is a uniformizer of $R_{1}$. Let $f$ be the monic
irreducible polynomial of $\alpha$ over $K_{1}$. Then one can write

$\displaystyle f=$ $\displaystyle x^{\nu(p)}+...+a_{1}x+a_{0}$
$\displaystyle=$ $\displaystyle(x-\alpha_{1})...(x-\alpha_{\nu(p)})$

where $\alpha=\alpha_{1}$. Let $\nu_{i}$ be the corresponding valuation of
$R_{i}$. We note that $\nu_{1}(p)=\nu_{2}(p)=\nu(p)$ since $\overline{R}$ has
no finite subfield as a subring. We consider the differentiation

$f^{\prime}=\sum_{i=1}^{\nu(p)}{\frac{f}{(x-\alpha_{i})}}.$

There are two cases.

  * •

Tame case: Suppose $L_{1}/K_{1}$ is tamely ramified. Hence, $\nu(\nu(p))=0$.
For all distinct $i$ and $j$, $\widetilde{\nu_{1}}(\alpha_{i})=1/\nu(p)$ and
hence $\widetilde{\nu_{1}}(\alpha_{i}-\alpha_{j})\geq 1/\nu(p)$. We obtain

$\displaystyle\widetilde{\nu_{1}}\big{(}f^{\prime}(\alpha_{1})\big{)}$
$\displaystyle=\widetilde{\nu_{1}}\left(\prod_{j\neq
1}(\alpha_{1}-\alpha_{j})\right)$ $\displaystyle=\sum_{j\neq
1}\widetilde{\nu_{1}}(\alpha_{1}-\alpha_{j})$
$\displaystyle\geq\frac{\nu(p)-1}{\nu(p)}.$

Since

$\widetilde{\nu_{1}}\big{(}f^{\prime}(\alpha_{1})\big{)}=\frac{\nu(p)-1}{\nu(p)}$

by Lemma 2.15, $\widetilde{\nu_{1}}(\alpha_{1}-\alpha_{j})=1/\nu(p)=M(R_{1})$
for $j\neq 1$. Hence we have

$\displaystyle\nu(p)+\nu(p)\nu\big{(}\nu(p)\big{)}$ $\displaystyle=\nu(p)$
$\displaystyle=M(R_{1})\nu(p)^{2}$

and Theorem 2.12 finishes the proof.

  * •

Wild case: Suppose $L_{1}/K_{1}$ is wildly ramified. Since
$\widetilde{\nu_{1}}(\alpha_{i}-\alpha_{j})\geq 1/\nu(p)$ for all distinct $i$
and $j$, we have $m:=\min\\{\widetilde{\nu_{1}}(\alpha_{1}-\alpha_{j})|\ j\neq
1\\}\geq 1/\nu(p)$ and

$\displaystyle M(R_{1})$
$\displaystyle=\max\left\\{\widetilde{\nu}\big{(}\alpha_{1}-\alpha_{j}\big{)}:j\neq
1\right\\}$ $\displaystyle\leq\widetilde{\nu_{1}}\left(\prod_{j\neq
1}(\alpha_{1}-\alpha_{j})\right)-(\nu(p)-2)m$
$\displaystyle\leq\widetilde{\nu_{1}}\left(\prod_{j\neq
1}(\alpha_{1}-\alpha_{j})\right)-\frac{\nu(p)-2}{\nu(p)}$
$\displaystyle=\widetilde{\nu_{1}}\big{(}f^{\prime}(\alpha_{1})\big{)}-\frac{\nu(p)-2}{\nu(p)}$
$\displaystyle\leq\frac{\nu(p)-1+\nu\big{(}\nu(p)\big{)}}{\nu(p)}-\frac{\nu(p)-2}{\nu(p)}$
$\displaystyle=\frac{1+\nu\big{(}\nu(p)\big{)}}{\nu(p)}$

by Lemma 2.15, and hence, $M(R_{1})\nu(p)^{2}\leq\nu(p)+\nu(p)\nu(\nu(p))$.
Again Theorem 2.12 finishes the proof.

∎

Note that the notation $\nu(p)$ in Theorem 2.16 is compatible with the
previously defined valuation. Suppose that a discrete valuation ring $R$ with
valuation $\nu$ and maximal ideal $\mathfrak{m}$ has $\overline{R}$ as its
residue ring. Then $\nu(p)$ is equal to the power of the maximal ideal
generated by $p$, that is, $Rp=\mathfrak{m}^{\nu(p)}$ as we noted in the proof
of Theorem 2.16.

##  3\. Functoriality

For a prime number $p$, let ${\mathcal{C}}_{p}$ be a category consisting of
the following data :

  * •

$\operatorname{Ob}({\mathcal{C}}_{p})$ is the family of absolutely unramified
complete discrete valuation rings of mixed characteristic having perfect
residue fields of characteristic $p$.

  * •

$\operatorname{Mor}_{{\mathcal{C}}_{p}}(R_{1},R_{2}):=\mathrm{Hom}(R_{1},R_{2})$
for $R_{1}$ and $R_{2}$ in $\operatorname{Ob}({\mathcal{C}}_{p})$.

Let ${\mathcal{R}}_{p}$ be a category consisting of the following data :

  * •

$\operatorname{Ob}({\mathcal{R}}_{p})$ is the family of perfect fields of
characteristic $p$.

  * •

$\operatorname{Mor}_{{\mathcal{R}}_{p}}({k_{1}},{k_{2}}):=\mathrm{Hom}(k_{1},k_{2})$
for $k_{1}$ and $k_{2}$ in $\operatorname{Ob}({\mathcal{R}}_{p})$.

Let $\Pr:\ {\mathcal{C}}_{p}\rightarrow{\mathcal{R}}_{p}$ be the canonical
projection functor. We restate Theorem 2.2 categorically as follows :

######  Theorem 3.1.

There exists a functor
$\operatorname{L}:{\mathcal{R}}_{p}\longrightarrow{\mathcal{C}}_{p}$ which
satisfies:

  * •

The composite functor $\Pr\circ\operatorname{L}$ is equivalent to the identity
functor $\operatorname{Id}_{{\mathcal{R}}_{p}}$.

  * •

The composite functor $\operatorname{L}\circ\Pr$ is equivalent to the identity
functor $\operatorname{Id}_{{\mathcal{C}}_{p}}$.

The main purpose of this section is to give a generalized version of Theorem
3.1 for the ramified case. For a prime number $p$ and a positive integer $e$,
let ${\mathcal{C}}_{p,e}$ be a category consisting of the following data :

  * •

$\operatorname{Ob}({\mathcal{C}}_{p,e})$ is the family of complete discrete
valuation rings of mixed characteristic having perfect residue fields of
characteristic $p$ and the ramification index $e$; and

  * •

$\operatorname{Mor}_{{\mathcal{C}}_{p,e}}(R_{1},R_{2}):=\mathrm{Hom}(R_{1},R_{2})$
for $R_{1}$ and $R_{2}$ in $\operatorname{Ob}({\mathcal{C}}_{p,e})$.

Let ${\mathcal{R}}_{p,e}^{n}$ be a category consisting of the following data :

  * •

For $n\leq e$, $\operatorname{Ob}({\mathcal{R}}_{p,e}^{n})$ is the family of
principal Artinian local rings $\overline{R}$ of length $n$ with perfect
residue fields of characteristic $p$, and for $n>e$,
$\operatorname{Ob}({\mathcal{R}}_{p,e}^{n})$ is the family of principal
Artinian local rings $\overline{R}$ of length $n$ with perfect residue fields
of characteristic $p$ such that
$p\in\overline{\mathfrak{m}}^{e}\setminus\overline{\mathfrak{m}}^{e+1}$ where
$\overline{\mathfrak{m}}$ is the maximal ideal of $\overline{R}$; and

  * •

$\operatorname{Mor}_{{\mathcal{R}}_{p,e}^{n}}(\overline{R_{1}},\overline{R_{2}}):=\mathrm{Hom}(\overline{R_{1}},\overline{R_{2}})$
for $\overline{R_{1}}$ and $\overline{R_{2}}$ in
$\operatorname{Ob}({\mathcal{R}}_{p,e,}^{n})$,

Note that for $e_{1},e_{2}\geq 1$ and for $n\leq e_{1},e_{2}$, two categories
${\mathcal{R}}_{p,e_{1}}^{n}$, ${\mathcal{R}}_{p,e_{2}}^{n}$ are the same. For
each $m>n$, let $\Pr_{n}:\
{\mathcal{C}}_{p,e}\rightarrow{\mathcal{R}}_{p,e}^{n}$ and $\Pr^{m}_{n}:\
{\mathcal{R}}_{p,e}^{m}\rightarrow{\mathcal{R}}_{p,e}^{n}$ be the canonical
projection functors respectively.

######  Definition 3.2.

Fix a prime number $p$ and a positive integer $e$.

  1. (1)

We say that the category ${\mathcal{C}}_{p,e}$ is _$n$ -liftable_ if there is
a functor
$\operatorname{L}:{\mathcal{R}}_{p,e}^{n}\longrightarrow{\mathcal{C}}_{p,e}$
which satisfies the following:

     * •

$(\Pr_{n}\circ\operatorname{L})(\overline{R})\cong\overline{R}$ for each
$\overline{R}$ in $\operatorname{Ob}({\mathcal{R}}_{p,e})$.

     * •

$\Pr_{1}\circ\operatorname{L}$ is equivalent to $\Pr^{n}_{1}$.

     * •

$\operatorname{L}\circ\Pr_{n}$ is equivalent to
$\operatorname{Id}_{{\mathcal{C}}_{p,e}}$.

We say that $\operatorname{L}$ is a _$n$ -th lifting functor_ of
${\mathcal{C}}_{p,e}$.

  2. (2)

The _lifting number_ for ${\mathcal{C}}_{p,e}$ is the smallest positive
integer $n$ such that ${\mathcal{C}}_{p,e}$ is $n$-liftable. If there is no
such $n$, we define the lifting number for ${\mathcal{C}}_{p,e}$ to be
$\infty$.

######  Remark 3.3.

  1. (1)

In _Definition 3.2_, the restriction of $\operatorname{L}$ to
$\mathrm{Iso}(R_{n})$ is a surjective group homomorphism from
$\mathrm{Iso}(R_{n})$ to $\mathrm{Iso}(R)$ for each
$R\in\operatorname{Ob}({\mathcal{C}}_{p,e})$.

  2. (2)

Suppose that there is a $n$-th lifting functor
$\operatorname{L}:{\mathcal{R}}_{p,e}^{n}\rightarrow{\mathcal{C}}_{p,e}$. For
any $\overline{R}$ in $\operatorname{Ob}({\mathcal{R}}_{p,e})$, up to
isomorphism, $\operatorname{L}(\overline{R})$ is a unique object in
$\operatorname{Ob}({\mathcal{C}}_{p,e})$ which has $\overline{R}$ as its
$n$-th residue ring. Suppose that $R$ in
$\operatorname{Ob}({\mathcal{C}}_{p,e})$ has $\overline{R}$ as its $n$-th
residue ring. Since $\operatorname{L}\circ\Pr_{n}$ is equivalent to the
identity functor $\operatorname{Id}_{{\mathcal{C}}_{p,e}}$,
$R=\operatorname{Id}_{{\mathcal{C}}_{p,e}}(R)$ is isomorphic to
$(\operatorname{L}\circ\Pr_{n})(R)=\operatorname{L}(\overline{R})$.

  3. (3)

The lifting number for ${\mathcal{C}}_{p}$ is $1$ by _Theorem 3.1_. We will
see that the lifting number for ${\mathcal{C}}_{p,e}$ is always larger than
$e$ whenever $e>1$ in _Corollary 3.17_. For $n\geq e$, we have that a functor
$\operatorname{L}_{n+1}:=\operatorname{L}_{n}\circ\Pr^{n+1}_{n}$ is a
$(n+1)$-th lifting functor of ${\mathcal{C}}_{p,e}$ for any $n$-th lifting
functor
$\operatorname{L}_{n}:{\mathcal{R}}_{p,e}^{n}\rightarrow{\mathcal{C}}_{p,e}$.
For $\overline{R}$ in $\operatorname{Ob}({\mathcal{R}}_{p,e}^{n+1})$, there
exists a ring $R$ in $\operatorname{Ob}({\mathcal{C}}_{p,e})$ which satisfies
$\Pr_{n+1}(R)=\overline{R}$ as noted in the proof of _Theorem 2.16_. Since
there is a unique object in $\operatorname{Ob}({\mathcal{C}}_{p,e})$ which has
$\Pr_{n}(R)$ as its $n$-th residue ring by _Remark 3.3.(2)_, we have

$({\Pr}_{n+1}\circ\operatorname{L}_{n+1})\left(\overline{R}\right)={\Pr}_{n+1}\circ(\operatorname{L}_{n}\circ{\Pr}^{n+1}_{n})\left(\overline{R}\right)={\Pr}_{n+1}(R)=\overline{R}.$

${\Pr}_{1}\circ{\operatorname{L}}_{n+1}=({\Pr}_{1}\circ{\operatorname{L}}_{n})\circ{\Pr}_{n}^{n+1}$
is equivalent to ${\Pr}_{1}^{n}\circ{\Pr}_{n}^{n+1}={\Pr}_{1}^{n+1}$ and
${\operatorname{L}}_{n+1}\circ{\Pr}_{n+1}=({\operatorname{L}}_{n}\circ{\Pr}_{n}^{n+1})\circ{\Pr}_{n+1}={\operatorname{L}}_{n}\circ{\Pr}_{n}$
is equivalent to $\operatorname{Id}_{{\mathcal{C}}_{p,e}}$.

######  Proposition 3.4.

For $1\leq i\leq 3$, let $R_{i}$ be a complete discrete valuation ring of
characteristic $0$ with perfect residue field of characteristic $p$. Let
$\mathfrak{m}_{i}$ be the maximal ideal of $R_{i}$ generated by $\pi_{i}$ and
$\nu_{i}$ corresponding valuation of $R_{i}$. For
$\phi^{1,2}:R_{1,n_{1}}\longrightarrow R_{2,n_{2}}$ and
$\phi^{2,3}:R_{2,n_{2}}\longrightarrow R_{3,n_{3}}$, suppose that there are
liftings $g^{1,2}:R_{1}\longrightarrow R_{2}$ and
$g^{2,3}:R_{2}\longrightarrow R_{3}$ of $\phi^{1,2}$ and $\phi^{2,3}$
respectively. If $\nu_{1}(p)=\nu_{2}(p)$, then $g=g^{2,3}\circ g^{1,2}$ is a
lifting of $\phi^{2,3}\circ\phi^{1,2}$. Moreover $g$ is a unique lifting of
$\phi^{2,3}\circ\phi^{1,2}$ when $n_{3}>M(R_{2})\nu_{2}(p)\nu_{3}(p)$ and
$n_{2}>M(R_{1})\nu_{1}(p)\nu_{2}(p)$.

###### Proof.

By Lemma 2.9, $M(R_{1})$ is equal to $M(R_{2})$, say $M$. Since $g^{1,2}$ is a
lifting of $\phi^{1,2}$, there is a representative $\beta_{1}$ of
$\phi^{1,2}(\pi_{1}+\mathfrak{m}_{1}^{n_{1}})$ such that
$\widetilde{\nu_{2}}(g^{1,2}(\pi_{1})-\beta_{1})>M$. We note that $\beta_{1}$
is a uniformizer of $R_{2}$. Since $g^{2,3}$ is a lifting of $\phi^{2,3}$,
there is a representative $\beta_{2}$ of
$(\phi^{2,3}\circ\phi^{1,2})(\pi_{1}+\mathfrak{m}_{1}^{n_{1}})=\phi^{2,3}(\beta_{1}+\mathfrak{m}_{2}^{n_{2}})$
such that $\widetilde{\nu_{3}}(g^{2,3}(\beta_{1})-\beta_{2})>M$. If we write
$g^{1,2}(\pi_{1})=\beta_{1}+x_{M}$ where $\widetilde{\nu_{2}}(x_{M})>M$, then
$g(\pi_{1})=g^{2,3}(g^{1,2}(\pi_{1}))=g^{2,3}(\beta_{1}+x_{M})$. Since
$\widetilde{\nu_{3}}(g^{2,3}(\beta_{1})-\beta_{2})>M$ and
$\widetilde{\nu_{3}}(g^{2,3}(x_{M}))=\widetilde{\nu_{2}}(x_{M})>M$,

$\widetilde{\nu_{3}}\big{(}g(\pi_{1})-\beta_{2}\big{)}=\widetilde{\nu_{3}}\big{(}g^{2,3}(\beta_{1})-\beta_{2}+g^{2,3}(x_{M})\big{)}>M.$

The equality
$(\phi^{2,3}\circ\phi^{1,2})_{red,1}\circ\operatorname{pr}_{1,1}=\operatorname{pr}_{3,1}\circ
g$ follows directly from $g=g^{2,3}\circ g^{1,2}$. By Definition 2.7 and
Proposition 2.10, $g$ is a lifting of $\phi^{2,3}\circ\phi^{1,2}$.

When $n_{3}>M(R_{2})\nu_{2}(p)\nu_{3}(p)=M(R_{1})\nu_{1}(p)\nu_{3}(p)$ and
$n_{2}>M(R_{1})\nu_{1}(p)\nu_{2}(p)$, $g$ is a unique lifting of
$\phi^{2,3}\circ\phi^{1,2}$ by Theorem 2.12.

∎

######  Corollary 3.5.

Let $e\geq 1$ and $R\in\operatorname{Ob}({\mathcal{C}}_{p,e})$. Suppose
$n>M(R)\nu(p)^{2}$. Let
$\operatorname{pr}^{n,n}|_{\mathrm{Iso}(R)}:\mathrm{Iso}(R)\rightarrow\mathrm{Iso}(R_{n})$
be the natural projection map. Then there exists a surjective group
homomorphism

$\operatorname{L}_{n}:\mathrm{Iso}(R_{n})\longrightarrow\mathrm{Iso}(R)$

which satisfies
$\operatorname{L}_{n}\circ(\operatorname{pr}^{n,n}|_{\mathrm{Iso}(R)})=\operatorname{Id}_{\mathrm{Iso}(R)}$.

######  Corollary 3.6.

Let $R_{1}$ be in $\operatorname{Ob}({\mathcal{C}}_{p,e_{1}})$ and $R_{2}$ in
$\operatorname{Ob}({\mathcal{C}}_{p,e_{2}})$. Suppose
$n_{2}>M(R_{1})\nu_{1}(p)\nu_{2}(p)$ and
$\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})$ is not empty. Then
$\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})$ and $\mathrm{Hom}(R_{1},R_{2})$ are
right $\mathrm{Iso}(R_{1,n_{1}})$-sets and there exists a surjective
$\mathrm{Iso}(R_{1,n_{1}})$-map

$\operatorname{L}_{n_{1},n_{2}}:\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})\longrightarrow\mathrm{Hom}(R_{1},R_{2})$

such that

$\operatorname{L}_{n_{1},n_{2}}\circ\operatorname{pr}^{n_{1},n_{2}}=\operatorname{Id}_{\mathrm{Hom}(R_{1},R_{2})}$

where
$\operatorname{pr}^{n_{1},n_{2}}:\mathrm{Hom}(R_{1},R_{2})\longrightarrow\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})$
is the natural projection map.

###### Proof.

It is clear that $\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})$ is a right
$\mathrm{Iso}(R_{1,n_{1}})$-set. By Lemma 2.5, we have
$n_{1}>M(R_{1})\nu_{1}(p)^{2}$. Since $\mathrm{Hom}(R_{1},R_{2})$ is a right
$\mathrm{Iso}(R_{1})$-set, $\mathrm{Hom}(R_{1},R_{2})$ is a right
$\mathrm{Iso}(R_{1,n_{1}})$-set via
$\operatorname{L}_{n}:\mathrm{Iso}(R_{n})\longrightarrow\mathrm{Iso}(R)$ by
Corollary 3.5. Moreover, by Proposition 3.4, the lifting map
$\operatorname{L}_{n_{1},n_{2}}$ is a $\mathrm{Iso}(R_{1,n_{1}})$-map. ∎

######  Lemma 3.7.

Let $R\subset S$ be discrete valuation rings and $S$ a finitely generated
$R$-module. The discriminant $\mathcal{D}_{S/R}$ of $S/R$ is equal to the norm
$\mathrm{Norm}(\mathfrak{D}_{S/R})$ of the different $\mathfrak{D}_{S/R}$ of
$S/R$.

###### Proof.

Chapter 3, Section 2 of [25]. ∎

Even though next corollary directly follows from Corollary 3.6, we state here
because it is useful for numerical calculations.

######  Corollary 3.8.

Let $R_{1}$ and $R_{2}$ be complete discrete valuation rings of characteristic
$0$ with perfect residue fields of characteristic $p$. Let $\mathfrak{m}_{i}$
be the maximal ideal of $R_{i}$ generated by $\pi_{i}$ and $\nu_{i}$
corresponding valuation of $R_{i}$ for $i=1,2$. Let $L_{i}$ and $K_{i}$ be the
fraction fields of $R_{i}$ and $W(k_{i})$ for $i=1,2$ respectively. Suppose
$\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})$ is not empty. Then there is a
surjective $\mathrm{Iso}(R_{1,n_{1}})$-map

$\operatorname{L}_{n_{1},n_{2}}:\mathrm{Hom}(R_{1,n_{1}},R_{2,n_{2}})\longrightarrow\mathrm{Hom}(R_{1},R_{2})$

such that

$\operatorname{L}_{n_{1},n_{2}}\circ\operatorname{pr}^{n_{1},n_{2}}=\operatorname{Id}_{\mathrm{Hom}(R_{1},R_{2})}$

if one of the following holds :

  * •

$L_{1}/K_{1}$ is a Galois extension.
$i$ is the least number such that the $i$-th ramification group $G_{i}$ of
$Gal(L_{1}/K_{1})$ vanishes.
$n_{2}>\nu_{2}(p)i$.

  * •

$n_{2}>\nu_{2}(p)+\nu_{2}(p)\nu_{1}(\nu_{1}(p))$.

  * •

$n_{2}>\nu_{2}(\mathcal{D}_{R_{1}/W(k_{1})})$ where
$\mathcal{D}_{R_{1}/W(k_{1})}$ is the discriminant of $R_{1}/W(k_{1})$. Here
$\nu_{2}(\mathcal{D}_{R_{1}/W(k_{1})})$ means $\nu_{2}(p^{a})$ where
$\mathcal{D}_{R_{1}/W(k_{1})}=p^{a}W(k_{1})$.

###### Proof.

  * •

We recall that $G_{i}$ is defined by $G_{i}=\\{\sigma\in
Gal(L_{1}/K_{1}):\nu_{1}(\sigma(\pi_{1})-\pi_{1})\geq i+1\\}$. Then $i$ is
equal to $M(R_{1})\nu_{1}(p)$.

  * •

By Lemma 2.15, one can obtain

$\displaystyle M(R_{1})$
$\displaystyle\leq\widetilde{\nu_{1}}\big{(}f^{\prime}(\alpha)\big{)}-\frac{\nu_{1}(p)-2}{\nu_{1}(p)}$
$\displaystyle\leq\frac{\nu_{1}(p)-1+\nu_{1}\big{(}\nu_{1}(p)\big{)}}{\nu_{1}(p)}-\frac{\nu_{1}(p)-2}{\nu_{1}(p)}$
$\displaystyle\leq\frac{1+\nu_{1}\big{(}\nu_{1}(p)\big{)}}{\nu_{1}(p)}$

as in the proof of Theorem 2.16. Hence,

$M(R_{1})\nu_{1}(p)\nu_{2}(p)\leq\nu_{2}(p)+\nu_{2}(p)\nu_{1}\big{(}\nu_{1}(p)\big{)}.$

  * •

Let $\mathfrak{D}_{R_{1}/W(k_{1})}$ be the different of $R_{1}/W(k_{1})$. Then
one can obtain that

$\displaystyle\nu_{2}(\mathcal{D}_{R_{1}/W(k_{1})})$
$\displaystyle=\nu_{2}(p)\widetilde{\nu_{2}}(\mathcal{D}_{R_{1}/W(k_{1})})$
$\displaystyle=\nu_{2}(p)\widetilde{\nu_{1}}(\mathcal{D}_{R_{1}/W(k_{1})})$
$\displaystyle=\nu_{2}(p)\nu_{1}(p)\widetilde{\nu_{1}}(\mathfrak{D}_{R_{1}/W(k_{1})})$
$\displaystyle=\nu_{2}(p)\nu_{1}(p)\widetilde{\nu_{1}}\big{(}f^{\prime}(\pi_{1})\big{)}$
$\displaystyle\geq\nu_{2}(p)\nu_{1}(p)M(R_{1}).$

The second equality follows from the fact that $\widetilde{\nu_{i}}$ is
normalized, the third equality follows from Lemma 3.7 and the fourth equality
follows from Lemma 2.15 where $f$ is the monic irreducible polynomial of
$\pi_{1}$ over $K_{1}$.

∎

######  Theorem 3.9.

The lifting number for ${\mathcal{C}}_{p,e}$ is finite. More precisely,
${\mathcal{C}}_{p,e}$ is $(e+e\nu(e)+1)$-liftable. Here $\nu(e)$ denotes the
exponent $n$ such that $e$ generates an ideal $\mathfrak{m}^{n}$ of $R$ in
$\operatorname{Ob}({\mathcal{C}}_{p,e})$ where $\mathfrak{m}$ denotes the
maximal ideal of $R$. $\nu(e)$ depends only on the prime number $p$ and the
ramification index $e$, in particular $\nu(e)$ is independent of the choice of
$R$ in $\operatorname{Ob}({\mathcal{C}}_{p,e})$.

###### Proof.

Suppose $n$ is bigger than $e+e\nu(e)$. For any
$\overline{R},\overline{R_{1}}$ and $\overline{R_{2}}$ in
$\operatorname{Ob}({\mathcal{R}}_{p,e}^{n})$, by Theorem 2.16, we define
$\operatorname{L}_{n}(\overline{R})$ to be a unique ring $R$ in
$\operatorname{Ob}({\mathcal{C}}_{p,e})$ which satisfies
$\Pr_{n}(R)=\overline{R}$. As in the proof of Theorem 2.16, $e+e\nu(e)\geq
M(R)e^{2}$. By Theorem 2.12, for any
$\phi:\overline{R_{1}}\longrightarrow\overline{R_{2}}$, there exists a unique
$n$-th lifting map
$\operatorname{L}(\phi):\operatorname{L}(\overline{R_{1}})\longrightarrow\operatorname{L}(\overline{R_{2}})$,
and hence we obtain a functor
$\operatorname{L}_{n}:{\mathcal{R}}_{p,e}^{n}\longrightarrow{\mathcal{C}}_{p,e}$
by Proposition 3.4. By Definition 2.7, $\operatorname{L}_{n}$ is a lifting
functor.

∎

######  Remark 3.10.

For a fixed absolutely unramified valued field $K$, $M(L)e(L)$ can be
arbitrarily large when extension degrees $[L:K]$ vary. For example, we can
take $L={\mathbb{Q}}_{p}(\zeta_{p^{n}})$ and $K={\mathbb{Q}}_{p}$. More
generally, if $L$ runs through subfields of a deeply ramified extension of a
local field $K$(see [10] for the definition of deeply ramified extensions),
then $M(L)e(L)$ can be arbitrarily large. But _Lemma 2.15_ and the proof of
_Theorem 2.16_ show that $M(L)e(L)$ must be bounded if we fix $[L:K]$. Hence
we deduce the finiteness of the lifting number for ${\mathcal{C}}_{p,e}$.

Example 2.13.(1) can be generalized as follows.

######  Proposition 3.11.

Let $R_{1}/W(k)$ and $R_{2}/W(k)$ be totally ramified extensions of degree
$e$. Then $R_{1,e}$ is isomorphic to $R_{2,e}$ as $W(k)$-algebras.

###### Proof.

Let $\pi_{i}$ be a uniformizer of $R_{i}$ and $\nu_{i}$ the valuation
corresponding to $R_{i}$ for $i=1,2$. By the theory of totally ramified
extensions(see Chapter 2 of [24] for example), the monic irreducible
polynomial $f_{i}$ of $\pi_{i}$ over $W(k)$ is an Eisenstein polynomial for
$i=1,2$. If we write $f_{i}=x^{e}+a_{i,e-1}x^{e-1}+...+a_{i,1}x+a_{i,0}$, then
$\nu_{i}(p)=\nu_{i}(a_{i,0})=e$ and $\nu_{i}(a_{i,j})\geq e$ for $i=1,2$ and
$j=1,2,...,e-1$. This shows

$\displaystyle R_{i,e}$ $\displaystyle=\frac{W(k)[\pi_{i}]}{(\pi_{i})^{e}}$
$\displaystyle\cong\frac{W(k)[x]}{(p,f_{i})}$
$\displaystyle=\frac{k[x]}{(x^{e}+...+a_{i,1}x+a_{i,0})}$
$\displaystyle=\frac{k[x]}{(x^{e})},$

and hence, $R_{1,e}$ is isomorphic to $R_{2,e}$ as $W(k)$-algebras.

∎

Now we focus on tamely ramified extensions. For the tame case, we can
calculate the lifting number.

######  Lemma 3.12.

For a perfect field $k$ of characteristic $p$, let $K$ be the fraction field
of the Witt ring $W(k)$ of $k$. For a positive integer $e$ prime to $p$,
suppose that there is a prime divisor $l$ of $e$ such that $\zeta_{l^{n}}$ is
in $k^{\times}$ and $\zeta_{l^{n+1}}$ is not in $k^{\times}$ for some $n$.
Then there are two totally ramified extensions $L_{1}$ and $L_{2}$ of degree
$e$ over $K$ which are not isomorphic over ${\mathbb{Q}}$.

###### Proof.

$\zeta_{l^{n}}$ is in $W(k)^{\times}$ and $\zeta_{l^{n+1}}$ is not in
$W(k)^{\times}$ by Hensel’s lemma. Then $L_{1}=K(\sqrt[e]{p})$ and
$L_{2}=K(\sqrt[e]{p\zeta_{l^{n}}})$ are totally ramified extensions of degree
$e$ over $K$. Suppose that there is an isomorphism
$\sigma:L_{2}\longrightarrow L_{1}$. Since Galois conjugates of $\sqrt[e]{p}$
and $\zeta_{el^{n}}$ over ${\mathbb{Q}}$ are $\sqrt[e]{p}\zeta_{e}^{i}$ and
$\zeta_{el^{n}}^{j}$ for each $i$ and $j$ where $j$ is prime to $e$
respectively,

$\sigma\left(\sqrt[e]{p\zeta_{l^{n}}}\right)=\sigma\left(\sqrt[e]{p}\zeta_{el^{n}}\right)=\sqrt[e]{p}\zeta_{el^{n}}^{k}$

for some $k$ prime to $l$. In particular, $L_{1}$ contains both $\sqrt[e]{p}$
and $\sqrt[e]{p}\zeta_{el^{n}}^{k}$, and hence, $\zeta_{l^{n+1}}$ is in
$L_{1}$. This is a contradiction since $L_{1}/K$ is totally ramified.

∎

######  Corollary 3.13.

Suppose that $p$ does not divide $e$ and $e>1$. Then $e+1$ is the lifting
number for ${\mathcal{C}}_{p,e}$.

###### Proof.

Since $\nu(p)=0$, $e+e\nu(e)+1=e+1$. By Theorem 3.9, ${\mathcal{C}}_{p,e}$ is
$(e+1)$-liftable. Let ${\mathbb{F}}_{p}$ be the prime field of $p$ elements.
Let $K$ be the fraction field of the Witt ring $W(k)$ of
$k={\mathbb{F}}_{p}(\zeta_{e})$. By Lemma 3.12, there are two totally ramified
extensions $L_{1}$ and $L_{2}$ of degree $e$ over $K$ such that there is no
isomorphism between $L_{1}$ and $L_{2}$. If ${\mathcal{C}}_{p,e}$ is
$e$-liftable, $L_{1}$ and $L_{2}$ are isomorphic over $K$ by Proposition 3.11
and it is a contradiction. ∎

######  Remark 3.14.

_Proposition 3.11_ and _Corollary 3.13_ show the difference between the
unramified case and the tamely ramified case. We can regard the absolutely
unramified valued fields of mixed characteristic as the absolutely tamely
ramified valued fields having the ramification index $e=1$. If we apply the
formula $e+1$ in _Corollary 3.13_ to ${\mathcal{C}}_{p}$, the lifting number
for ${\mathcal{C}}_{p}$ should be $1+1=2$. But the argument in the proof of
_Corollary 3.13_ does not work for ${\mathcal{C}}_{p}$. For an absolutely
unramified complete discrete valued field $K$, there is a unique totally
ramified extension of degree $1$ over $K$, that is, $K$ itself. Hence the fact
that the lifting number for ${\mathcal{C}}_{p}$ is $1$ does not disagree with
_Corollary 3.13_.

For the wild case, we have the following example. Let
$R_{1}={\mathbb{Z}}_{2}[\sqrt{2}]$ and $R_{2}={\mathbb{Z}}_{2}[\sqrt{10}]$.
There is no homomorphism between $R_{1}$ and $R_{2}$ by Kummer theory. But
there is an isomorphism between $R_{1,6}$ and $R_{2,6}$ since

$\displaystyle R_{1,6}$
$\displaystyle=\frac{{\mathbb{Z}}_{2}[\sqrt{2}]}{({\sqrt{2}}^{6})}\cong\frac{{\mathbb{Z}}_{2}[x]}{(x^{2}-2,8)}$
$\displaystyle=\frac{{\mathbb{Z}}_{2}[x]}{(x^{2}-10,8)}\cong\frac{{\mathbb{Z}}_{2}[\sqrt{10}]}{({\sqrt{2}}^{6})}=R_{2,6}.$

This shows that the lifting number for ${\mathcal{C}}_{2,2}$ is
$2+2\nu(2)+1=7>\nu(2)$ by Theorem 3.9. In general, we have the lower bound
$e+1$ of the lifting number for the wild case. For proving this, we need the
following lemma.

######  Lemma 3.15.

For a perfect field $k$ of characteristic $p$, let $K$ be the fraction field
of the Witt ring $W(k)$ of $k$. Let $e$ be a positive integer divided by $p$.
Then there are two totally ramified extensions $L_{1}$ and $L_{2}$ of degree
$e$ over $K$ which are not isomorphic over ${\mathbb{Q}}$.

###### Proof.

We write $e=sp^{r}$ for some positive integers $s$ and $r$ where $s$ is prime
to $p$. Let ${\mathbb{Q}}_{\infty}/{\mathbb{Q}}$ be the cyclotomic
${\mathbb{Z}}_{p}$-extension, in particular
$Gal({\mathbb{Q}}_{\infty}/{\mathbb{Q}})\cong{\mathbb{Z}}_{p}$. Let $M_{r}$ be
a unique subfield of ${\mathbb{Q}}_{\infty}$ such that
$[M_{r}:{\mathbb{Q}}]=p^{r}$. By the theory of cyclotomic fields(See for
example Chapter 1 of [25]), the Galois extension $M_{r}/{\mathbb{Q}}$ is
totally ramified at the place above $p$. Let $\alpha$ be a uniformizer of
$M_{r}$ corresponding to the place above $p$. Since, $M_{r}/{\mathbb{Q}}$ is a
Galois extension, $M_{r}={\mathbb{Q}}(\alpha)={\mathbb{Q}}(\sigma(\alpha))$
for any embedding $\sigma$. We fix an embedding ${\mathbb{Q}}^{alg}\subset
K^{alg}$.

Let $L_{1}=K(p^{1/e})=K(p^{1/s},p^{1/p^{r}})$ and $L_{2}=K(p^{1/s},\alpha)$.
Then $L_{1}$ and $L_{2}$ are totally ramified extensions of degree $e$ over
$K$. If there is an isomorphism $\sigma:L_{2}\longrightarrow L_{1}$, $L_{1}$
contains both $\sigma(\alpha)$ and $p^{1/p^{r}}$. Since
${\mathbb{Q}}(\alpha)={\mathbb{Q}}(\sigma(\alpha))$,
$K(\sigma(\alpha))=K(\alpha)$ is contained in $L_{1}$. We note that
$[K(p^{1/p^{r}},\alpha):K(p^{1/p^{r}})]$ divides $[K(\alpha):K]=p^{r}$ since
$K(\alpha)/K$ is a Galois extension. Since

$s=\left[L_{1}:K\left(p^{1/p^{r}}\right)\right]=\left[L_{1}:K\left(p^{1/p^{r}},\alpha\right)\right]\left[K\left(p^{1/p^{r}},\alpha\right):K(p^{1/p^{r}})\right],$

$[K(p^{1/p^{r}},\alpha):K(p^{1/p^{r}})]$ divides $s$. Hence we obtain
$[K(p^{1/p^{r}},\alpha):K(p^{1/p^{r}})]=\mathrm{gcd}(s,p^{r})=1$. This shows
$K(p^{1/p^{r}})=K(\alpha)$ since $[K(p^{1/p^{r}}):K]=[K(\alpha):K]$. This is a
contradiction, and hence, $L_{1}$ and $L_{2}$ are not isomorphic. ∎

######  Proposition 3.16.

Let $p$ be a prime number and $e$ be a natural number divided by $p$. Then the
lifting number for ${\mathcal{C}}_{p,e}$ is bigger than $e$.

###### Proof.

By Lemma 3.15, there are two totally ramified extensions $L_{1}$ and $L_{2}$
of degree $e$ over ${\mathbb{Q}}_{p}$ such that there is no isomorphism over
${\mathbb{Q}}_{p}$ between $L_{1}$ and $L_{2}$. If ${\mathcal{C}}_{p,e}$ is
$e$-liftable, $L_{1}$ and $L_{2}$ are isomorphic over ${\mathbb{Q}}_{p}$ by
Proposition 3.11 and it is a contradiction. Hence, the lifting number for
${\mathcal{C}}_{p,e}$ is bigger than $e$.

∎

######  Corollary 3.17.

The lifting number for ${\mathcal{C}}_{p,e}$ is bigger than $e$ whenever
$e>1$.

Although we have the lower bound $e+1$ and the upper bound $e+e\nu(e)+1$ of
the lifting number for ${\mathcal{C}}_{p,e}$, we have no clue to calculate the
lifting number explicitly for the wild case.

######  Question 3.18.

What is the lifting number for the wild case ?

######  Remark 3.19.

When $e>1$, for a lifting functor
$\operatorname{L}_{n}:{\mathcal{R}}_{p,e}^{n}\longrightarrow{\mathcal{C}}_{p,e}$
and $\overline{R}$ in ${\mathcal{R}}_{p,e}^{n}$, any complete discrete
valuation ring $R$ which has $\overline{R}$ as its $n$-th residue ring
necessarily has ramification index $e$ and equal to
$\operatorname{L}_{n}(\overline{R})$ by _Corollary 3.17_. But for the lifting
functor $\operatorname{L}:{\mathcal{R}}_{p}\longrightarrow{\mathcal{C}}_{p}$
in _Theorem 3.1_, there is no information on ramification indices in
$\operatorname{Ob}({\mathcal{R}}_{p})$. Really there are many complete
discrete valuation rings with different ramification indices which have the
same residue field. For example, $R_{1}={\mathbb{Z}}_{3}[\sqrt[3]{3}]$ and
$R_{2}={\mathbb{Z}}_{3}[\sqrt[2]{3}]$ have the same residue field, but their
ramification indices are different.

For a fixed set $\mathcal{N}=\\{n_{e}\in{\mathbb{N}}\\}_{e\in{\mathbb{N}}}$,
if we try to make a unified lifting functor from
${\mathcal{R}}_{\mathcal{N}}:=\bigcup_{e}{\mathcal{R}}_{p,e}^{n_{e}}$ to
${\mathcal{C}}:=\bigcup_{e}{\mathcal{C}}_{p,e}$, we can not apply our method
to get such a functor since $M(R_{e})e$ is unbounded as varying $e$ and
$R_{e}\in\operatorname{Ob}(C_{p,e})$. But we have a unified functor for a
finite set of ramification indices.

######  Corollary 3.20.

For a finite set $\\{e_{1},...,e_{s}\\}$ of ramification indices, there is a
finite set of natural numbers $\\{n_{1},..,n_{s}\\}$ such that there exists a
lifting functor

$\operatorname{L}:\bigcup_{1\leq k\leq
s}{\mathcal{R}}_{p,e_{k}}^{n_{k}}\longrightarrow\bigcup_{1\leq k\leq
s}{\mathcal{C}}_{p,e_{k}}^{n_{k}}.$

###### Proof.

The proof follows from Lemma 2.5, Proposition 3.4, Corollary 3.8 and Theorem
3.9. ∎

##  4\. Ax-Kochen-Ershov principle for finitely ramified valued fields

Our main goal in this section is to prove a strengthened version of Basarab’s
result on the AKE-principle for finitely ramified valued fields of mixed
characteristic with perfect residue fields. Firstly, we quickly review the
basic results in model theory of valued fields, concentrating on the AKE-
principle. We take the language of valued fields, which consists of three
types of sorts for valuation fields, residue fields, and value groups. Let
${\mathcal{L}}_{K}=\\{+,-,\cdot;0,1\\}$ be the ring language for valued
fields,
${\mathcal{L}}_{k}=\\{+^{\prime},-^{\prime},\cdot^{\prime};0^{\prime},1^{\prime}\\}$
be the ring language for residue fields, and
${\mathcal{L}}_{\Gamma}=\\{+^{*};0^{*};<\\}$ be the ordered group language for
value groups. Let
${\mathcal{L}}_{val}={\mathcal{L}}_{K}\cup{\mathcal{L}}_{k}\cup{\mathcal{L}}_{\Gamma}$
be the language of valued fields. Next, we consider an extended language of
${\mathcal{L}}_{val}$ by adding the ring languages for the $n$-th residue
rings. For each $n\leq 1$, let
${\mathcal{L}}_{R_{n}}=\\{+_{n},-_{n},\cdot_{n};0_{n},1_{n}\\}$ be the ring
language for the $n$-th residue ring. For $n=1$, we identify
${\mathcal{L}}_{R_{1}}={\mathcal{L}}_{k}$. We get an extended language
${\mathcal{L}}_{val,R}={\mathcal{L}}_{val}\cup\bigcup_{n\geq
1}{\mathcal{L}}_{R_{n}}$ for valued fields. Let
$(K_{1},\nu_{1},k_{1},\Gamma_{1})$ and $(K_{2},\nu_{2},k_{2},\Gamma_{2})$ be
valued fields, and let $R_{1,n}$ and $R_{2,n}$ be the $n$-th residue rings of
$(K_{1},\nu_{1})$ and $(K_{2},\nu_{2})$ respectively. We say $(K_{1},\nu_{1})$
and $(K_{2},\nu_{2})$ are elementarily equivalent if they are elementarily
equivalent in ${\mathcal{L}}_{val}$. If $(K_{1},\nu_{1})$ and
$(K_{2},\nu_{2})$ are elementarily equivalent, then they are elementarily
equivalent in ${\mathcal{L}}_{val,R}$ because the $n$-th residue rings are
interpretable in ${\mathcal{L}}_{val}$. For $(K_{1},\nu_{1})$ and
$(K_{2},\nu_{2})$ which are elementarily equivalent, it necessarily implies
that

  * •

$k_{1}$ and $k_{2}$ are elementarily equivalent in ${\mathcal{L}}_{k}$;

  * •

$\Gamma_{1}$ and $\Gamma_{2}$ are elementarily equivalent in
${\mathcal{L}}_{\Gamma}$; and

  * •

$R_{1,n}$ and $R_{2,n}$ are elementarily equivalent in ${\mathcal{L}}_{R_{n}}$
for each $n\leq 1$.

Ax and Kochen in [4], and Ershov in [14] proved the fact that these conditions
on the residue fields and the value groups imply elementary equivalence for
unramified valued fields:

######  Theorem 4.1.

[4, 14](The Ax-Kochen-Ershov principle) Let $(K_{1},\nu_{1},k_{1},\Gamma_{1})$
and $(K_{2},\nu_{2},k_{2},\Gamma_{2})$ be unramified henselian valued fields
of characteristic zero.

$K_{1}\equiv K_{2}$ if and only if $k_{1}\equiv k_{2}$ and
$\Gamma_{1}\equiv\Gamma_{2}$.

Basarab in [6] extended Theorem 4.1 for henselian valued fields of finite
ramification indices, including local fields of characteristic zero.

######  Theorem 4.2.

[6] Let $(K_{1},\nu_{1},k_{1},\Gamma_{1})$ and
$(K_{2},\nu_{2},k_{2},\Gamma_{2})$ be henselian valued fields of mixed
characteristic having finite absolute ramification indices. The following are
equivalent :

  1. (1)

$K_{1}\equiv K_{2}$.

  2. (2)

$R_{1,n}\equiv R_{2,n}$ for each $n\leq 1$ and $\Gamma_{1}\equiv\Gamma_{2}$.

Next we review on the coarse valuations. For the coarse valuations, we refer
to [13, 20, 27, 31].

######  Remark/Definition 4.3.

[27] Suppose $(K,\nu,k,\Gamma)$ has the finite absolute ramification index so
that the value group has the minimum positive element, and let $\pi$ be a
uniformizer so that $\nu(\pi)$ is the smallest positive element in $\Gamma$.
Let $\Gamma^{\circ}$ be the convex subgroup of $\Gamma$ generated by
$\nu(\pi)$ and
$\dot{\nu}:K\setminus\\{0\\}\longrightarrow\Gamma/\Gamma^{\circ}$ be a map
sending $x(\neq 0)\in K$ to $\nu(x)+\Gamma^{\circ}\in\Gamma/\Gamma^{\circ}$.
The map $\dot{\nu}$ is a valuation, called the coarse valuation. The residue
field $K^{\circ}$, called the core field of $(K,\nu)$, of $(K,\dot{\nu})$
forms a valued field equipped with a valuation $\nu^{\circ}$ induced from
$\nu$ and the value groups $\Gamma^{\circ}$. More precisely, the valuation
$\nu^{\circ}$ is defined as follows: Let
$\operatorname{pr}_{\dot{\nu}}:R_{\dot{\nu}}\longrightarrow K^{\circ}$ be the
canonical projection map and let $x\in R_{\dot{\nu}}$. If
$x^{\circ}:=\operatorname{pr}_{\dot{\nu}}(x)\in K^{\circ}\setminus\\{0\\}$,
then $\nu^{\circ}(x^{\circ}):=\nu(x)$. And $x^{\circ}=0\in K^{\circ}$ if and
only if $\nu(x)>\gamma$ for all $\gamma\in\Gamma^{\circ}$.

######  Lemma 4.4.

  1. (1)

Let $R_{\nu}$, $R_{\dot{\nu}}$, and $R_{\nu^{\circ}}$ be the valuation rings
of $(K,\nu)$, $(K,\dot{\nu})$, and $(K^{\circ},\nu^{\circ})$ respectively.
Then $(\operatorname{pr}_{\dot{\nu}})^{-1}(R_{\nu^{\circ}})=R_{\nu}$.

  2. (2)

Let $R_{n}$ and $R^{\circ}_{n}$ be the $n$-th residue rings of $(K,\nu)$ and
$(K^{\circ},\nu^{\circ})$ respectively. Then there is a canonical isomorphism
$\theta_{n}:R_{n}\longrightarrow R^{\circ}_{n}$ such that
$\operatorname{pr}_{n}^{\nu^{\circ}}\circ(\operatorname{pr}_{\dot{\nu}}|_{R_{\nu}})=\theta_{n}\circ\operatorname{pr}_{n}$,
where $\operatorname{pr}_{n}:R_{\nu}\longrightarrow R_{n}$ and
$\operatorname{pr}_{n}^{\nu^{\circ}}:R_{\nu^{\circ}}\longrightarrow
R^{\circ}_{n}$ are the canonical projection map.

  3. (3)

If $(K,\nu)$ is henselian, then $(K,\dot{\nu})$ is henselian.

  4. (4)

If $(K,\nu)$ is $\aleph_{1}$-saturated, then $(K^{\circ},\nu^{\circ})$ is
complete.

###### Proof.

(1) Note that $R_{\dot{\nu}}:=\\{x\in K|\ \dot{\nu}(x)\geq 0\\}=\\{x\in K|\
\nu(x)\geq\gamma\ \mbox{for some }\gamma\in\Gamma^{\circ}\\}$. Let $x\in
R_{\dot{\nu}}$ be such that $\operatorname{pr}_{\dot{\nu}}(x)=:x^{\circ}\in
R_{\nu^{\circ}}$, that is, $\nu^{\circ}(x^{\circ})(\in\Gamma^{\circ})\geq 0$.
If $x^{\circ}=0$, $\nu(x)>\gamma$ for all $\gamma\in\Gamma^{\circ}$ and $x\in
R_{\nu}$. If $x^{\circ}\neq 0$, then $\nu^{\circ}(x^{\circ})=\nu(x)\geq 0$ in
$\Gamma^{\circ}$ ,and hence $\nu(x)\geq 0$ in $\Gamma$. Thus $x\in R_{\nu}$.
Therefore, for $x\in R_{\dot{\nu}}$, $x\in R_{\nu}$ if and only if
$x^{\circ}\in R_{\nu^{\circ}}$.

(2) Note that each $\theta_{n}$ is induced from
$\operatorname{pr}_{\dot{\nu}}|_{R_{\nu}}:R_{\nu}\longrightarrow
R_{\nu^{\circ}}$. It is easy to see that each $\theta_{n}$ is surjective. To
show that $\theta_{n}$ is injective, it is enough to show that $\nu(x)\geq n$
if and only if $\nu^{\circ}(x^{\circ})\geq n$ for $x\in R_{\nu}$. It clearly
comes from the definition of $\nu^{\circ}$ in (1).

(3)-(4) Section 5 of [20]. ∎

######  Proposition 4.5.

Let $(K_{1},\nu_{1},\Gamma_{1})$ and $(K_{2},\nu_{2},\Gamma_{2})$ be valued
fields. Let $R_{1,n}$ and $R_{2,n}$ be the $n$-th residue rings of $K_{1}$ and
$K_{2}$ respectively. Suppose

  * •

$R_{1,n}\equiv R_{2,n}$ for each $n\geq 1$;

  * •

$\Gamma_{1}\equiv\Gamma_{2}$

Then there are $\aleph_{1}$-saturated elementary extensions
$(K_{1}^{\prime},\nu_{1}^{\prime},\Gamma_{1}^{\prime})$ and
$(K_{2}^{\prime},\nu_{2}^{\prime},\Gamma_{2}^{\prime})$ of $K_{1}$ and $K_{2}$
such that

  * •

$R_{1,n}^{\prime}\cong R_{2,n}^{\prime}$ for $n\geq 1$;

  * •

$\Gamma_{1}^{\prime}\cong\Gamma_{2}^{\prime}$

, where $R_{1,n}^{\prime}$ and $R_{2,n}^{\prime}$ are the $n$-th residue rings
of $K_{1}^{\prime}$ and $K_{2}^{\prime}$ respectively.

###### Proof.

We inductively construct chains of valued fields
$(K_{1}^{i},\Gamma_{1}^{i})_{i\in\omega}$ and
$(K_{2}^{i},\Gamma_{2}^{i})_{i\in\omega}$, and isomorphisms
$\xi_{j}^{i}:R_{1,j}^{i}\longrightarrow R_{2,j}^{i}$ for $0<i$ and $1\leq
j\leq i$, where $R_{1,j}^{i}$ and $R_{2,j}^{i}$ are the $j$-th residue rings
of $K_{1}^{i}$ and $K_{2}^{i}$ respectively such that for $i\in\omega$,

  * $(1)_{i}$

$K_{1}^{i}\prec K_{1}^{i+1}$ and $K_{2}^{i}\prec K_{2}^{i+1}$;

  * $(2)_{i}$

$\xi_{j}^{i}\subset\xi_{j}^{i+1}$ for $1\leq j\leq i$;

  * $(3)_{i}$

$\Gamma_{1}^{i}\cong\Gamma_{2}^{i}$.

Recall the Keisler-Shelah isomorphism theorem :

######  Theorem 4.6.

(Keisler-Shelah Isomorphism Theorem) Let ${\mathcal{M}}$ and ${\mathcal{N}}$
be two first order structures. If ${\mathcal{M}}\equiv{\mathcal{N}}$, then
there is a ultrafilter ${\mathcal{U}}$ on an infinite set $I$ such that

${\mathcal{M}}^{{\mathcal{U}}}\cong{\mathcal{N}}^{{\mathcal{U}}},$

where ${\mathcal{M}}^{{\mathcal{U}}}$ and ${\mathcal{N}}^{{\mathcal{U}}}$ are
the ultrapowers of ${\mathcal{M}}$ and ${\mathcal{N}}$ with respect to
${\mathcal{U}}$.

###### Proof.

See [11]. ∎

Since $\Gamma_{1}\equiv\Gamma_{2}$, by Theorem 4.6, there is an ultrafilter
${\mathcal{U}}_{0}$ such that
$\Gamma_{1}^{{\mathcal{U}}_{0}}\cong\Gamma_{2}^{{\mathcal{U}}_{0}}$. Set
$(K_{1}^{0},\Gamma_{1}^{0})=(K_{1}^{{\mathcal{U}}_{0}},\Gamma_{1}^{{\mathcal{U}}_{0}})$
and
$(K_{2}^{0},\Gamma_{2}^{0})=(K_{2}^{{\mathcal{U}}_{0}},\Gamma_{2}^{{\mathcal{U}}_{0}})$.
Assume we construct sequences of valued fields
$(K_{1}^{i},\Gamma_{1}^{i})_{i\leq m}$ and $(K_{2}^{i},\Gamma_{2}^{i})_{i\leq
m}$ with isomorphisms $\xi_{j}^{i}:\ R_{1,j}^{i}\longrightarrow R_{2,j}^{i}$
for $1\leq j\leq i\leq m$ for some $m\geq 0$ satisfying the conditions
$(1)_{i}$, $(2)_{i}$, and $(3)_{i}$ for $i\leq m$. Since $R_{1,m+1}^{m}\equiv
R_{2,m+1}^{m}$, from Theorem 4.6, there is an ultrafilter ${\mathcal{U}}$ such
that
$\xi_{m+1}^{m+1}:(R_{1,m+1}^{m})^{{\mathcal{U}}}\cong(R_{2,m+1}^{m})^{{\mathcal{U}}}$.
Set $K_{1}^{m+1}=(K_{1}^{m})^{{\mathcal{U}}}$ and
$K_{2}^{m+1}=(K_{2}^{m})^{{\mathcal{U}}}$, and set
$\xi_{j}^{m+1}=(\xi_{j}^{m})^{{\mathcal{U}}}:\ R_{1,j}^{m+1}\longrightarrow
R_{2,j}^{m+1}$ for each $j\leq m$, where
$R_{1,j}^{m+1}=(R_{1,j}^{m})^{{\mathcal{U}}}$ and
$R_{2,j}^{m+1}=(R_{2,j}^{m})^{{\mathcal{U}}}$. Set
$\Gamma_{1}^{m+1}=(\Gamma_{1}^{m})^{{\mathcal{U}}}\cong(\Gamma_{2}^{m})^{{\mathcal{U}}}=\Gamma_{2}^{m+1}$.
Then the sequences of valued fields $(K_{1}^{i},\Gamma_{1}^{i})_{i\leq m+1}$
and $(K_{2}^{i},\Gamma_{2}^{i})_{i\leq m+1}$ with isomorphisms
$\xi_{j}^{i}:R_{1,j}^{i}\longrightarrow R_{2,j}^{i}$ for $1\leq j\leq i\leq
m+1$ satisfying $(1)_{i}$, $(2)_{i}$, and $(3)_{i}$ for $i\leq m+1$. By
induction, we get chains of valued fields, $(K_{1}^{i})_{i\geq 0}$ and
$(K_{2}^{i})_{i\geq 0}$ with isomorphisms
$\xi_{n}^{i}:R_{1,n}^{i}\longrightarrow R_{2,n}^{i}$ for $1\leq n\leq i$
satisfying $(1)_{i}$, $(2)_{i}$, and $(3)_{i}$ for $i\geq 0$.

Next consider the unions $K_{1}^{\omega}:=\bigcup K_{1}^{i}$ and
$K_{2}^{\omega}:=\bigcup K_{2}^{i}$, which are elementary extensions of
$K_{1}$ and $K_{2}$ respectively. Each $n$-th residue ring $R_{k,n}^{\omega}$
is the union of $R_{k,n}^{i}$’s for $k=1,2$. For each $n\geq 1$, define
$\xi_{n}^{\omega}:=\bigcup\xi_{n}^{i}:R_{1,n}^{\prime}\longrightarrow
R_{2,n}^{\prime}$. By $(2)_{i}$, the map $\xi_{n}^{\omega}$ is a well-defined
isomorphism from $R_{1,n}^{\omega}$ to $R_{2,n}^{\omega}$. Hence
$K_{1}^{\omega}$ and $K_{2}^{\omega}$ have the isomorphic $n$-th residue rings
for each $n$. At last to get $\aleph_{1}$-saturated valued fields, consider an
ultrapower $K_{k}^{\omega+1}:=(K_{k}^{\omega})^{{\mathcal{U}}^{\prime}}$ with
respect to a nonprincipal ${\mathcal{U}}^{\prime}$ on $\omega$ for $k=1,2$,
and $K_{1}^{\omega+1}$ and $K_{2}^{\omega+1}$ are desired valued fields. ∎

By combining Theorem 4.1 and Lemma 4.4, Proposition 4.5, we reduce the problem
on elementary equivalence between henselian valued fields of mixed
characteristic having finite ramification indices to the problem on
isometricity between complete discrete valued fields of mixed characteristic
whose the $n$-th residue rings are isomorphic for each $n\geq 1$. Now we
improve Theorem 4.2.

######  Theorem 4.7.

Let $(K_{1},\nu_{1},k_{1},\Gamma_{1})$ and $(K_{2},\nu_{2},k_{2},\Gamma_{1})$
be henselian valued fields of mixed characteristic with finite ramification
indices. Suppose $k_{1}$ and $k_{2}$ are perfect fields of characteristic
$p>0$. For $n\geq 1$, let $R_{1,n}$ and $R_{2,n}$ be the $n$-th residue rings
of $K_{1}$ and $K_{2}$ respectively. Let
$n_{0}>\max\\{e_{\nu_{1}}(p)(1+e_{\nu_{1}}(e_{\nu_{1}}(p)),e_{\nu_{2}}(p)(1+e_{\nu_{2}}(e_{\nu_{2}}(p))\\}$.
The following are equivalent:

  1. (1)

$K_{1}\equiv K_{2}$;

  2. (2)

$\Gamma_{1}\equiv\Gamma_{2}$ and $R_{1,n}\equiv R_{2,n}$ for each $n\leq 1$;
and

  3. (3)

$\Gamma_{1}\equiv\Gamma_{2}$ and $R_{1,n_{0}}\equiv R_{2,n_{0}}$.

###### Proof.

Let $(K_{1},\nu_{1},k_{1},\Gamma_{1})$ and $(K_{2},\nu_{2},k_{2},\Gamma_{2})$
be henselian valued fields of characteristic zero with the finite ramification
indices so that $\Gamma_{1}$ and $\Gamma_{2}$ have the minimum positive
elements. Suppose $k_{1}$ and $k_{2}$ are perfect fields of characteristic
$p>0$. It is easy to check $(1)\Rightarrow(2)\Rightarrow(3)$. We show
$(3)\Rightarrow(1)$.

$(3)\Rightarrow(1)$. Suppose $R_{1,n_{0}}\equiv R_{2,n_{0}}$ and
$\Gamma_{1}\equiv\Gamma_{2}$. By the proof of Proposition 4.5, we may assume
that $R_{1,n_{0}}\cong R_{2,n_{0}}$ and $\Gamma_{1}\cong\Gamma_{2}$, and that
$(K_{1},\nu_{1},\Gamma_{1})$ and $(K_{2},\nu_{2},\Gamma_{2})$ are
$\aleph_{1}$-saturated. Consider the coarse valuations $\dot{\nu}_{1}$ and
$\dot{\nu}_{2}$ of $\nu_{1}$ and $\nu_{2}$ respectively and the valued fields
$(K_{1},\dot{\nu}_{1},\Gamma_{1}/\Gamma^{\circ}_{1})$ and
$(K_{2},\dot{\nu}_{2},\Gamma_{2}/\Gamma^{\circ}_{2})$, where
$\Gamma^{\circ}_{i}$ is the convex subgroup of $\Gamma_{i}$ generated by the
minimum positive element in $\Gamma_{i}$ for $i=1,2$. Since $(K_{1},\nu_{1})$
and $(K_{2},\nu_{2})$ are $\aleph_{1}$-saturated, by Lemma 4.4.(4), the core
fields $(K^{\circ}_{1},\nu^{\circ}_{1})$ and $(K^{\circ}_{2},\nu^{\circ}_{2})$
are complete discrete valued fields, where $\nu^{\circ}_{1}$ and
$\nu^{\circ}_{2}$ are the valuation induced from $\nu_{1}$ and $\nu_{2}$
respectively. Since the $n_{0}$-th residue rings of $(K_{1},\nu_{1})$ and
$(K_{2},\nu_{2})$ are isomorphic, by Lemma 4.4.(2), the $n_{0}$-th residue
rings of $(K^{\circ}_{1},\nu^{\circ}_{1})$ and
$(K^{\circ}_{2},\nu^{\circ}_{2})$ are isomorphic.

By Theorem 2.12 and the proof of Theorem 2.16, $K^{\circ}_{1}$ and
$K^{\circ}_{2}$ are isomorphic. Since $\Gamma_{1}\cong\Gamma_{2}$,
$\Gamma_{1}/\Gamma^{\circ}_{1}\cong\Gamma_{2}/\Gamma^{\circ}_{2}$. Therefore
by Theorem 4.1, $(K_{1},\dot{\nu}_{1})\equiv(K_{2},\dot{\nu}_{2})$. To get
that $(K_{1},\nu_{1})\equiv(K_{2},\nu_{2})$, it is enough to show that the
valuation rings $R_{\nu_{1}}$ of $(K_{1},\nu_{1})$ and $R_{\nu_{2}}$ of
$(K_{2},\nu_{2})$ are definable in $(K_{1},\dot{\nu}_{1})$ and
$(K_{2},\dot{\nu}_{2})$ by the same formula. We need the following lemma on a
definability of valuation ring in the ring language.

######  Lemma 4.8.

Let $(K,\nu)$ be a complete field of characteristic zero. Suppose the residue
field $k$ is perfect and has prime characteristic $p$. Then the valuation ring
$R_{\nu}$ of $(K,\nu)$ is definable by the formula

$\phi_{q}(x)\equiv\exists y\ y^{q}=1+px^{q}$

for some $q>0$ such that $p\not|q$ and $q>e_{\nu}(p)$. For example, we can
take $q$ as $p^{l}+1$ for sufficiently large $l>0$.

###### Proof.

See [8]. ∎

Take $l>0$ large enough so that
$q:=p^{l}+1>\max\\{e_{\nu_{1}}(p),e_{\nu_{2}}(p)\\}$. By Lemma 4.8,
$\phi_{q}(x)$ defines the residue rings $R_{\nu^{\circ}_{1}}$ and
$R_{\nu^{\circ}_{2}}$ of $(K^{\circ}_{1},\nu^{\circ}_{1})$ and
$(K^{\circ}_{2},\nu^{\circ}_{2})$. By Lemma 4.4.(1), the valuation rings
$R_{\nu_{1}}$ and $R_{\nu_{2}}$ are definable by the same formula in
$(K_{1},\dot{\nu}_{1})$ and $(K_{2},\dot{\nu}_{1})$ so that
$(K_{1},\nu_{1})\equiv(K_{2},\nu_{2})$. ∎

We give some corollaries of Theorem 4.7. At first, we improve the result in
[5] on a decidability of henselian valued fields of finite absolute
ramification indices in the case of perfect residue fields.

######  Corollary 4.9.

Let $(K,\nu,\Gamma)$ be a henselian valued field of mixed characteristic
having finite absolute ramification index and the perfect residue field. Let
$R_{n}$ be the $n$-th residue ring of $(K,\nu)$ for each $n\geq 1$. Let
$n_{0}>e_{\nu}(p)(1+e_{\nu}(e_{\nu}(p))$. Let $\operatorname{Th}(K,\nu)$ be
the theory of $(K,\nu)$, $\operatorname{Th}(\Gamma)$ be the theory of
$\Gamma$, and $\operatorname{Th}(R_{n})$ be the theory of $R_{n}$. The
following are equivalent :

  1. (1)

$\operatorname{Th}(K,\nu)$ is decidable.

  2. (2)

$\operatorname{Th}(\Gamma)$ is decidable, and $\operatorname{Th}(R_{n})$ is
decidable for each $n\geq 1$.

  3. (3)

$\operatorname{Th}(\Gamma)$ is decidable, and $\operatorname{Th}(R_{n_{0}})$
is decidable.

###### Proof.

(1) $\Leftrightarrow$ (2) It was already given by Basarab in [5].

(1) $\Leftrightarrow$ (3) Let $(K,\nu,\Gamma,k)$ be a henselian valued fields
of mixed characteristic having a perfect residue field $k$. Let $p>0$ be the
characteristic of $k$ and let $e(:=e_{\nu}(p))$ be the absolute ramification
index of $(K,\nu)$. Suppose $e$ is finite. Consider the following theory
$T_{p,e}$ consisting of the following statements, which can be expressed by
the first order logic;

  * •

$(K,\nu)$ is a henselian valued field of characteristic zero;

  * •

$\Gamma$ is an abelian ordered group having the minimum positive element;

  * •

$k$ is a perfect field of characteristic $p>0$;

  * •

$(K,\nu)$ has the absolute ramification index $e$.

By Theorem 4.7, the theory
$T_{p,e}\cup\operatorname{Th}(\Gamma)\cup\operatorname{Th}(R_{n_{0}})$ is
complete. Thus $\operatorname{Th}(K,\nu)$ is decidable if and only if
$\operatorname{Th}(\Gamma)$ and $\operatorname{Th}(R_{n_{0}})$ are decidable.
∎

Thus we get the following results on local fields of mixed characteristic.

######  Corollary 4.10.

[5][6] Let $(K_{1},\nu_{1})$ and $(K_{2},\nu_{2})$ be local fields of mixed
characteristic.

  1. (1)

$(K_{1},\nu_{1})\equiv(K_{2},\nu_{2})\Leftrightarrow f:K_{1}\cong K_{2}$.

  2. (2)

$\operatorname{Th}(K_{1},\nu_{1})$ is decidable.

Next we recall the following definition introduced in [6]:

######  Definition 4.11.

[6] Let $T$ be the theory of a henselian valued field $(K,\nu,\Gamma)$ of
mixed characteristic having finite absolute ramification index $e$. Let
$\lambda(T)\in{\mathbb{N}}\cup\\{\infty\\}$ be defined as the smallest
positive integer $n$ (if such a number exists) such that for every henselian
valued field $(K^{\prime},\nu^{\prime},\Gamma^{\prime})$ of mixed
characteristic having the same absolute ramification index as
$(K,\nu,\Gamma)$, the following are equivalent:

  1. (1)

$(K^{\prime},\nu^{\prime},\Gamma^{\prime})\models T$.

  2. (2)

$\Gamma\equiv\Gamma^{\prime}$ and the $n$-th residue rings of $(K,\nu)$ and
$(K^{\prime},\nu^{\prime})$ are elementarily equivalent.

Otherwise, $\lambda(T)=\infty$.

In [6], Basarab posed the following question.

######  Question 4.12.

[6] Let $T$ be the theory of a henslian valued field of mixed characteristic
having finite absolute ramification index. Is $\lambda(T)$ finite ?

Basarab in [6] gave a positive answer to Question 4.12 for the case of local
fields of mixed characteristic. Lifting number, which was defined in Section
3, gives a more general answer to the question. Basarab’s invariant
$\lambda(T)$ is always finite and smaller than or equal to the lifting number
for arbitrary finitely ramified henselian valued fields of mixed
characteristic with perfect residue fields.

######  Corollary 4.13.

Let $(K,\nu)$ be a henselian valued field of mixed characteristic $(0,p)$
having finite absolute ramification index $e=e_{\nu}(p)$ with perfect residue
field. Let $T$ be the theory of $(K,\nu)$. Then

  1. (1)

$\lambda(T)$ is smaller than or equal to the lifting number for
${\mathcal{C}}_{p,e}$.

  2. (2)

$\lambda(T)\leq e_{\nu}(p)(1+e_{\nu}(e_{\nu}(p))+1$.

###### Proof.

The proof follows from the proof of Theorem 4.7. ∎

We compute explicitly $\lambda(T)$ for the theories $T$ of some tamely
ramified valued fields. We say that an abelian group $G$ is $e$-divisible
(respectively, uniquely $e$-divisible) when the multiplication by $e$ map,
$e:G\longrightarrow G$ is surjective (respectively, bijective). We denote the
unit group of a ring $R$ by $R^{\times}$.

######  Lemma 4.14.

Let $(K,W(k),\mathfrak{m},k)$ be an absolutely unramified complete discrete
valued field of mixed characteristic $(0,p)$ with perfect residue field $k$.
Suppose that $k^{\times}$ is $e$-divisible for a positive integer $e$ prime to
$p$.

  1. (1)

If $\zeta_{e}$ is contained in $W(k)$, then there exists a unique totally
tamely ramified extension $L$ of degree $e$ over $K$.

  2. (2)

If $\zeta_{e}$ is not contained in $W(k)$, then there exists a unique totally
tamely ramified extension $L$ of degree $e$ over $K$ up to $K$-isomorphism.

###### Proof.

Let $S^{\prime}$ be the group of nonzero Teichmüller representatives of $W(k)$
and $U^{(n)}=1+\mathfrak{m}^{n}$ the $n$-th principal unit group of $W(k)$ for
each $n\geq 1$. Since

$U^{(n)}=\ker\left(W(k)^{\times}\longrightarrow\left(\frac{W(k)}{\mathfrak{m}^{n}}\right)^{\times}\right)$

and

$W(k)=\varprojlim_{n}\left(\frac{W(k)}{\mathfrak{m}^{n}}\right),$

we have

$\displaystyle
W(k)^{\times}=\varprojlim_{n}\left(\frac{W(k)}{\mathfrak{m}^{n}}\right)^{\times}=\varprojlim_{n}\left(\frac{W(k)^{\times}}{U^{(n)}}\right).$

Since $W(k)^{\times}=S^{\prime}\times U^{(1)}$, we obtain

$U^{(1)}=\varprojlim_{n}\left(\frac{U^{(1)}}{U^{(n)}}\right).$

Since $U^{(n)}/U^{(n+1)}\cong k$ for each $n\geq 1$, a short exact sequence

$0\longrightarrow\frac{U^{(n+1)}}{U^{(n+2)}}\longrightarrow\frac{U^{(n)}}{U^{(n+2)}}\longrightarrow\frac{U^{(n)}}{U^{(n+1)}}\longrightarrow
0$

shows that $U^{(1)}/U^{(n)}$ is a $p$-group, and hence, uniquely $e$-divisible
for each $n$. Hence, $U^{(1)}$ is uniquely $e$-divisible and
$W(k)^{\times}=S^{\prime}\times U^{(1)}$ is $e$-divisible since
$k^{\times}\cong S^{\prime}$.

(1) Suppose that $\zeta_{e}$ is contained in $S^{\prime}$. Then there is a
unique totally tamely ramified extension of degree $e$ over $K$ by Kummer
theory since

$\frac{K^{\times}}{(K^{\times})^{e}}=\frac{p^{{\mathbb{Z}}}\times
W(k)^{\times}}{p^{e{\mathbb{Z}}}\times\left(W(k)^{\times}\right)^{e}}\cong\frac{{\mathbb{Z}}}{e{\mathbb{Z}}}.$

(2) Suppose that $\zeta_{e}$ is not contained in $S^{\prime}$. For a totally
tamely ramified extension $L$ of degree $e$ over $K$, there is $u$ in
$W(k)^{\times}$ such that $L=K(\sqrt[e]{pu})$ by the theory of tamely ramified
extensions(see Chapter 2 of [24] for example). Since $W(k)^{\times}$ is
$e$-divisible, there is $v$ in $W(k)^{\times}$ such that $v^{e}=u$. Hence,
$\sqrt[e]{pu}=\sqrt[e]{p}v\zeta_{e}^{i}$ for some $i$. This shows that
$L=K(\sqrt[e]{pu})=K(\sqrt[e]{p}\zeta_{e}^{i})$ is isomorphic to
$K(\sqrt[e]{p})$ over $K$ since the irreducible polynomial of $\sqrt[e]{p}$
over $K$ is $x^{e}-p$.

∎

######  Proposition 4.15.

Let $(K,\nu,\Gamma,k)$ be a finitely tamely ramified henselian valued field of
mixed characteristic with perfect residue field. Let $e\geq 2$ be the absolute
ramification index of $(K,\nu)$. Let $T$ be the theory of $(K,\nu)$.

  1. (1)

If $k^{\times}$ is $e$-divisible, then $\lambda(T)=1$.

  2. (2)

If there is a prime divisor $l$ of $e$ such that $\zeta_{l^{n}}\in k^{\times}$
and $\zeta_{l^{n+1}}\notin k^{\times}$ for some $n$, and $\Gamma$ is a
${\mathbb{Z}}$-group, then $\lambda(T)=e+1$.

###### Proof.

(1) Suppose $k^{\times}$ is $e$-divisible. Let
$(K^{\prime},\nu^{\prime},\Gamma^{\prime},k^{\prime})$ be a henselian valued
field of mixed characteristic with a perfect residue field having absolute
ramification index $e$. Suppose $k\equiv k^{\prime}$ and
$\Gamma\equiv\Gamma^{\prime}$. By the proof of Proposition 4.5, we may assume
that $k\cong k^{\prime}$, $\Gamma\cong\Gamma^{\prime}$, and both $K$ and
$K^{\prime}$ are $\aleph_{1}$-saturated. Consider the core fields
$(K^{\circ},\nu^{\circ},k^{\circ})$ and
$((K^{\prime})^{\circ},(\nu^{\prime})^{\circ},(k^{\prime})^{\circ})$ of
$(K,\nu)$ and $(K^{\prime},\nu^{\prime})$ respectively. Since $k^{\times}$ is
$e$-divisible, so is $(k^{\circ})^{\times}$. Then by Lemma 4.14,
$(K^{\circ},\nu^{\circ})\cong((K^{\prime})^{\circ},(\nu^{\prime})^{\circ})$.
By the proof of Theorem 4.7, we have $(K,\nu)\equiv(K^{\prime},\nu^{\prime})$.
Thus $\lambda(T)=1$.

(2) Suppose there is a prime divisor $l$ of $e$ and a natural number $n$ such
that $\zeta_{l^{n}}\in k^{\times}$ and $\zeta_{l^{n+1}}\notin k^{\times}$, and
$\Gamma\equiv{\mathbb{Z}}$. Let $p$ be the characteristic of $k$ and $e$ be
the absolute ramification index of $(K,\nu)$. Let $T_{p,e}$ be the theory
introduced in the proof of Corollary 4.9. Set
$T_{0}=T_{p,e}\cup\operatorname{Th}(\Gamma)\cup\operatorname{Th}(R_{e})$.
Consider the following theories:

  * •

$T_{1}=T_{0}\cup\\{\exists x(x^{e}-p=0)\\}$;

  * •

$T_{2}=T_{0}\cup\left\\{\exists
xy\big{(}(x^{e}-py=0)\wedge\Phi_{l^{n}}(y)=0\big{)}\right\\}$;

  * •

$T_{3}=T_{0}\cup\left\\{\neg\exists x(x^{e}-p=0),\neg\exists
xy\big{(}(x^{e}-py=0)\wedge\Phi_{l^{n}}(y)=0\big{)}\right\\}$,

where $\Phi_{l^{n}}(X)\in{\mathbb{Z}}[X]$ is the $l^{n}$-th cyclotomic
polynomial. By the proof of Lemma 3.12, we have

  * •

each pairwise union of $T_{1}$, $T_{2}$, and $T_{3}$ is inconsistent;

  * •

$T_{1}$ and $T_{2}$ are consistent;

and for a finitely tamely ramified henselian valued field
$(K^{\prime},\nu^{\prime},\Gamma^{\prime},k^{\prime})$ of mixed characteristic
$(0,p)$ having absolute ramification index $e$, if $k^{\prime}\equiv k$,
$\Gamma^{\prime}\equiv\Gamma$, and $R_{e}^{\prime}\equiv R_{e}$ for the $e$-th
residue ring $R_{e}^{\prime}$ of $(K^{\prime},\nu^{\prime})$, then there is
$i\in\\{1,2,3\\}$

  * •

$(K^{\prime},\nu^{\prime})\models T_{i}$.

Since $(K,\nu)\models T_{0}$ and there are at least two different complete
theories containing $T_{0}$, we have $\lambda(T)\geq e+1$. By Corollary 4.13,
we conclude that $\lambda(T)=e+1$.

∎

For some wild cases, we have a lower bound for $\lambda(T)$.

######  Proposition 4.16.

Let $p$ be a prime number and $e$ be a positive integer divided by $p$. Let
$(K,\nu,\Gamma,k)$ be a finitely ramified henselian valued field of mixed
characteristic $(0,p)$ having absolute ramification index $e\geq 2$. Suppose
$k$ is perfect and $\Gamma$ is ${\mathbb{Z}}$-group. Then $\lambda(T)\geq e+1$
for $T=\operatorname{Th}(K,\nu)$.

###### Proof.

The proof is similar to the proof of Proposition 4.15. Let $T_{p,e}$ and
$T_{0}$ be the theory introduced in the proof of Proposition 4.15. We write
$e=sp^{r}$ for positive integers $s$ and $r$ where $s$ is prime to $p$. Let
$\alpha\in{\mathbb{Q}}^{alg}$ be in the proof of Lemma 3.15 such that $\alpha$
is a uniformizer of $M_{r}$ corresponding to the place above $p$ where
$M_{r}={\mathbb{Q}}(\alpha)$ is the $r$-th subfield of the cyclotomic
${\mathbb{Z}}_{p}$-extension ${\mathbb{Q}}_{\infty}$ of degree $p^{r}$ over
${\mathbb{Q}}$. Let $f(X)$ be the minimal polynomial of $\alpha$ over
${\mathbb{Q}}$. Consider the following theories:

  * •

$T_{1}=T_{0}\cup\\{\exists x(x^{e}-p=0)\\}$;

  * •

$T_{2}=T_{0}\cup\\{\exists x(x^{s}-p=0),\ \exists x(f(x)=0)\\}$;

  * •

$T_{3}=T_{0}\cup\left\\{\neg\exists x(x^{e}-p=0),\neg\big{(}\exists
x(x^{s}-p=0)\wedge\exists x(f(x)=0)\big{)}\right\\}$

By the proof of Lemma 3.15, we have

  * •

each pairwise union of $T_{1}$, $T_{2}$, and $T_{3}$ is not consistent;

  * •

$T_{1}$ and $T_{2}$ are consistent;

and for a ramified henselian valued field
$(K^{\prime},\nu^{\prime},\Gamma^{\prime},k^{\prime})$ of mixed characteristic
$(0,p)$ having absolute ramification index $e$, if $k^{\prime}\equiv k$,
$\Gamma^{\prime}\equiv\Gamma$, and $R_{e}^{\prime}\equiv R_{e}$ for the $e$-th
residue ring $R_{e}^{\prime}$ of $(K^{\prime},\nu^{\prime})$, then there is
$i\in\\{1,2,3\\}$

  * •

$(K^{\prime},\nu^{\prime})\models T_{i}$.

Since $(K,\nu)\models T_{0}$ and there are at least two different complete
theories containing $T_{0}$, we have $\lambda(T)\geq e+1$. ∎

We list some special cases of Proposition 4.15 and Proposition 4.16. For a
positive integer $s$, we say that $s^{\infty}$ divides $[k:{\mathbb{F}}_{p}]$
if there is a subfield $k_{n}$ of $k$ such that $[k_{n}:{\mathbb{F}}_{p}]$ is
finite and $s^{n}$ divides $[k_{n}:{\mathbb{F}}_{p}]$ for each $n\geq 1$.

######  Corollary 4.17.

Suppose $(K,\nu,\Gamma,k)$ is a finitely ramified henselian valued field of
mixed characteristic $(0,p)$ having absolute ramification index $e\geq 2$. Let
$T$ be the theory of $K$. Let $s$ be the order of the group $\mu_{e}\cap
k^{\times}$ where $\mu_{e}$ is the group generated by $\zeta_{e}$.

Case $p\not|e$.

  * •

$\lambda(T)=1$ when $k=k^{alg}$;

  * •

$\lambda(T)=1$ when $K$ is a subfield of ${\mathbb{C}}_{p}$ and $s^{\infty}$
divides $[k:{\mathbb{F}}_{p}]$;

  * •

$\lambda(T)=e+1$ when $K$ is a subfield of ${\mathbb{C}}_{p}$ and $s^{\infty}$
does not divide $[k:{\mathbb{F}}_{p}]$.

Case $p|e$.

  * •

$\lambda(T)\geq e+1$ when $K$ is a subfield of ${\mathbb{C}}_{p}$.

Propositon 4.15.(1) shows that Basarab’s invariant $\lambda(T)$ can be
strictly smaller than the bound in Corollary 4.13 for the tame case. In the
following example, the same thing can happen for the wild case.

######  Example 4.18.

Let
$(K,R,\nu)=({\mathbb{Q}}_{3}(\sqrt[3]{3}),{\mathbb{Z}}_{3}[\sqrt[3]{3}],\nu)$,
$f(x)=x^{3}-3$ and $\alpha_{1}=\sqrt[3]{3}$,
$\alpha_{2}=\sqrt[3]{3}\zeta_{3}$, and $\alpha_{3}=\sqrt[3]{3}\zeta_{3}^{2}$.
Since
$f(x)=(x-\sqrt[3]{3})(x-\sqrt[3]{3}\zeta_{3})(x-\sqrt[3]{3}\zeta_{3}^{2})=(x-\alpha_{1})(x-\alpha_{2})(x-\alpha_{3})$
and
$[{\mathbb{Q}}_{3}(\sqrt[3]{3},\zeta_{3}):{\mathbb{Q}}_{3}(\sqrt[3]{3})]=2$,

$\frac{x^{3}-3}{x-\sqrt[3]{3}}=(x-\sqrt[3]{3}\zeta_{3})(x-\sqrt[3]{3}\zeta_{3}^{2})=(x-\alpha_{2})(x-\alpha_{3})$

is irreducible over ${\mathbb{Q}}_{3}(\sqrt[3]{3})$, that is, $\alpha_{2}$ and
$\alpha_{3}$ are conjugate each other over ${\mathbb{Q}}_{3}(\sqrt[3]{3})$. It
follows that
$\widetilde{\nu}(\alpha_{1}-\alpha_{2})=\widetilde{\nu}(\alpha_{1}-\alpha_{3})$.
By _L_ emma 2.15,

$\widetilde{\nu}(f^{\prime}(\alpha_{1}))=\widetilde{\nu}((\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3}))=2\widetilde{\nu}(\alpha_{1}-\alpha_{2})\leq\frac{\nu(3)-1+\nu\big{(}\nu(3)\big{)}}{\nu(3)}.$

Hence we have the following bound

$\displaystyle M(R)$
$\displaystyle=\max\left\\{\widetilde{\nu}\big{(}\alpha_{1}-\alpha_{j}\big{)}:j\neq
1\right\\}$
$\displaystyle=\widetilde{\nu}(\alpha_{1}-\alpha_{2})=\widetilde{\nu}(\alpha_{1}-\alpha_{3})$
$\displaystyle=\frac{\widetilde{\nu}(f^{\prime}(\alpha_{1}))}{2}$
$\displaystyle\leq\frac{\nu(3)-1+\nu\big{(}\nu(3)\big{)}}{2\nu(3)}$
$\displaystyle=\frac{3-1+\nu(3)}{6}$ $\displaystyle=\frac{5}{6}.$

Note that $M(R)=5/6$ in this case. So we have

$M(R)\nu(3)^{2}\leq\frac{5}{6}3^{2}=\frac{15}{2}\leq
8\leq\nu(3)+\nu(3)\nu(\nu(3))=3+3\nu(3)=12.$

Since the sentence $\exists X(X^{3}-3=0)$ belongs to the theory of the $n$-th
residue ring of $R$ for all $n\geq 1$, _Theorem 2.12_ shows that Basarab’s
invariant $\lambda(T)$ for $K$ is smaller than or equal to $8$, which is
strictly smaller than $\nu(3)(1+\nu(\nu(3)))+1=12$.

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