Residue field domination in real closed valued fields

TL;DR

This paper introduces a notion of residue field domination in real closed valued fields, generalizing stable domination, and characterizes forking behavior in these fields and algebraically closed valued fields.

Contribution

It defines residue field domination for valued fields and proves domination by residue field sorts over the value group, extending to geometric sorts and laying groundwork for broader theories.

Findings

Real closed valued fields are dominated by residue field sorts over the value group.

Characterization of forking and h-forking in real closed and algebraically closed valued fields.

Foundations for extending results to power-bounded T-convex theories.

Abstract

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field sort and in the geometric sorts. These results characterize forking and \th-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded $T$-convex theory.

Equations

$$1\to k^{\times }\to \mathrm{RV}\to \Gamma \to 0,$$

$$B_{\gamma }^{\mathrm{o}p}(a)=\{x\in K:v(x-a)>\gamma \},$$

$$B_{\gamma }^{\mathrm{c}l}(a)=\{x\in K:v(x-a)\ge \gamma \}$$

$$\mathrm{rk}(S/R)\ge \mathrm{rk}(k_S/k_R)+{\dim }_F({\Gamma }_S/{\Gamma }_R).$$

$${\mathrm{kInt}}_S^B=\mathrm{acl}(k(B)\{{\mathrm{RV}}_{\gamma }(B){\}}_{\gamma \in S}).$$

$${\mathrm{kInt}}_S^B=\mathrm{acl}(k_B\{d_{\gamma }{\}}_{\gamma \in S}).$$

$$\mathrm{res}(a_1/e_1),\dots ,res(a_r/e_r),\mathrm{res}(b_1),\dots ,\mathrm{res}(b_s)$$

$$\Gamma (C(d_i{)}_{i\ne i_0}(b_j))=\Gamma (C)\oplus F((v(a_i{)}_{i\ne i_0}).$$

$$k(C(d_i)(b_j{)}_{j\ne j_0})=k(C)⟨\mathrm{res}(b_j{)}_{j\ne j_0}⟩.$$

$$\mathrm{rk}(\Gamma (C{b}_1,\dots ,b_s)/\Gamma (C))=0.$$

$$v(\sum _{i=1}^n{c}_i{m}_i)=\min \{v(c_i)+v(m_i):1\le 1\le n\}$$

$$C^n\cdot m=\{\sum _{i=1}^n{c}_i{m}_i:c_i\in C\}$$

$$\displaystyle v(L^{n}\cdot m)\cap\Gamma(M)=\bigcup_{1\leq i\leq n}\big{(}v(m_{i})+\Gamma(C)\big{)}.$$

$$\mathrm{tp}(M/C)⊢\mathrm{tp}(M/L).$$

$$\mathrm{res}(a_1/e_1),\dots ,\mathrm{res}(a_r/e_r),\mathrm{res}(b_1),\dots ,\mathrm{res}(b_s)$$

$$R^{(j)}=\mathrm{dcl}(k_M,\mathrm{res}(a_1/e_1),\dots ,\mathrm{res}(a_j/e_j),\mathrm{res}(b_1),\dots ,\mathrm{res}(b_s)).$$

$$R^{(0)}$$

$$R^{(r)}$$

$$p_{v^{\prime }}=p^{(0)}\circ \cdots \circ p^{(r-1)}\circ p^{\ast }\circ p_v.$$

$$0<v^{\prime }(a_1/e_1)\ll \cdots \ll v^{\prime }(a_r/e_r)\ll v^{\prime }(x),$$

$$v^{\prime }(\ell )=\sum _{i=1}^r{p}_i{v}^{\prime }(a_i)+\gamma =\sum _{i=1}^r{p}_i{\delta }_i+\sum _{i=1}^r{p}_i{ϵ}_i+\gamma ,$$

$$\{{\delta }_1,\dots ,{\delta }_r,{ϵ}_1,\dots ,{ϵ}_t\}$$

$$v^{\prime }(m)=\sum _{i=1}^r{p}_i^{\prime }{ϵ}_i+\sum _{i=1}^t{q}_i{\mu }_i+{\gamma }^{\prime },$$

$$v^{\prime }(x)=\text{min}\{v^{\prime }({\tilde{\ell }}_1{m}_1),\dots ,v^{\prime }({\tilde{\ell }}_n{m}_n)\}=\text{min}\{{\delta }_1+v({\tilde{\ell }}_1{m}_1),\dots ,{\delta }_n+v({\tilde{\ell }}_n{m}_n)\}.$$

$$0<v({\tilde{\ell }}_j{m}_j)-v({\tilde{\ell }}_i{m}_i)\le {\delta }_i-{\delta }_j,$$

$$v(a)=\text{min}\{v({\ell }_1{m}_1),\dots ,v({\ell }_n{m}_n)\}.$$

$$v(m_i)=-v({\ell }_i)={\lambda }_i\in {\Gamma }_L.$$

$${\ell }_1(m_1+d_1)+\cdots +{\ell }_n(m_n+d_n)=a+({\ell }_1{d}_1+\cdots +{\ell }_n{d}_n)$$

$$x_1{B}_{{\lambda }_1}^{\mathrm{o}p}(m_1)+\cdots +x_n{B}_{{\lambda }_n}^{\mathrm{o}p}(m_n)>0$$

$$\mathrm{kInt}_{\Gamma_{L}}^{M}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\text{\th}}_{\mathrm{kInt}_{\Gamma_{L}}^{C}}\mathrm{kInt}_{\Gamma_{L}}^{L}.$$

$$\mathrm{tp}(L/C{\Gamma }_L{\mathrm{kInt}}_{{\Gamma }_L}^M)⊢\mathrm{tp}(L/M).$$

Full text

The second author was partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada, and wishes to thank MSRI, the CIMS of New York University and the Mathematics Department of the University of California Berkeley for hospitality during some of the work.

The third author was supported by a grant from the Simons Foundation (318364, J.M.).

1. Introduction

The notion of domination of a theory by its stable part was developed by Haskell, Hrushovski and Macpherson in [HHM08] and illustrated in the case of an algebraically closed valued field. It follows from the elimination of imaginaries in [HHM06] that the stable part of an algebraically closed valued field consists essentially of vector spaces over the algebraically closed residue field. In [HHM08, Chapter 12] it is shown that an algebraically closed valued field is dominated over its value group by its residue field. Our goal here is to prove an appropriate analogue of these results for a certain class of ordered valued fields. In an ordered valued field (that is, an ordered field with a convex valuation) both the value group and the residue field are ordered, so there is no stable part. Instead we consider a notion of domination by the residue field over the value group. We prove the following theorems (the terminology is defined in the subsequent pages).

Over a maximal base, an ordered valued field is dominated by the value group and residue field (Corollary 2.6). 2. 2)

Over a maximal base and its value group, a real closed valued field is dominated by the sorts internal to the residue field (Corollary 2.10). 3. 3)

Over a maximal base, forking and þ-forking are determined by the value group and residue field (Theorem 3.4). 4. 4)

An expansion of a maximal base in the geometric sorts is dominated by the value group and residue field, and over its value group is dominated by the sorts internal to the residue field (Theorem 4.6).

We hope to be able to extend these results to the more general situation of substructures of power-bounded $T$-convex structures in some future work. For this reason we work in the $T$-convex setting (see [vdDL]) whenever possible.

Let $R$ be any o-minimal field (that is, an o-minimal expansion of a real closed field) in a language $\mathcal{L}_{o}$. Expand $\mathcal{L}_{o}$ to $\mathcal{L}$ by adding a unary predicate $V$ for a convex subring (which we will also refer to as $V$), and consider $R$ as an $\mathcal{L}$-structure. When the reduct $R|_{\mathcal{L}_{o}}$ is a pure real closed field, $\mathrm{Th}(R)$ is the theory of real closed valued fields, RCVF. This theory has elimination of quantifiers in an appropriate language [CD], and elimination of imaginaries in an appropriate sorted language [M]. In the more general setting where $R|_{\mathcal{L}_{o}}$ is an o-minimal field with theory $T$, one gets similar results if the structure $R$ is $T$-convex; that is, if $f(V)\subseteq V$ for all continuous functions $f\colon R\to R$ that are $\emptyset$-definable in $\mathcal{L}_{o}$. In this case, $\mathrm{Th}(R)$ is universally axiomatisable (in $\mathcal{L}$ expanded by a constant for an element of $R^{>V}$), has quantifier elimination relative to quantifier elimination in $R|_{\mathcal{L}_{o}}$, and there is a natural way to make the residue field into a model of $T$ (see [vdDL]). It is not yet known what is a minimal sorted language in which $\mathrm{Th}(R)$ has elimination of imaginaries (see [HHM13]). Stronger statements about the definable sets can be made when $R$ is power-bounded; that is, if for each definable $f\colon R\to R$ there is $\lambda$ in the field of exponents of $R$ such that $|f(x)|<x^{\lambda}$ for all sufficiently large $x$. Furthermore, if an o-minimal field is power-bounded, then every model of its theory is power-bounded (see for example [vdD, p. 23] for more details on power-boundedness).

For most of the paper, we work in a monster model $\mathcal{R}^{\mathrm{eq}}$, where $\mathcal{R}$ is an $\mathcal{L}$-structure such that its $\mathcal{L}_{o}$-reduct is a power-bounded o-minimal field, and $\mathcal{V}$ is a $T$-convex subring (although for most results we will restrict to the special case of a pure RCVF). In what follows we shall refer to such structures $\mathcal{R}$ as power-bounded $T$-convex structures. The quotient $k\colon=\mathcal{V}/\mathfrak{m}$, where $\mathfrak{m}$ is the maximal ideal of $\mathcal{V}$, is the residue field of $\mathcal{R}$, and we write $\mathrm{res}\colon\mathcal{V}\to k$ for the natural map. The ordered abelian group $\mathcal{R}^{\times}/(\mathcal{V}\setminus\mathfrak{m})$ is the value group $\Gamma$, and we write $v\colon\mathcal{R}^{\times}\to\Gamma$ for the natural map. The valuation is also well-defined on $\mathrm{RV}:=\mathcal{R}^{\times}/(1+\mathfrak{m})$, which is viewed as a multiplicative group. Associated to $\mathrm{RV}$ is an exact sequence of abelian groups (note that while $k^{\times}$ and $\mathrm{RV}$ are multiplicative groups, $\Gamma$ is additive):

[TABLE]

where the map $k^{\times}\to\mathrm{RV}$ is inclusion (note that $x+\mathfrak{m}=x(1+\mathfrak{m})$ for $x\in\mathcal{R}$ with $v(x)=0$), and $\mathrm{RV}\to\Gamma$ is the map $x(1+\mathfrak{m})\mapsto v(x)$, where $x\in\mathcal{R}^{\times}$. Setwise, $x(1+\mathfrak{m})\in\mathrm{RV}$ is just the open ball around $x$ of radius $v(x)$. For any $\gamma\in\Gamma$, we write $\mathrm{RV}_{\gamma}$ for the fiber above $\gamma$, and observe that $\mathrm{RV}_{0}$ is definably isomorphic to $k^{\times}$.

For any substructure $K$ of $\mathcal{R}$, we write $V_{K}$ for its valuation ring, $\Gamma_{K}$ for its value group and $k_{K}$ for its residue field. Algebraic closure and definable closure are always taken in the model-theoretic sense and in the sorted structure $\mathcal{R}^{\mathrm{eq}}$, unless stated otherwise. Note that since $\mathcal{R}$, $k$, and $\Gamma$ are ordered, algebraic closure in these sorts is equal to definable closure. For sets $A$, $B$ and a subfield $C$ of $\mathcal{R}$, we write $C[A,B]$ for the ring generated by $A$ and $B$ over $C$, $C(A,B)$ for the field generated by $A$ and $B$ over $C$, and $C\langle A,B\rangle$ for the o-minimal structure generated by $A$ and $B$ over $C$. We shall also write $\Gamma(A)=\mathrm{dcl}(A)\cap\Gamma$ and $k(A)=\mathrm{dcl}(A)\cap k$. In general, for any collection of sorts $\mathcal{S}$, we write $\mathcal{S}(A)=\mathrm{dcl}(A)\cap\mathcal{S}$.

Given a valued field $K$, we shall write

[TABLE]

for the open (respectively closed) ball of radius $\gamma$ around $a$, where $a\in K$, $\gamma\in\Gamma_{K}$. If we do not want to specify whether the ball is open or closed, we write $\mathrel{B_{\gamma}(a)}$.

We use the following results about definable sets in $T$-convex theories.

Fact 1.1**.**

[T, Theorem 12.10], [vdD, Proposition 7.6] Let $R$ be $T$-convex with $T$ power-bounded and let $S\subseteq R$ be a definable set. Then $S$ is a finite boolean combination of points, intervals, and open and closed balls defined over $R$.

Fact 1.2**.**

[vdD, Theorem A] Let $R$ be $T$-convex, and let $S\subseteq R^{n}$ be a definable set. Then $\mathrm{res}(S)\subseteq k^{n}$ is definable in $k$, considered as a model of $T$.

Fact 1.3**.**

[vdD, Theorem B] Let $R$ be $T$-convex with $T$ power-bounded, and let $S\subseteq{(R^{\times})}^{n}$ be a definable set. Then $v(S)\subseteq{\Gamma_{R}}^{n}$ is definable in $\Gamma$, considered as an ordered vector space over the field of exponents.

In a pure valued field, the dimension inequality relates the transcendence degree of a field extension to the degrees of the extensions of the value group and residue field. In a power-bounded $T$-convex theory, the proof of the corresponding result, called the Wilkie Inequality, is much more subtle.

Fact 1.4**.**

[vdD, Theorem C] Let $T$ be power-bounded with field of exponents $F$, and let $R\preceq S$ be $T$-convex structures. Then

[TABLE]

Now suppose that $R\preceq R\langle a\rangle$ are power-bounded $T$-convex structures, and $a\in\mathcal{R}$ is a singleton. If the extension $R\langle a\rangle$ is not immediate, then by Fact 1.4, exactly one of the value group and residue field increases. There is a precise description of the increased value group or residue field.

Fact 1.5**.**

Let $R\preceq R\langle a\rangle$ be models of a power-bounded $T$-convex theory.

1. i)

If $\mathrm{res}(a)\notin k_{R}$ then $k_{R\langle a\rangle}=k_{R}\langle\mathrm{res}(a)\rangle$ [vdDL, Lemma 5.1]. 2. ii)

Assume $T$ is power bounded with field of exponents $F$. If $v(a)\notin\Gamma_{R}$ then $\Gamma_{R\langle a\rangle}=\Gamma_{R}\oplus F\cdot v(a)$ [vdD, Lemma 5.4].

Furthermore, in a power-bounded $T$-convex theory, the residue field and value group are orthogonal, in the sense of the following statement.

Fact 1.6**.**

[vdD, Proposition 5.8] Let $R$ be a model of a power-bounded $T$-convex theory. Any definable function from $k_{R}$ to $\Gamma_{R}$ or from $\Gamma_{R}$ to $k_{R}$ has finite image.

Our general goal is to study the extent to which types in the valued field are controlled by their restriction to different sorts in the structure. We call this property domination. It generalizes the notion of stable domination in [HHM08] and is related to the notion of compact domination [HPP]. The reader will note the definition of domination is reminiscent of the uniqueness of nonforking extensions in a stable theory. Thus, in an extremely imprecise sense, the existence of domination is an instance of stable-like behavior in the structure.

Let $\mathcal{S}$ and $\mathcal{T}$ be collections of sorts which are stably embedded in all models of some theory. Assume that $\mathcal{S}$ and $\mathcal{T}$ are rosy and hence have some notion of independence.

Definition 1.7**.**

Let $C\subseteq A$ be sets of parameters. We say that $\mathrm{tp}(A/C)$ is dominated by $\mathcal{S}$ if whenever $B\supset C$ with $\mathcal{S}(B)$ independent from $\mathcal{S}(A)$ over $\mathcal{S}(C)$, one has that $\mathrm{tp}(A/C\mathcal{S}(B))$ implies $\mathrm{tp}(A/CB)$. Further, we say that $\mathrm{tp}(A/C)$ is dominated by $\mathcal{S}$ over $\mathcal{T}$ if $\mathrm{tp}(A/C\mathcal{T}(A))$ is dominated by $\mathcal{S}$.

We will also express the property of domination using automorphisms, as follows.

Fact 1.8**.**

Let $A$, $B$, $C$, $\mathcal{S}$, and $\mathcal{T}$ be as above, with $\mathcal{S}(B)$ independent from $\mathcal{S}(A)$ over $\mathcal{S}(C)$. Then the following are equivalent:

1. i)

$\mathrm{tp}(A/C\mathcal{T}(A)\mathcal{S}(B))$ implies $\mathrm{tp}(A/C\mathcal{T}(A)B)$; 2. ii)

given an automorphism $\sigma$ of $\mathcal{R}$ fixing $C\mathcal{T}(A)\mathcal{S}(B)$, there is an automorphism agreeing with $\sigma$ on $A$ and fixing $C\mathcal{T}(A)B$.

In Corollary 2.6, we use this terminology with $\mathcal{S}=\Gamma\cup k$. In Corollary 2.10, $\mathcal{S}$ consists of (some of) the sorts internal to the residue field and $\mathcal{T}$ is the value group. In both cases, the notion of independence is given by þ-forking (we recall the definition in Section 3, or see [A]). We will only need to use þ-independence applied to parameters from $k$, $\Gamma$, or $\mathrm{RV}_{\gamma}$, all of which are o-minimal, which allows us, for most of the paper, to use the more concrete description below [EO, Section 6].

Definition 1.9**.**

Given any (possibly infinite) tuple of parameters $(a_{i})$ from a definable set whose induced structure is o-minimal, and a set of parameters $C$, a subtuple $(a_{i_{j}})$ is said to be an o-minimal basis of $(a_{i})$ over $C$ if $(a_{i})\in\mathrm{acl}(C(a_{i_{j}}))$ and the $a_{i_{j}}$ are algebraically independent over $C$. When the length of $(a_{i_{j}})$ is finite, this length is called the rank of the tuple $(a_{i})$ over $C$. Any definable set has rank equal to the maximal rank of a tuple from the definable set, and a type has rank equal to the rank of any tuple realizing the type.

Fact 1.10**.**

The following are equivalent when the parameters $A$ come from a set with induced o-minimal structure:

1. i)

A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}B; 2. ii)

for every finite tuple $a$ from $A$, the rank of $\mathrm{tp}(a/CB)$ equals the rank of $\mathrm{tp}(a/C)$.

Example 1.11**.**

In a real closed field, algebraic independence in the sense of model theory is the same as field theoretic algebraic independence. Thus A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}B if and only if any tuple from $\mathrm{acl}(CA)$ which is algebraically dependent over $\mathrm{acl}(CB)$ is also algebraically dependent over $\mathrm{acl}(C)$. It is also equivalent to say that $\mathrm{acl}(CA)$ is linearly disjoint from $\mathrm{acl}(CB)$ over $\mathrm{acl}(C)$; the algebraic closure of the base is necessary here. Clearly, in a divisible ordered abelian group, A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}B if and only if $\mathrm{acl}(CA)\cap\mathrm{acl}(CB)=\mathrm{acl}(C)$.

Combining Definition 1.7 with Fact 1.8, we see that $\mathrm{tp}(A/C)$ is dominated by $k$ over $\Gamma$ if, whenever one has $B$ with k(B)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{k(C)}k(A), and an automorphism $\sigma:A\to A^{\prime}$ fixing $C\Gamma(A)k(B)$, then there is an automorphism $\tau:A\to A^{\prime}$ fixing $C\Gamma(A)B$. Note that $k(C)=k(C\Gamma(A))=k(C\Gamma(A^{\prime}))$ by Fact 1.6.

Recall that a definable set $E$ is internal to a definable set $D$ if there is a finite set $A$ such that $E\subseteq\mathrm{dcl}(D\cup A)$.

Definition 1.12**.**

Given a set $B$, and $S\subseteq\Gamma$, we define

[TABLE]

As $k(B)$ and $\mathrm{RV}_{\gamma}(B)$, where $\gamma\in S$, are all internal to the residue field, so is $\mathrm{kInt}_{S}^{B}$. In the ACVF case, $\mathrm{acl}(C\mathrm{kInt}_{\Gamma_{L}}^{M})$ can be shown to be precisely the part of $M^{\mathrm{eq}}$ which is internal to the residue field and contained in sets definable over $C$ and $\Gamma(L)$ [HHM08, 12.9]. This may also be true in the RCVF case, but we have not investigated it. Since the residue field is stably embedded, and the function witnessing the internality of $RV_{\gamma}$ uses parameters from within $RV_{\gamma}$, $\mathrm{kInt}_{S}^{B}$ is also stably embedded.

In the case where $B$ is a model, any element of $\mathrm{RV}_{\gamma}(B)$ is definable over $k_{B}$ from any other element. This is clearly true for $\gamma=0$, and is obtained for arbitrary $\gamma$ using the $b(1+\mathfrak{m})$-definable bijection $\mathrm{RV}_{0}(B)\to\mathrm{RV}_{\gamma}(B)$ given by $x(1+\mathfrak{m})\mapsto bx(1+\mathfrak{m})$, where $v(b)=\gamma$ and $v(x)=0$. Thus, if for each $\gamma\in S$, we let $d_{\gamma}$ be an element of $\mathrm{RV}_{\gamma}(B)$, then

[TABLE]

The following proposition corresponds to [HHM08, 12.9, 12.10] in the context of algebraically closed valued fields. The proof given there uses Morley rank, which is not available to us in the ordered field context.

Proposition 1.13**.**

Let $\mathcal{R}$ be a power-bounded $T$-convex structure with field of exponents $F$, and let $L$ and $M$ be substructures of $\mathcal{R}$. Let $C$ be a common substructure of $L$ and $M$, and suppose that $\Gamma_{L}\subseteq\Gamma_{M}$. The following are equivalent.

1. i)

For some (equivalently any) choice of elements $(a_{i})$ and $(b_{j})$ from $L$ and $(e_{i})$ from $M$ such that $(v(a_{i}))$ is an $F$-basis of $\Gamma(L)$ over $\Gamma(C)$, $(\mathrm{res}(b_{j}))$ is an o-minimal basis of $k(L)$ over $k(C)$, and for each $i$, $v(e_{i})=v(a_{i})$, and for all finite subtuples $a_{1},\dots,a_{r}$, $b_{1},\dots,b_{s}$, the sequence

[TABLE]

is algebraically independent over $k_{M}$ (in the model-theoretic sense); 2. ii)

$\mathrm{kInt}_{\Gamma_{L}}^{L}$* is algebraically independent from $\mathrm{kInt}_{\Gamma_{L}}^{M}$ over $C\Gamma_{L}$;* 3. iii)

\mathrm{kInt}_{\Gamma_{L}}^{L}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C\Gamma_{L}}\mathrm{kInt}_{\Gamma_{L}}^{M}.

Proof.

Let $(a_{i})$, $(b_{j})$, and $(e_{i})$ be sequences as in the hypothesis of the proposition. For each $i$, we let $d_{i}=\mathrel{B^{\mathrm{o}p}_{v(a_{i})}(a_{i})}$.

Claim*.*

The sequence $d_{1},\ldots,d_{r},\mathrm{res}(b_{1}),\ldots,\mathrm{res}(b_{s})$ is algebraically independent over $C\Gamma_{L}$.

• Proof of Claim.  We first show algebraic independence over $C$. Assume to the contrary that $d_{i_{0}}\in\mathrm{acl}(C(d_{i})_{i\neq i_{0}}(b_{j}))$. Then $v(a_{i_{0}})\in\Gamma(C(d_{i})_{i\neq i_{0}}(b_{j}))$. However, it follows from Fact 1.5 ii) that

[TABLE]

This contradicts the independence of $(a_{i})$. Similarly, it follows from Fact 1.5 i) that

[TABLE]

If some $\mathrm{res}(b_{j_{0}})\in\mathrm{acl}(C,(d_{i})(b_{j})_{j\neq j_{0}})$ then $\mathrm{res}(b_{j_{0}})\in k(C)\langle\mathrm{res}(b_{j})_{j\neq j_{0}}\rangle$, which would contradict the assumption of independence on the $\mathrm{res}(b_{j})$.

We now show that $d_{1},\ldots,d_{r},\mathrm{res}(b_{1}),\ldots,\mathrm{res}(b_{s})$ is algebraically independent over $C\Gamma_{L}$. If not, then there is some $k$ with $1\leq k\leq r$, an element $x\in\{d_{1},\ldots,d_{r},\mathrm{res}(b_{1}),\ldots,\mathrm{res}(b_{s})\}$ and a non-constant function $f$ definable with parameters in $\{d_{1},\ldots,d_{r},\mathrm{res}(b_{1}),\ldots,\mathrm{res}(b_{s})\}\setminus\{x\}\cup\{v(a_{1}),\dots,v(a_{k-1})\}$ with $f(v(a_{k}))=x$. This would contradict Fact 1.6.

$\Box$ (Claim)

Now we prove the equivalence of the three statements. The equivalence of ii) and iii) is Fact 1.10. Note that the o-minimal rank of $k(Cb_{1},\ldots,b_{s})$ over $k(C)$ equals $s$, so by the Wilkie inequality,

[TABLE]

By the above claim, $\mathrm{kInt}_{\Gamma_{L}}^{L}$ is algebraically independent from $\mathrm{kInt}_{\Gamma_{L}}^{M}$ over $C\Gamma_{L}$ if and only if some (equivalently every) sequence $d_{1},\ldots,d_{r},$ $\mathrm{res}(b_{1}),$ $\ldots,\mathrm{res}(b_{s})$ has rank $r+s$ over $\mathrm{kInt}_{\Gamma_{L}}^{M}$, which, by stable embeddedness of $\mathrm{kInt}_{\Gamma_{L}}^{M}$, happens if and only if the sequence $d_{1},\ldots,d_{r},$ $\mathrm{res}(b_{1})$, $\ldots,\mathrm{res}(b_{s})$ has rank $r+s$ over $M$. As each $d_{i}$ is interdefinable over $M$ with $\mathrm{res}(a_{i}/e_{i})$, the sequence $d_{1},\ldots,d_{r},$ $\mathrm{res}(b_{1})$, $\ldots,\mathrm{res}(b_{s})$ has rank $r+s$ over $M$ if and only if $\mathrm{res}(a_{1}/e_{1}),\ldots,$ $\mathrm{res}(a_{r}/e_{r}),\mathrm{res}(b_{1}),$ $\dots,\mathrm{res}(b_{s})$ has rank $r+s$ over $M$, which is equivalent to i). ∎

2. Isomorphism theorems

We begin with some remarks about separated bases over a maximal subfield.

Definition 2.1**.**

Let $C\subseteq\mathcal{R}$ be a valued field. We say that a finite sequence $m_{1},\ldots,m_{n}\in\mathcal{R}$, is separated over $C$ if

[TABLE]

for all $c_{1},\ldots,c_{n}\in C$.

Definition 2.2**.**

A valued field $C$ is maximal if it has no proper immediate extension. It is spherically complete if every non-empty chain of balls defined over $C$ has an element in $C$.

The fact that a field is maximal if and only if it is spherically complete dates to Kaplansky and Ostrovski (see [K, Theorem 8.28]). The Hahn field $\mathbb{R}((\mathbb{Q}))$ is an example of a maximal real closed field. It follows from results in [T] that any power-bounded $T$-convex theory has a maximal model; details of the proof are not relevant for this article. The definition of spherical completeness can be extended to a finite vector space over the valued field $C$ in the following way.

Lemma 2.3**.**

Let $C$ be a spherically complete valued field, and assume that the tuple $m=(m_{1},\ldots,m_{n})$ is separated over $C$. Then the valued group

[TABLE]

is spherically complete, in the sense that every non-empty chain of balls with centers in $C^{n}\cdot m$ and radii in $v(C^{n}\cdot m)$ has an element in $C^{n}\cdot m$.

Proof.

The proof is a straight-forward calculation. ∎

It is also well-known that a finite-dimensional vector space over a maximal field has a separated basis (see [H, Lemma 2.6] for a proof). Indeed, the following stronger statement is true, as shown in the proof of [HHM08, Proposition 12.11].

Fact 2.4**.**

Let $L$ and $M$ be valued field extensions of the maximal valued field $C$ such that $\Gamma(L)\cap\Gamma(M)=\Gamma(C)$ and $k_{L}$ and $k_{M}$ are linearly disjoint over $k_{C}$. Then, for any finite-dimensional subspace $U$ of $M$ over $C$, there is a separated basis $m=(m_{1},\ldots,m_{n})$ for $U$ over $C$ which is also separated over $L$. Furthermore,

[TABLE]

For the rest of this section we assume that $\mathcal{R}$ is a pure real closed valued field. We now prove our first domination statement, in the formulation involving automorphisms.

Theorem 2.5**.**

Let $C$, $L$, $M$ be ordered valued fields with $C$ maximal and $C\subseteq L\cap M$. Assume that $\Gamma_{L}\cap\Gamma_{M}=\Gamma_{C}$ and $k_{L}$ is linearly disjoint from $k_{M}$ over $k_{C}$. Suppose there are automorphisms $\sigma$ and $\tau$ of $\mathcal{R}$ fixing $C\Gamma_{L}k_{L}$ with $\sigma(L)=L^{\prime}$, $\tau(M)=M^{\prime}$. Then there is an automorphism of $\mathcal{R}$ which maps $N=C(L,M)$ to $N^{\prime}=C(L^{\prime},M^{\prime})$ and restricts to $\sigma$ on $L$ and $\tau$ on $M$. Furthermore, $\Gamma_{N}$ is the group generated by $\Gamma_{L}$ and $\Gamma_{M}$, and $k_{N}$ is the field generated by $k_{L}$ and $k_{M}$.

Proof.

We first assume that $\tau=\mathrm{id}$. We extend $\sigma|_{L}$ to $C[L,M]$ by setting $\sigma(\sum_{i=1}^{n}\ell_{i}m_{i})=\sum_{i=1}^{n}\sigma(\ell_{i})m_{i}$, and then to the field of fractions $N$ in the only possible way. We need to show that the extended function, also called $\sigma$, preserves the valuation and the ordering. It will then extend to an automorphism of $\mathcal{R}$ by quantifier elimination. The fact that $\sigma$ is a valued field isomorphism is proved in [HHM08, Proposition 12.11], and it is also shown there that $\Gamma_{N}$ is the group generated by $\Gamma_{L}$ and $\Gamma_{M}$, and that $k_{N}$ is the field generated by $k_{L}$ and $k_{M}$. Here we need to show that $\sigma$ also preserves the ordering. We thank Tom Scanlon for pointing out the argument below, which is much simpler than our original proof.

Suppose for contradiction that there is $a\in C[L,M]$ with $a>0$, $\sigma(a)<0$. First suppose that $a=\ell m$ for some $\ell\in L$ and $m\in M$. Dividing by $m$, we see that this contradicts the assumption that $\sigma$ preserves the ordering on $L$. Now let $a=\sum_{i=1}^{n}\ell_{i}m_{i}$. As $\Gamma_{N}$ is the group generated by $\Gamma_{L}$ and $\Gamma_{M}$, $v(a)=v(\ell m)$ for some $\ell\in L$, $m\in M$. Write $a=\ell ma_{1}$, where $a_{1}\in C[L,M]$ and $v(a_{1})=0$. As $\sigma(a)=\sigma(\ell m)\sigma(a_{1})$ and $\sigma(\ell m)$ does not change sign (as already noted), it must be that $a_{1}$ and $\sigma(a_{1})$ have opposite sign. But then $\mathrm{res}(a_{1})$ and $\mathrm{res}(\sigma(a_{1}))=\sigma(\mathrm{res}(a_{1}))$ have opposite sign, which contradicts the fact that $\sigma$ is the identity on $k_{N}$, as it is generated by $k_{L}$ and $k_{M}$.

Now suppose $\tau$ is any automorphism fixing $Ck_{L}\Gamma_{L}$ and such that $\tau:M\to M^{\prime}$. Apply $\tau^{-1}$ to $N^{\prime}=C(L^{\prime},M^{\prime})$ to get a valued field $\widetilde{N}=C(\widetilde{L},M)$ for $\widetilde{L}=\tau^{-1}(L^{\prime})$. As $\Gamma_{L}=\Gamma_{L^{\prime}}=\Gamma_{\widetilde{L}}$ and $k_{L}=k_{L^{\prime}}=k_{\widetilde{L}}$, the hypotheses of the theorem apply to $\widetilde{L}$ and $M$, so we may apply the case of the theorem that we have already proven to deduce that there is an isomorphism from $N$ to $\widetilde{N}$ which restricts to $\tau^{-1}$ on $L$ and the identity on $M$. Composing this isomorphism with $\tau$ gives us the desired map $N\to N^{\prime}$. ∎

We restate the theorem in the language of domination.

Corollary 2.6**.**

Let $C\subseteq M$ be substructures of $\mathcal{R}$ with $C$ maximal, $k_{M}$ a regular extension of $k_{C}$ and $\Gamma_{M}/\Gamma_{C}$ torsion free.Then $\mathrm{tp}(M/C)$ is dominated by the value group and residue field.

Proof.

Let $L\supseteq C$ be another substructure and assume that k_{L}\Gamma_{L}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{k_{C}\Gamma_{C}}k_{M}\Gamma_{M}. We need to show that $\mathrm{tp}(M/C\Gamma_{L}k_{L})\vdash\mathrm{tp}(M/L)$. That is, as in Fact 1.8, for any automorphism $\tau$ of $\mathcal{R}$ fixing $C\Gamma_{L}k_{L}$ there is an automorphism agreeing with $\tau$ on $M$ and fixing $L$. This is the conclusion of Theorem 2.5. The hypotheses of the theorem are satisfied because the þ-independence of the residue fields implies their algebraic independence over $\mathrm{acl}(k_{C})$ (and thus over $k_{C}$), and hence their linear disjointness over $k_{C}$ by the regularity assumption (see [L, Theorem VIII 4.12]). Similarly, the þ-independence of the value groups implies, as observed in Example 1.11, that $\Gamma_{L}\cap\Gamma_{M}\subseteq\mathrm{dcl}(\Gamma_{C})$. But the definable closure of $\Gamma_{C}$ is just the divisible hull of $\Gamma_{C}$, and by our assumption on torsion $\mathrm{dcl}(\Gamma_{C})\cap\Gamma_{M}=\Gamma_{C}$. Thus $\Gamma_{L}\cap\Gamma_{M}=\Gamma_{C}$. ∎

Corollary 2.6 has the following consequence in the special case that $L$ is an immediate extension of the real closure $C^{\mathrm{rc}}$ of $C$. Despite the fact that $C$ is not necessarily dense in its real closure, the order type of $M$ with respect to $L$ is determined by its order type with respect to $C$. This applies in particular when $L$ is a maximal extension of $C^{\mathrm{rc}}$.

Corollary 2.7**.**

Let $C\subseteq M$ be substructures of $\mathcal{R}$ with $C$ maximal, $k_{M}$ a regular extension of $k_{C}$ and $\Gamma_{M}/\Gamma_{C}$ torsion free. Let $L$ be an immediate extension of the real closure $C^{\mathrm{rc}}$ of $C$. Then

[TABLE]

Proof.

The fact that $\mathrm{tp}(M/C)\vdash\mathrm{tp}(M/Ck_{C^{\mathrm{rc}}}\Gamma_{C^{\mathrm{rc}}})$ is immediate. By the previous corollary, $\mathrm{tp}(M/Ck_{L}\Gamma_{L})\vdash\mathrm{tp}(M/L)$. The conclusion follows as $k_{C^{\mathrm{rc}}}=k_{L}$ and $\Gamma_{C^{\mathrm{rc}}}=\Gamma_{L}$. ∎

We now move on to proving our stronger domination result: over a maximal base, the ordered valued field is dominated by the $k$-internal sorts over its value group. Those readers familiar with [HHM08, 12.15] may safely skip to the final claim of the proof of Theorem 2.9 below. However, as the proof in [HHM08] incorrectly defines the parameters $e_{i}$, tacitly uses the uniqueness of non-forking extensions in a stable theory (not available to us), and, rather than using [HHM08, 12.11] directly, uses the following equivalent formulation not explicitly stated in [HHM08], we hesitate to refer new readers to the proof in [HHM08], and have instead reproduced it here.

Corollary 2.8**.**

Let $C$, $L$, and $M$ be substructures of a large algebraically (respectively real) closed valued field. Assume that $C$ is a maximal substructure of both $L$ and $M$ with $\Gamma_{L}\cap\Gamma_{M}=\Gamma_{C}$, and $k_{L}$, $k_{M}$ linearly disjoint over $k_{C}$. If $L^{\prime}\models\mathrm{tp}(L/C\Gamma_{M}k_{M})$ then $L^{\prime}\models\mathrm{tp}(L/M)$. Furthermore, $\Gamma_{C(L,M)}$ is the group generated by $\Gamma_{L}$ and $\Gamma_{M}$, and $k_{C(L,M)}$ is the field generated by $k_{L}$ and $k_{M}$.

Proof.

Let $\sigma:L\to L^{\prime}$ witness $L^{\prime}\models\mathrm{tp}(L/C\Gamma_{M}k_{M})$. We wish to extend $\sigma$ to an isomorphism $\widetilde{\sigma}:C(L,M)\to C(L^{\prime},M)$ which is the identity on $M$. Suppose that $M=C(m)$ where $m$ is a possibly infinite tuple and that $\sigma(m)=\widetilde{m}$. We need to find $\tau$ such that $\tau:\widetilde{m}\mapsto m$ while fixing $L$, $k_{M}$ and $\Gamma_{M}$. Letting $\widetilde{M}=C(\widetilde{m})$ we see that $M\models\mathrm{tp}(\widetilde{M}/C\Gamma_{M}k_{M})$. We apply [HHM08, 12.11] (resp. Theorem 2.5) with the roles of $L$ and $M$ reversed. This gives us both the desired $\tau$ and proves the desired statements about the value group and residue field of $C(L,M)$. Then we define $\widetilde{\sigma}=\tau\circ\sigma$. ∎

Below, $\mathcal{R}$ is a real closed valued field.

Theorem 2.9**.**

Let $C$ be a maximal real closed valued field. Let $L$, $M$ be real closed valued fields with $C\subseteq L\cap M$ and $\Gamma_{L}\subseteq\Gamma_{M}$. Assume that $\mathrm{kInt}_{\Gamma_{L}}^{L}$ is algebraically independent from $\mathrm{kInt}_{\Gamma_{L}}^{M}$ over $C\Gamma_{L}$. Suppose $\sigma$ is an automorphism of $\mathcal{R}$ fixing $C\Gamma_{L}\mathrm{kInt}_{\Gamma_{L}}^{M}$ and such that $\sigma(L)=L^{\prime}$. Then there is an automorphism of $\mathcal{R}$ which restricts to $\sigma$ on $L$, fixes $M$ pointwise, and maps $C(L,M)$ to $C(L^{\prime},M)$.

Proof.

In outline, we begin, as in [HHM08, Proposition 12.15], by perturbing the valuation to a finer one, $v^{\prime}$, which satisfies the hypothesis that $\Gamma_{(L,v^{\prime})}\cap\Gamma_{(M,v^{\prime})}=\Gamma_{(C,v^{\prime})}$. We can then apply Corollary 2.8 to extend $\sigma|_{L}$ to a valued field isomorphism from $C(L,M)=N$ to $C(L^{\prime},M)=N^{\prime}$ which extends the identity on $M$. Finally we show that $\sigma$ also preserves the ordering on the fields. This last step requires some details of the construction of the perturbation of the valuation, so it is worth repeating (and correcting) the proof from [HHM08] here. We use the language of places for this construction; information about places can be found in [ZS, Chapter VI].

The assumption that $L$ and $L^{\prime}$ satisfy the same type over $C\Gamma_{L}\mathrm{kInt}_{\Gamma_{L}}^{M}$ allows us to assume that $\sigma$ fixes $C\Gamma_{M}\mathrm{kInt}_{\Gamma_{L}}^{M}$. For suppose that there is some $\mu\in\Gamma_{M}$ with $\sigma(\mu)=\mu^{\prime}$. We wish to show there is $\tau:\mu^{\prime}\mapsto\mu$ fixing $L^{\prime}\mathrm{kInt}_{\Gamma_{L}}^{M}$ so that we may replace $\sigma$ with $\tau\circ\sigma$. By stable embeddedness of $\Gamma$, we must show that $\mu$ and $\mu^{\prime}$ realize the same type over $\Gamma(L^{\prime}\mathrm{kInt}_{\Gamma_{L}}^{M})$, so it suffices to show $\Gamma(L^{\prime}\mathrm{kInt}_{\Gamma_{L}}^{M})=\Gamma_{L}$. Suppose that there is an $L^{\prime}$-definable function $f$ from $\mathrm{kInt}_{\Gamma_{L}}^{M}$ to $\Gamma$. By the orthogonality of $k$ and $\Gamma$, for every $\lambda\in\Gamma_{L}$, $f$ takes only finitely many values on $RV_{\lambda}(M)$, and thus these values are algebraic over $L^{\prime}$ and hence in $\Gamma_{L^{\prime}}=\Gamma_{L}$.

Choose $a_{1},\ldots,a_{r}$ from $L$ and $e_{1},\ldots,e_{r}$ from $M$ such that, for each $1\leq i\leq r$, $v(a_{i})=v(e_{i})$ and $\{v(a_{i})\}$ forms a $\mathbb{Q}$-basis for $\Gamma_{L}$ modulo $\Gamma_{C}$. Choose $b_{1},\ldots,b_{s}$ from $L$ such that $\{\mathrm{res}(b_{1}),\ldots,\mathrm{res}(b_{s})\}$ is a transcendence basis for $k_{L}$ over $k_{C}$. By Proposition 1.13, the elements

[TABLE]

are algebraically independent over $k_{M}$. For $0\leq j\leq r$, let

[TABLE]

In particular,

[TABLE]

For each $0\leq j\leq r-1$, choose a place $p^{(j)}:R^{(j+1)}\to R^{(j)}$ fixing $R^{(j)}$ and such that $p^{(j)}(\mathrm{res}(a_{j+1}/e_{j+1}))=0$, which is possible by the algebraic independence of $\mathrm{res}(a_{1}/e_{1}),\dots,\mathrm{res}(a_{r}/e_{r})$ over $k_{M}$. Also choose a place $p^{*}:k(N)\to R^{(r)}$ fixing $R^{(r)}$. Write $p_{v}:\mathrm{dcl}(N)\to k(N)$ for the place corresponding to our given valuation $v$. Define $p_{v^{\prime}}:\mathrm{dcl}(N)\to R^{(0)}$ to be the composition

[TABLE]

Let $v^{\prime}$ be a valuation associated to the place $p_{v^{\prime}}$. Notice that all the places $p^{(j)}$ and $p^{*}$ are the identity on $k_{M}$, so we may identify $(M,v)$ and $(M,v^{\prime})$, including identifying the value groups $\Gamma_{M}$ and $\Gamma_{(M,v^{\prime})}$. Similarly, the places are all the identity on $k_{L}$, so the value groups $\Gamma_{L}$ and $\Gamma_{(L,v^{\prime})}$ are isomorphic, but we shall see that we cannot simultaneously identify $\Gamma_{M}$ with $\Gamma_{(M,v^{\prime})}$ and $\Gamma_{L}$ with $\Gamma_{(L,v^{\prime})}$. Also notice that we cannot expect $v^{\prime}$ to be convex with respect to the ordering.

We now have two valuations $v$ and $v^{\prime}$ on $N$. If $x\in M\subseteq N$, then $v(x)=v^{\prime}(x)$, and if $x,y\in L\subseteq N$ then $v(x)\leq v(y)$ implies $v^{\prime}(x)\leq v^{\prime}(y)$. Furthermore, the construction has ensured that for any $x\in M$ with $v(x)>0$,

[TABLE]

where $\gamma\ll\delta$ means that $n\gamma<\delta$ for any $n\in\mathbb{N}$ (and hence $\Gamma_{(L,v^{\prime})}\not=\Gamma_{L}$). Let $\Delta$ be the subgroup of $\Gamma_{(N,v^{\prime})}$ generated by $v^{\prime}(a_{1}/e_{1}),\ldots,v^{\prime}(a_{r}/e_{r})$. Then $\Delta$ is a convex subgroup of $\Gamma_{(N,v^{\prime})}$ and $\Gamma_{(N,v^{\prime})}=\Delta\oplus\Gamma(N)$ (where the right-hand group is ordered lexicographically).

To see that $\Gamma_{(L,v^{\prime})}\cap\Gamma_{(M,v^{\prime})}=\Gamma_{(C,v^{\prime})}$, let $m\in M$ and $\ell\in L$ be such that $v^{\prime}(m)=v^{\prime}(\ell)$. Set $v^{\prime}(\frac{a_{i}}{e_{i}})=\delta_{i}$ and $v^{\prime}(e_{i})=\epsilon_{i}$. As $(v(a_{i}))$ generates $\Gamma_{L}$ over $\Gamma_{C}$, and $\Gamma_{L}$ and $\Gamma_{(L,v^{\prime})}$ are isomorphic,

[TABLE]

where $p_{i}\in\mathbb{Q}$ and $\gamma\in\Gamma_{C}$. The set

[TABLE]

is algebraically independent over $\Gamma_{C}$ since $\Gamma_{(N,v^{\prime})}=\Delta\oplus\Gamma_{N}$. Next, note that since $v^{\prime}(e_{i})=v(e_{i})$, $\{v^{\prime}(e_{i})\}$ forms a $\mathbb{Q}$-basis of $\Gamma_{L}\subseteq\Gamma_{M}=\Gamma_{(M,v^{\prime})}$ over $\Gamma_{C}$. Let $\mu_{1},\dots,\mu_{t}$ be such that $\{\epsilon_{i}\}\cup\{\mu_{j}\}$ forms a $\mathbb{Q}$-basis of $\Gamma_{M}$ over $\Gamma_{C}$. Then

[TABLE]

where $q_{i}\in\mathbb{Q}$ and $\gamma^{\prime}\in\Gamma_{C}$. It follows that each $p_{i}=p^{\prime}_{i}=0$ and each $q_{i}=0$, hence $v^{\prime}(\ell)=v^{\prime}(m)\in\Gamma_{C}$.

Next we must check that $k_{v^{\prime}}(L)$ and $k_{v^{\prime}}(M)$ are linearly disjoint. But note that the place $p^{(0)}\circ\cdots\circ p^{(r-1)}\circ p^{*}:k(N)\to\mathrm{dcl}(k(M),k(L))$ is the identity on $k_{M}$ and $k_{L}$ and thus also on their compositum. Thus $k_{L}$ and $k_{M}$ being linearly disjoint over $k_{C}$ implies linear disjointness of $k_{(L,v^{\prime})}$ and $k_{(M,v^{\prime})}$ over $k_{(C,v^{\prime})}$.

Hence we can apply Corollary 2.8 to deduce that the isomorphism $\sigma|_{L}$ extends to a valued field isomorphism $\sigma^{\prime}\colon(N,v^{\prime})\to(N^{\prime},v^{\prime})$ which is the identity on $M$. As $v^{\prime}$ is a refinement of $v$, $\sigma^{\prime}$ is also an isomorphism of $(N,v)$.

It remains to show that $\sigma^{\prime}$ preserves the ordering on $N$. Assume to the contrary, and let $n$ be minimal with $\sum_{i=1}^{n}\ell_{i}m_{i}>0$ and $\sigma^{\prime}(\sum_{i=1}^{n}\ell_{i}m_{i})<0$, where $m_{i}\in M$ and $\ell_{i}\in L$. Since $\Gamma_{(LM)}=\Gamma_{M}$, we may assume that $v(\sum_{i=1}^{n}\ell_{i}m_{i})=0$. Let $U=C^{n}\cdot m$ where $m=(m_{1},\dots,m_{n})$. Modifying $m$ if needed we may assume that $m$ forms a separated basis for $U$ over $C$ with respect to $v^{\prime}$, and hence, by Fact 2.4, is also separated over $L$ with respect to $v^{\prime}$. In fact, $m$ is also separated over $L$ with respect to $v$: for consider an element $x=\tilde{\ell}_{1}m_{1}+\dots+\tilde{\ell}_{n}m_{n}$ of $L^{n}\cdot m$. Since $\Gamma_{(N,v^{\prime})}=\Delta\oplus\Gamma_{N}$ and by the construction of $v^{\prime}$, $v^{\prime}(x)=\delta+v(x)$ for some $\delta\in\Delta$. On the other hand,

[TABLE]

Suppose the minimum is achieved at $j$. Then $v(x)=v(\tilde{\ell}_{j}m_{j})$. To show that $v(\tilde{\ell}_{j}m_{j})\leq v(\tilde{\ell}_{i}m_{i})$ for all $i$, suppose for a contradiction that it is not. Then $\delta_{j}+v(\tilde{\ell}_{j}m_{j})\leq\delta_{i}+v(\tilde{\ell}_{i}m_{i})$ implies

[TABLE]

a contradiction with the convexity of $\Delta$.

Since $m$ forms a separated basis of $L^{n}\cdot m$ over $L$, we have

[TABLE]

By the minimality of $n$, $v(\ell_{1}m_{1})=\dots=v(\ell_{n}m_{n})$, since if, say, $v(\ell_{1}m_{1})>v(a)$, then subtracting $\ell_{1}m_{1}$ from $a$ would not change the sign of $a$, nor would subtracting $\sigma^{\prime}(\ell_{1}m_{1})$ from $\sigma^{\prime}(a)$. Thus for each $i$, $v(\ell_{i}m_{i})=0$, so

[TABLE]

This implies $\ell_{1}\mathrel{B^{\mathrm{o}p}_{\lambda_{1}}(m_{1})}+\dots+\ell_{n}\mathrel{B^{\mathrm{o}p}_{\lambda_{n}}(m_{n})}>0$, since an element thereof is of the form

[TABLE]

for some $d_{1},\ldots,d_{n}$ with $v(d_{i})>\lambda_{i}$. As $v(\ell_{1}d_{1}+\dots+\ell_{n}d_{n})\geq\min\{v(\ell_{i}d_{i})\}>0$, we see that adding it to $a$ cannot change the sign of $a$. Thus

[TABLE]

is a formula in $\mathrm{tp}(\ell_{1},\dots,l_{n}/\mathrm{kInt}_{\Gamma_{L}}^{M})$, which $\sigma^{\prime}$ preserves. So $\sigma^{\prime}(a)>0$.

∎

As for Theorem 2.5, we can restate the theorem in terms of domination.

Corollary 2.10**.**

Let $C\subseteq L$ be elementary substructures of $\mathcal{R}$ with $C$ maximal. Then $\mathrm{tp}(L/C)$ is dominated over its value group by the $k$-internal sorts.

Proof.

Let $M\supseteq C\Gamma_{L}$ be another substructure of $\mathcal{R}$ and assume that

[TABLE]

We need to show that $\mathrm{tp}(L/C\Gamma_{L}\mathrm{kInt}_{\Gamma_{L}}^{M})\vdash\mathrm{tp}(L/C\Gamma_{L}M)$. That is, as in Fact 1.8, for any automorphism $\sigma$ of $\mathcal{R}$ fixing $C\Gamma_{L}\mathrm{kInt}_{\Gamma_{L}}^{M}$, there is an automorphism agreeing with $\sigma$ on $L$ and fixing $M$. This is the conclusion of Theorem 2.9 (for $M^{\mathrm{rc}}$ and hence also for $M$). The hypothesis holds by Proposition 1.13, as $\mathrm{kInt}_{\Gamma_{C}}^{M^{\mathrm{rc}}}=\mathrm{kInt}_{\Gamma_{C}}^{M}$. ∎

Corollary 2.11**.**

Assume $C$, $L$, $M$ are as in Theorem 2.9, except not necessarily real closed, but with $k_{L}$ a regular extension of $k_{C}$ and $\Gamma_{L}/\Gamma_{C}$ torsion free. Then

[TABLE]

Proof.

We may assume $M$ is maximal and real-closed, so that $\bar{C}$, an immediate maximal extension of $C^{rc}$, may be constructed within $M$. As an aside, note that $\bar{C}$ is in fact real closed. For otherwise there would be an element $c$ algebraic over $\bar{C}$. But $c$ could not generate a ramified extension of $\bar{C}$, as it would necessarily give a new element in the divisible hull of $\Gamma_{\bar{C}}$, which already is divisible. Likewise it cannot be a residual extension, as $k(C)$ is already real closed. Nor can it be immediate, as $\bar{C}$ is maximal.

Without loss of generality, $L$ is $C(\ell)$ for a finite tuple $\ell$. Let $\bar{L}$ be $\bar{C}(\ell)$ and let $\bar{L}^{\prime}$ be $\bar{C}(\ell^{\prime})$ (where $\ell^{\prime}$ is $\sigma(\ell)$). It suffices to show that $\bar{L},\bar{L}^{\prime},M$ and $\bar{C}$ satisfy the hypotheses of Theorem 2.9, for then the conclusion of the theorem is that $\sigma$ extends by the identity on $M$, and hence that $L$ and $L^{\prime}$ realize the same type over $M$ as desired.

Applying Theorem 2.5, we see that $\Gamma_{\bar{L}}$ is generated by $\Gamma_{L}$ and $\Gamma_{\bar{C}}$, and that $k_{\bar{L}}$ is generated by $k_{L}$ and $k_{\bar{C}}$. Since $\Gamma_{L}$ and $\Gamma_{\bar{C}}$ are both contained in $\Gamma_{M}$, we have established $\Gamma(\bar{L})\subseteq\Gamma(M)$. Furthermore, there is a subtuple $(\ell_{n_{i}})$ of $\ell$ in $L$ such that $(v(\ell_{n_{i}}))$ generates both $\Gamma_{L}$ over $\Gamma_{C}$ and $\Gamma_{\bar{L}}$ over $\Gamma_{\bar{C}}$. Also there is a subtuple $(\ell_{n_{j}})$ of $\ell$ such that $(\mathrm{res}(\ell_{n_{j}}))$ generates both $k_{L}$ over $k_{C}$ and $k_{\bar{L}}$ over $k_{\bar{C}}$. Taking any $e_{i}$ in $M$ with $v(e_{i})=v(\ell_{n_{i}})$, we see that the hypothesis \mathrm{kInt}_{\Gamma_{L}}^{L}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C\Gamma_{L}}\Gamma_{L}\mathrm{kInt}_{\Gamma_{L}^{M}} implies by Proposition 1.13 that the sequence $\{\mathrm{res}(\ell_{n_{i}}/e_{i}),\mathrm{res}(\ell_{n_{j}})\}_{i,j}$ is algebraically independent over $k_{M}$, and the existence of such elements implies that \mathrm{kInt}_{\Gamma_{\bar{L}}}^{\bar{L}}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{\bar{C}\Gamma_{\bar{L}}}\mathrm{kInt}_{\Gamma_{\bar{L}}}^{M}.

Finally, we need to verify that $\bar{L}$ and $\bar{L^{\prime}}$ satisfy the same type over $\bar{C}\mathrm{kInt}_{\Gamma_{\bar{L}}}^{M}$. We have that $\ell$ and $\ell^{\prime}$ satisfy the same type over $C\Gamma_{L}\mathrm{kInt}_{\Gamma_{L}}^{M}$. Choose a tuple $m$ so that the $\mathrm{res}(m_{i})$ generate $k(M)$ over $k(C)$ (and each $m_{i}$ generates a residual extension). As in Corollary 2.6, the hypothesis that $k_{L}$ is a regular extension of $k(C)$ means that the independence of $k_{L}$ and $k_{M}$ over $C$ implies linear disjointness. Similarly, as $\Gamma_{L}/\Gamma_{C}$ is torsion free, and $\Gamma_{\bar{C}(m)}=\Gamma_{\bar{C}}$, we have that $\Gamma_{L}\cap\Gamma_{\bar{C}(m)}=\Gamma_{C}$. Thus we may assume that $\sigma(\bar{C}(m))=\bar{C}(m)$. Now we may apply Theorem 2.5 to $L=C(\ell)$ and $\bar{C}(m)$ and conclude that we have an automorphism mapping $L$ to $L^{\prime}$ fixing $\bar{C}$ and $\mathrm{kInt}_{\Gamma_{\bar{L}}}^{M}\subseteq\bar{C}(l,m)$, as required.

∎

3. Forking and þ-forking

Theorem 2.5, especially as expressed in the form of Corollary 2.8, has the pleasing consequence that forking and þ-forking over a maximal base are controlled by the value group and residue field.

We begin by recalling the definitions and a few basic properties of forking and þ-forking.

Definition 3.1**.**

A formula $\varphi(x,b)$ divides over $C$ if there is a sequence $(b_{i})_{i<\omega}$ in $\mathrm{tp}(b/C)$ with $b=b_{0}$ and $\{\varphi(x,b_{i})\}$ $m$-inconsistent. A formula $\varphi(x,b)$ þ-divides over $C$ if there is $d$ such that $\{\varphi(x,\tilde{b}):\tilde{b}\models\mathrm{tp}(b/Cd)\}$ is infinite and $m$-inconsistent. A formula (þ-)forks over $C$ if it implies a disjunction of formulas which (þ-)divide over $C$. We say that $\mathrm{tp}(a/bC)$ (þ-)forks over $C$ if it contains a formula which (þ-)forks over $C$. If $\mathrm{tp}(a/bC)$ does not (þ-)fork over $C$, we say $a$ is *(þ-)independent * from $b$ over $C$ and write this as a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b (respectively a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b). Clearly, þ-forking implies forking.

The difference between forking and dividing can sometimes be an issue, but in a large class of theories, including weakly o-minimal theories and algebraically closed valued fields, forking and dividing are the same [CS, the remarks preceding Proposition 2.6 together with Corollary 5.5].

Forking is not transitive in a non-simple theory, but the following partial left transitivity (sometimes called the pairs lemma) holds in all theories. Often stated for dividing, it can be seen to hold for forking as well. The corresponding property of þ-forking also holds in all theories.

Fact 3.2**.**

In any theory

1. i)

[O, Lemma 2.1.6] if a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{A}c and b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{Aa}c then ab\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{A}c; and 2. ii)

[S, Lemma 1.5] if a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{A}c and b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{Aa}c then ab\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{A}c.

Lemma 3.3**.**

In either ACVF or RCVF,

1. i)

k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b* if and only if k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}k(Cb)\Gamma(Cb); and* 2. ii)

k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b* if and only if k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}k(Cb)\Gamma(Cb).*

Proof.

In both i) and ii) the left to right implication is clear, as k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b implies k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}\mathrm{acl}(Cb) and $\mathrm{acl}(Cb)$ contains $k(Cb)\Gamma(Cb)$ (and similarly for þ-independence).

Since $k$ and $\Gamma$ are orthogonal, k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b if and only if k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b and \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b. Furthermore, one has k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}k(Cb)\Gamma(Cb) if and only if k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}k(Cb) and \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}\Gamma(Cb). Thus we need to show that k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}k(Cb) implies k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b and also that \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}\Gamma(Cb) implies \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b.

Assume that k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b. Then, as forking is dividing, there is a formula $\varphi(x,a)\in\mathrm{tp}(k(Ca)/bC)$, and there is $m<\omega$, and $b=b_{0},b_{1},b_{2},\dots$ satisfying $\mathrm{tp}(b/C)$ such that $\{\varphi(x,b_{i})|i<\omega\}$ is $m$-inconsistent. By stable embeddedness of $k$, the subset of $k^{n}$ defined by $\varphi(x,b)$ is also defined by $\psi(x,e)$ for $e$ a tuple from the residue field. As $k$ eliminates imaginaries, we may assume $e$ is the canonical parameter for the set defined by $\varphi(x,b)$. Thus, as this set is definable over $Cb$, $e$ is contained in $k(Cb)$. Suppose that $\sigma_{i}$ is an automorphism of $\mathcal{R}$ that maps $b$ to $b_{i}$ and fixes $C$. Then, letting $e_{i}=\sigma_{i}(e)$, one sees that $\{\psi(x,e_{i})|i<\omega\}$ is $m$-inconsistent and witnesses k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}k(Cb).

The proof that \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}\Gamma(Cb) implies \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b is similar.

Now suppose k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b. We first consider the case in which $\mathrm{tp}(k(Ca)/Cb)$ contains a formula that þ-divides over $C$.

Claim*.*

Suppose $\varphi$ is a formula over $C$ such that $\varphi(x,b)$ defines a subset of $k^{n}$ and þ-divides over $C$. Then there is $\psi(x,e)$, defining the same subset of $k^{n}$, which also þ-divides over $C$, and where $e\in k(Cb)$.

• Proof of Claim.  We have $d$ such that $\{\varphi(x,\tilde{b})\colon\tilde{b}\models\mathrm{tp}(b/Cd)\}$ is $m$-inconsistent and infinite. As above, we can replace $\varphi(x,b)$ with $\psi(x,e)$ defining the same set and with $e$ in $\mathrm{dcl}(Cb)$. Thus $\psi(x,e)$ þ-divides over $C$. $\Box$ (Claim)

Now suppose $\mathrm{tp}(k(Ca)/Cb)$ contains $\varphi(x,b)$ which implies $\bigvee\varphi_{i}(x,b_{i})$, where each $\varphi_{i}(x,b_{i})$ þ-divides over $C$. By stable embeddedness of $k$, we may replace $\varphi(x,b)$ with $\psi(x,e)$ defining the same set and with $e\in k(Cb)$. By the claim, we may replace each $\varphi_{i}(x,b_{i})$ with a $\psi_{i}(x,e_{i})$ defining the same set and with $e_{i}\in k(Cb_{i})$. Since $\varphi$ and $\psi$ define the same set, as do $\varphi_{i}$ and $\psi_{i}$, we have $\psi(x,e)$ implies $\bigvee\psi_{i}(x,e_{i})$. Thus k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mathchar 12854\relax\kern 8.00134pt\hss}\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mathchar 12854\relax\kern 3.92064pt\hss}\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mathchar 12854\relax\kern 2.00034pt\hss}\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}k(Cb).

The proof that \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}\Gamma(Cb) implies \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b is similar. ∎

Theorem 3.4**.**

Let $C$ be a substructure of a model $\mathcal{R}$ of either RCVF or ACVF, and let $a,b\in\mathcal{R}$. Assume that $C$ is a model (or just that $k(Ca)$ is a regular extension of $k_{C}$ and $\Gamma(Ca)/\Gamma_{C}$ is torsion free) and maximal. Then

1. i)

a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b* if and only if k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}k(Cb)\Gamma(Cb);* 2. ii)

a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b* if and only if k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}k(Cb)\Gamma(Cb).*

Proof.

In both i) and ii), the left to right direction is clear. Now assume that k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}k(Cb)\Gamma(Cb), that is, k(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}k(Cb) and \Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}\Gamma(Cb). The former (together with $C$ being a model or our regularity assumption) implies that $k(Ca)$ and $k(Cb)$ are linearly disjoint over $C$ while the latter (together with $C$ being a model or our torsion free assumption) implies that $\Gamma(Ca)\cap\Gamma(Cb)=\Gamma(C)$. Thus, we may apply Corollary 2.8 in the following fashion to see that a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{Ck(Ca)\Gamma(Ca)}b: choose an indiscernible sequence $b=b_{0},b_{1},\dots$ in $\mathrm{tp}(b/Ck(Ca)\Gamma(Ca))$. Let $p(x,Ck(Ca)\Gamma(Ca)b)=\mathrm{tp}(a/Ck(Ca)\Gamma(Ca)b)$. Since, by the corollary, a partial isomorphism mapping $b$ to $b_{i}$ and fixing $Ck(Ca)\Gamma(Ca)$ may be extended to one fixing $a$ as well, we have that $a\models\bigcap_{i}p(x,Ck(Ca)\Gamma(Ca)b_{i})$ and thus $\mathrm{tp}(a/Ck(Ca)\Gamma(Ca)b)$ does not divide (and hence does not fork) over $Ck(Ca)\Gamma(Ca)$.

By Lemma 3.3, k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b if and only if k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}k(Cb)\Gamma(Cb). Likewise, k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b if and only if k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}k(Cb)\Gamma(Cb). Using Fact 3.2, a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{Ck(Ca)\Gamma(Ca)}b and k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b imply ak(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b. Since $k(Ca)\Gamma(Ca)$ is in $\mathrm{acl}(Ca)$, this is equivalent to a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{C}b.

Likewise (recalling independence implies þ-independence), a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}_{Ck(Ca)\Gamma(Ca)}b and k(Ca)\Gamma(Ca)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b together imply a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b. ∎

As $k\times\Gamma$ is rosy (and hence þ-forking is symmetric), we have the following corollary.

Corollary 3.5**.**

a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}b* if and only if b\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox to0.0pt{\hss\displaystyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\displaystyle\smile\hss}\kern 5.71527pt}{\textstyle\kern 5.71527pt\hbox to0.0pt{\hss\textstyle\mid\hss}\lower 3.87495pt\hbox to0.0pt{\hss\textstyle\smile\hss}\kern 5.71527pt}{\scriptstyle\kern 2.80048pt\hbox to0.0pt{\hss\scriptstyle\mid\hss}\lower 1.89871pt\hbox to0.0pt{\hss\scriptstyle\smile\hss}\kern 2.80048pt}{\scriptscriptstyle\kern 1.42882pt\hbox to0.0pt{\hss\scriptscriptstyle\mid\hss}\lower 0.96873pt\hbox to0.0pt{\hss\scriptscriptstyle\smile\hss}\kern 1.42882pt}}^{\text{\th}}_{C}a when $C$ is a maximal model.*

Of course, þ-forking is symmetric in general if and only if the theory is rosy, so one cannot expect this to hold without $C$ being a maximal model.

4. Geometric sorts in a $T$-convex theory

We now turn to the problem of extending the above domination results to the geometric sorts. This requires developing the work on prime resolutions from [HHM08] to apply to the case of ordered valued fields. Much of the structure below comes from [HHM08, Chapter 11], however there are enough differences in the ordered case to make it worthwhile to reproduce it here.

4.1. Prime resolutions

Our goal in this subsection is to prove the existence of prime resolutions of definable sets in the geometric language in a power-bounded $T$-convex theory. Recall that the geometric sorts in a valued field $K$ are a collection of definable $V_{K}$-submodules of $K^{n}$ and their torsors. In a pure algebraically closed valued field or real closed valued field, the theory eliminates imaginaries with respect to these sorts [HHM06, M]. This collection of sorts does not suffice to eliminate imaginaries in a richer language which includes a function symbol for the restricted exponential function [HHM13]. Nevertheless, it is still of interest to understand domination in the context of this sorted language.

The geometric sorts in a valued field can be identified with coset spaces of matrix groups over the field, as follows (full details of this identification can be found in [HHM08, Chapter 7]). Let $B_{n}(K)$ ($B_{n}(V_{K})$ and $B_{n}(k_{K})$ respectively) be the (multiplicative) group of upper triangular invertible $n\times n$ matrices over $K$ ($V_{K}$ and $k_{K}$ respectively). Further, let $B_{n,m}(k_{K})$ be the set of elements of $B_{n}(k_{K})$ whose $m$th column has a $1$ as the $m$th entry and [math] elsewhere. Let $B_{n,m}(V_{K})$ be the set of matrices in $B_{n}(V_{K})$ which reduce modulo $\mathfrak{m}$ componentwise to elements of $B_{n,m}(k_{K})$. The sort $S_{n}$ of $V_{K}$-submodules of $K^{n}$ can be identified with the set of codes for left cosets of $B_{n}(V_{K})$ in $B_{n}(K)$ and the sort $T_{n}$ of torsors of elements of $S_{n}$ can be identified with a set of codes for elements of $\bigcup_{m=1}^{n}B_{n}(K)/B_{n,m}(V_{K})$.

We set \mathcal{G}=\bigcup_{n=1}^{\infty}\big{(}\mathcal{S}_{n}\cup\mathcal{T}_{n}\big{)}.

Definition 4.1**.**

Let $A$ be a substructure of $\mathrel{\mathcal{R}^{\mathrm{eq}}}$. A pre-resolution of $A$ is a substructure $D$ of $\mathrel{\mathcal{R}^{\mathrm{eq}}}$ such that $A\subseteq\mathrm{acl}(D\cap\mathcal{R})$. A resolution of $A$ is a substructure $D$ of $\mathcal{R}$ which is algebraically closed in $\mathcal{R}$ and is such that $A\subseteq\mathrm{dcl}{D}$. The resolution $D$ is prime over $A$ if it embeds over $A$ into any other resolution.

The proof of existence of prime resolutions goes via properties of opaque equivalence relations. We summarize this briefly from [HHM08].

Definition 4.2**.**

[HHM08, Definition 11.5]

1. i)

Let $E$ be a $C$-definable equivalence relation on a $C$-definable set $D$. We say that $E$ is opaque if, for each $C$-definable $F\subseteq D$, $F=F^{\prime}\cup F^{\prime\prime}$, where $F^{\prime}$ is a union of $E$-equivalence classes and $F^{\prime\prime}$ is contained in a finite union of $E$-equivalence classes. 2. ii)

We say that $\mathrm{tp}(a/C)$ is opaquely layered if there are sequences $a_{1},\ldots,a_{N}$ of (imaginary) elements and $E_{1},\ldots,E_{N}$ of opaque equivalence relations such that each $a_{i}$ is an $E_{i}$-equivalence class, each $E_{i}$ is defined over $C\cup\{a_{j}:j<i\}$, and $\mathrm{dcl}(Ca)=\mathrm{dcl}(C,a_{1},\ldots,a_{N})$. 3. iii)

The $C$-definable equivalence relation $E$ on $D$ is opaquely layered over $C$ if for every $d\in D$, $\mathrm{tp}(d/E)$ is opaquely layered over $C$. 4. iv)

Let $G$ be a $C$-definable group and $F$ a $C$-definable subgroup. Then $G/F$ is opaquely layered (opaque) over $C$ if the equivalence relation $xF=yF$ is opaquely layered (opaque) over $C$.

Proposition 4.3**.**

[HHM08, Lemma 11.7]** Let $C\subseteq\mathcal{R}$ and suppose that $a$ is a finite tuple of imaginaries such that $\mathrm{tp}(a/C)$ is opaquely layered, witnessed by $a_{0},\ldots,a_{N}$ and $E_{0}\ldots,E_{N}$. Then $\mathrm{dcl}(Ca)$ has a pre-resolution $D$ which is atomic over $C\langle a\rangle$. If $D^{\prime}$ is another pre-resolution that contains $C$ and an element from each $E_{i}$-class, then $D$ embeds elementarily into $D^{\prime}$.

In the situation of the present paper, where the elements of $\mathrel{\mathcal{R}^{\mathrm{eq}}}$ are equivalence classes in the home sort $\mathcal{R}$, the statement can be strengthened to say that $D$ is a resolution, not just a pre-resolution.

We now apply these concepts to build prime resolutions for the geometric sorts in $T$-convex theories.

Proposition 4.4**.**

Let $K$ be a model of a power-bounded $T$-convex theory.

1. i)

The additive groups $K/V_{K}$ and $V/\mathfrak{m}_{K}$ are opaque, as is the multiplicative group $K^{\times}/(1+\mathfrak{m}_{K})$. 2. ii)

The multiplicative group $K^{\times}/V^{\times}_{K}$ is opaquely layered. 3. iii)

The groups $B_{n}(K)/B_{n}(V_{K})$ and $B_{n}(K)/B_{n,m}(V_{K})$ are opaquely layered.

Proof.

Part i) is an immediate consequence of the property arising from Fact 1.1, that the definable subsets of $K$ have finitely many convex components and that the equivalence classes of each of the given quotients are convex subsets of $K$.

To prove part ii), let $E$ be the equivalence relation

[TABLE]

on $K^{\times}$. Then the fact that the type of the $E$-equivalence class of $b\in K^{\times}$ is opaquely layered is witnessed by the equivalence relations

[TABLE]

and by the sequence of imaginary elements $b/E_{1},b/E_{2}$.

iii) This is exactly the same as the proof of the corresponding statement in [HHM08, Lemma 11.13]. ∎

Theorem 4.5**.**

Let $K$ be a power-bounded $T$-convex structure, and let $C\subseteq K$. Let $e$ be a finite set of imaginaries from $\mathcal{G}$. Then $\mathrm{dcl}(Ce)$ admits a resolution $D$ which is minimal, prime and atomic over $Ce$. Up to isomorphism over $Ce$, $D$ is the unique prime resolution of $\mathrm{dcl}(Ce)$. Furthermore, $k_{D}=k(Ce)$, $\Gamma_{D}=\Gamma(Ce)$ and $\mathrm{kInt}_{\Gamma(D)}^{D}=\mathrm{kInt}_{\Gamma(Ce)}^{\mathrm{acl}(Ce)}$.

Proof.

The proof is essentially the same as that of [HHM08, Theorem 11.14]. We include it here in order to clarify a few points in the argument, and make a few simplifications. In particular, parts of the argument are made easier due to having definable Skolem functions in the main sort. Furthermore, the use of [HHM08, Theorem 10.15] is unnecessary.

As in the proof of [HHM08, Theorem 11.14], we may assume that $e$ has the same definable closure over $C$ as some pair $(a,b)$ with $a\in B_{n}(K)/B_{n}(V_{K})$ and $b\in B_{m}(K)/B_{mm}(V_{K})$. By Proposition 4.4 iii), these groups are opaquely layered, and hence so is $\mathrm{tp}(ab/C)$ (using also [HHM08, Lemma 11.6]). By Proposition 4.3, $C\langle e\rangle$ has an atomic resolution. This resolution is prime because $\mathcal{R}$ has definable Skolem functions.

In either of the above cases, the resolution $D$ is also minimal. For suppose there is another resolution $D^{\prime}$ with $D^{\prime}\subseteq D$. By primality, $D$ embeds into $D^{\prime}$ over $C\langle e\rangle$. As $D$ has finite transcendence degree over $C$ (this follows from the proof of Proposition 4.3), $D^{\prime}=D$. The fact that $D$ is unique up to isomorphism over $C\langle e\rangle$ is immediate from primality and minimality.

Finally, we show that $k_{D}=k(Ce)$ and $\Gamma_{D}=\Gamma(Ce)$. The inclusions $\Gamma(Ce)\subseteq\Gamma_{D}$ and $k(Ce)\subseteq k_{D}$ are obvious. To see the containment in the other direction, consider $d\in D$. As $D$ is atomic, $\mathrm{tp}(d/Ce)$ is isolated, and hence also $\mathrm{tp}(\mathrm{res}(d)/Ce)$ and $\mathrm{tp}(v(d)/Ce)$ are both isolated. As $k$ is real closed and $\Gamma$ is divisible abelian, the only isolated types are algebraic, so both $\mathrm{res}(d)$ and $v(d)$ are in $\mathrm{dcl}(Ce)$. Thus $\Gamma_{D}\subseteq\Gamma(Ce)$ and $k_{D}\subseteq k(Ce)$. All that remains is to note that

[TABLE]

∎

4.2. Domination in the sorted structure

We can now state our domination results in the sorted language $\mathcal{G}$.

Theorem 4.6**.**

Let $C$ be a substructure of $\mathcal{R}$ which is maximal as a valued field. Let $A$ be a definably closed subset of $\mathcal{R}\cup\mathcal{G}$, with $A=\mathrm{dcl}(Ce)$ for a countable tuple of imaginaries $e\in\mathcal{G}$. Then

1. i)

$\mathrm{tp}(A/C)$* is dominated by the value group and residue field;* 2. ii)

$\mathrm{tp}(A/C)$* is dominated by the $k$-internal sorts over the value group.*

Proof.

i) Let $M$ be any substructure of $\mathcal{R},$ containing $C$ with $k_{A}$, $k_{M}$ independent over $k_{C}$ and $\Gamma_{A}$, $\Gamma_{M}$ independent over $\Gamma_{C}$. We need to show that

[TABLE]

Take $\sigma\in\mathrm{Aut}(\mathcal{R}/Ck_{M}\Gamma_{M})$ with $\sigma(A)=A^{\prime}$. By Theorem 4.5 (and [HHM08, Corollary 11.15]), there is a resolution $L$ of $A$ with $k_{L}=k_{A}$ and $\Gamma_{L}=\Gamma_{A}$. Let $L^{\prime}$ be $\sigma(L)$. Assume $\sigma(L)=L^{\prime}$ has the same residue field and value group as $A$. As the hypotheses of Corollary 2.8 are met, we may find an automorphism $\tau$ agreeing with $\sigma$ on $L$ and restricting to the identity on $M$. Since $A\subseteq\mathrm{dcl}(L)$, $\tau$ maps $A$ to $A^{\prime}$. Thus $A$ and $A^{\prime}$ realize the same type over $M$.

ii) Let $C^{+}=\mathrm{dcl}(C\Gamma_{A})$ and assume that $M$ is a substructure containing $C^{+}$ such that $\mathrm{kInt}_{C^{+}}^{M}$ is independent from $\mathrm{kInt}_{C^{+}}^{A}$. Apply Theorem 4.5 to obtain a resolution $L$ of $A$ with $\mathrm{kInt}^{L}_{C^{+}}=\mathrm{kInt}^{A}_{C^{+}}$ and $\Gamma_{L}=\Gamma_{A}$. Apply Corollary 2.11 to see that $\mathrm{tp}(L/C^{+}\mathrm{kInt}_{C^{+}}^{M})\vdash\mathrm{tp}(L/M)$ and reason as in (i) to see that $\mathrm{tp}(A/C^{+}\mathrm{kInt}_{C^{+}}^{M})\vdash\mathrm{tp}(A/M)$, as required.

∎