# Corrigendum to "Valued Fields, Metastable Groups"

Ehud Hrushovski, Silvain Rideau-Kikuchi, Paul Z Wang

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Ehud Hrushovski, Silvain Rideau-Kikuchi, Paul Z Wang. Corrigendum to "Valued Fields, Metastable Groups". 2024. hal-04601852

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CORRIGENDUM TO "VALUED FIELDS, METASTABLE GROUPS"

EHUD HRUSHOVSKI, SILVAIN RIDEAU-KIKUCHI, AND PAUL Z. WANG

We wish to correct an error pointed out by the third author in the paper "Valued fields, Metastable groups" [HRK19] by the first two authors. Numbering and notation follow that of the original paper. In particular, we work in a large model $U$ of complete theory $T$ that eliminates imaginaries.

As far as we know, [HRK19, Lemma 2.24] does not suffice to prove the main theorem and we need a more general version (Corollary 2.45 of the present note). This requires a slightly strengthened version of the hypothesis (FD$_{ω}$) of [HRK19], namely:

**Definition 2.21.** $T$ has (FD$_{ω}$) if, in addition to (FD), any *countable* $C$ is contained in a metastability basis $M$ which is an $\kappa_1$-saturated model and such that for any finitely acl-generated $G \subseteq \Gamma$ and $S \subseteq \text{St}_{M \cup G}$ over $M$, isolated types over $M \cup S$ are dense.

This stronger hypothesis also holds in ACVF. Note that a maximally complete model of ACVF is $\kappa_1$-saturated if and only if its value group and residue fields are.

**Definable directed partial orders and cofinal types.** The main new results consist in showing that nicely behaved definable filters can be completed to a definable type (see Proposition 2.44). This in turn implies that definable directed orders admit definable cofinal types, generalizing [HRK19, Lemma 2.24].

**Definition 2.42.** A *definable filter basis* on a definable set $X$ is a definable family $\mathcal{B}$ of definable subsets of $X$, forming a filter basis; i.e. if $U, V \in \mathcal{B}$ then there exists $W \in \mathcal{B}$ with $W \subseteq U \cap V$. We also assume $\emptyset \notin \mathcal{B}$.

This is a strengthening of the notion of definable filter (see [HRK19, Section 2.1]): if $\mathcal{B}$ is a definable filter basis, then the filter generated by $\mathcal{B}$ is definable. The converse does not hold in general.

We will be needing two operations on filters bases:

* Let $f : X \to Y$ be definable, and let $\mathcal{B}$ be a definable filter basis on $X$. Then the pushforward $f_*\mathcal{B} = \{f(U) : U \in \mathcal{B}\}$ is a definable filter basis.

* Let $\mathcal{B}$ be a filter basis on $X \times Y$ for some (pro)-definable set $X$ and $Y$ (over some model $M$) and let $a \in X$. Assume that, for every $U \in \mathcal{B}$, $U_a = U \cap \{a\} \times Y \neq \emptyset$ — we say that $\text{tp}(a/M)$ is consistent with (the filter generated by) $\mathcal{B}$. Then $\mathcal{B}_a = \{U_a : U \in \mathcal{B}\}$ is a filter basis.

Let $M$ be an $\kappa_1$-saturated model and let $\Gamma$ be a stably embedded $o$-minimal $M$-definable set. Further assume that, for any finite tuple $a$, $\gamma = \Gamma(Ma)$ is countably dcl-generated over $M$ and $\text{tp}(a/M\gamma)$ is definable: for any formula $\phi(x, y, z)$, the set $\{c \in M : \phi(x, c, \gamma) \in \text{tp}(a/M\Gamma(Ma))\}$ is definable.

Assuming (FD$_{ω}$), any countable set $C$ is contained in such an $M$.

**Lemma 2.43.** Let $X$ be a countably pro-$M$-definable set. Let $\pi$ be a filter on $X \times \Gamma$ which is generated by countably many $M$-definable filter bases. Let $a \in X$ be such that $\text{tp}(a/M)$ is definable and consistent with $\pi$. Then there exists an $M$-definable type $q \supset p$ concentrating on $X \times \Gamma$ which is consistent with $\pi$.
CORRIGENDUM TO "VALUED FIELDS, METASTABLE GROUPS"

EHUD HRUSHOVSKI, SILVAIN RIDEAU-KIKUCHI, AND PAUL Z. WANG

We wish to correct an error pointed out by the third author in the paper “Valued fields, Metastable groups” [HRK19] by the first two authors. Numbering and notation follow that of the original paper. In particular, we work in a large model $U$ of complete theory $T$ that eliminates imaginaries.

As far as we know, [HRK19, Lemma 2.24] does not suffice to prove the main theorem and we need a more general version (Corollary 2.45 of the present note). This requires a slightly strengthened version of the hypothesis (FD$_{ω}$) of [HRK19], namely:

**Definition 2.21.** $T$ has (FD$_{ω}$) if, in addition to (FD), any *countable* $C$ is contained in a metastability basis $M$ which is an $\kappa_1$-saturated model and such that for any finitely acl-generated $G \subseteq \Gamma$ and $S \subseteq \text{St}_{M \cup G}$ over $M$, isolated types over $M \cup S$ are dense.

This stronger hypothesis also holds in ACVF. Note that a maximally complete model of ACVF is $\kappa_1$-saturated if and only if its value group and residue fields are.

**Definable directed partial orders and cofinal types.** The main new results consist in showing that nicely behaved definable filters can be completed to a definable type (see Proposition 2.44). This in turn implies that definable directed orders admit definable cofinal types, generalizing [HRK19, Lemma 2.24].

**Definition 2.42.** A *definable filter basis* on a definable set $X$ is a definable family $\mathcal{B}$ of definable subsets of $X$, forming a filter basis; i.e. if $U, V \in \mathcal{B}$ then there exists $W \in \mathcal{B}$ with $W \subseteq U \cap V$. We also assume $\emptyset \notin \mathcal{B}$.

This is a strengthening of the notion of definable filter (see [HRK19, Section 2.1]): if $\mathcal{B}$ is a definable filter basis, then the filter generated by $\mathcal{B}$ is definable. The converse does not hold in general.

We will be needing two operations on filters bases:

* Let $f : X \to Y$ be definable, and let $\mathcal{B}$ be a definable filter basis on $X$. Then the pushforward $f_*\mathcal{B} = \{f(U) : U \in \mathcal{B}\}$ is a definable filter basis.

* Let $\mathcal{B}$ be a filter basis on $X \times Y$ for some (pro)-definable set $X$ and $Y$ (over some model $M$) and let $a \in X$. Assume that, for every $U \in \mathcal{B}$, $U_a = U \cap \{a\} \times Y \neq \emptyset$ — we say that $\text{tp}(a/M)$ is consistent with (the filter generated by) $\mathcal{B}$. Then $\mathcal{B}_a = \{U_a : U \in \mathcal{B}\}$ is a filter basis.

Let $M$ be an $\kappa_1$-saturated model and let $\Gamma$ be a stably embedded $o$-minimal $M$-definable set. Further assume that, for any finite tuple $a$, $\gamma = \Gamma(Ma)$ is countably dcl-generated over $M$ and $\text{tp}(a/M\gamma)$ is definable: for any formula $\phi(x, y, z)$, the set $\{c \in M : \phi(x, c, \gamma) \in \text{tp}(a/M\Gamma(Ma))\}$ is definable.

Assuming (FD$_{ω}$), any countable set $C$ is contained in such an $M$.

**Lemma 2.43.** Let $X$ be a countably pro-$M$-definable set. Let $\pi$ be a filter on $X \times \Gamma$ which is generated by countably many $M$-definable filter bases. Let $a \in X$ be such that $\text{tp}(a/M)$ is definable and consistent with $\pi$. Then there exists an $M$-definable type $q \supset p$ concentrating on $X \times \Gamma$ which is consistent with $\pi$.
*Proof.* If there exists a $\gamma \in \text{dcl}(Ma)$ such that $a\gamma \models \pi$, we can choose $q = \text{tp}(a\gamma/M)$. Now, assume that there is not such $\gamma$. For every family of definable functions $g_m : X \to \Gamma$, we define the following equivalence relation : for all $m, n \in M$, we say that $m \sim_g n$ if there exists $U \in \pi$ such that $[g_m(a), g_n(a)] \cap U_a = \emptyset$.

**Claim 2.43.1.** The relation $\sim_g$ has finitely many classes which are all definable.

*Proof.* Let $U = (U_b)_b$ be a definable filter basis contained in $\pi$. We say that $m \sim_{g,U} n$ if there exists a $b \in M$ with $[g_m(a), g_n(a)] \cap U_{b,a} = \emptyset$. This is an equivalence relation on the set of $m$ such that there exists a $b$ with $g_m(a) \notin U_{b,a}$ — whose complement we can see as another class. Since $\Gamma$ is $o$-minimal, each $U_{b,a}$ is union of at most $n$ intervals. Then, since the $U_{b,a}$ form a filter basis, there are at most $n+2$ classes. Moreover, the relation $\sim_g$ is the intersection of all the $\sim_{g,U}$. Choosing (countably many) representatives for all the classes of the $\sim_{g,U}$, we see that all the classes of $\sim_g$ are represented in the $\aleph_1$-saturated model $M$.

Now, for any $m \in M$, the class of $m$ is the union, as $U$ ranges over definable filter bases contained in $\pi$, of the sets $\{m' : \exists b [g_m(a), g_m'(a)] \cap U_{b,a} = \emptyset\}$ which are definable. So all the classes are (countably) ind-$M$-definable. Hence, by compactness and $\aleph_1$-saturation of $M$, there are finitely many classes and they are all definable. $\square$

Let $E_g$ denote the $\sim_g$-class of tuples $m$ such that there exists a $U \in \pi$ such that $(-\infty, g_m(a)] \cap U_a = \emptyset$ — or $E_g = \emptyset$ if no such $m$ exists. Let $q \supset p$ be the type such that $q(x, y) \vdash g_m(x) < y$ if and only of $m \in E_g$. By construction, it is definable and consistent with $\pi$. $\square$

**Proposition 2.44.** Let $\mathcal{B}$ be an $M$-definable filter basis on some $M$-definable set $X$. Then $\mathcal{B}$ is consistent with an $M$-definable type.

*Proof.* Let $\mathfrak{F}$ consist of all $M$-definable functions $X \to \Gamma$, seen as a pro-definable function. Using Lemma 2.43 iteratively, we find an $M$-definable type $\text{tp}(\gamma/M)$ of tuples from $\Gamma$ which is consistent with $\mathfrak{F}_*\pi$.

Note that, at stage $\alpha$, we have a definable type $\text{tp}(\gamma/M)$ of tuples from $\Gamma$ consistent with $\mathfrak{F}_*\pi$. So there exists some $a \models \pi|_M$ with $\gamma \in \Gamma(\text{acl}(Ma))$ (via the first $\alpha$ functions in $\mathfrak{F}$). So $\gamma$ is countably dcl-generated over $M$, we may assume that $\gamma$ is a countable tuple to apply Lemma 2.43. In the end, we have $a \models \pi|_M$ with $\mathfrak{F}(a) = \gamma$. Then $\text{tp}(a/M\gamma)$ is definable and, by transitivity, $\text{tp}(a/M)$ is definable. $\square$

**Corollary 2.45.** Let $\leq$ be an $M$-definable directed partial order on an $M$-definable set $P$. Then there exists an $M$-definable type $p$ cofinal in $P$ : for any $c \in P$, we have $\models (d_p x)x \geq c$.

*Proof.* Consider the $M$-definable filter basis of all cones $\{x : b \leq x\}$. $\square$

**Discussion.** If there exists a definable weakly order preserving map $j : \Gamma \to P$ with cofinal image, then we can use the definable type at $\infty$ of $\Gamma$, $r_\infty$, to obtain a cofinal definable type of $P$, namely $j_*r_\infty$.

In general, it is not always possible to find a one-dimensional cofinal subset of $P$. For instance, when $\Gamma$ is a divisible ordered Abelian group, consider the product of two closed intervals of incommensurable sizes; or the subdiagonal part of a square.

## 4. STABLY DOMINATED GROUPS

We also take this opportunity to correct an error in the statement of [HRK19, Corollary 4.12].

**Corollary 4.12.** Let $G$ be a stably dominated pro-$C$-definable group, $\dim_{\text{st}}(G) < \infty$. Then the generic types of $G$ over $C$ are precisely the types $\text{tp}(c/C)$ such that for h generic in $G$ over $C_c$, $\dim_{\text{st}}(\text{St}_C(h \cdot c)/C_h)$ is maximal (as $c$ varies).
*Proof.* If there exists a $\gamma \in \text{dcl}(Ma)$ such that $a\gamma \models \pi$, we can choose $q = \text{tp}(a\gamma/M)$. Now, assume that there is not such $\gamma$. For every family of definable functions $g_m : X \to \Gamma$, we define the following equivalence relation : for all $m, n \in M$, we say that $m \sim_g n$ if there exists $U \in \pi$ such that $[g_m(a), g_n(a)] \cap U_a = \emptyset$.

**Claim 2.43.1.** The relation $\sim_g$ has finitely many classes which are all definable.

*Proof.* Let $U = (U_b)_b$ be a definable filter basis contained in $\pi$. We say that $m \sim_{g,U} n$ if there exists a $b \in M$ with $[g_m(a), g_n(a)] \cap U_{b,a} = \emptyset$. This is an equivalence relation on the set of $m$ such that there exists a $b$ with $g_m(a) \notin U_{b,a}$ — whose complement we can see as another class. Since $\Gamma$ is $o$-minimal, each $U_{b,a}$ is union of at most $n$ intervals. Then, since the $U_{b,a}$ form a filter basis, there are at most $n+2$ classes. Moreover, the relation $\sim_g$ is the intersection of all the $\sim_{g,U}$. Choosing (countably many) representatives for all the classes of the $\sim_{g,U}$, we see that all the classes of $\sim_g$ are represented in the $\aleph_1$-saturated model $M$.

Now, for any $m \in M$, the class of $m$ is the union, as $U$ ranges over definable filter bases contained in $\pi$, of the sets $\{m' : \exists b [g_m(a), g_m'(a)] \cap U_{b,a} = \emptyset\}$ which are definable. So all the classes are (countably) ind-$M$-definable. Hence, by compactness and $\aleph_1$-saturation of $M$, there are finitely many classes and they are all definable. $\square$

Let $E_g$ denote the $\sim_g$-class of tuples $m$ such that there exists a $U \in \pi$ such that $(-\infty, g_m(a)] \cap U_a = \emptyset$ — or $E_g = \emptyset$ if no such $m$ exists. Let $q \supset p$ be the type such that $q(x, y) \vdash g_m(x) < y$ if and only of $m \in E_g$. By construction, it is definable and consistent with $\pi$. $\square$

**Proposition 2.44.** Let $\mathcal{B}$ be an $M$-definable filter basis on some $M$-definable set $X$. Then $\mathcal{B}$ is consistent with an $M$-definable type.

*Proof.* Let $\mathfrak{F}$ consist of all $M$-definable functions $X \to \Gamma$, seen as a pro-definable function. Using Lemma 2.43 iteratively, we find an $M$-definable type $\text{tp}(\gamma/M)$ of tuples from $\Gamma$ which is consistent with $\mathfrak{F}_*\pi$.

Note that, at stage $\alpha$, we have a definable type $\text{tp}(\gamma/M)$ of tuples from $\Gamma$ consistent with $\mathfrak{F}_*\pi$. So there exists some $a \models \pi|_M$ with $\gamma \in \Gamma(\text{acl}(Ma))$ (via the first $\alpha$ functions in $\mathfrak{F}$). So $\gamma$ is countably dcl-generated over $M$, we may assume that $\gamma$ is a countable tuple to apply Lemma 2.43. In the end, we have $a \models \pi|_M$ with $\mathfrak{F}(a) = \gamma$. Then $\text{tp}(a/M\gamma)$ is definable and, by transitivity, $\text{tp}(a/M)$ is definable. $\square$

**Corollary 2.45.** Let $\leq$ be an $M$-definable directed partial order on an $M$-definable set $P$. Then there exists an $M$-definable type $p$ cofinal in $P$: for any $c \in P$, we have $\models (d_p x)x \geq c$.

*Proof.* Consider the $M$-definable filter basis of all cones $\{x : b \leq x\}$. $\square$

**Discussion.** If there exists a definable weakly order preserving map $j : \Gamma \to P$ with cofinal image, then we can use the definable type at $\infty$ of $\Gamma$, $r_\infty$, to obtain a cofinal definable type of $P$, namely $j_*r_\infty$.

In general, it is not always possible to find a one-dimensional cofinal subset of $P$. For instance, when $\Gamma$ is a divisible ordered Abelian group, consider the product of two closed intervals of incommensurable sizes; or the subdiagonal part of a square.

## 4. STABLY DOMINATED GROUPS

We also take this opportunity to correct an error in the statement of [HRK19, Corollary 4.12].

**Corollary 4.12.** Let $G$ be a stably dominated pro-$C$-definable group, $\dim_{\text{st}}(G) < \infty$. Then the generic types of $G$ over $C$ are precisely the types $\text{tp}(c/C)$ such that for h generic in $G$ over $C_c$, $\dim_{\text{st}}(\text{St}_C(h \cdot c)/C_h)$ is maximal (as $c$ varies).
*Proof.* Let $\theta$ and $\mathfrak{g}$ be as in [HRK19, Proposition 4.6]. Since $h \cdot c$ is generic in $G$ over $C$, $\mathrm{acl}(\mathrm{St}_C(h \cdot c)) = \mathrm{acl}(C\theta(h \cdot c)) = \mathrm{acl}(C\theta(h)\theta(c))$. So $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(h \cdot c)/Ch) = \mathrm{dim}_{\mathrm{st}}(\theta(h) \cdot \theta(c)/Ch) = \mathrm{dim}_{\mathrm{st}}(\theta(c)/Ch) \le \mathrm{dim}_{\mathrm{st}}(\theta(c)/C) \le \mathrm{dim}_{\mathrm{st}}(\mathfrak{g})$.

If $c \models p\mathrm{acl}(C)$ for some stably dominated generic $p$, by symmetry, $c \models p\mathrm{acl}(Ch)$ and hence $h \cdot c$ is generic in $G$ over $Ch$. Thus $\theta(h \cdot c)$ is generic in $\mathfrak{g}$ and $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(h \cdot c)/Ch) = \mathrm{dim}_{\mathrm{st}}(\mathfrak{g})$ is maximal. Conversely, if $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(h \cdot c)/Ch)$ is maximal, $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(h \cdot c)/Ch) = \mathrm{dim}_{\mathrm{st}}(\theta(c)/C) = \mathrm{dim}_{\mathrm{st}}(\mathfrak{g})$. It follows that $\theta(c)$ is generic in $\mathfrak{g}$ over $C$ and, by [HRK19, Lemma 4.9], $c$ is generic in $G$ over $C$. $\square$

## 5. ABELIAN GROUPS

We can now amend [HRK19, Section 5] using the new material.

**Lemma 5.5. (FD)** Let $C$ be a metastability basis which is a model and $A$ be a pro-limit of $C$-definable Abelian groups with $\mathrm{dim}_{\mathrm{st}}(A) = n < \infty$. Then $A$ contains a stably dominated pro-$C\Gamma$-definable subgroup $S$ with stable homomorphic image of dimension $n$.

*Proof.* Since $C$ is a model and Morley rank is definable, we can find a $g \in A$ such that $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(g)/C) = n$. The proof now follows as in [HRK19, Lemma 5.5].

Note that, since Morley rank is definable, it is subadditive. Indeed, it suffices to check that, by induction, for every definable function $f: X \to Y$ whose fibers have constant Morley rank $n$, $\mathrm{MR}(X) = \mathrm{MR}(f(Y)) + n$. Assume not, then there exist $(S_i)_{i \in \omega} \subseteq X$ definable pairwise disjoint with $\mathrm{MR}(S_i) \ge \mathrm{MR}(f(Y)) + n$. We may assume that, for all $i$, the Morley rank of the fibers of $f$ restricted to $S_i$ is constant. If, for some $i$, $\mathrm{MR}(f(S_i)) < \mathrm{MR}(f(Y))$, then, by induction, $\mathrm{MR}(S_i) < \mathrm{MR}(f(Y)) + n$, a contradiction. So, for all $i$, we have $\mathrm{MR}(f(S_i)) = \mathrm{MR}(f(Y))$ and hence we may assume that there is some $a \in \bigcap_i f(S_i) \ne \emptyset$. Then, the fibers $S_{i,a} = S_i \cap f^{-1}(a)$ are all disjoint and $\mathrm{MR}(S_{i,a}) < n$. By induction, it follows that $\mathrm{MR}(S_i) < \mathrm{MR}(f(Y)) + n$, a contradiction once again. $\square$

### 5.1. Limit stably dominated groups.

**Definition 5.6.** Let $G$ be a pro-$C$-definable group and $q$ be a (potentially infinitary) type over $C$. For all $t \models q$, let $S_t$ be a pro-$Ct$-definable subgroup of $G$ (uniformly in $t$). We call $(S_t)_{t \models q}$ a limit stably dominated family for $G$ if:

(1) $S_t$ is a connected stably dominated subgroup of $G$.

(2) If $W \subseteq G$ is connected and stably dominated, pro-$C$-definable, then $W \subseteq S_t$ for some $t \models q$.

(3) The family $(S_t)_{t \models q}$ is directed: any small set of realizations of $q$ has an upper bound in the order defined by $t_1 \le t_2$ if $S_{t_1} \subseteq S_{t_2}$.

The group $H = \bigcup_{t \models q} S_t$ is called the *limit stably dominated subgroup* of $G$. If $G = H$, we say that $G$ is *limit stably dominated*.

**Proposition 5.9.** Let $A$ be a pro-limit of $C$-definable Abelian groups. Assume $A$ has bounded weight. Then, a limit stably dominated family for $A$ exists over $C$.

*Proof.* Let $C_1$ be a sufficiently saturated model containing $C$ and let $C_2 \supseteq C_1$ be a metastability basis which is a model. Let $(A_i)_{i \in I}$ be the family of all connected stably dominated pro-$C_1$-definable subgroups of $A$. Let $S$ be as in [HRK19, Proposition 5.3]. By [HRK19, Lemma 3.13], $S = S_t$ is pro-$Ct$-definable for some small tuple $t$. Let $q = \mathrm{tp}(t/C)$.

If $W \subseteq A$ is a connected stably dominated group, we must show that $W \subseteq S_{t'}$, for some $t' \models q$. For this purpose we can replace $W$ by a conjugate, under the group of automorphisms of the universal domain over $C$. Thus, we may assume $W$ is defined over $C_1$. In this case, $W \subseteq S_t$.
*Proof.* Let $\theta$ and $\mathfrak{g}$ be as in [HRK19, Proposition 4.6]. Since $h \cdot c$ is generic in $G$ over $C$, $\mathrm{acl}(\mathrm{St}_C(h \cdot c)) = \mathrm{acl}(C\theta(h \cdot c)) = \mathrm{acl}(C\theta(h)\theta(c))$. So $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(h \cdot c)/Ch) = \mathrm{dim}_{\mathrm{st}}(\theta(h) \cdot \theta(c)/Ch) = \mathrm{dim}_{\mathrm{st}}(\theta(c)/Ch) \le \mathrm{dim}_{\mathrm{st}}(\theta(c)/C) \le \mathrm{dim}_{\mathrm{st}}(\mathfrak{g})$.

If $c \models p$ $\mathrm{acl}(C)$ for some stably dominated generic $p$, by symmetry, $c \models p$ $\mathrm{acl}(Ch)$ and hence $h \cdot c$ is generic in $G$ over $Ch$. Thus $\theta(h \cdot c)$ is generic in $\mathfrak{g}$ and $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(h \cdot c)/Ch) = \mathrm{dim}_{\mathrm{st}}(\mathfrak{g})$ is maximal. Conversely, if $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(h \cdot c)/Ch)$ is maximal, $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(h \cdot c)/Ch) = \mathrm{dim}_{\mathrm{st}}(\theta(c)/C) = \mathrm{dim}_{\mathrm{st}}(\mathfrak{g})$. It follows that $\theta(c)$ is generic in $\mathfrak{g}$ over $C$ and, by [HRK19, Lemma 4.9], $c$ is generic in $G$ over $C$. $\square$

## 5. ABELIAN GROUPS

We can now amend [HRK19, Section 5] using the new material.

**Lemma 5.5. (FD)** Let $C$ be a metastability basis which is a model and $A$ be a pro-limit of $C$-definable Abelian groups with $\mathrm{dim}_{\mathrm{st}}(A) = n < \infty$. Then $A$ contains a stably dominated pro-$C\Gamma$-definable subgroup $S$ with stable homomorphic image of dimension $n$.

*Proof.* Since $C$ is a model and Morley rank is definable, we can find a $g \in A$ such that $\mathrm{dim}_{\mathrm{st}}(\mathrm{St}_C(g)/C) = n$. The proof now follows as in [HRK19, Lemma 5.5].

Note that, since Morley rank is definable, it is subadditive. Indeed, it suffices to check that, by induction, for every definable function $f: X \to Y$ whose fibers have constant Morley rank $n$, $\mathrm{MR}(X) = \mathrm{MR}(f(Y)) + n$. Assume not, then there exist $(S_i)_{i \in \omega} \subseteq X$ definable pairwise disjoint with $\mathrm{MR}(S_i) \ge \mathrm{MR}(f(Y)) + n$. We may assume that, for all $i$, the Morley rank of the fibers of $f$ restricted to $S_i$ is constant. If, for some $i$, $\mathrm{MR}(f(S_i)) < \mathrm{MR}(f(Y))$, then, by induction, $\mathrm{MR}(S_i) < \mathrm{MR}(f(Y)) + n$, a contradiction. So, for all $i$, we have $\mathrm{MR}(f(S_i)) = \mathrm{MR}(f(Y))$ and hence we may assume that there is some $a \in \bigcap_i f(S_i) \ne \emptyset$. Then, the fibers $S_{i,a} = S_i \cap f^{-1}(a)$ are all disjoint and $\mathrm{MR}(S_{i,a}) < n$. By induction, it follows that $\mathrm{MR}(S_i) < \mathrm{MR}(f(Y)) + n$, a contradiction once again. $\square$

### 5.1. Limit stably dominated groups.

**Definition 5.6.** Let $G$ be a pro-$C$-definable group and $q$ be a (potentially infinitary) type over $C$. For all $t \models q$, let $S_t$ be a pro-$Ct$-definable subgroup of $G$ (uniformly in $t$). We call $(S_t)_{t \models q}$ a limit stably dominated family for $G$ if:

(1) $S_t$ is a connected stably dominated subgroup of $G$.

(2) If $W \subseteq G$ is connected and stably dominated, pro-$C$-definable, then $W \subseteq S_t$ for some $t \models q$.

(3) The family $(S_t)_{t \models q}$ is directed: any small set of realizations of $q$ has an upper bound in the order defined by $t_1 \le t_2$ if $S_{t_1} \subseteq S_{t_2}$.

The group $H = \bigcup_{t \models q} S_t$ is called the *limit stably dominated subgroup* of $G$. If $G = H$, we say that $G$ is *limit stably dominated*.

**Proposition 5.9.** Let $A$ be a pro-limit of $C$-definable Abelian groups. Assume $A$ has bounded weight. Then, a limit stably dominated family for $A$ exists over $C$.

*Proof.* Let $C_1$ be a sufficiently saturated model containing $C$ and let $C_2 \supseteq C_1$ be a metastability basis which is a model. Let $(A_i)_{i \in I}$ be the family of all connected stably dominated pro-$C_1$-definable subgroups of $A$. Let $S$ be as in [HRK19, Proposition 5.3]. By [HRK19, Lemma 3.13], $S = S_t$ is pro-$Ct$-definable for some small tuple $t$. Let $q = \mathrm{tp}(t/C)$.

If $W \subseteq A$ is a connected stably dominated group, we must show that $W \subseteq S_{t'}$, for some $t' \models q$. For this purpose we can replace $W$ by a conjugate, under the group of automorphisms of the universal domain over $C$. Thus, we may assume $W$ is defined over $C_1$. In this case, $W \subseteq S_t$.
This proves (2) of the definition of a limit stably dominated family. Directness of the family $(S_t)_{t=q}$ follows from [HRK19, Proposition 5.3], together with (2). □

Note that if $C = C_2$ can be assumed to be a sufficiently saturated metastability basis, then we can choose $q$ to be a type concentrating on some power of $\Gamma$ as is the case in [HRK19, Definition 5.6].

**Lemma 5.10.** Let $C$ be a metastability basis which is a model. Let $A$ be a $C$-definable Abelian group and let $H$ be a connected stably dominated $\infty$-C-definable subgroup of $A$.

(FD) $H$ is contained in a CΓ-definable stably dominated subgroup.

(FD_ω) $H$ is contained in a CΓ-definable connected stably dominated subgroup.

The proof is the same as [HRK19, Lemma 5.10], except that we use the new version of Lemma 5.5, requiring $C$ to be a model.

**Proposition 5.12.** (FD_ω) Let $A$ be a C-definable Abelian group. Then there exist C-definable families $H^\nu$ of definable subgroups $H_t^\nu$ of $A$, such that:

(1) Any $H_t^\nu$ is stably dominated.

(2) Any connected stably dominated $\infty$-definable subgroup of $A$ (over any set of parameters) is contained in some $H_t^\nu$.

*Proof.* By Lemma 5.10, to prove (2), it suffices to consider definable connected stably dominated subgroups of $A$.

For a definable Abelian group $B$, define invariants $n$ and $k$ as follows. First, set $n = \dim_{\mathrm{st}}(B)$. Let $Z(B)$ be the collection of definable subgroups $S \le B$ with stable homomorphic images of dimension $n$; by Lemma 5.5 and [HRK19, Corollary 4.16], $Z(B) \ne \emptyset$. Let $Z_2(B) = \{(S,T) : S \in Z(B), T \le S \text{ CΓ-definable, } S/T \text{ Γ-internal}\}$. Let $k = \max\{\dim_o(S/T) : (S,T) \in Z_2(B)\}$; by (FD), such a maximum exists. The pairs $(n,k)$ are ordered lexicographically.

Pick any definable connected stably dominated definable $B \le A$. If $(S/B, T/B)$ attains the maximum for $A/B$, then the pullbacks to $A$ show that $(n,k)(A) \ge (n,k)(A/B)$. Thus increasing $B$ has the effect of decreasing $(n,k)(A/B)$. Let $B_0$ be such that $(n,k)(A/B_0)$ is minimal. For any stably dominated $H \le A$, $H + B_0$ is also stably dominated and if $S_t$ is a family of stably dominated subsets of $A/B_0$, its lifting to $A$ is a family of stably dominated subsets of $A$; so it suffices to find families $\{S_t^\nu\}$ for $A/B_0$. Thus, we may assume $(n,k)(A/B) = (n,k)(A)$ for any connected stably dominated CΓ-definable $B \le A$. Let $(n,k) = (n,k)(A)$.

**Claim 5.12.1.** Let $S$ be a definable subgroup of $A$ admitting a definable surjective homomorphism $\theta : S \to g$ to a definable stable group of Morley rank $n$ and a definable surjective homomorphism $\xi : S \to W$ to a definable $\Gamma$-internal group of $\alpha$-minimal dimension $k$. Then there exists a $C$-definable family $H_t$ of stably dominated subgroups of $A$ such that $\ker(\xi) = H_t$ for some $t$.

*Proof.* Clearly $S, \xi, g, \theta, W$ lie in a $C$-definable family $(S_t, \xi_t, g_t, \theta_t, W_t)$ such that $S_t \le A$, $g_t$ is stable, $\dim_{\mathrm{st}}(g_t) = n$, $W_t$ is a $\Gamma$-internal, $\dim_o(W_t) = k$, $\theta_t : S_t \to g$ is a surjective homomorphism, $\xi_t : S_t \to W_t$ is a surjective homomorphism. Let $H_t = \ker(\xi_t)$. Let $T_t \le S_t$ be as in [HRK19, Proposition 4.14]. By [HRK19, Lemma 4.15], $T_t$ is $Ct$-definable. By [HRK19, Lemma 2.26], $S_t/T_t$ is $\Gamma$-internal. The group $T_t/H_t \cap T_t$ is $\Gamma$-internal and stably dominated, it is therefore finite and hence trivial, since $T_t$ is connected. So $T_t \le H_t$. By maximality of $k$, $H_t/T_t$ is $\Gamma$-internal of $\alpha$-minimal dimension $0$, so it is finite. It follows that $H_t$ is stably dominated. □

Let $B$ be any definable stably dominated connected subgroup of $A$. Let $(S,T) \in Z_2(A/B)$ be such that $S$ has a stable homomorphic image of dimension $n$ and $S/T$ is $\Gamma$-internal of $\alpha$-minimal
This proves (2) of the definition of a limit stably dominated family. Directness of the family $(S_t)_{t=q}$ follows from [HRK19, Proposition 5.3], together with (2). □

Note that if $C = C_2$ can be assumed to be a sufficiently saturated metastability basis, then we can choose $q$ to be a type concentrating on some power of $\Gamma$ as is the case in [HRK19, Definition 5.6].

**Lemma 5.10.** Let $C$ be a metastability basis which is a model. Let $A$ be a $C$-definable Abelian group and let $H$ be a connected stably dominated $\infty$-C-definable subgroup of $A$.

(FD) $H$ is contained in a CΓ-definable stably dominated subgroup.

(FD_ω) $H$ is contained in a CΓ-definable connected stably dominated subgroup.

The proof is the same as [HRK19, Lemma 5.10], except that we use the new version of Lemma 5.5, requiring $C$ to be a model.

**Proposition 5.12.** (FD_ω) Let $A$ be a C-definable Abelian group. Then there exist C-definable families $H^\nu$ of definable subgroups $H_t^\nu$ of $A$, such that:

(1) Any $H_t^\nu$ is stably dominated.

(2) Any connected stably dominated $\infty$-definable subgroup of $A$ (over any set of parameters) is contained in some $H_t^\nu$.

*Proof.* By Lemma 5.10, to prove (2), it suffices to consider definable connected stably dominated subgroups of $A$.

For a definable Abelian group $B$, define invariants $n$ and $k$ as follows. First, set $n = \dim_{\mathrm{st}}(B)$. Let $Z(B)$ be the collection of definable subgroups $S \le B$ with stable homomorphic images of dimension $n$; by Lemma 5.5 and [HRK19, Corollary 4.16], $Z(B) \ne \emptyset$. Let $Z_2(B) = \{(S,T) : S \in Z(B), T \le S \text{ CΓ-definable, } S/T \text{ Γ-internal}\}$. Let $k = \max\{\dim_o(S/T) : (S,T) \in Z_2(B)\}$; by (FD), such a maximum exists. The pairs $(n,k)$ are ordered lexicographically.

Pick any definable connected stably dominated definable $B \le A$. If $(S/B, T/B)$ attains the maximum for $A/B$, then the pullbacks to $A$ show that $(n,k)(A) \ge (n,k)(A/B)$. Thus increasing $B$ has the effect of decreasing $(n,k)(A/B)$. Let $B_0$ be such that $(n,k)(A/B_0)$ is minimal. For any stably dominated $H \le A$, $H + B_0$ is also stably dominated and if $S_t$ is a family of stably dominated subsets of $A/B_0$, its lifting to $A$ is a family of stably dominated subsets of $A$; so it suffices to find families $\{S_t^\nu\}$ for $A/B_0$. Thus, we may assume $(n,k)(A/B) = (n,k)(A)$ for any connected stably dominated CΓ-definable $B \le A$. Let $(n,k) = (n,k)(A)$.

**Claim 5.12.1.** Let $S$ be a definable subgroup of $A$ admitting a definable surjective homomorphism $\theta : S \to g$ to a definable stable group of Morley rank $n$ and a definable surjective homomorphism $\xi : S \to W$ to a definable $\Gamma$-internal group of $\alpha$-minimal dimension $k$. Then there exists a $C$-definable family $H_t$ of stably dominated subgroups of $A$ such that $\ker(\xi) = H_t$ for some $t$.

*Proof.* Clearly $S, \xi, g, \theta, W$ lie in a $C$-definable family $(S_t, \xi_t, g_t, \theta_t, W_t)$ such that $S_t \le A$, $g_t$ is stable, $\dim_{\mathrm{st}}(g_t) = n$, $W_t$ is a $\Gamma$-internal, $\dim_o(W_t) = k$, $\theta_t : S_t \to g$ is a surjective homomorphism, $\xi_t : S_t \to W_t$ is a surjective homomorphism. Let $H_t = \ker(\xi_t)$. Let $T_t \le S_t$ be as in [HRK19, Proposition 4.14]. By [HRK19, Lemma 4.15], $T_t$ is $C\ell$-definable. By [HRK19, Lemma 2.26], $S_t/T_t$ is $\Gamma$-internal. The group $T_t/H_t \cap T_t$ is $\Gamma$-internal and stably dominated, it is therefore finite and hence trivial, since $T_t$ is connected. So $T_t \le H_t$. By maximality of $k$, $H_t/T_t$ is $\Gamma$-internal of $\alpha$-minimal dimension $0$, so it is finite. It follows that $H_t$ is stably dominated. □

Let $B$ be any definable stably dominated connected subgroup of $A$. Let $(S,T) \in Z_2(A/B)$ be such that $S$ has a stable homomorphic image of dimension $n$ and $S/T$ is $\Gamma$-internal of $\alpha$-minimal
dimension $k$. Let $(S', T')$ be the pullbacks of $(S, T)$ to $A$. Then $B \le T'$. Clearly $S'$ admits a stable homomorphic image of dimension $n$ and $S'/T'$ is $\Gamma$-internal of $\mathcal{O}$-minimal dimension $k$. By the Claim 5.12.1, there exists a $C$-definable family $H_t$ of stably dominated groups such that $T' = H_t$ for some $t$. $\square$

**Corollary 5.13.** (*FD*<sub>ω</sub>) Let $A$ be a $C$-definable Abelian group. There exists a definable certifiably stably dominated family $H_t$ such that any connected stably dominated $\infty$-definable subgroup of $A$ is contained in some $H_t$.

*Proof.* Let $H_t^\nu$ be as in Proposition 5.12 and let $p'$ and $S_{t'}$ be as in Proposition 5.9. Let $a \models p'$, then, for some $\nu$ and $t$, we have $S_a \le H_t^\nu$. Let $p = \text{tp}(t/C)$. Then, for $t' \models p'$, for some $t \models p$, $S_{t'} \le H_t^\nu$. In particular, any $\infty$-definable connected stably dominated subgroup of $A$ is contained in some $H_t^\nu$, with $t \models p$.

For any $t \models p$, $H_t^{\nu,0}$ is definable and has finite index, say $l$, in $H_t^\nu$. Let $L_t$ be a $C$-definable family of subgroups of $H_t^\nu$ such that $[H_t^\nu : L_t] \le l$ and whenever $t \models p$, $L_t = H_t^{\nu,0}$. Then every $L_t$ is stably dominated and any $\infty$-definable connected stably dominated subgroup of $A$ is contained in some $L_t$ with $t \models p$.

Finally, let $Q$ be the set of $t$ such that for every $s$, if $L_t/L_s \cap L_t$ is finite then it is trivial. This set is definable by [HRK19, Lemma 2.20]. If $t \models p$, then $L_t$ is connected and hence $t \in Q$. Moreover, for any $s \in Q$, $L_s^0 \le L_t$ for some $t \models p$. But then $L_s^0 \le L_s \cap L_t$, so $L_s/L_s \cap L_t$ is finite and hence trivial, by definition of $Q$. It follows that $L_s \le L_t$, concluding the proof. $\square$

**Theorem 5.16.** Let $A$ be a pro-limit of interpretable Abelian groups. Assume $A$ has bounded weight. Then the limit stably dominated subgroup $H$ exists and $A/H$ is almost internal to $\Gamma$.

If (*FD*<sub>ω</sub>) holds and *A* is interpretable, then *A*/*H* is Γ-internal and *H* is definable, it admits a generic type *p* and it is connected.

*Proof.* The existence and properties of $H$ are all established as in [HRK19, Theorem 5.16], expect for the existence of the definable cofinal type on $Q$ (which might not be a subset of some $\Gamma^n$) which follows from Corollary 2.45. $\square$

## 6. VALUED FIELDS: STABLY DOMINATED GROUPS AND ALGEBRAIC GROUPS

We conclude by pointing out the cursory modifications induced on [HRK19, Section 6].

**Lemma 6.18.** Let $G$ be a bounded $C$-definable subgroup of an algebraic group $\tilde{G}$ over $F$. Let $(H_t)_t$ be a certifiably stably dominated $C$-definable family of subgroups of $G$ forming a directed system under inclusion and let $H := \bigcup_t H_t$. Assume $G/H$ is $\Gamma$-internal. Then $H$ is stably dominated. Moreover $G/H$ is definably compact.

The proof is identical as that of [HRK19, Lemma 6.18], except that we now use Corollary 2.45 instead of [HRK19, Lemma 2.24].

The following is an immediate consequence of Theorem 5.16 and Lemma 6.18.

**Corollary 6.19.** Let $A$ be an Abelian variety over $K$. Then there is a definably compact group $C$ defined over $\Gamma$, and a definable homomorphism $\phi: A \to C$ with stably dominated kernel $H$. In particular $A$ has a unique maximal stably dominated connected $\infty$-definable subgroup—which is definable.

## REFERENCES

[HRK19] Ehud Hrushovski and Silvain Rideau-Kikuchi. Valued fields, metastable groups. Sel. Math., New Ser., 25(3):58, 2019. Id/No 47.
dimension $k$. Let $(S', T')$ be the pullbacks of $(S, T)$ to $A$. Then $B \le T'$. Clearly $S'$ admits a stable homomorphic image of dimension $n$ and $S'/T'$ is $\Gamma$-internal of $\mathcal{O}$-minimal dimension $k$. By the Claim 5.12.1, there exists a $C$-definable family $H_t$ of stably dominated groups such that $T' = H_t$ for some $t$. $\square$

**Corollary 5.13.** (*FD*<sub>ω</sub>) Let $A$ be a $C$-definable Abelian group. There exists a definable certifiably stably dominated family $H_t$ such that any connected stably dominated $\infty$-definable subgroup of $A$ is contained in some $H_t$.

*Proof.* Let $H_t^\nu$ be as in Proposition 5.12 and let $p'$ and $S_{t'}$ be as in Proposition 5.9. Let $a \models p'$, then, for some $\nu$ and $t$, we have $S_a \le H_t^\nu$. Let $p = \text{tp}(t/C)$. Then, for $t' \models p'$, for some $t \models p$, $S_{t'} \le H_t^\nu$. In particular, any $\infty$-definable connected stably dominated subgroup of $A$ is contained in some $H_t^\nu$, with $t \models p$.

For any $t \models p$, $H_t^{\nu,0}$ is definable and has finite index, say $l$, in $H_t^\nu$. Let $L_t$ be a $C$-definable family of subgroups of $H_t^\nu$ such that $[H_t^\nu : L_t] \le l$ and whenever $t \models p$, $L_t = H_t^{\nu,0}$. Then every $L_t$ is stably dominated and any $\infty$-definable connected stably dominated subgroup of $A$ is contained in some $L_t$ with $t \models p$.

Finally, let $Q$ be the set of $t$ such that for every $s$, if $L_t/L_s \cap L_t$ is finite then it is trivial. This set is definable by [HRK19, Lemma 2.20]. If $t \models p$, then $L_t$ is connected and hence $t \in Q$. Moreover, for any $s \in Q$, $L_s^0 \le L_t$ for some $t \models p$. But then $L_s^0 \le L_s \cap L_t$, so $L_s/L_s \cap L_t$ is finite and hence trivial, by definition of $Q$. It follows that $L_s \le L_t$, concluding the proof. $\square$

**Theorem 5.16.** Let $A$ be a pro-limit of interpretable Abelian groups. Assume $A$ has bounded weight. Then the limit stably dominated subgroup $H$ exists and $A/H$ is almost internal to $\Gamma$.

If (*FD*<sub>ω</sub>) holds and *A* is interpretable, then *A*/*H* is *Γ*-internal and *H* is definable, it admits a generic type *p* and it is connected.

*Proof.* The existence and properties of $H$ are all established as in [HRK19, Theorem 5.16], expect for the existence of the definable cofinal type on $Q$ (which might not be a subset of some $\Gamma^n$) which follows from Corollary 2.45. $\square$

## 6. VALUED FIELDS: STABLY DOMINATED GROUPS AND ALGEBRAIC GROUPS

We conclude by pointing out the cursory modifications induced on [HRK19, Section 6].

**Lemma 6.18.** Let $G$ be a bounded $C$-definable subgroup of an algebraic group $\tilde{G}$ over $F$. Let $(H_t)_t$ be a certifiably stably dominated $C$-definable family of subgroups of $G$ forming a directed system under inclusion and let $H := \cup_t H_t$. Assume $G/H$ is $\Gamma$-internal. Then $H$ is stably dominated. Moreover $G/H$ is definably compact.

The proof is identical as that of [HRK19, Lemma 6.18], except that we now use Corollary 2.45 instead of [HRK19, Lemma 2.24].

The following is an immediate consequence of Theorem 5.16 and Lemma 6.18.

**Corollary 6.19.** Let $A$ be an Abelian variety over $K$. Then there is a definably compact group $C$ defined over $\Gamma$, and a definable homomorphism $\phi: A \to C$ with stably dominated kernel $H$. In particular $A$ has a unique maximal stably dominated connected $\infty$-definable subgroup—which is definable.

## REFERENCES

[HRK19] Ehud Hrushovski and Silvain Rideau-Kikuchi. Valued fields, metastable groups. Sel. Math., New Ser., 25(3):58, 2019. Id/No 47.
MATHEMATICAL INSTITUTE, UNIVERSITY OF OXFORD, ANDREW WILES BUILDING, OXFORD, OX2 6GG, UNITED KINGDOM.

Email address: ehud.hrushovski@maths.ox.ac.uk

DÉPARTEMENT DE MATHÉMATIQUES ET APPLICATIONS, ÉCOLE NORMALE SUPÉRIEURE ET CNRS, 45 RUE D'ULM, 75230 PARIS CEDEX 05, FRANCE

Email address: silvain.rideau-kikuchi@ens.fr

DÉPARTEMENT DE MATHÉMATIQUES ET APPLICATIONS, ÉCOLE NORMALE SUPÉRIEURE, 45 RUE D'ULM, 75230 PARIS CEDEX 05, FRANCE

Email address: paul.wang@ens.fr
MATHEMATICAL INSTITUTE, UNIVERSITY OF OXFORD, ANDREW WILES BUILDING, OXFORD, OX2 6GG, UNITED KINGDOM.

Email address: ehud.hrushovski@maths.ox.ac.uk

DÉPARTEMENT DE MATHÉMATIQUES ET APPLICATIONS, ÉCOLE NORMALE SUPÉRIEURE ET CNRS, 45 RUE D'ULM, 75230 PARIS CEDEX 05, FRANCE

Email address: silvain.rideau-kikuchi@ens.fr

DÉPARTEMENT DE MATHÉMATIQUES ET APPLICATIONS, ÉCOLE NORMALE SUPÉRIEURE, 45 RUE D'ULM, 75230 PARIS CEDEX 05, FRANCE

Email address: paul.wang@ens.fr