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PAC STRUCTURES AS INVARIANTS OF FINITE GROUP ACTIONS

Published online by Cambridge University Press:  20 October 2023

DANIEL MAX HOFFMANN
Affiliation:
INSTITUT FÜR GEOMETRIE TECHNISCHE UNIVERSITÄT DRESDEN DRESDEN GERMANY and INSTYTUT MATEMATYKI UNIWERSYTET WARSZAWSKI WARSZAWA POLAND E-mail: daniel.max.hoffmann@gmail.com URL: https://sites.google.com/site/danielmaxhoffmann/home
PIOTR KOWALSKI*
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI WROCŁAW POLAND URL: http://www.math.uni.wroc.pl/~pkowa/
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Abstract

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property is first order.

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Article
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

1. Introduction

In this paper, we consider the notion of a pseudo algebraically closed (PAC) substructure of a stable structure. This notion originates from the theory of pseudo algebraically closed fields, which were first considered by Ax in 1960s while he worked on pseudofinite fields [Reference Ax2]. Studying PAC structures beyond the case of fields was initiated by Hrushovski [Reference Hrushovski26] in the strongly minimal context. Pillay and Polkowska considered the PAC property in the stable case [Reference Pillay and Polkowska41], there are slight differences with the approach we take here. PAC structures also appeared in Afshordel’s thesis [Reference Afshordel1]. Recently, PAC structures were analyzed by the first author [Reference Hoffmann21, Reference Hoffmann22] and also by Dobrowolski, the first author, and Lee [Reference Dobrowolski, Hoffmann and Lee15].

Here, we are working with a (complete) stable theory T which admits quantifier elimination and then focus on its universal part $T_{\forall }$ . In other words, a typical situation looks as follows. We have a universal theory $T_{\forall }$ with a stable model completion T, so T has quantifier elimination and T axiomatizes existentially closed models of $T_{\forall }$ . Then, intuitively, the class of PAC structures in T lies in between the class of existentially closed structures (models of T) and the class of all the structures considered (models of $T_{\forall }$ ). There are several possible definitions of the notion of PAC, we adopt here the definition from [Reference Hoffmann21] (expressed in terms involving stationary types), which is a slight modification of the definition from [Reference Pillay and Polkowska41], and which is equivalent to Afshordel’s definition from [Reference Afshordel1] in the case of stable theories. To define the notion of a PAC structure, one needs to use an appropriate notion of irreducibility. In the classical case of PAC fields, a topological notion is used coming from the Zariski topology. Hrushovski used in [Reference Hrushovski26] “Morley irreducibility”, that is he considered definable sets of Morely degree one. Pillay and Polkowska used [Reference Pillay and Polkowska41] stationary types and we proceed similarly here (however, we avoid any saturation requirements as given in [Reference Pillay and Polkowska41]). We say that a structure $F\models T_{\forall }$ is PAC in T (see Definition 2.3) if all stationary types (in the sense of the theory T) over F are finitely satisfiable in F. Let us point out that in the case of the theory of algebraically closed fields, all the irreducibility notions mentioned above are essentially the same. However, this is not the case for other theories of interest as the theory of differentially closed fields of characteristic $0$ or the theory of compact complex manifolds (see Section 4.1.2). Nevertheless, we show in Section 4.1 that all these irreducibility notions lead to the same notion of a PAC structure.

For an extension $F\subseteq K$ of models of $T_{\forall }$ , we obtain relative notions of K-strongly PAC and algebraically K-strongly PAC (see Definition 2.4). They are meaningful and can be though of as measuring the distance between being PAC and being a model of T (K-strongly PAC) or between being definably closed and algebraically closed (algebraically K-strongly PAC) (see Remark 2.5).

Our main motivation for considering PAC structures comes from model theory of group actions. In the set-up above, we consider actions of a fixed group G on models of $T_{\forall }$ by automorphisms. Clearly, such actions are first-order expressible in an appropriate language and we aim to describe existentially closed actions and check whether a model companion of the theory of such actions exists. The result below may be considered as an abstract generalization of our theorem about finite group actions on fields (see [Reference Hoffmann and Kowalski23, Theorem 3.29]) and as a continuation of studies from [Reference Hoffmann21].

Theorem 3.13. Let G be a finite group and T be a stable theory coding finite sets, which has quantifier elimination and eliminates strong types (that is, types over algebraically closed sets are stationary). Assume that G acts faithfully on $K=\operatorname {\mathrm {dcl}}(K)\models T_{\forall }$ . Then, the following are equivalent.

  1. (1) The action of G on K is existentially closed.

  2. (2) The structure of invariants $K^G$ is K-strongly PAC.

  3. (3) The structure of invariants $K^G$ is PAC and algebraically K-strongly PAC.

The above theorem gives a description of existentially closed finite group actions, but it is not clear whether this description is first-order, so this theorem does not settle the question of the existence of a model companion of the theory of finite actions. We can show the following implication.

Theorem 3.23. Let G be a finite group and T be as in the statement of Theorem 3.13. If the class of T-PAC structures is elementary, then the model companion of the theory of G-actions on models of $T_{\forall }$ exists.

After the abstract description of existentially closed actions (Theorem 3.13) and giving a criterion for existence of a model companion of the theory of finite actions (Theorem 3.23), we focus on particular examples of theories. We discuss the following three stable theories of fields of positive characteristic (p is a prime and e is a positive integer):

  1. (1) The theory $\mathrm {SCF}_{p,e}$ of separably closed fields of characteristic p and inseparability degree e.

  2. (2) The theory $\mathrm {SCF}_{p,\infty }$ of separably closed fields of characteristic p and infinite inseparability degree.

  3. (3) The theory $\mathrm {DCF}_{p}$ of differentially closed fields of characteristic p.

In the most interesting cases of the theories $\mathrm {SCF}_{p,\infty }$ and $\mathrm {DCF}_{p}$ , we do not have elimination of imaginaries, however we still have its weaker versions (coding finite sets and eliminating strong types), which are enough for the set-up from Theorems 3.13 and 3.23. For these theories, we describe PAC structures in a first-order way using a result of Tamagawa (see Theorem 4.21) about positive characteristic PAC fields. We finish with some general questions regarding the PAC property and existence of a model companion of the theory of finite actions. It should be mentioned that after replacing a finite group G with the infinite cyclic group $({\mathbb {Z}},+)$ , then the model theory of actions of $({\mathbb {Z}},+)$ has been thoroughly studied (see, e.g., [Reference Chatzidakis and Hrushovski12] and [Reference Chatzidakis and Pillay14]). We compare these two situation in Section 4.4.

This paper is organized as follows: In Section 2, we introduce several versions of the notion of a PAC structure and show the basic results about them. In Section 3, we put the group action to the picture and prove the main two abstract results stated above (Theorems 3.13 and 3.23). In Section 4, we consider some particular theories (mostly theories of fields of positive characteristic) and give a first-order characterization of PAC structures with respect to these theories.

2. Preliminaries

2.1. Set-up

Let T be a complete first-order theory with a monster model $\operatorname {\mathrm {\mathfrak {C}}}\models T$ (i.e., a strongly $\bar {\kappa }$ -homogeneous and $\bar {\kappa }$ -saturated model of T for a very big cardinal $\bar {\kappa }$ ). Throughout the paper, $\operatorname {\mathrm {acl}}$ and $\operatorname {\mathrm {dcl}}$ mean the algebraic closure and the definable closure in $\operatorname {\mathrm {\mathfrak {C}}}$ . Usually, x stands for a (finite) tuple of variables. Moreover, for the rest of this paper, let G be a group such that $|G|<\bar {\kappa }$ .

Bearing in mind any future applications, we try in this paper to formulate each result with a minimal list of assumptions. Therefore, we organize our general model-theoretic assumptions in the following list (we are aware that there are some overlaps, but we preferred more transparent exposition):

  1. (QE) T has quantifier elimination.

  2. (FS) T codes finite tuples (i.e., eliminates finite imaginaries).

  3. (FS+) T has (FS) and for every $k<\omega $ , for every variable x corresponding to a real sort and the $0$ -definable equivalence relation E on $S_x^k$ given by

    $$ \begin{align*} E(\bar{x},\bar{x}')\quad\iff\quad\{x_1,\ldots,x_k\}=\{x_1^{\prime},\ldots,x_k^{\prime}\}, \end{align*} $$
    there exists a $0$ -definable in L function $f:S_x^k\to S_w$ such that E is a fibration of f.
  4. (ST+) T is stable and types over algebraically closed sets are stationary (elimination of strong types).

Convention: If a statement starts with any combination of the above properties, it means that we assume the properties given in this particular combination. For example, the following remark assumes property (FS):

Remark 2.1. (FS) The condition (FS $+$ ) is equivalent to:

  • on each sort there is at least one $0$ -definable element, and

  • there is a sort with at least two $0$ -definable elements.

Proof Similarly as in the proof of Lemma 8.4.7 from [Reference Tent and Ziegler48], but, here, we allow many sorted structures.

Remark 2.2. Let us discuss what one can do to meet the above requirements if starting from arbitrary stable $L_0$ -theory $T_0$ . As we would like to work under assumptions of quantifier elimination and elimination of imaginaries, we pass to the language $L:=(L_0^{\operatorname {\mathrm {eq}}})^m$ and L-theory $T:=(T_0^{\operatorname {\mathrm {eq}}})^m$ (we add imaginary sorts and then do the Morleysation). This new theory T is stable, has quantifier elimination and elimination of imaginaries. On top of that, every $0$ -definable equivalence relation E on $\operatorname {\mathrm {\mathfrak {C}}}^n$ is the fibration of the canonical projection $\pi _E:\operatorname {\mathrm {\mathfrak {C}}}^n\to \operatorname {\mathrm {\mathfrak {C}}}^n/E$ which is build-in in the language $(L_0^{\operatorname {\mathrm {eq}}})^m$ , thus a $0$ -definable function. Strong types in any stable theory are stationary, and for any b and A. Therefore T enjoys all the properties: (QE), (FS), (FS $+$ ), and (ST $+$ ).

2.2. Notion of PAC structure and auxiliary facts

In this subsection, we recall several definitions and useful facts from [Reference Hoffmann21] and [Reference Hoffmann22]. We also provide a few new notions closely related to the old definitions. The reader may also consult [Reference Pillay and Polkowska41] and [Reference Polkowska43] for more on PAC structures in general model theoretic framework. Also [Reference Afshordel1] provides a nice of exposition of the notion of a PAC structure and related topics. A well-written survey on different variants of the notion of elimination of imaginaries and related concepts from the Galois theory in [Reference Casanovas and Farré10].

Definition 2.3. (Let T be stable.) A substructure F of $\operatorname {\mathrm {\mathfrak {C}}}$ is pseudo-algebraically closed (PAC) if every stationary type over F (in the sense of the $L(F)$ -theory of $\operatorname {\mathrm {\mathfrak {C}}}$ ) is finitely satisfiable in F.

The above definition appears in [Reference Hoffmann21] (see also Definition 5.29 in [Reference Afshordel1]). In subsection 3.1 of [Reference Hoffmann21], there is a discussion on possible choices of the definition of a PAC substructure and a comparison of Definition 2.3 to definitions of PAC structures given in [Reference Hrushovski26] and [Reference Pillay and Polkowska41]. In short, Definition 2.3 coincides with the definition of a PAC substructure in the strongly minimal context of [Reference Hrushovski26] and relaxes the saturation assumption from the definition of a PAC substructure from [Reference Pillay and Polkowska41]. Note that every PAC substructure is automatically definably closed. Thus PAC substructures for $T=\operatorname {\mathrm {ACF}}$ coincide with perfect pseudo-algebraically closed fields (as defined in, e.g., [Reference Fried and Jarden18]).

Definition 2.4. Let $F=\operatorname {\mathrm {dcl}}(F)\subseteq K\subseteq \operatorname {\mathrm {\mathfrak {C}}}$ .

  1. (1) We say that F is K-strongly PAC if each type $p(x)\in S(F)$ , which has a unique non-forking extension over K, is finitely satisfiable in F.

  2. (2) We say that F is algebraically K-strongly PAC if each algebraic type $p(x)\in S(F)$ , which has a unique non-forking extension over K, is finitely satisfiable (thus realized) in F.

Note that being K-strongly PAC for $F\subseteq K$ implies being algebraically K-strongly PAC for F. Moreover, being K-strongly PAC for F implies being a PAC substructure for F.

Remark 2.5. It should help to understand the relative notions of (algebraically) K-strongly PAC by considering the ultimate cases of $K=F$ and $K\models T$ . It is quite easy to see the following.

  1. (1) A structure F is F-strongly PAC if and only if $F\models T$ .

  2. (2) (T is stable) A structure F is K-strongly PAC for $K\models T$ if and only if F is PAC.

  3. (3) A structure F is algebraically F-strongly PAC if and only if $F=\operatorname {\mathrm {acl}}(F)$ .

  4. (4) A structure F is algebraically K-strongly PAC for $K\models T$ if and only if $F=\operatorname {\mathrm {dcl}}(F)$ .

Definition 2.6.

  1. (1) Let $F\subseteq K$ be small subsets of $\mathfrak {C}$ . We say that $F\subseteq K$ is primary if

    $$ \begin{align*}\operatorname{\mathrm{dcl}}(K)\cap\operatorname{\mathrm{acl}}(F)=\operatorname{\mathrm{dcl}}(F).\end{align*} $$
  2. (2) Let $F\subseteq K$ be small subsets of $\mathfrak {C}$ . We say that $F\subseteq K$ is regular if $F\subseteq K$ is primary and $F=\operatorname {\mathrm {dcl}}(F)$ .

  3. (3) Let F be a small definably closed substructure of $\mathfrak {C}$ . We say that F is regularly closed if for every small substructure $F'$ of $\mathfrak {C}$ , which is a regular extension of F, it follows $F\preceq _1 F'$ (i.e., F is existentially closed in $F'$ ).

The above notion of a primary extension was previously (e.g., [Reference Hoffmann21, Reference Hoffmann22]) called “regular”. It corresponds to regular extensions in $T=$ ACF provided the smaller field is perfect (equivalently, definably closed). Here, we decided to follow closer the terminology from the theory of fields and distinguish between “primary” and “regular” extensions. We plan to refine even more the notion of the model-theoretic “regular” extension after studying a possible notion of the model-theoretic separable extension in the future.

Now, we will sharpen facts from earlier articles that lead to the main results in this manuscript. The majority of [Reference Ax2 Reference Afshordel1] was written under the assumption of (full) elimination of imaginaries, elimination of quantifiers and stability. This is fine if we are interested in an abstract approach to the subject. However, as we are interested in applications of our results to particular theories, which do not enjoy elimination of imaginaries (see Section 4), we need to relax this assumption. Moreover, the assumption on stability was not crucial in several useful facts from [Reference Hoffmann21], making them applicable in a broader context. Therefore we take the opportunity to provide the following results with minimal assumptions. The proofs of the following facts remain almost the same as the proofs of their counterparts from [Reference Hoffmann21]. Recall that “regular” extensions from [Reference Hoffmann21] are now “primary” extensions.

All the proper subsets, substructures and tuples of the monster model $\operatorname {\mathrm {\mathfrak {C}}}$ are, if not stated otherwise, small in comparison to the saturation of $\operatorname {\mathrm {\mathfrak {C}}}$ . Here, upper case letters, like E or A, are denoting proper subsets, and lower case letters, like a, stand for tuples.

Fact 2.7 (Fact 3.32 in [Reference Hoffmann21])

(FS) If $E\subseteq A$ is primary then for every $a\in \operatorname {\mathrm {acl}}(E)$ there exists a unique extension of $\operatorname {\mathrm {tp}}(a/E)$ over A.

Fact 2.8 (Fact 3.33 in [Reference Hoffmann21])

(FS) If $E\subseteq A$ is primary, $f_1,f_2\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}})$ and $f_1|_E=f_2|_E$ , then there exists $h\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}})$ such that $h|_A=f_1|_A$ and $h|_{\operatorname {\mathrm {acl}}(E)}=f_2|_{\operatorname {\mathrm {acl}}(E)}$ .

Fact 2.9 (Corollary 3.34 in [Reference Hoffmann21])

(FS) If $E\subseteq A$ is primary and $A_0\subseteq A$ then $\operatorname {\mathrm {tp}}(A_0/E)$ has a unique extension over $\operatorname {\mathrm {acl}}(E)$ .

The following definition is taken from page 21 of [Reference Afshordel1].

Definition 2.10. We say that a type $p(x)\in S(A)$ is $\operatorname {\mathrm {acl}}$ -stationary if it has a unique extension over $\operatorname {\mathrm {acl}}(A)$ .

Lemma 2.11. (FS) Consider $p\in S(E)$ . The following are equivalent:

  1. (1) p is $\operatorname {\mathrm {acl}}$ -stationary,

  2. (2) $E\subseteq \operatorname {\mathrm {dcl}}(Ea)$ is primary for some $a\models p$ ,

  3. (3) $E\subseteq \operatorname {\mathrm {dcl}}(Ea)$ is primary for every $a\models p$ .

Proof The proof is similar to the proof of Lemma 3.35 in [Reference Hoffmann21], but a few steps require sharper reasoning, thus we include it here.

The equivalence (2) $\iff $ (3) follows by definition. First, we argue for (1) $\Rightarrow $ (2): assume (1) and suppose that (2) does not hold. As p is $\operatorname {\mathrm {acl}}$ -stationary, there exists a unique extension $p|_{\operatorname {\mathrm {acl}}(E)}$ of p over E. Let $a\models p|_{\operatorname {\mathrm {acl}}(E)}$ , then $a\models p$ and $E\subseteq Ea$ is not primary. Take

$$ \begin{align*}c\in\operatorname{\mathrm{dcl}}(Ea)\cap\operatorname{\mathrm{acl}}(E)\setminus\operatorname{\mathrm{dcl}}(E).\end{align*} $$

Since $c\not \in \operatorname {\mathrm {dcl}}(E)$ , there exists $f\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}}/E)$ such that $f(c)\neq c$ . We see that $f(a)\models p|_{\operatorname {\mathrm {acl}}(E)}$ , so there exists $h\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}}/\operatorname {\mathrm {acl}}(E))$ such that $h(a)=f(a)$ . Note that $h^{-1}f\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}}/Ea)$ and, because $c\in \operatorname {\mathrm {dcl}}(Ea)$ and $c\in \operatorname {\mathrm {acl}}(E)$ ,

$$ \begin{align*}c=h^{-1}f(c)=f(c)\neq c,\end{align*} $$

so a contradiction. The implication (2) $\Rightarrow $ (1) is contained in Fact 2.9.

Fact 2.12 (Lemma 3.35 in [Reference Hoffmann21])

(FS, ST $+$ ) Consider $p\in S(E)$ . The following are equivalent:

  1. (1) p is stationary,

  2. (2) p is $\operatorname {\mathrm {acl}}$ -stationary,

  3. (3) $E\subseteq Ea$ is primary for some $a\models p$ ,

  4. (4) $E\subseteq Ea$ is primary for every $a\models p$ .

Fact 2.13 (Corollary 3.36 in [Reference Hoffmann21])

(QE, FS, ST $+$ ) For any small substructure N there exists a non-algebraic stationary type over N in any finitely many variables.

Fact 2.14 (Corollary 3.38 in [Reference Hoffmann21])

(FS, ST $+$ ) Assume that $A,B\subseteq \operatorname {\mathrm {\mathfrak {C}}}$ , $E\subseteq A$ is primary, $f_1,f_2\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}})$ and $f_1|_E=f_2|_E$ . If and then there exists $h\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}})$ such that $h|_A=f_1|_A$ and $h|_B=f_2|_B$ .

Fact 2.15 (Lemma 3.39 in [Reference Hoffmann21])

(FS, ST $+$ ) If $E\subseteq A\cap B$ , $E\subseteq A$ is primary and then $B\subseteq BA$ is primary.

Fact 2.16 (Corollary 3.40 in [Reference Hoffmann21])

(FS, ST $+$ ) If $E\subseteq A$ and $E\subseteq B$ are primary, and then also $E\subseteq BA$ is primary.

Remark 2.17.

  1. (1) (FS, ST $+$ ) $F\subseteq K$ is primary if and only if for every tuple b from $\operatorname {\mathrm {dcl}}(K)$ , the type $\operatorname {\mathrm {tp}}(b/F)$ is stationary (Fact 2.12).

  2. (2) (QE, FS, ST $+$ ) Using the item (1), a substructure F is PAC if and only if it is definably closed and regularly closed.

Definition 2.18.

  1. (1) Assume that $F\subseteq K$ are substructures of $\operatorname {\mathrm {\mathfrak {C}}}$ . We say that K is normal over F (or we say that $F\subseteq K$ is a normal extension) if $\sigma (K)\subseteq K$ for every $\sigma \in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}}/K)$ . (Note that if K is small and $F\subseteq K$ is normal, then it must be $K\subseteq \operatorname {\mathrm {acl}}(F)$ .)

  2. (2) Assume that $F\subseteq K\subseteq \operatorname {\mathrm {acl}}(F)$ are small substructures of $\mathfrak {C}$ such that $F=\operatorname {\mathrm {dcl}}(F)$ , $K=\operatorname {\mathrm {dcl}}(K)$ and K is normal over F. In this situation we say that $F\subseteq K$ is a Galois extension.

Definition 2.19. Assume that $F\subseteq K$ is an extension of substructures in $\operatorname {\mathrm {\mathfrak {C}}}$ . We define the Galois group of the extension $F\subseteq K$ as

$$ \begin{align*}G(K/F):=\operatorname{\mathrm{Aut}}(K/F)=\{f|_K\;|\;f\in\operatorname{\mathrm{Aut}}(\operatorname{\mathrm{\mathfrak{C}}}/F),\,f(K)=K \}.\end{align*} $$

Moreover B is any subset of $\operatorname {\mathrm {\mathfrak {C}}}$ , then the extension $\operatorname {\mathrm {dcl}}(B)\subseteq \operatorname {\mathrm {acl}}(B)$ is Galois and we speak about the absolute Galois group of B which is the following profinite group:

$$ \begin{align*}G(B):=G(\operatorname{\mathrm{acl}}(B)/\operatorname{\mathrm{dcl}}(B)).\end{align*} $$

Note that the above definition of $G(K/F)$ is often expressed in terms of the automorphisms of K as an L-structure on its own, but as we will work under the assumption of the quantifier elimination, both variants of the definition coincide and it just the matter of taste.

The following useful fact is standard and its proof is straightforward.

Lemma 2.20. Assume that $F\subseteq K$ is a Galois extension and $p(x)\in S(F)$ . Then the Galois group $G(K/F)$ acts transitively on the set of extensions of p over K.

The following definition and example are taken from [Reference Dobrowolski, Hoffmann and Lee15] and [Reference Pillay and Polkowska41]. A more detailed discussion of examples of PAC structures and the property from Definition 2.21 will be given in Section 4.

Definition 2.21 (Let T be stable)

We say that PAC is a first-order property in T ( $=\operatorname {\mathrm {Th}}(\operatorname {\mathrm {\mathfrak {C}}})$ ) if there exists a set $\Sigma $ of $\mathcal {L}$ -sentences such that for any $P\subseteq \operatorname {\mathrm {\mathfrak {C}}},$

$$ \begin{align*}P\models \Sigma\qquad\iff\qquad P\text{ is PAC}.\end{align*} $$

Example 2.22.

  1. (1) PAC is a first-order property in ACF $_p$ for $p=0$ and for p being a prime number, see Proposition 11.3.2 in [Reference Fried and Jarden18].

  2. (2) The axioms given in Proposition 5.6 from [Reference Pillay and Polkowska41] show that PAC is a first-order property (in the above sense) in DCF $_0$ which is formulated in a different way than the condition “PAC is a first-order property” appearing in [Reference Pillay and Polkowska41].

3. Finite group actions

The main goal of this section is to describe existentially closed substructures with a finite group action in first-order terms. The general strategy is as follows. First, characterize their structure by the structure of the invariants of the group action, then answer which properties of the invariants correspond to the existential closedness of the whole substructure with group action. Finally, express these properties as first-order statements.

3.1. Basic facts

We introduce the language $L_G$ being the language L extended by a unary function symbol $\sigma _g$ for each $g\in G$ , i.e., $L_G=L\,\cup \,\{\sigma _g\;|\;g\in G\}$ . Often, “ $\sigma _g$ ” will denote also the interpretation of the symbol $\sigma _g$ in a given $L_G$ -structure. Moreover, we set $\bar {\sigma }:=(\sigma _g)_{g\in G}$ . We consider the collection of sentences in the language $L_G$ , say $A_G$ , which precisely expresses the following:

  • $\sigma _g$ is an automorphism of the L-structure for every $g\in G$ ,

  • $\sigma _g\circ \sigma _h=\sigma _{g\cdot h}$ for all $g,h\in G$ .

In other words, if K is an L-structure, and there exists an $L_G$ -structure $(K,\bar {\sigma })$ living on K, we have that $(K,\bar {\sigma })\models A_G$ if and only if for each $g\in G$ we have that $\sigma _g\in \operatorname {\mathrm {Aut}}(K)$ and the map

$$ \begin{align*}G\ni g\mapsto\sigma_g\in\operatorname{\mathrm{Aut}}(K)\end{align*} $$

is a group homomorphism.

Definition 3.1.

  1. (1) Let $(K,\bar {\sigma })$ be an $L_G$ -structure. We say that $\bar {\sigma }$ is a G-action on K if $(K,\bar {\sigma })\models A_G$ .

  2. (2) If $T'$ is an L-theory, then by $(T')_G$ we denote the set of consequences of $T'\,\cup \,A_G$ .

  3. (3) If $(K,\bar {\sigma })\models (T_{\forall })_G$ , where K is of cardinality smaller than the saturation of $\operatorname {\mathrm {\mathfrak {C}}}$ , then we call it a substructure with G-action. Note that, without loss of generality, $K\subseteq \operatorname {\mathrm {\mathfrak {C}}}$ , thus the name “substructure”.

  4. (4) We say that a substructure with G-action $(K,\bar {\sigma })$ is existentially closed if $(K,\bar {\sigma })$ is an existentially closed model of the theory $(T_{\forall })_G$ .

  5. (5) If the existentially closed models of the theory $(T_{\forall })_G$ form an elementary class, we denote the theory of this class by $G-T$ .

Definition 3.2. Assume that $(K,\bar {\sigma })$ is a substructure with G-action. Then we denote

$$ \begin{align*}K^G:=\{a\in K\;|\;(\forall g\in G)\,(\sigma_g(a)=a)\,\}\end{align*} $$

and call it the substructure of invariants.

Remark 3.3. (QE) Let $(K,\bar {\sigma })$ be a substructure with G-action. If $(K,\bar {\sigma })$ is existentially closed then $K=\operatorname {\mathrm {dcl}}(K)$ . If $K=\operatorname {\mathrm {dcl}}(K)$ then $K^G=\operatorname {\mathrm {dcl}}(K^G)$ . For the standard proofs, the reader may consult Remarks 3.24 and 3.26 in [Reference Hoffmann21].

Lemma 3.4. (QE) Let $(K,\bar {\sigma })$ be a substructure with G-action and let $p(x)\in S(K)$ be a G-invariant type (i.e., $\sigma _g(p)=p$ for every $g\in G$ ). Then for any $a\models p$ the set $\operatorname {\mathrm {dcl}}(K,a)$ might be equipped with a G-action extending $(K,\bar {\sigma })$ and acting trivially on a.

Proof Let $a\models p$ and let $\bar {k}$ be some enumeration of K. Then $\bar {k}a\equiv \sigma _g(\bar {k})a$ for any $g\in G$ . This implies that, for each $g\in G$ , there exists $\sigma _g^{\prime }\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}})$ such that $\sigma ^{\prime }_g|_K=\sigma _g$ and $\sigma ^{\prime }_g(a)=a$ . Naturally, $(K,(\sigma _g)_{g\in G})\subseteq (\operatorname {\mathrm {dcl}}(K,a),(\sigma ^{\prime }_g)_{g\in G})$ .

Fact 3.5 (Lemma 2.10 from [Reference Hoffmann22])

(QE, FS) If G is finite and $(K,\bar {\sigma })$ is a substructure with G-action such that $\operatorname {\mathrm {dcl}}(K)=K$ and the action of G on K is faithful (i.e., if $g\neq h$ then there is $a\in K$ such that $\sigma _g(a)\neq \sigma _h(a)$ ), then:

  • $K\subseteq \operatorname {\mathrm {acl}}(K^G)$ ,

  • $K^G\subseteq K$ is a Galois extension,

  • $G(K/K^G)\cong G$ .

Proof By Lemma 2.10 from [Reference Hoffmann22], Fact 3.7 and Proposition 4.7 from [Reference Casanovas and Farré10]. Being more precise, we obtain the two first bullets as in Lemma 3.23 from [Reference Hoffmann21] and then we repeat the proof of Lemma 2.10(4) from [Reference Hoffmann21] using a variant of the finite Galois correspondence stated in Proposition 4.7 in [Reference Casanovas and Farré10].

Lemma 3.6. (QE, FS, ST $+$ ) If $(K,\bar {\sigma })$ is an existentially closed substructure with G-action, then the group action if faithful.

Proof Consider any enumeration of G, say $(g_i)_{i\in I}$ where $(I,<)$ is a linear order. Let $p(x)\in S(K^G)$ be a non-algebraic stationary type (existing by Fact 2.13), and let $\bar {b}=(b_i)_{i\in I}\models p^{\otimes I}|_{K^G}$ be such that . Let F denote $\operatorname {\mathrm {dcl}}(K^G,\bar {b})$ , and let $F'$ denote $\operatorname {\mathrm {dcl}}(K,\bar {b})$ .

As the type $p^{\otimes I}|_K$ is also stationary, the extension $K^G\subseteq F$ is regular. For each $g\in G$ , let $\theta _g$ be a bijection of I such that $g\cdot g_i=g_{\theta _g(i)}$ holds for each $i\in I$ . As the set $\{b_i\;|\;i\in I\}$ is $K^G$ -indiscernible, for each $g\in G$ there exists $\tau _g\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}}/K^G)$ such that $\tau _g(b_i)=b_{\theta _g(i)}$ .

Now, Corollary 3.38 from [Reference Hoffmann21], allows us to simultaneously extend each $\sigma _g$ (over K) and $\tau _g$ (over F) to an automorphism $\sigma ^{\prime }_g\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}})$ , for each $g\in G$ . We have that $(K,(\sigma _g)_{g\in G})\subseteq (F',(\sigma ^{\prime }_g)_{g\in G})$ , thus $(K,(\sigma _g)_{g\in G})$ is existentially closed in $(F',(\sigma ^{\prime }_g)_{g\in G})$ . If $g\neq h$ , then $\sigma ^{\prime }_g(b)\neq \sigma ^{\prime }_h(b)$ for some $b\in F'$ , and so there will be $a\in K$ such that $\sigma _g(a)\neq \sigma _h(a)$ .

Lemma 3.7. (QE) If G is finitely generated and $(K,\bar {\sigma })$ is an existentially closed substructure with G-action, then $K^G$ is K-strongly PAC.

Proof Consider $p(x)\in S(K^G)$ which has a unique non-forking extension over K, say $\tilde {p}(x)\in S(K)$ . As $p(x)$ is invariant under action of automorphisms $\sigma _g|_{K^G}$ , where $g\in G$ , we have that $\tilde {p}(x)$ is invariant under action of automorphisms $\sigma _g$ , where $g\in G$ (otherwise, we would get distinct non-forking extensions of p over K).

Let $b\models \tilde {p}$ , by Lemma 3.4 there exists an extension of substructures with G-action,

$$ \begin{align*}(K,(\sigma_g)_{g\in G})\subseteq (K',(\sigma^{\prime}_g)_{g\in G})\end{align*} $$

such that $b\in (K')^G$ . By our assumption, we have that $(K,(\sigma _g)_{g\in G})$ is existentially closed in $(K',(\sigma _g^{\prime })_{g\in G})$ .

Now, let $\varphi (a,x)\in p(x)$ . As T has quantifier elimination, we may assume that $\varphi (y,x)$ is quantifier-free, what we do. Of course $\models \varphi (a,b)$ and so

$$ \begin{align*}(K',(\sigma^{\prime}_g)_{g\in G})\models (\exists x)\,(\varphi(a,x)\,\wedge\,\bigwedge\limits_{g\in X}\sigma_g(x)=x),\end{align*} $$

where X denotes the finite set of generators of G. Hence

$$ \begin{align*}(K,(\sigma_g)_{g\in G})\models (\exists x)\,(\varphi(a,x)\,\wedge\,\bigwedge\limits_{g\in X}\sigma_g(x)=x)\end{align*} $$

and for some $b_0\in K^G$ we have that $\models \varphi (a,b_0)$ .

Therefore we see that an existentially closed substructure with G-action has a quite tame substructure of invariants. The next subsection is dedicated to the converse of this implication, so we would like to show that “if the substructure of invariants is tame then the whole substructure with G-action is existentially closed”.

Remark 3.8. In Proposition 3.56 from [Reference Hoffmann21], it was shown that if $(K,\bar {\sigma })$ is an existentially closed substructure with G-action, then K is PAC. However, the aforementioned proposition assumes quantifier elimination, elimination of imaginaries and stability (but G there can be arbitrary).

3.2. Invariants of existentially closed actions

Lemma 3.9. (QE, FS) Assume that G is finite, $(K,(\sigma _g)_{g\in G})\subseteq (K',(\sigma ^{\prime }_g)_{g\in G})$ is an extension of substructures with G-action, the group action of G on K is faithful and $\operatorname {\mathrm {dcl}}(K)=K$ . If $K^G$ is algebraically K-strongly PAC, then $K^G\subseteq (K')^G$ is regular.

Proof If $\operatorname {\mathrm {dcl}}(K)=K$ then also $\operatorname {\mathrm {dcl}}(K^G)=K^G$ . Moreover, $K^G\subseteq (K')^G$ is regular if and only if $K^G\subseteq \operatorname {\mathrm {dcl}}((K')^G)$ is regular and there is a unique way of extending G-action from $K'$ over $\operatorname {\mathrm {dcl}}(K')$ . Therefore, without loss of generality, we assume that $K'=\operatorname {\mathrm {dcl}}(K')$ and so $\operatorname {\mathrm {dcl}}((K')^G)=(K')^G$ . We need to show that $(K')^G\cap \operatorname {\mathrm {acl}}(K^G)=K^G$ .

Let $a\in (K')^G\cap \operatorname {\mathrm {acl}}(K^G)\setminus K^G$ . Because for every $g\in G$ , we have that $\sigma _g\big (\operatorname {\mathrm {tp}}(a/K)\big )=\operatorname {\mathrm {tp}}\big (\sigma _g(a)/K\big )$ and $a\in (K')^G$ , we see that $\operatorname {\mathrm {tp}}(a/K)$ is a G-invariant type. By Fact 3.5 and Lemma 2.20, we see that $\operatorname {\mathrm {tp}}(a/K)$ is a unique extension of $\operatorname {\mathrm {tp}}(a/K^G)$ over K.

As $a\in \operatorname {\mathrm {acl}}(K^G)$ and (e.g., Remark 5.3 in [Reference Casanovas9]), $\operatorname {\mathrm {tp}}(a/K^G)\subseteq \operatorname {\mathrm {tp}}(a/K)$ is a non-forking extension. Because $K^G$ is algebraically K-strongly PAC, $\operatorname {\mathrm {tp}}(a/K^G)$ is finitely satisfiable in $K^G$ . As $a\in \operatorname {\mathrm {acl}}(K^G)$ , this means that it must be $a\in K^G$ .

Definition 3.10. Assume that $C\subseteq K\subseteq \operatorname {\mathrm {\mathfrak {C}}}$ and that G is finite. We call the pair $(C,K)$ G-closed if $C\subseteq K$ is a Galois extension, $G(K/C)\cong G$ and there is no $K'\subseteq \operatorname {\mathrm {acl}}(K)$ , $K\subsetneq K'$ , such that the action of $G(K/C)$ extends over $K'$ .

Lemma 3.11. (QE, FS) Assume that G is finite, $(K,\bar {\sigma })$ is a substructure with G-action such that action of G on K is faithful and $\operatorname {\mathrm {dcl}}(K)=K$ . Then $(K^G,K)$ is G-closed if and only if $K^G$ is algebraically K-strongly PAC.

Proof By Fact 3.5, $K\subseteq \operatorname {\mathrm {acl}}(K^G)$ , $K^G\subseteq K$ is Galois and $G(K/K^G)\cong G$ .

Assume that $(K^G,K)$ is G-closed and let $p(x)\in S(K^G)$ be algebraic with a unique extension $\tilde {p}(x)$ over K (being a non-forking extension follows naturally from , e.g., Remark 5.3 in [Reference Casanovas9]). We have that $\tilde {p}$ is G-invariant and so, by Lemma 3.4, if $b\models \tilde {p}$ then there exists an extension of substructures with a G-action,

$$ \begin{align*}(K,(\sigma_g)_{g\in G})\subseteq (K',(\sigma^{\prime}_g)_{g\in G})\end{align*} $$

such that $K'=\operatorname {\mathrm {dcl}}(K,b)$ and $b\in (K')^G$ . As $K'=\operatorname {\mathrm {dcl}}(K,b)\subseteq \operatorname {\mathrm {acl}}(K^G)=\operatorname {\mathrm {acl}}(K)$ , it must be that $K=K'$ , so $b\in K$ and finally $b\in K^G$ .

Now, we show the right-to-left implication. Assume that $K'\subseteq \operatorname {\mathrm {acl}}(K)$ and there is an extension of substructures with G-action:

$$ \begin{align*}(K,(\sigma_g)_{g\in G})\subseteq (K',(\sigma^{\prime}_g)_{g\in G}).\end{align*} $$

By Lemma 3.9, $K^G\subseteq (K')^G$ is regular. As $(K')^G\subseteq K'\subseteq \operatorname {\mathrm {acl}}(K)=\operatorname {\mathrm {acl}}(K^G)$ it must be $(K')^G\subseteq \operatorname {\mathrm {dcl}}(K^G)=K^G$ , so $K^G=(K')^G$ . By the proof of Proposition 4.1 from [Reference Dobrowolski, Hoffmann and Lee15] and the Galois correspondence for finite extensions (e.g., Theorem 12 in [Reference Medvedev and Takloo-Bighash32]), there exists a finite tuple b from K such that $K=\operatorname {\mathrm {dcl}}(K^G,b)$ . Moreover, by the same proof of Proposition 4.1 from [Reference Dobrowolski, Hoffmann and Lee15], we also have that $K'=\operatorname {\mathrm {dcl}}((K')^G,b)$ . Because $K^G=(K')^G$ , we have that $K=\operatorname {\mathrm {dcl}}(K^G,b)=\operatorname {\mathrm {dcl}}((K')^G,b)=K'$ .

The following remark is not important for the main results of this paper and its purpose is mainly to generalize Theorem 3.25 from [Reference Hoffmann and Kowalski23]. As we use in its proof the Elementary Equivalence for PAC structures ([Reference Dobrowolski, Hoffmann and Lee15]), we need to add more assumptions.

Remark 3.12. Let T be stable with elimination of quantifiers and elimination of imaginaries. Assume that PAC is a first-order property. Suppose that $(C,K)\subseteq (C',K')$ is an extension of G-closed substructures such that C and $C'$ are PAC. Then $C\preceq C'$ .

Proof It is enough to reproduce the proof of Theorem 3.25 from [Reference Hoffmann and Kowalski23], but in this more general context. By the proof of Theorem 3.22 from [Reference Hoffmann and Kowalski23] or more similar Lemma 3.54 from [Reference Hoffmann21], we have that C and $C'$ are bounded PAC structures. Thus, by Corollary 3.11 from [Reference Dobrowolski, Hoffmann and Lee15], it is enough to show that the restriction map $r:G(C')\to G(C)$ is an isomorphism. After combining Lemmas 3.11 and 3.9, we obtain that $C\subseteq C'$ is regular, so r is an epimorphism.

By Theorem 4.4 from [Reference Hoffmann22], $G(C)$ is projective, which means that there exists embedding h as in the following diagram.

But then $\mathcal {G}_0:=h[G(C)]\leqslant G(C')$ is a closed subgroup such that $r|_{\mathcal {G}_0}:\mathcal {G}_0\to G(C)$ is an isomorphism.

Because $K\subseteq \operatorname {\mathrm {acl}}(C)$ and $K'\subseteq \operatorname {\mathrm {acl}}(C')$ , the restriction maps $G(C)\to G$ and $G(C')\to G$ lead to the following commutative diagram:

and so $\mathcal {G}_0\mathcal {N}=G(C')$ for $\mathcal {N}:=\ker \big (G(C')\to G\big )$ . By Lemma 3.31 from [Reference Hoffmann21], this implies that $\mathcal {G}_0=G(C')$ as expected.

Theorem 3.13. (QE, FS, ST $+$ ) Assume that G is finite, say $|G|=l$ . Let $(K,\bar {\sigma })$ be a substructure with G-action such that G acts faithfully on K and $\operatorname {\mathrm {dcl}}(K)=K$ . The following are equivalent:

  1. (1) $(K,\bar {\sigma })$ is existentially closed,

  2. (2) $K^G$ is K-strongly PAC,

  3. (3) $K^G$ is PAC and algebraically K-strongly PAC,

  4. (4) $K^G$ is PAC and $(K^G,K)$ is G-closed.

Proof By Lemma 3.11, (3) $\iff $ (4). (1) $\Rightarrow $ (2) follows by Lemma 3.7. The implication (2) $\Rightarrow $ (3) follows by definitions. To get the theorem, we will show that (3) $\Rightarrow $ (1).

Assume that $\operatorname {\mathrm {dcl}}(K)=K$ , the group action is faithful and that $K^G$ is PAC and algebraically K-strongly PAC. Using Fact 3.5, we obtain the following:

  • $K\subseteq \operatorname {\mathrm {acl}}(K^G)$ ,

  • $K^G\subseteq K$ is a Galois extension,

  • $G(K/K^G)\cong G$ .

The proof of Proposition 4.1 from [Reference Dobrowolski, Hoffmann and Lee15] gives us existence of a finite tuple $\bar {b}=(b_0,\ldots ,b_{l-1})$ from K such that $K=\operatorname {\mathrm {dcl}}(K^G,\bar {b})$ .

Consider $(K,(\sigma _g)_{g\in G})\subseteq (K',(\sigma ^{\prime }_g)_{g\in G})$ . Without loss of generality, we may assume that $(K',(\sigma ^{\prime }_g)_{g\in G})$ is existentially closed, in particular $\operatorname {\mathrm {dcl}}(K')=K'$ . We have that the group action of G on $K'$ is faithful, thus by Fact 3.5, we have that $K'\subseteq \operatorname {\mathrm {acl}}((K')^G)$ , $(K')^G\subseteq K'$ is Galois, and $G(K'/(K')^G)\cong G$ . Lemma 3.9 gives us that $K^G\subseteq (K')^G$ is regular, which means that the restriction map $G(K'/(K')^G)\to G(K/K^G)$ is onto, and so it is an isomorphism of finite groups. The last thing implies

$$ \begin{align*}K'=\operatorname{\mathrm{dcl}}((K')^G,\bar{b}).\end{align*} $$

Let $\bar {B}$ be some enumeration of $\{\sigma _g(b_i)\;|\;g\in G,\;i<l\}$ . We have that $K'=\operatorname {\mathrm {dcl}}((K')^G,\bar {b})=\operatorname {\mathrm {dcl}}((K')^G,\bar {B})$ . Assume that

$$ \begin{align*}(K',(\sigma^{\prime}_g)_{g\in G})\models \phi(a)\end{align*} $$

for some tuple a from $K'$ and some quantifier-free formula $\phi (x)\in L_G(K)$ . First, we may present $\phi (a)$ as $\varphi _0(\sigma ^{\prime }_{g_0}(a),\ldots ,\sigma ^{\prime }_{g_{l-1}}(a))$ , where $\varphi _0(x_0,\ldots ,x_{l-1})\in L(K)$ is quantifier-free. Second, since $K=\operatorname {\mathrm {dcl}}(K^G,\bar {B})$ , we may present $\varphi _0(\sigma ^{\prime }_{g_0}(a),\ldots ,\sigma ^{\prime }_{g_{l-1}}(a))$ as $\varphi (\sigma ^{\prime }_{g_0}(a),\ldots ,\sigma ^{\prime }_{g_{l-1}}(a),\bar {B})$ , where $\varphi (x_0,\ldots ,x_{l-1},\bar {y})\in L(K^G)$ is quantifier-free.

Let $\sigma ^{\prime }_{g_0}=\operatorname {\mathrm {id}}_L$ , so $\sigma ^{\prime }_{g_0}(a)=a$ . Because $a\in K'=\operatorname {\mathrm {dcl}}((K')^G,\bar {B})$ , there exists a finite tuple $\bar {c}\subseteq (K')^G$ and a quantifier-free formula $\psi _0(\bar {z},\bar {y},x)\in L$ such that:

  • $\psi _0(\bar {c},\bar {B},\operatorname {\mathrm {\mathfrak {C}}})=\{a\}$ ,

  • $\models (\forall \bar {z},\bar {y},x,x')\,\big (\psi _0(\bar {z},\bar {y},x)\,\wedge \,\psi _0(\bar {z},\bar {y},x')\,\longrightarrow \,x=x'\big )$ .

Because $\sigma _{g_i}$ permutes $\bar {B}$ , there exists a permutation $s_i$ such that $\sigma _{g_i}(\bar {B})=s_i(\bar {B})$ . We define $\psi _i(\bar {z},\bar {y},x)$ as $\psi _0(\bar {z},s_i(\bar {y}),x)$ . Note that $\psi _i(\bar {c},\bar {B},\operatorname {\mathrm {\mathfrak {C}}})=\{\sigma ^{\prime }_{g_i}(a)\}$ and

$$ \begin{align*}(K',(\sigma_g')_{g\in G})\models\end{align*} $$
$$ \begin{align*}(\forall\bar{z},x,x')\,\bigg(\bigwedge\limits_{g\in G}\sigma_g(\bar{z})=\bar{z}\,\wedge\,\psi_0(\bar{z}, \bar{B},x)\,\wedge\,\psi_i(\bar{z},\bar{B},x')\,\rightarrow\,\sigma_{g_i}(x)=x'\bigg).\end{align*} $$

To see the last line, let $\bar {d}\subseteq (K')^G$ , $m,m'\in K'$ be such that

$$ \begin{align*}\models \psi_0(\bar{d},\bar{B},m)\,\wedge\,\psi_i(\bar{d},\bar{B},m').\end{align*} $$

We do know that $\psi _0(\bar {d},\bar {B},\operatorname {\mathrm {\mathfrak {C}}})=\{m\}$ , which after applying an extension $\tilde {\sigma }_{g_i}\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}})$ of $\sigma ^{\prime }_{g_i}$ changes it into $\psi _0(\bar {d},s_i(\bar {B}),\operatorname {\mathrm {\mathfrak {C}}})=\{\sigma ^{\prime }_{g_i}(m)\}$ . We have that

$$ \begin{align*}m'\in\psi_i(\bar{d},\bar{B},\operatorname{\mathrm{\mathfrak{C}}})=\{\sigma^{\prime}_{g_i}(m)\}.\end{align*} $$

Since the whole formula is universal and has only parameters from K, it follows that

$$ \begin{align*}(K,(\sigma_g)_{g\in G})\models (\forall\bar{z},x,x')\,\bigg(\bigwedge\limits_{g\in G}\sigma_g(\bar{z})=\bar{z}\,\wedge\,\psi_0(\bar{z},\bar{B},x)\, \wedge\,\psi_i(\bar{z},\bar{B},x')\,\rightarrow\,\sigma_{g_i}(x)=x'\bigg), \end{align*} $$

where $i<l$ .

Consider $p(\bar {z}):=\operatorname {\mathrm {tp}}(\bar {c}/K^G)$ . Because $K^G\subseteq (K')^G$ is regular (thus also primary) and $\bar {c}\subseteq (K')^G$ , Fact 2.12 implies that $p(\bar {z})$ is stationary. As $K^G$ is PAC, the type $p(\bar {z})$ is finitely satisfiable in $K^G$ . The tuple $\bar {B}\subseteq K$ is algebraic over $K^G$ , hence there exists a quantifier-free $\theta (\bar {y})\in L(K^G)$ such that $\theta (\bar {y})\vdash \operatorname {\mathrm {tp}}(\bar {B}/K^G)$ . The following formula:

$$ \begin{align*} (\exists\,\bar{y},\,x_0,\ldots,x_{l-1})\,\bigg( \bigwedge\limits_{i<l}\psi_i(\bar{z},\bar{y},x_i)\, \wedge\,\varphi(x_0,\ldots,x_{l-1},\bar{y})\, \wedge\,\theta(\bar{y})\bigg) \end{align*} $$

belongs to $p(\bar {z})$ , thus there exists $\bar {d}\subseteq K^G$ such that

$$ \begin{align*} \models(\exists\,\bar{y},\,x_0,\ldots,x_{l-1})\,\bigg( \bigwedge\limits_{l<e}\psi_i(\bar{d},\bar{y},x_i)\, \wedge\,\varphi(x_0,\ldots,x_{l-1},\bar{y})\, \wedge\,\theta(\bar{y})\bigg). \end{align*} $$

It means that there are $\bar {B}'\subseteq \operatorname {\mathrm {\mathfrak {C}}}$ and $a^{\prime }_0,\ldots ,a^{\prime }_{l-1}\in \operatorname {\mathrm {\mathfrak {C}}}$ such that

$$ \begin{align*}\models \bigwedge\limits_{i<l}\psi_i(\bar{d},\bar{B}',a^{\prime}_i)\, \wedge\,\varphi(a^{\prime}_0,\ldots,a^{\prime}_{l-1},\bar{B}')\, \wedge\,\theta(\bar{B}').\end{align*} $$

Since $\models \theta (\bar {B})\,\wedge \,\theta (\bar {B}')$ , there exists $f\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}}/K^G)$ such that $f(\bar {B}')=\bar {B}$ . By applying f, we obtain

$$ \begin{align*}\models \bigwedge\limits_{i<l}\psi_i(\bar{d},\bar{B},f(a^{\prime}_i))\, \wedge\,\varphi(f(a^{\prime}_0),\ldots,f(a^{\prime}_{l-1}),\bar{B}).\end{align*} $$

We have that $\psi _0(\bar {d},\bar {B},\operatorname {\mathrm {\mathfrak {C}}})=\{f(a^{\prime }_0)\}$ . Since, for each $i<l$ , the subset $\psi _i(\bar {d},\bar {B},\operatorname {\mathrm {\mathfrak {C}}})=\psi _0(\bar {d},s_i(\bar {B}),\operatorname {\mathrm {\mathfrak {C}}})$ is an automorphic image of $\psi _0(\bar {d},\bar {B},\operatorname {\mathrm {\mathfrak {C}}})$ , it must be that $|\psi _i(\bar {d},\bar {B},\operatorname {\mathrm {\mathfrak {C}}})|=1$ and so $f(a^{\prime }_i)\in \operatorname {\mathrm {dcl}}(K^G,\bar {B})=K$ for each $i<l$ . Moreover, we have that $\sigma _{g_i}(f(a^{\prime }_0))=f(a^{\prime }_i)$ for each $i<l$ . Therefore $\models \varphi (f(a^{\prime }_0),\ldots ,f(a^{\prime }_{l-1}),\bar {B})$ leads to

$$ \begin{align*}(K,(\sigma_g)_{g\in G})\models \varphi\big(\sigma_{g_0}(f(a^{\prime}_0)),\ldots,\sigma_{g_{l-1}}(f(a^{\prime}_{0})),\bar{B}\big),\end{align*} $$

as expected.

Corollary 3.14. (QE, FS, ST $+$ ) Let G be finite and let $(K,\bar {\sigma })$ be a substructure with G-action. Then $(K,\bar {\sigma })$ is existentially closed if and only if:

  1. (1) $\operatorname {\mathrm {dcl}}(K)=K$ ,

  2. (2) the group action of G on K is faithful, and

  3. (3) $K^G$ is K-strongly PAC.

Proof If the conditions (1)–(3) hold then $(K,\bar {\sigma })$ is existentially closed by Theorem 3.13.

If $(K,\bar {\sigma })$ is existentially closed, then Remark 3.3 gives us that $\operatorname {\mathrm {dcl}}(K)=K$ , Lemma 3.6 gives that the group action is faithful, and the fact that $K^G$ is K-strongly PAC follows from Lemma 3.7.

In a very similar way, we conclude the following.

Corollary 3.15. (QE, FS, ST $+$ ) Let G be finite and let $(K,\bar {\sigma })$ be a substructure with G-action. Then $(K,\bar {\sigma })$ is existentially closed if and only if:

  1. (1) $\operatorname {\mathrm {dcl}}(K)=K$ ,

  2. (2) the group action of G on K is faithful, and

  3. (3) $K^G$ is PAC and algebraically K-strongly PAC.

3.3. Existence of model companion

Remark 3.16. Assume that $A\subseteq C$ is a Galois extension (e.g., if QE, FS, G is finite and G acts faithfully on $K=\operatorname {\mathrm {dcl}}(K)$ , we can take $A=K^G$ and $C=K$ ). Let $p(x)\in S(A)$ . The following are equivalent:

  1. (1) There exists unique extension of p over C.

  2. (2) There exists a $G(C/A)$ -invariant extension of p over C.

Proof If there is only one extension of $p(x)$ over C it is automatically $G(C/A)$ -invariant. Assume that $p(x)$ has a $G(C/A)$ -invariant extension over C, say $p_1(x)\in S(C)$ and led $p_2(x)\in S(C)$ be also an extension of $p(x)$ . As $p_1|_{A}=p_2|_{A}$ , there exists $f\in \operatorname {\mathrm {Aut}}(\operatorname {\mathrm {\mathfrak {C}}}/A)$ such that $f(p_1)=p_2$ . Since $A\subseteq C$ is Galois, we know that $f|_C=\sigma $ for some $\sigma \in G(C/A)$ . Then, $p_2=f(p_1)=f|_C(p_1)=\sigma (p_1)=p_1$ .

Definition 3.17. Let $\pi (x)$ be a partial type over A. We say that $\pi $ is A-irreducible if there exists $p(x)\in S(A)$ such that $\pi \vdash p$ .

Remark 3.18. Let $\varphi (x)\in L(A)$ be a consistent formula. Then, $\varphi (x)$ is A-irreducible if and only if $\{\varphi (x)\}$ is A-irreducible, which is equivalent to saying that there are no formulae $\varphi _1(x),\varphi _2(x)\in L(A)$ such that $\varphi (\operatorname {\mathrm {\mathfrak {C}}})\,\cap \,\varphi _1(\operatorname {\mathrm {\mathfrak {C}}})\neq \emptyset $ , $\varphi (\operatorname {\mathrm {\mathfrak {C}}})\,\cap \,\varphi _2(\operatorname {\mathrm {\mathfrak {C}}})\neq \emptyset $ and $\varphi (\operatorname {\mathrm {\mathfrak {C}}})\,\cap \,\varphi _1(\operatorname {\mathrm {\mathfrak {C}}})\,\cap \,\varphi _2(\operatorname {\mathrm {\mathfrak {C}}})=\emptyset $ . We use this characterization in the crucial Remark 3.20.

Lemma 3.19. (QE, FS) Assume that G is finite and G acts faithfully on $K=\operatorname {\mathrm {dcl}}(K)$ . The following are equivalent:

  1. (1) $K^G$ is algebraically K-strongly PAC.

  2. (2) Each algebraic type $p(x)\in S(K^G)$ , which has a G-invariant extension over K, is satisfiable in $K^G$ .

  3. (3) Each G-invariant algebraic type $\tilde {p}(x)\in S(K)$ is satisfiable in K.

  4. (4) For each $\theta (x)\in L(K^G)$ , if $0<|\theta (\operatorname {\mathrm {\mathfrak {C}}})|<\omega $ and $\theta (\operatorname {\mathrm {\mathfrak {C}}})$ is K-irreducible then $\theta (K^G)\neq \emptyset $ .

Proof The proof is easy, so we only sketch it. By Remark 3.16, we immediately obtain (1) $\iff $ (2). For the (2) $\iff $ (3), it is enough to observe that a G-invariant algebraic type is isolated by a $L(K^G)$ -formula, so its restriction to $K^G$ is also algebraic. We argue similarly on (3) $\iff $ (4): a G-invariant algebraic type over K is isolated by an $L(K^G)$ -formula, which is consistent, algebraic (i.e., has finitely many realizations) and K-irreducible.

Remark 3.20. (QE, FS $+$ ) In this remark, we investigate in what way being a K-irreducible formula may be expressed as a first-order statement. Assume that $K=\operatorname {\mathrm {dcl}}(K)$ and consider a quantifier-free formula $\varphi (y,x)\in L$ and a tuple $a\in K^y$ . Moreover assume that $0<|\varphi (a,\operatorname {\mathrm {\mathfrak {C}}})|=n<\omega $ . Recall that we can use the technical condition (FS $+$ ), listed at the very beginning of the paper. Let $S_x$ be the sort related to the variable x and let $E(\bar {x},\bar {x}')$ be a $\emptyset $ -definable equivalence relation given by a formula expressing that “ $\{x_1,\ldots ,x_{n-1}\}=\{x^{\prime }_1,\ldots ,x^{\prime }_{n-1}\}$ ”, where $\bar {x}=(x_1,\ldots ,x_{n-1})$ and $\bar {x}'=(x^{\prime }_1,\ldots ,x^{\prime }_{n-1})$ are tuples of variables from $S_x$ . As we assume, E is the fibration of a $0$ -definable function $f:(S_x)^{n-1}\to S_w$ . Note that the elements of the image of f correspond to nonempty subsets of $S_x(\operatorname {\mathrm {\mathfrak {C}}})$ of the size at most $n-1$ .

One more thing before coming to the point. Assume that the element d belongs to the sort $S_w(\operatorname {\mathrm {\mathfrak {C}}})$ , then the formula

$$ \begin{align*} (\exists\, x_1,\ldots,x_{n-1}\in S_x)\,\bigg( f(x_1,\ldots,x_{n-1})=d\,\wedge\,\bigwedge\limits_{i=1}^{n-1}\varphi(a,x_i)\bigg) \end{align*} $$

is modulo T equivalent to a quantifier-free formula, say $\xi _{\varphi ,n}(a,d)$ .

Now, $\varphi (a,x)$ is K-irreducible if and only if there is no proper subset $\emptyset \neq X\subsetneq \varphi (a,\operatorname {\mathrm {\mathfrak {C}}})$ , such that X is K-definable. In other words, for each proper subset $\emptyset \neq X\subsetneq \varphi (a,\operatorname {\mathrm {\mathfrak {C}}})$ , we have that the code $\ulcorner X\urcorner $ does not belong to $\operatorname {\mathrm {dcl}}(K)=K$ . We can express this last sentence, in the structure K, as follows:

$$ \begin{align*}K\models\neg(\exists\,w\in S_w)\,(\xi_{\varphi,n}(a,w)).\end{align*} $$

We will use the above in the proof of Theorem 3.23.

Now, we want to show that being a definably closed (in L) subset of $\operatorname {\mathrm {\mathfrak {C}}}$ is a first-order statement.

Remark 3.21. (QE) Consider any $\theta (y,x)\in L$ . The formula

$$ \begin{align*}\theta(y,x)\,\wedge\,(\forall\, x_1,x_2)\,\big(\theta(y,x_1)\,\wedge\,\theta(y,x_2)\,\rightarrow\,x_1=x_2\big)\end{align*} $$

is equivalent modulo T to some quantifier-free $\psi _{\theta }(y,x)\in L$ . Moreover, also the formula $(\exists \,x)\,(\psi _{\theta }(y,x))$ is equivalent modulo T to a quantifier-free formula $\psi ^0_{\theta }(y)\in L$ . Consider

$$ \begin{align*}\Sigma:=\{\psi^0_{\theta}(y)\,\rightarrow\,(\exists\,x)\,(\psi_{\theta}(y,x)\big)\;|\;\theta(y,x)\in L\}.\end{align*} $$

Lemma 3.22 (QE, in the notation of Remark 3.21)

For a substructure $B\subseteq \operatorname {\mathrm {\mathfrak {C}}}$ , $B\models \Sigma $ if and only if $\operatorname {\mathrm {dcl}}(B)=B$ .

Proof Assume that $B\models \Sigma $ and let $b\in \operatorname {\mathrm {dcl}}(B)$ . There exists a formula $\theta (y,x)\in L$ and $a\in B$ such that $\theta (a,\operatorname {\mathrm {\mathfrak {C}}})=\{b\}$ . Then $\models \psi _{\theta }(a,b)$ and $\models \psi ^0_{\theta }(a)$ . As $\psi ^0_{\theta }$ is quantifier-free, also $B\models \psi ^0_{\theta }(a)$ , thus $B\models (\exists \,x)\,(\psi _{\theta }(a,x))$ . It means that there exists $b'\in B$ such that $\models \psi _{\theta }(a,b')$ . We see that $b'=b$ and so $b\in B$ .

Now, let $B=\operatorname {\mathrm {dcl}}(B)$ . Assume that $B\models \psi ^0_{\theta }(a)$ for some $a\in B$ and $\theta (y,x)\in L$ . We have $\models \psi ^0_{\theta }(a)$ , so there exists some $b\in \operatorname {\mathrm {\mathfrak {C}}}$ such that $\models \psi _{\theta }(a,b)$ . This implies that $b\in \operatorname {\mathrm {dcl}}(a)\subseteq \operatorname {\mathrm {dcl}}(B)=B$ . Therefore there exists $b\in B$ such that $B\models \psi _{\theta }(a,b)$ .

The following theorem is an answer towards Question 2.9 (also Question 5.1) and Conjecture 5.2 from [Reference Hoffmann21].

Theorem 3.23. (QE, FS $+$ , ST $+$ ) Let G be finite. The model companion of the theory of substructures with G-action exists provided PAC is a first-order property.

Proof By Corollary 3.15 and Lemma 3.19, we need to write down as first-order statements the following conditions:

  1. (0) $(K,(\sigma _g)_{g\in G})$ is a substructure with G-action,

  2. (1) $\operatorname {\mathrm {dcl}}(K)=K$ ,

  3. (2) the group action of G on K is faithful,

  4. (3) $K^G$ is PAC,

  5. (4) for each $\theta (x)\in L(K^G)$ , if $0<|\theta (\operatorname {\mathrm {\mathfrak {C}}})|<\omega $ and $\theta (\operatorname {\mathrm {\mathfrak {C}}})$ is K-irreducible then $\theta (K^G)\neq \emptyset $ .

We are working in the language $L_G$ . The condition (0) is naturally a first-order statement, similarly the condition (2). Lemma 3.22 shows that also the condition (1) is a first-order statement. By the assumptions, condition (3) is a first-order statement. To finish the proof of the theorem, we need to show that the condition (4) is also a first-order statement.

There is no harm in assuming that the formula $\theta (x)$ is $\varphi (a,x)$ for some tuple a from $K^G$ and some quantifier-free formula $\varphi (y,x)\in L$ . The condition (4) will be expressed as an axiom scheme running over all quantifier-free formulae $\varphi (y,x)\in L$ and all $0<n<\omega $ .

Fix a quantifier-free formula $\varphi (y,x)\in L$ and a natural number $n>0$ . There exists a quantifier-free L-formula $\psi _{\varphi }(y)$ equivalent modulo T to the formula $(\exists ^{=n}\,x)\,(\varphi (y,x))$ . We are in situation of Remark 3.20, so we can involve the formula $\xi _{\varphi ,n}(y,w)$ . Our axiom scheme may be written as

$$ \begin{align*}(\forall\,y)\,\bigg(\bigwedge\limits_{g\in G}\sigma_g(y)=y\,\wedge\,\psi_{\varphi}(y)\,\wedge\, \neg(\exists\,w\in (S_x)^{n-1}/E)\,(\xi_{\varphi,n}(y,w))\end{align*} $$
$$ \begin{align*}\rightarrow\;(\exists\,x)\bigg(\bigwedge\limits_{g\in G}\sigma_g(x)=x\,\wedge\,\varphi(y,x)\bigg)\bigg).\\[-43pt]\end{align*} $$

Question 3.24. (QE, FS $+$ , ST $+$ ) Can we obtain a converse of Theorem 3.23? More precisely, does the following equivalence hold: the model companion of the theory of substructures with G-action exists for every finite group G if and only if PAC is a first-order property?

Remark 3.25. After writing the proofs of Theorems 3.13 and 3.23, we have noticed (but we have not checked all the details) that this result holds in a much greater generality, that is, if in the definition of PAC we replace “stationary” with “acl-stationary” (a unique extension over algebraic closure of the parameters), then the assumptions of stability, coding finite sets, and eliminating strong types may be skipped in Theorem 3.23. However, in this case it is unclear how useful such a result would be in terms of axiomatizing existentially closed finite group actions in this case, since there is no guarantee that faithful actions of a finite groups exist at all in general (consider, for example, the theory of linear orders) and the faithfulness in the stable was guaranteed by Lemma 3.6.

4. PAC structures in particular theories

In this section, we discuss the PAC property in some specific cases as well as some general methods for understanding PAC structures with respect to a given theory. As we are going to consider the notions of a regular extension and of a PAC structure in different theories, we plan to write “T-regular” and “T-PAC” instead of “regular in T” and “PAC in T”, respectively.

We will often refer to several particular stable theories as: the theory of compact complex manifolds CCM (for background, the reader is referred to [Reference Moosa, Baaz, Friedman and Krajíček35]) and the theories of differentially closed fields of characteristic $0$ denoted $\mathrm {DCF}_0$ (see, e.g., [Reference Marker, Marker, Messmer and Pillay29]) and its positive characteristic version $\mathrm {DCF}_p$ (see, e.g., [Reference Wood49] and [Reference Wood50]), and the theories of separably closed fields of positive characteristic $\mathrm {SCF}_{p,e}$ and $\mathrm {SCF}_{p,\infty }$ (see, e.g., [Reference Messmer, Marker, Messmer and Pillay33]).

4.1. General methods

In this subsection, we focus on two general contexts in which the PAC property is well understood. However, in both these cases showing that PAC is a first-order property requires some extra work.

4.1.1. Totally transcendental theories

In this part, we assume that the theory T is $\omega $ -stable. As before, let us fix for convenience a monster model $\mathfrak {C}$ of T and an arbitrary small substructure $K\subset \mathfrak {C}$ . It is well-known that stationary types in $\omega $ -stable theories are determined by the formulas of Morley degree one belonging to them. In particular, we have the following result, which actually coincides with Hrushovski’s definition of the PAC property in the strongly minimal case (see [Reference Hrushovski26, Definition 1.2] and [Reference Hoffmann21, Proposition 3.10]).

Proposition 4.1. If T is a $\omega $ -stable theory, then K is T-PAC if and only if for any formula $\varphi \in L(K)$ of multiplicity (Morley degree) one, we have that $\varphi (K)\neq \emptyset $ .

We recall that “DMP” stands for “Definable Multiplicity Property” and it says that for any formula $\phi (x;a)\in L(K)$ , there is a formula $\theta (y)\in \operatorname {\mathrm {tp}}(a)$ such that whenever $\mathfrak {C}\models \theta (a')$ then we have:

$$ \begin{align*}\mathrm{RM}\left(\phi(x;a')\right)=\mathrm{RM}\left(\phi(x;a)\right),\ \ \ \ \deg_M\left(\phi(x;a')\right)=\deg_M\left(\phi(x;a)\right)\end{align*} $$

(see, e.g., [Reference Kikyo and Pillay27, Definition 1.1]). Some $\omega $ -stable theories have DMP and some do not (see Remark 4.3). We get the following obvious conclusion, which was also stated in [Reference Afshordel1] under the assumption of finiteness of the Morley rank.

Proposition 4.2. If T is $\omega $ -stable with quantifier elimination and has DMP, then being T-PAC is first-order.

Proof Since T has DMP, for each $\phi (x;y)\in L$ , there is $\theta _{\phi }(y)$ such that for all $c\in \mathfrak {C}^{|y|}$ , we have:

$$ \begin{align*}\mathfrak{C}\models \theta_{\phi}(c)\ \ \ \ \ \text{if and only if}\ \ \ \ \ \deg_M\left(\phi(x;c)\right)=1.\end{align*} $$

Therefore, it is easy to write down a first-order axiom scheme expressing the T-PAC property.

Remark 4.3. We comment here on several particular $\omega $ -stable theories.

  1. (1) Proposition 4.2 applies to the case of $T=\mathrm {ACF}_p$ , that is to the classical notion of PAC.

  2. (2) It is known that Morley degree is not definable in the theory DCF $_0$ (see [Reference Freitag16, Question 1.2] and [Reference Freitag, Leon Sanchez and Li17]). However, DCF $_0$ -PAC is still first-order as it was shown in [Reference Pillay and Polkowska41].

  3. (3) It is open whether the theory of compact complex manifolds has DMP, however another approach towards the PAC property works here, which will be discussed in the next part. Partial results towards DMP for the theory CCM were obtained in [Reference Radin45].

4.1.2. Noetherian theories

In this part, we assume that models of T are naturally equipped with an extra topological structure. This assumptions is modelled on the case of $T=\mathrm {ACF}_p$ and the Zariski topology. Such issues were thoroughly discussed in [Reference Zilber52]. We diverge here a bit from the set-up of [Reference Zilber52] to cover the case of the theory DCF $_0$ as well.

We start from a purely topological context. Assume that S is a Noetherian topological space and let $\mathbb {B}$ be the Boolean algebra of constructible sets in S. In this part, “irreducible” always refers to the topological irreducibility with respect to a given Noetherian topology. The following properties are folklore and they can be easily checked.

  • If V is a non-empty closed irreducible subset of S, then

    $$ \begin{align*}p_V:=\{C\in \mathbb{B}\ |\ \mathrm{int}_V(C\cap V)\neq \emptyset\}\end{align*} $$
    is an ultrafilter on $\mathbb {B}$ .
  • The map $V\mapsto p_V$ is a bijection between the set of closed irreducible subsets of S and the set of ultrafilters on $\mathbb {B}$ .

We specify now our model-theoretic context.

Definition 4.4. By a Noetherian theory, we mean a pair $(T,\sum )$ , where T is a complete L-theory and $\sum $ consists of L-formulas of the form $\varphi (x;y)$ , where the variables $x,y$ vary, such that for any $M\models T$ and any $A\subseteq M$ , we have the following:

  • A subset $V\subseteq M^{|x|}$ is said to be A-closed if and only if there is $a\subset A$ and $\varphi (x;y)\in \sum $ such that $V=\varphi (M;a)$ .

  • The family of A-closed sets constitutes the family of closed sets of a Noetherian topology, which we call the A-topology.

  • Constructible sets with respect to the A-topology coincide with A-definable subsets (in Cartesian powers of M).

Remark 4.5.

  1. (1) It should automatically follow (possibly after adding some light assumptions such as the equality being in $\sum $ ) that models of our Noetherian theories are topological structures in the sense of [Reference Bays, Gavrilovich and Hils4, Definition 5.1] and [Reference Zilber52, Section 2].

  2. (2) The referee has pointed out to us that a very similar notion of a Noetherian theory was recently introduced by Martin-Pizarro and Ziegler (see [Reference Martin-Pizarro and Ziegler30, Definition 2.18]).

Example 4.6. We discuss several examples and non-examples of the above situation.

  1. (1) The theory of algebraically closed fields (of a given characteristic) is Noetherian by considering the Zariski topology.

  2. (2) The theory of compact complex manifolds (CCM) is also Noetherian, where the (Zariski) Noetherian topology is given by closed analytic subsets (see [Reference Zilber52, Section 3.4.2]).

  3. (3) In the case of differential fields, we have the Kolchin topology.

  4. (4) The theory $\mathrm {SCF}_{p,e}$ with the $\lambda $ -topology is not an example, since the $\lambda $ -topology is not Noetherian (see [Reference Messmer, Marker, Messmer and Pillay33, Section 4.6]).

For a fixed $A\models T_{\forall }$ and $n>0$ , it is clear that the map $V\mapsto p_V$ is a bijection between the set of appropriate A-closed A-irreducible sets and the Stone space $S_n(A)$ of n-types over A. In particular, any Noetherian theory is $\omega $ -stable. We still need to have a connection between the topology and forking, which is given by the following.

Proposition 4.7. Assume that $A\subseteq M\models T$ and $p_V\in S_n(M)$ . Then, the type $p_V$ does not fork over A if and only if V is definable over $\operatorname {\mathrm {acl}}(A)$ .

Proof Since $p_V$ does not fork over A if and only if it does not fork over $\operatorname {\mathrm {acl}}(A)$ , we can and will assume that $A=\operatorname {\mathrm {acl}}(A)$ .

$(\Rightarrow )$ Let $V=V_b$ and assume that $V_b$ is not definable over A. Let us define

$$ \begin{align*}V_0:=\bigcap_{\mathrm{tp}(c/A)=\mathrm{tp}(b/A)}V_c.\end{align*} $$

Since $V_b$ is not definable over A, we get that $V_0\subsetneq V$ . By Noetherianity, V is definable and closed. Since $V_0$ is A-invariant, we get that $V_0$ is A-definable. In particular, the formula “ $x\in V\setminus V_0$ ” belongs to $p_V$ . Since the formula “ $x\in V\setminus V_0$ ” forks over A (see, e.g., the characterization of forking from [Reference Pillay40, Lemma 2.16(c)]), the type $p_V$ forks over A.

$(\Leftarrow )$ We assume that V is A-definable. It is enough to show that for any proper M-closed $W=W_b\subset V$ , we have that the formula “ $x\in V\setminus W$ ” does not fork over A. If this formula forks over A, then by (the logical) compactness, there is a finite set of A-conjugates $b=b_1,\ldots ,b_n$ such that

$$ \begin{align*}\left(V\setminus W_{b_1}\right)\cap \ldots \cap \left(V\setminus W_{b_n}\right)=\emptyset.\end{align*} $$

But then $V=W_{b_1}\cup \cdots \cup W_{b_n}$ and each $W_{b_i}$ is a proper M-closed subset of V, which contradicts the M-irreducibility of V.

We obtain the expected description of stationary types.

Corollary 4.8. Let $A\models T_{\forall }$ , and $p_W\in S_n(A)$ . Then, $p_W$ is stationary if and only if W is absolutely irreducible, that is for any $M\models T$ containing A as a substructure, W is irreducible in the M-topology.

Proof Let $W=W_1\cup \cdots \cup W_n$ be the decomposition of W into the M-irreducible M-closed components. By uniqueness, each $W_i$ is defined over $\operatorname {\mathrm {acl}}(A)$ . By Proposition 4.7, each type $p_{W_i}$ does not fork over A. Since for each i, we have $\mathrm {cl}_A(W_i)=W$ , we get that each $p_{W_i}$ extends $p_W$ . It is easy to see now that $W_i$ ’s correspond exactly to non-forking extensions of $p_W$ , which concludes the proof.

Similarly as in the case of Proposition 4.1, we get the following result.

Proposition 4.9. For any $K\models T_{\forall }$ , we have that K is T-PAC if and only if for any absolutely irreducible K-closed set V and any non-empty relatively K-open $U\subseteq V$ , we have that $U(K)\neq \emptyset $ .

Remark 4.10.

  1. (1) In the cases of $T=\mathrm {ACF}_p$ and $T=\mathrm {DCF}_{0,m}$ , we can just consider the condition “ $V(K)\neq \emptyset $ ” in Proposition 4.9, since these topologies have basis of open sets being definably isomorphic to affine closed sets. It looks like there is no similar simplification for the theory CCM, since (at least in the category of complex manifolds) being isomorphic to a compact complex manifold would imply being closed.

  2. (2) Proposition 4.9 together with item $(1)$ above directly generalizes the classical case of $T=\mathrm {ACF}_p$ .

  3. (3) For $T=\mathrm {DCF}_{0,m}$ the description from Proposition 4.9 (together with item $(1)$