23

PAC STRUCTURES IN NUTSHELLDANIEL MAX HOFFMANN †Instytut Matematyki, Uniwersytet Warszawski
1. INTRODUCTION

This is an overwiev of recent results about pseudo‐algebraically closed substruc‐tures (PAC substructures) of some stable ambient structure. Our aim is a compact
presentation of the main results from [16], [15], [11] and [18], so a reader interested
in studying PAC substructures might quickly get into the subject. The content of
the text covers all topics from our talk at the RIMS Model Theory Workshop 2018
and the arrangement of the text is based on the order of the mentioned talk.
We fix some stable theory T in a language \mathcal{L} , such that T has quantifier elimi‐
nation and elimination of imaginaries, then we choose a monster model of T , say\mathfrak{C} , and a group G.
We are interested in providing new examples of structures which may serve inthe studies on neo‐stability. So the general rule for our approach is to start with astable structure and then “break” the stability in a controlled manner. It turns outthat PAC substructures suit perfectly for our purposes, since they are controlled
by absolute Galois groups. In other words, we want to “break” stability and obtain
PAC substructures with understandable absolute Galois groups.
We thank organizers of the RIMS Model Theory Workshop 2018‐ especially, weare grateful to Hirotaka Kikyo for his support and kindness.
2. PSEUDO‐ALGEBRAICALLY CLOSED STRUCTURES

2.1. Definitions. In the theory of fields an extension of fields E\subseteq F is regular if

it is separable and E is relatively algebraically closed in F(E^{alg}\cap F=E) . Thereis no reasonable generalization of the notion of separability for abstract structures,
therefore, if we want to introduce a general notion of regularity, we need to restrict
our considerations to some subclass of substructures of \mathfrak{C} . If the field E is perfect(i.e. definably closed in some monster model of ACF), then E\subseteq F is separable.
Passing to definable closure seems to be a solution:

Definition 2.1. Let E\subseteq A be small subsets of \mathfrak{C} . We say that E\subseteq A is regular if

dc1(A)nacl(E)=dcl(E) .
The above notion of regularity coincides with the notion of stationarity given in[21, Definition 5.17] and in [20]:Fact 2.2 (Lemma 3.35 in [16]). For a small set E\subseteq \mathfrak{C} and a complete type p overE it follows:p\dag er

SDG.

The

is stationary \Leftrightarrow(\forall A\models p) ( E\subseteq EA is regular)\Leftrightarrow(\exists A\models p) ( E\subseteq EA is regular).

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supportedbythe2016/21/N/ST1/01465 , and 2015/19/B/ST1/01150.

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D. M. HOFFMANN

Note that regularity is invariant under the action of automorphisms, every ex‐tension of an algebraically closed subset is regular, and regularity is transitive (ifE\subseteq A and A\subseteq B are regular, then also E\subseteq B is regular). By Lemma 3.3 in[16], if A is an existentially closed substructure of B (where A and B are small
substructures of \mathfrak{C} ), then A\subseteq B is regular, so regular extensions are related to ex‐
istential closedness. A substructure is existentially closed if it is existentially closed

in all extensions (i.e. all small extensions living inside \mathfrak{C} ). Quite interesting notion
occurs, if we relax conditions in this definition by requiring existential closedness
only in regular extensions:Definition 2.3. Assume that M\preceq \mathfrak{C} . Let

N be a substructure of M .

We saythat N is pseudo‐algebraically closed (PAC) in M if for every small (in the senseof \mathfrak{C}) substructure N' of M , which is a regular extension of N , it follows that N isexistentially closed in N' . If M=\mathfrak{C} we say “PAC substructure” instead of “PAC
substructure in \mathfrak{C} ”

Note that PAC substructures are definably closed (so PAC substructures in thecase of fields coincide with perfect PAC fields). For some reasons it will be betterto call PAC substructures “regularly closed substructures” (e.g. it may happen
that algebraically closed substructure is not pseudo‐algebraically closed), but weintroduced this definition as an analogon of the well known (in field theory) notionof PAC fields. A field K is pseudo‐algebraically closed (is a PAC field) if each
non‐empty absolutely irreducible K ‐variety has a K ‐rational point. It might bedifficult to generalize such a definition of PAC fields, but luckily for us there isa theorem saying that PAC fields are exactly these fields which are existentiallyclosed in every regular extension (Proposition 11.3.5 in [14]). Hence “regularly
closed substructures”’ are called “PAC substructures” Introducing PAC substruc‐tures (called also PAC structures) was motivated by key role of PAC fields in the
research of field theory in the second half of the 20th century. In short, they were“discovered” in [1] and [2] as a generalization of pseudofinite fields, but the name
“PAC fields5’ was given in [13]. Reader interested in results about PAC fields may
consult: [6], [7]), [8], [9], [10], [12]. The most important property of PAC fields‐
so‐called “Elementarily Equivalence Theorem” (Theorem 20.3.3 in [14], Proposition
33 in [9]) ‐ implies that, to a large extent, logical properties of PAC fields are con‐trolled by their absolute Galois groups. A desired generalization ‐ “Elementarily
Equivalence Theorem for Structures”’ ‐ will be provided in an upcoming subsection(two PAC substructures have the same first order theory if they have isomorphic
Galois groups), but before this let us discuss basic properties of PAC structures.
In our approach, we define PAC substructures as a generalization of existentiallyclosed substructures. However, we are not first in studying PAC substructures.

Studying PAC substructures has begun (to our knowledge) in [19], where the author
attempted to develop “geometric simplicity theory” by developing an analogon ofmodel theory of bounded PAC fields (which are simple) in the context of strongly
minimal ambient model (in our case the ambient model is stable). He defined PAC
substructures in the following way:

Definition 2.4 (Definition 1.2 in [19]). Let T' be strongly minimal and has quan‐tifier elimination and elimination of imaginaries. Let M\models T' satisfy the definablemultiplicity property (consult “Framework” at page 8. in [19]). A subset P of M

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PAC STRUCTURES IN NUTSHELL

is a PAC_{Hru} subset of M , if every multiplicity 1 formula with parameters fromP

has a solution in P.

By Proposition 3.10 in [16], both notions of a PAC substructure (Definitions 2.3
and 2.4) coincide in the strongly minimal context. Many important facts about
PAC structures are provided in [31] and in [32]. Authors of [31] and [32] work witha stronger notion of a PAC substructure (in the stable context):
Definition 2.5 (Definition 3.1 in [31]). Let T' be stable, \kappa\geq|T'|^{+} a cardinal,
M\models T' , and let P be an \mathcal{L} ‐substructure of M . We say that P is a \kappa-PAC_{PP}substructure of M if whenever A\subseteq P has cardinality smaller than \kappa and p(x) is
complete stationary type over A (in the sense of M ), then p has a realization in P.
This definition of a PAC substructure assumes some saturation, which ‐ as we

will see soon ‐ is crucial if we want to avoid serious obstructions (e.g. in the
Elementary Equivalence Theorem for Structures). For the reader convenience wevisualize relations between all three definitions. In the set‐up of a strongly minimal
theory:PAC_{Hru}+\kappa-saturated\Rightarrow\kappa-PAC_{PP}\Rightarrow

PAC_{Hru}=PAC.

In the set‐up of a stable theory:
PAC+\kappa-saturated\Rightarrow\kappa-PAC_{PP}\Rightarrow

PAC.

It was mentioned that working with somehow saturated PAC structures might bemore convenient. Nevertheless, passing to an elementary extension which is PAC
substructure is not “free of charge”’ :Lemma 2.6 (Lemma 2.12 in [11]). Suppose PAC is a first order property and puresaturation over P is a first order property. Let P be PAC in \mathfrak{C} . Let \kappa be an infinite
cardinal. Then, there is a \kappa ‐saturated elementary extension P^{*} of P , which is PAC
in \mathfrak{C}.

Where the missing notions are defined as follows:
Definition 2.7. Assume that M\preceq \mathfrak{C} and P is a substructure of M.

(1) We say that M is purely saturated over P if every type over P is realized
in M.

(2) We say that PAC is a first order property if there exists a set \Sigma of \mathcal{L}\cup\{P\}‐formulas such that

(M, P)\models T\cup\Sigma\Leftrightarrow

M\models T and P is PAC in M.
P is a first order property if there exists(3) We say that pure saturation overa set \Sigma of \mathcal{L}\cup\{P\} ‐formulas such that(i) if M\models T is purely saturated over P , then (M, P)\models T\cup\Sigma,(ii) if (M, P)\models T\cup\Sigma and (M, P) is |T|^{+} ‐saturated, then M is purely
saturated over P.

If T has no finite cover property (an assumption stronger than stability, see Defi‐
nition 4.1 and Theorem 4.2 in Chapter II of [34]), then pure saturation over P is a
first order property (see Remark 2.10 in [11] or Remark 3.6 in [31]). Therefore ourimpression is that it might be very hard to pass to a saturated elementary extension

which remains PAC (which is not the case for existentially closed substructures)and so results for non‐saturated PAC substructures are desired, even if they require

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D. M. HOFFMANN

more complicated techniques. Some of results require a weaker variant of saturation
which we introduce right now.
For a small substructure P of \mathfrak{C} , let us define

ST(P, \kappa, \lambda) := {qftp (\overline{d}/A)

A\subseteq P, |A|<\kappa,

P\subseteq dc1(P,\overline{d}) is regular and |\overline{d}|<\lambda }.
If
is PAC, then each element of ST(F, \kappa, \lambda) is a partial type in the senseof Th (P) . We say that P is \kap \lambda ‐regularly saturated if every element of ST(P, \kappa, \lambda)
is realized in P . We use \kappa ‐regularly saturated” for \kap \kap ‐regularly saturated” (seeDefinition 3.2 in [11]). If P is PAC and \kappa ‐saturated (in the sense of Th (P)), thenP

P is \omega\kap a ‐saturated.

On the other hand passing to elementary substructure preserves being PAC (byFact 2.4 and Fact 2.6 in [11]).
2.2. A bite of Galois theory. First of all, let us clarify what do we understandby Galois extension.

Definition 2.8.
(1) Assume that A\subseteq C are substructures of \mathfrak{C} . We say that
C is normal over A (or we say that A\subseteq C is a normal extension) if
Aut(\mathfrak{C}/A)\cdot C\subseteq C. (Note that if C is small and A\subseteq C is normal, then it
must be C\subseteq ac1(A). )(2) Assume that A\subseteq C\subseteq ac1(A) are small substructures of \mathfrak{C} such that A=
dc1(A), C=dc1(C) and C is normal over A . In this situation we say that
A\subseteq C is a Galois extension.

If A\subseteq C is a Galois extension, then Aut(C/A) is a profinite group. Many facts
from classical Galois theory have their counterparts in the model‐theoretic variantof Galois theory. Some of these generalization might be found in [29], in Section 3.
in [16] or in [15]. To convince the reader that a lot of the classical Galois theory
generalizes quite straightforward, let us provide some highlights.

Fact 2.9 (The Galois correspondence). Let A\subseteq C be a Galois extension, introduce\mathcal{B} :=\{B|A\subseteq B=dc1(B)\subseteq C\},
\mathcal{H}:=\{H|H\leq Aut(C/A) is closed \}.
Then \alpha(B) :=Aut(C/B) is a mapping between \mathcal{B} and \mathcal{H}, \beta(H) :=C^{H} is a mapping
between \mathcal{H} and \mathcal{B} and it follows\alpha 0\beta=id, \beta\circ\alpha=id.

Lemma 2.10 (Lemma 2.10 in [15]). Assume that N is a small definably closed sub‐structure of \mathfrak{C} equipped with a G ‐action (\tau_{g})_{g\in G}(|G| is smaller than the saturationof \mathfrak{C}) . Let i:Garrow Aut(N/N^{G}) be given by i(g) :=\tau_{g}.
(1) If for every b\in N the orbit G\cdot b is definable, then N^{G}\subseteq N is normal.
(2) If N\subseteq ac1(N^{G}) , then N^{G}\subseteq N is a Galois extension.
(3) If G is finite, then N\subseteq ac1(N^{G}) , hence also the second point follows.
(4) (Artin’s theorem) If G is finite and the G ‐action (\tau_{g})_{g\in G} is faithful, theni:G\cong Aut(N/N^{G})(5) Assume that.

G is profinite,

the G ‐action (\tau_{g})_{g\in G} is faithful and for everym\in N the stabiliser Stab (m)=\{g\in G|\tau_{g}(m)=m\}

is an open subgroupof G. Then N^{G}\subseteq N is Galois and i : G\cong Aut(N/N^{G}) (as profinitegroups).

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PAC STRUCTURES IN NUTSHELL

Absolute Galois groups and Galois groups of extensions play a major role in our
description of PAC substructures of \mathfrak{C} :G(A) :=Aut(acl(A)/dcl(A))
G(B/A) := Aut (dcl(B)/dcl(A))(the absolute Galois group of A )(the Galois group of extension A\subseteq B )For example, A\subseteq B is regular extension if and only if the restriction map G(B)arrow
G(A) is onto. Moreover, one may find interesting that all profinite groups occur in
our stable theory T as Galois groups of some Galois extensions:

Fact 2.11 (Corollary 3.3 in [15]). A group G is profinite if and only if there exista Galois extension A\subseteq B of small substructures of \mathfrak{C} such that G\cong Aut(B/A) .
An important property of the absolute Galois group of a PAC substructure is

projectivity, which was known in the strongly minimal context (Lemma 1.17 in [19])and then generalized to the stable context (Theorem 4.4 in [15]):
Theorem 2.12 (Theorem 4.4 in [15]). If a small N is PAC, then G(N) is projec‐tive.

A natural question is whether any projective profinite group occurs as the abso‐lute Galois group of some PAC substructure. The following result puts some lighton this problem.
Theorem 2.13. Assume that A\subseteq B is a Galois extension of small substructures of\mathfrak{C} and assume that there is an epimorhism of profinite groups \alpha : Garrow Aut(B/A) ,and G is projective. there exists a definably closed substructure P\supseteq A of \mathfrak{C} and an
isomorphism of profinite groups \gamma : Garrow G(P) such that\gammaG(P)\downarow|_{N}

Aut (B/A)

is commuting. Moreover, if any type over A has only finitely many extensions overac1(A) , then P is PAC.

For example, if T is \omega ‐stable, then a profinite group G is projective if and only
if G is isomorphic to the absolute Galois group of some PAC substructure of \mathfrak{C}(see Corollary 4.10 in [15]). How about the same for any stable T ? We do notknow it due to the use of a generalization of the Ax‐Roquette theorem in the proofof Theorem 2.13. There are problems with generalizing the Ax‐Roquette theorem(algebraic extension of a PAC field is a PAC field), more precisely: there is an
example of a stable theory

T

where algebraic closure of a PAC substructure is

not a PAC substructure (see Section 5. in [31]). The best (to our knowledge)
generalization of the Ax‐Roquette theorem was provided in Proposition 3.9 in [31]:Proposition 2.14. Suppose P is \kappa-PAC_{PP} substructure of \mathfrak{C}.
i) Any finite algebraic extension of P is also a \kappa-PAC_{PP} substructure of \mathfrak{C}.

ii) Suppose that in T every (finitary) complete type over any set has finite
multiplicity. Then any algebraic extension of P is \kappa-PAC_{PP}.

In Lemma 4.5 in [15], the above proposition was formulated for PAC substructures:
if P is a PAC substructure and any type over P has only finitely many (non‐forking)28
D. M. HOFFMANN

extensions over ac1(P) , then any algebraic extension of P is a PAC substructure.

Actually, formulation of Lemma 4.5 in [15], which is rather complicated, gives us
also that any finite Galois extension of a PAC substructure (finite” means that
the Galois group of this extension is finite) is a PAC substructure (to show this you
may use “Primitive Element Theorem Theorem 5.3 in [11]).
2.3. Elementarily Equivalence Theorem. The idea that the theory of a PAC
substructure might be controlled by its absolute Galois group comes from the theory

of PAC fields (see works of Fried, Jarden and Kiehne). However, some results for
PAC substructures were obtained even without generalizing the Embedding Lemma(Lemma 2.1 in [22]) or Elementarily Equivalence Theorem (Theorem 3.2 in [22]).
For example, in [32], it was shown that PAC substructures with small absolute
Galois group (i.e. bounded PAC substructures) are simple (Corollary 3.22 in [32]).
This result was obtained under assumption that “PAC is a first order property
which seems to be natural assumption in studying PAC substructures.
The Embedding Lemma was crucial in the proof of the Elementarily Equiva‐

lence Theorem for PAC fields and its generalized form (Lemma 3.5 in [11]) plays
important role in the proof of the Elementarily Equivalence Theorem for PAC sub‐structures (Proposition 3.8 in [11]). Moreover, the generalized Embedding Lemmamight be used in a description of types in PAC substructures and is used in the

proof of the upcoming Weak Indpendence Theorem ([18]). Since the generalized
Embedding Lemma is involved in several techniques, we provide it here in the mostgeneral formulation.

Lemma 2.15 (Embedding Lemma, Lemma 3.5 in [11]). Assume that
L\subseteq L', M\subseteq M', E\subseteq E', F\subseteq F' are small Galois extensions in \mathfrak{C},L\subseteq E, M\subseteq F, M'\subseteq F',\bullet\bullet\bullet\bullet\bullet\bullet\bulletL'\subseteq E' is regular,
F is \kappa ‐regularly saturated, where \kappa\geq(\max\{|E|, |T|\})^{+},F is PAC in \mathfrak{C},\Phi_{0}\in Aut(\mathfrak{C}) is such that \Phi_{0}(L)=M and \Phi_{0}(L')=M',
G(F'/F)arrow G(E'/E) is a continuous group homomorphism such that\varphi :G(F'/F)arrow^{\varphi}G(E'/E)

res1 \downarrow res

G(M'/M)arrow^{\varphi 0}G(L'/L)
where \varphi_{0}(\sigma) :=\Phi_{0}^{-1}0\sigma 0\Phi_{0} for each \sigma\in G(F'/F) , commutes.
Then there exists\Phi\in Aut(\mathfrak{C}) such that\bullet\Phi|_{L'}=\Phi_{0}|_{L'},\bullet\Phi(E)\subseteq F, \Phi(E')\subseteq F',\bullet\varphi(\sigma)=\Phi^{-1}0\sigma\circ\Phi for any \sigma\in G(F'/F)

Moreover, if \varphi is onto and E'=ac1(E) , then \Phi(E)\subseteq F is regular.
On the other hand, we provide here only a consequence, which has rather sim‐ple and elegant formulation, from the Elementarily Equivalence Theorem for PAC

substructures (Proposition 3.8 in [11]).

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PAC STRUCTURES IN NUTSHELL

Theorem 2.16 (Elementarily Equivalence Theorem, Corollary 3.9 in [11]). If E
and F are \kappa ‐regularly saturated PAC substructures of \mathfrak{C}(\kappa\geq|T|) , and for somedefinably closed L\subseteq F\cap E of size strictly smaller than \kappa there exists a continuous
isomorphism \varphi : G(F)arrow G(E) such that
(D

G(F

(E)

G(L)
is commuting, then E\equiv LF.

It is remarkable that there is no (to our knowledge) easy way to relax the assump‐tion about saturation in the above Theorem 2.16. We can assume that “PAC is a

first order property” and “pure saturation over P is a first order property” and thentry to pass to saturated elementary extensions of our given PAC substructures E
and F . The problem in this approach is that the isomorphism \varphi of profinite groups
G(E) and G(F) does not carry enough data about “model‐theoretic” structure of
E and F , and therefore there are serious problems in lifting up \varphi to the level ofsaturated elementary extensions of E and F . The issue shows up if we consider
many sorted structures:
Example 2.17. Consider two sorted structure (\mathbb{C}, \mathbb{C}) , where \mathb {C} is the field of com‐plex numbers \mathb {C} and there is no interaction between sorts. Note that (\mathbb{Q}, acl(\mathbb{Q}))\not\equiv(acl(\mathbb{Q}), \mathbb{Q}) , but

G(\mathbb{Q}, ac1(\mathbb{Q}))\cong G(\mathbb{Q})\cross G(acl(\mathbb{Q}))\cong G(acl(\mathbb{Q}))\cross G(\mathbb{Q})\cong G(ac1(\mathbb{Q}), \mathbb{Q}).

We may even pass to (\mathbb{C}, \mathbb{C})^{eq} to get elimination of imaginaries, and then do theMorleyisation, to obtain quantifier elimination, but absolute Galois groups willremain isomorphic despite substructures do not share the same first order theory.
Clearly, the above example is based on fact that the isomorphism between abso‐lute Galois groups is “mixing” sorts. To avoid such phenomena, the notion of sorted

isomorphism is introduced (in [11]) and developed into notion of sorted profinitegroups (in [18]), which form a category. In short: sorted structure on G(F) codeson which sorts live primitive elements of finite Galois extensions of F . Sorted profi‐
nite groups can be presented as first order structures in some abstract many sorted

language, so called (sorted) compelete systems, and, what is more surprising, inter‐
preted in (\mathfrak{C}, F) (structure \mathfrak{C} considered with a predicate for set F ). The reader
interested in more details may find them in [18] (which will be published on arxivesoon).
The Elemetarily Equivalence Theorem for non‐saturated PAC structures, The‐

orem 5.8 in [11], assumes “PAC is a first order property”’ and “pure saturation
over P is a first order property and requires that isomorphism between G(F) andG(E) preserves sorts. So the price for relaxing the assumption about saturation (inThereom 2.16) is high and to avoid paying it, in the following text, we will focus
mostly on saturated PAC substructures.
2.4. Weak Independence Theorem. This subsection is only to mention someideas and describe results without precise formulation of them. Elementarily Equiv‐

alence Theorem characterizes the first order theory of a PAC substructure/field.

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D. M. HOFFMANN

Usually model theory aspires to make one more step and try to find some geomet‐
rical behavior in models of a given theory. We would like to imitate methods fromgeometric stability theory in non‐stable theories which are lying not to far fromstability. To do this we need to consider a notion of independence, which inherits
some properties from well‐known geometrical notions of independence.

In [5], the author describes a very interesting concept of inducing a notion of
independence in a (saturated) PAC field K by combining forking independence
present in the separable closure of K and forking independence given in the complete

system corresponding to G(K) . The main ingredient is Theorem 2.1 from [5], whichsays that if there is “a solution of the independence theorem problem” on thelevel of complete system corresponding to G(K) , then there is “a solution of the
independence theorem problem” in the field K . Using this result, Nick Ramsey inhis doctoral dissertation (Corollary 7.2.7 in [33]) shows that for a PAC field K thefollowing holds: if the theory of the complete system related to G(K) is NSOP_{1},then K is NSOP_{1} . In Ramsey s result the notion of independence in K is combinedfrom forking independence in the separable closure of K and from Kim‐dividing on
the level of complete system corresponding to G(K) .
We were interested in generalizing the aforementioned results to the level of PAC
substructures of stable monster \mathfrak{C} and we plan to provide such generalizations in[18]. For example, we already generalized Theorem 2.1 from [5], which is the main
ingredient also on the level of PAC substructures. Now, generalizing Theorem 7.2.6

and Corollary 7.2.7 from [33] should be straightforward and we are working on it.
One of our goals is to provide new examples of theories with NSOP_{1} . One could
ask: could not we construct such examples in a more direct way, e.g. structures
given by more combinatorial approach? Yes, we could, but in the case of NSOP_{1}
PAC substructures, we already dispose a tool to study model‐theoretical propertiesof these structures‐ the general Galois theory. And as we will see in a moment,
there is a way to construct PAC substructures with a desired absolute Galois group.
3. SUBSTRUCTURES W1TH G‐ACT1ON

This part borrows results from [16] and [15]. The reader should treat [16] as apublished version of author’s doctoral dissertation. The initial idea behind [16] wasto obtain a way of “breaking”, in a controlled manner, stability of a given stable
theory T by considering additional group action by automorphisms. The desired
goals were achieved and under some assumptions the resulting theory remains sim‐ple but not stable. As side product of studies on existentially closed substructureswith a group action, it was observed that such substructures are PAC substructures

and the definition of PAC substructures was slightly reformulated (see Section 3.1
in [16]) so we obtained Definition 2.3. It was also observed that there is a link
between group G which acts on considered substructures and their absolute Galois
groups. Therefore we may adjust G to control PAC substructures given as reductsof existentially closed substructures with a group action of G.

3.1. Basics. The research in [16], which is a base for the following lines, was mo‐tivated by results of [8] and [17]. We fix a group G and set language \mathcal{L}^{G} to be thelanguage \mathcal{L} extended by unary function symbols \overline{\sigma}=(\sigma_{g})_{g\in G}.

Definition 3.1 (Definition 2.5 in [16]).

(1) We introduce set of \mathcal{L}^{G} ‐sentences

A_{G} , which contains exactly the following axioms:

31
31
PAC STRUCTURES IN NUTSHELLi) \sigma_{g} is an automorphism of \mathcal{L} ‐structure for every g\in G,ii) \sigma_{g}\circ\sigma_{h}=\sigma_{g*h} for every g, h\in G.
(2) Let (M, (\sigma_{g})_{g\in G}) be an \mathcal{L}^{G} ‐structure. We say that (\sigma_{g})_{g\in G} is a G ‐action
on M if (M, (\sigma_{g})_{g\in G})\models A_{G}.
(3) If T' is an \mathcal{L} ‐theory, then T_{G}' is an \mathcal{L}^{G} ‐theory equal to the set of conse‐
quences of T'\cup A_{G} , i.e. T_{G}'=Cn(T'\cup A_{G}) .
There is an interesting question whether a class of existentially closed models of

T_{G}' , for a given theory T' , is an elementary class, in other words whether modelcompanion of T_{G}' exists (for some reasons it is more reasonable to ask about theexistence of the model companion for (T_{\forall}')_{G} ). The question is even more interesting,since the answer for the case of G=\mathbb{Z} is related to nfcp. For research in this line,

the reader may consult [3], [23], [24], [25] and [26].
For our purposes, we will focus on the class of existentially closed models of (T_{\forall})_{G}(recall that T is stable and allows to eliminate quantifiers and imaginaries), so small
substructures of \mathfrak{C} which are equipped with a G ‐action and which are existentiallyclosed among the class of substructures equipped with a G ‐action. By Proposition

3.56 in [15], such substructures are PAC substructures in \mathfrak{C} . Moreover, if G isfinitely generated, then also the substructure of invariants is PAC (by Proposition
3.51 in [16]) and it is bounded (Corollary 3.50 in [16]). More precisely, if (M,\overline{\sigma})
is an existentially closed model of (T_{\forall})_{G} , then by invariants of the G ‐action weunderstand elements of the set M^{G} := { m\in M|\sigma_{g}(m)=m for all g\in G }.
By Corollary 3.22 in [32], the set M^{G} for finitely generated G , constitutes an \mathcal{L} ‐substructure which is, as bounded PAC substructure, simple (see Theorem 4.40 in[16]). Now, we proceed to the simplicity of a prospective model companion of thetheory (T_{\forall})_{G}.
3.2. Independence Theorem and simplicity. In this subsection we assume
that the model companion of the theory (T_{\forall})_{G} , say T_{G}^{mc} , exists. We will describe
assumptions needed to get simplicity of T_{G}^{mc} . First, let us recall that if a PAC

substructure is bounded (its absolute Galois group is small as a profinite group)
then its theory is simple (Corollary 3.22 in [32]). In the case of PAC fields, we havea stronger fact: a PAC field K is simple if and only if it is bounded (Fact 2.6.7 in[27]). If (K,\overline{\sigma}) is an existentially closed field with a G ‐action, then K is PAC and
if K is not bounded then (K,\overline{\sigma}) can not be simple (since it reduct is not simple).
Therefore, if we attempt to show simplicity of some monster model (\mathfrak{M},\overline{\sigma}) of T_{G}^{mc},
it is natural to assume that \mathfrak{M} is bounded in \mathfrak{C} (i.e. G(\mathfrak{M}) is a small profinitegroup, e.g. if G is finite then \mathfrak{M} is bounded, see Proposition 4.25 in [16]):

Theorem 3.2 (Theorem 4.21 and Proposition 4.26 in [16]). Assume that \mathfrak{M} isbounded in \mathfrak{C} . Let (M,\overline{\sigma})\preceq(\mathfrak{M},\overline{\sigma}) and let p_{1}(x_{1}), p_{2}(x_{2}), p_{3}(x_{3}), p_{12}(x_{1}, x_{2}) ,p_{23}(x_{2}, x_{3}) and p_{13}(x_{1}, x_{3}) be complete \mathcal{L}^{G} ‐types over M which satisfy p_{i}(x_{i}),p_{j}(x_{j})\subseteq
p_{ij}(x_{i}, x_{j}) and if a_{i}a_{j}\models p_{ij}(x_{i}, x_{j}) thena_{j}\downar ow a_{i}\circ.
Then, there exists a complete \mathcal{L}^{G} ‐type p_{123}(x_{1}, x_{2}, x_{3}) which extends each p_{ij}(x_{i}, x_{j})and such that if a_{1}a_{2}a_{3}\models p_{123}(x_{1}, x_{2}, x_{3}) thena_{3)}\downar ow a_{1}a_{2}M\circ.

32
D. M. HOFFMANN

Ternary relationL^{\circ} which occurs in the above theorem is defined as follows:A)\rflo r_{\lrcorner}E\circ B \Leftrightar ow G\cdot A_{\backslash } \rflo r_{\lrcorner}G\cdot BG\cdot E’

where A, B, E are small subsets of \mathfrak{M} . By Proposition 4.16 from [16], this ternary
relation is an independence relation, hence it is forking independence in (\mathfrak{M},\overline{\sigma})and so (\mathfrak{M},\overline{\sigma}) is simple (Theorem 4.22 in [16]) ‐ under the assumption that \mathfrak{M} isbounded. Moreover, boundedness of \mathfrak{M} implies also that the theory of (\mathfrak{M},\overline{\sigma}) has
geometric elimination of imaginaries (Theorem 4.36 in [16]).
There are several examples of (T_{\forall})_{G} which have model companions:\bulletACFA is supersimple, not stable,\bulletif G is finite thenG- TCF (e.c.

fields with group action of G , see [17]) is

supersimple of SU‐rank equal to |G| , not stable\bullet\mathb {Q} ACFA (e.c. fields with group action of \mathb {Q} , see [28]) is simple, not super‐
simple, not stable\bulletCCMA (Complex Compact Manifolds with Automorphism, see [4]) is super‐
simple, not stable

We end this part with an unexpected result binding together simplicity of a fieldequipped with a group action with simplicity of its reduct to the structure of thepure field.

Corollary 3.3 (Corollary 4.31 in [16]). Let T be the theory of fields, \mathcal{L} be thelanguage of rings and let G be a group (not necessarily finite). Assume that T_{G}^{mc}( =G —TCF) exists and let (K, (\sigma_{g})_{g\in G})\models T_{G}^{mc} . The following are equivalent.
(1) The theory of K in the language \mathcal{L} is simple.
(2) The field K is a bounded field.
(3) The theory of (K, (\sigma_{g})_{g\in G}) in the language \mathcal{L}^{G} is simple.
3.3. Frattini cover. As we already mentioned, we can adjust absolute Galois

groups (in the sense of \mathfrak{C} ) of existentially closed substructures of \mathfrak{C} equipped witha G ‐action by changing the group G . Assume that (M,\overline{\sigma}) is such a substructure of\mathfrak{C} . Now, we will describe some correlations between G and Galois groups related
to substructure M . This description involves the following notion:

Definition 3.4. Let H, H' be profinite groups and \pi : Harrow H' be a continuous
epimorphism. The mapping \pi is called a Frattini cover if for each closed subgroupH_{0} of H , the condition \pi(H_{0})=H' implies that H_{0}=H . If moreover the profinitegroup H is projective, the map \pi is the unique universal Frattini cover (consult
Proposition 22.6.1 in [14]).
The intuition behind the following results is that the G ‐action on M is determinedby its restriction to F :=ac1(M^{G})\cap M . Note that M^{G}\subseteq F is a Galois extension(by Lemma 3.28 in [16]), hence G(F/M^{G}) is a profinite group and it makes sense to

study this group by standard for profinite groups methods. For example, G(F/M^{G})is generated as a profinite group by (\sigma_{g}|_{F})_{g\in G} (Proposition 3.30 in [16]), which
implies that if G is finitely generated, then also G(M^{G}) is finitely generated (asa profinite group, see Proposition 3.49 in [16]). The following result generalizes asimilar one for fields with group action from [35], the proof in [15] uses Theorem
1.1 from [30] which is based on the classification of finite simple groups.

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PAC STRUCTURES IN NUTSHELL

Proposition 3.5 (Proposition 5.1 in [15]). If G is finitely generated then G(F/M^{G})\cong\hat{G} (the profinite completion of G ).
Let us assume that G is finitely generated. We know that M^{G} and M are PAC

substructures of \mathfrak{C} . After using some results from [15] and Theorem 2.12, we may
conclude:Corollary 3.6 (Corollary 5.5 in [15]). The restriction map

G(M^{G})arrow G(Mnac1(M^{G})/M^{G})is the universal Frattini cover.

By Proposition 3.5 we obtain the following short exact sequence

G(Mnac1(M^{G}))arrow \mathcal{G}(M^{G})arrow G(Mnac1(M^{G})/M^{G})\cong\hat{G}which gives us a generalization of a result which was known in the case of fields([35], [17]):
Corollary 3.7 (Corollary 5.6 in [15]). It follows that
(1) G(M^{G})\cong Fratt(\hat{G}) (the domain of the universal Frattini cover of the group\hat{G})

(2) (
G

,

Mn

acl (M^{G}) )\cong ker(Fratt (\hat{G})arrow\hat{G} ).

Finally, we see that M^{G} is a PAC substructure of \mathfrak{C} controlled by its absolute
Galois group which is isomorphic to Fratt (\hat{G}) . If G is finite, then M\subseteq ac1(M^{G})(by Lemma 2.10.(2) ) so Mnac1(M^{G})=M andcontrolled by its absolute Galois group given askerM

is a PAC substructure of \mathfrak{C}(Fratt (G)arrow G ).

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\dag er

INSTYTUT MATEMATYKI, UNIWERSYTET WARSZAWSKI, WARSZAWA, POLAND

Email address: daniel. \max .hoffmann@gmail.com