Model theory of Galois actions of torsion Abelian groups
aa r X i v : . [ m a t h . L O ] M a r MODEL THEORY OF GALOIS ACTIONS OF TORSIONABELIAN GROUPS ¨OZLEM BEYARSLAN ♣ AND PIOTR KOWALSKI ♠ Abstract.
We show that the theory of Galois actions of a torsion Abeliangroup A is companionable if and only if for each prime p , the p -primary partof A is either finite or it coincides with the Pr¨ufer p -group. We also provide amodel-theoretic description of the model companions we obtain. Contents
1. Introduction 12. General results about G -fields 42.1. Pseudo-existentially closed G -fields and PAC fields 42.2. Existentially closed G -fields 62.3. Chains of theories 123. Absolute Galois groups 144. Negative results 205. Positive results 226. Miscellaneous results 316.1. Another negative argument 316.2. Groups containing Z × Z Introduction A difference field is a field with an endomorphism. If this endomorphism isinvertible, then a difference field is the same as an action of the group ( Z , +) byfield automorphisms. Model theory of difference fields has been extensively studiedfor more than 20 years (see e.g. [14, 3, 4]). It is also natural to study model theoryof fields with actions of an arbitrary (fixed) group, instead of the infinite cyclicgroup. This topic had not been considered much until recently, we give a shortaccount of earlier works below. • Besides the theory ACFA corresponding to the action of Z , model theory offields with free group actions has been also considered, which resulted in thetheory ACFA n , see e.g. [10], [12], [20, Theorem 16], and [16, Proposition4.12]. ♠ Supported by the Narodowe Centrum Nauki grant no. 2018/31/B/ST1/00357.2010
Mathematics Subject Classification
Primary 03C60; Secondary 12H10, 11S20, 20E18.
Key words and phrases . Difference field, Model companion, Pr¨ufer group, Frattini cover. • Actions of the group Z × Z were considered by Hrushovski, who provedthat a model companion does not exist in this case (see [11]). • Actions of finite groups were considered first by Sj¨ogren in [20]. • Model theory of actions of ( Q , +) on fields were studied in [15].For a fixed group G , the first natural question to be considered is the following: doesa model companion of the theory of fields with G -actions exist? In the examplesgiven above, the corresponding model companions exist, except for the case of thegroup Z × Z . If such a model companion exists, then we call this model companion G -TCF and we say that “ G -TCF exists”.More recently, Daniel Hoffmann and the second author considered in [8] thecase of finite groups (being unaware then of Sj¨ogren’s work from [20]). In [1], theauthors of this paper extended some of the results from [3] and [8] into a verynatural common context of virtually free groups. This work is a continuation of thegeneral line of research from [1], however, it goes in a different direction, that is weconsider infinite torsion Abelian groups. Let A be a torsion Abelian group. Thispaper is almost exclusively devoted to the proof of the following result. Theorem 1.1.
The theory A − TCF exists if and only if for each prime p , the p -primary part of A is either finite or it is isomorphic with the Pr¨ufer p -group.Moreover, if the theory A − TCF exists, then it is simple; and it is strictly simple(that is: simple, not stable, and not supersimple) when A is infinite. Regarding the question of the existence of the theory G -TCF for G coming from agiven class of groups, the theorem above is a rare instance of a situation when a full answer is given. For example, we are still far from obtaining a corresponding answerfor the class of all finitely generated groups: we showed in [1] that for virtually freegroups the corresponding model companions exist, and we only conjectured in [1]the opposite implication (this conjecture is confirmed in [1] in the case of finitelygenerated commutative groups). It should be also noted that Theorem 1.1 disprovesour own [1, Conjecture 5.12], that is G -TCF need not exists for a locally virtuallyfree (even locally finite) group G .Let us fix our (standard) notation here. We denote the set of all prime numbersby P . For n >
0, the cyclic group of order n is denoted by C n and C p ∞ = lim −→ n C p n is the Pr¨ufer p -group. For any group G and any ordinal number α , by G ( α ) wedenote the direct sum of α copies of G . If a group G acts on a set X , then by X G we denote the set of invariants of this action. For a field K , Gal( K ) denotes the(profinite) absolute Galois group of K , that is the group Gal( K sep /K ), where K sep is the separable closure of K .By a G -field , we mean a field with an action of the group G by field automor-phisms. Similarly, we consider G -rings, G -field extensions, etc. By L G , we denotethe natural language of G -fields, where the elements of G serve as unary functionsymbols. Remark 1.2.
We give here two conditions on an Abelian group A , which areequivalent to the condition appearing in the statement of Theorem 1.1.(1) The group A does not contain (up to an isomorphism) any of the followingtwo “forbidden subgroups”: • C ( ω ) p , ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 3 • C p ⊕ C p ∞ .(2) There is no p ∈ P such that there exists an infinite strictly increasingsequence C p ∼ = P < P < P < . . . , where each P i is a finite p -subgroup of A .We would like to say a few words about the shape of the axioms of the theory A -TCF from the statement of Theorem 1.1. The axioms of ACFA from [3] are geometric , that is they describe the intersections of algebraic varieties with thegraph of the generic automorphism. In the case of a finite group, geometric axiomswere used in [8] as well, which was the main difference with the approach taken in[20]. Using the Bass-Serre theory, the geometric axioms from [8] were “glued” in [1]to obtain geometric axioms for actions of arbitrary finitely generated virtually freegroups. Let us now go back to the axioms from [20]. They are not geometric in theabove sense, since they describe the properties of certain absolute Galois groups.This is why we call them Galois axioms and we formalize this notion below.
Definition 1.3.
We say that the theory of a G -field K is axiomatised by Galoisaxioms , if G is the union of its finitely generated subgroups ( G i ) i ∈ I (for conveniencewe assume that 0 is the smallest element of I and G = { } ) such that the theoryof the G -field K is implied by the following statements:(1) the action of G on K is faithful (we say that the G -field K is strict );(2) K is a perfect field;(3) for each i ∈ I , K G i is PAC;(4) for each i ∈ I , we have: Gal (cid:0) K G i (cid:1) ∼ = G i , where ( G i ) i ∈ I is a fixed collection of small profinite groups.Clearly, Items (1) and (2) are first-order. By [5, Chapter 11.3], Item (3) is afirst-order condition as well. Since the set of extensions of a field F (inside a fixedalgebraic closure of F ) of a fixed degree n is “uniformly definable” in F (see e.g. [18,Remark 2.6(i)]) and there are finitely many of them in the situation of Definition1.3 (by the smallness assumption), we see that Item (4) is also first-order. We wouldlike to point out that in the case of a group G which is not finitely generated, thefield of constants K G is not definable in the G -field K (it is merely type-definable).Hence, there is not much chance for any statement about Gal( K ) to be first-order.As we have said above, the theory G − TCF is axiomatised by Galois axioms fora finite G (by [20]), which we will also point out in Lemma 2.17. In this paper, weprove a version of this result for torsion Abelian groups satisfying the equivalentconditions from Remark 1.2.This paper is organized as follows. In Section 2, we collect some general results(originating often from [20]) about existentially closed G -fields and we also discussHrushovski’s notion of pseudo-existentially closed G -fields. In Section 3, we showa crucial technical result about pronilpotency of certain absolute Galois groups.Section 4 is concerned with the negative (or non-existence) results. More precisely,we show there the left-to-right implication from Theorem 1.1. Section 5 is about thepositive results, that is we show there the right-to-left implication from Theorem1.1. In Section 6, we collect several miscellaneous results and observations regardingthe model theory of G -fields. ¨O. BEYARSLAN AND P. KOWALSKI General results about G -fields In this section, we discuss Hrushovski’s notion of pseudo-existentially closed G -fields and we also collect the results about existentially closed G -fields and PACfields, which will be important in the sequel. We finish this section with some well-known results about chains of theories and we give examples of such chains in thecase of group actions on fields.2.1. Pseudo-existentially closed G -fields and PAC fields. The following no-tion we learnt from Udi Hrushovski (private communication). It originated fromour attempts to show that if G has a subgroup isomorphic to Z × Z , then G -TCFdoes not exist (those attempts will be discussed in Section 6.2). Definition 2.1. A G -field F is pseudo-existentially closed (abbreviated p.e.c. ), iffor any G -field extension F ⊆ K such that the pure field extension F ⊆ K isregular, the G -field F is existentially closed in the G -field K .We would like to point out that if G = { } , then p.e.c. G -fields are exactly PACfields, which justifies the choice of the term “pseudo-existentially closed” above.We will also use in the sequel the abbreviation “e.c.” for “existentially closed”. Werecall that a G -field is G -closed , if it has no proper algebraic (as pure fields) G -fieldextensions.The crucial good property of p.e.c. G -fields is the result below, which is clearlyfalse for e. c. G -fields (consider H = { } ). This result and its proof was pointedout to us by Udi Hrushovski. Proposition 2.2.
Suppose that M is a p.e.c. G -field and H G . Then M is ap.e.c. H -field as well.Proof. Assume that F is a G -field and F ⊆ K is an H -field extension, which isregular (as an extension of pure fields). To conclude the proof, it is enough toconstruct a field extension of K ⊆ L and an action of G on L such that F ⊆ L is a G -field extension and K ⊆ L is an H -field extension.Let W of be a set of representatives for G/H such that 1 ∈ W . For each r ∈ W we fix a set rK such that:(i) 1 K = K ;(ii) for all r, r ′ ∈ W , if r = r ′ , then rK ∩ r ′ K = F ;(iii) there is a bijection (denoted by r as well) r : K → rK such that for all x ∈ F , we have r ( x ) = r · x (the action of G on F ).For any g ∈ G and r ∈ W , there are unique r ′ ∈ W and h ∈ H such that gr = r ′ h and we define a bijection g : rK → r ′ K by the following commutative diagram: rK g / / r − (cid:15) (cid:15) r ′ KK · h / / K, r ′ O O where · h comes from the given action of H on K . Let us also define: Z := [ r ∈ W rK. It is easy to see that the above diagram defines an action of G on the set Z , whichextends the action of H on K and the action of G on F . For each r ∈ W , we define ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 5 a field structure on rK in such a way that r : K → rK is a field isomorphism. Thenfor each g, r, r ′ as above, the map g : rK → r ′ K is a field isomorphism as well.We define now: R := O r ∈ W rK, where the tensor product is taken over the field F . By the universal property ofthe tensor product (in the category of F -algebras), the action of G on Z uniquelyextends to an action of G on R by ring automorphisms. Since the field extension F ⊆ K is regular, the ring R is a domain by [2, Proposition 2 in §
17, A.V.141].Hence, we can take as L the field of fractions of R with the induced action of G . (cid:3) Remark 2.3.
We collect here several observations about e.c., p.e.c, and G -closedfields. Let H G .(1) It is easy to see that a G -field K is e.c. if and only if K is both p.e.c. and G -closed.(2) It is also clear that if K is a G -field, which is H -closed, then K is G -closedas well.(3) Proposition 2.2 says that the opposite happens with the notion of p.e.c: if K is a p.e.c. G -field, then K is a p.e.c. H -field.(4) It is easy to find examples of e.c. G -fields, which are not e.c. H -fields(taking H = { } ).(5) We still do not know whether the existence of G -TCF implies the existenceof H -TCF (see Conjecture 6.6 for a special case).(6) The notion of a p.e.c. G -field and Proposition 2.2 should generalize to thecontext of an arbitrary theory (instead of the theory of fields) having astable completion, which is the context considered in [7].The following result will be crucial in the sequel. Proposition 2.4. If K is a p.e.c. G -field and G is finitely generated, then the field K G (the field of constants) is PAC.Proof. Let us denote K G by F , and we take an absolutely irreducible variety V over F . Then, the field extension F ⊆ F ( V ) is regular (see e.g. the remark above[5, Lemma 2.6.4]). Therefore, the ring R := C ( V ) ⊗ C K is a domain. We define a G -ring structure on R such that K ⊆ R is a G -ring extension and R G = C ( V ) in theobvious way. Then V ( R G ) = ∅ , since there is an R G -rational point correspondingto the identity map. Let L be the fraction field of R . Then the G -action on R extends uniquely to a G -action on L by field automorphisms.Therefore, we have: • the extension K ⊆ L is a G -field extension; • the field extension K ⊆ L is again regular; • the statement “ V ( L G ) = ∅ ” is first-order (since G is finitely generated).Since K is a p.e.c. G -field, we obtain that V ( K G ) = ∅ , which finishes the proof. (cid:3) Corollary 2.5.
Suppose that K is a p.e.c G -field and H G is a finitely generatedsubgroup. Then the field K H is PAC. In particular, K = K { } is a PAC field.Proof. It follows directly from Propositions 2.2 and 2.4. (cid:3)
Remark 2.6.
We would like to comment here how the results of this subsectionare related to [20]. ¨O. BEYARSLAN AND P. KOWALSKI (1) It is stated in [20, Theorem 2] that if G is an arbitrary group and K is anexistentially closed G -field, then K G is PAC. The proof of [20, Theorem 2]is basically the same as the proof of Proposition 2.4 above, and we believethat one still has to assume that G is finitely generated for this proof towork (although, we do not have a counterexample for the statement withan arbitrary group G ).(2) In [20, Theorem 3], it is stated that if K is an existentially closed G -field,then K is PAC. Corollary 2.5 is more general and its proof is simpler, sinceit is using Proposition 2.2.2.2. Existentially closed G -fields. In this subsection, we go through several re-sults which originally appeared in [20], namely: Theorems 4, 5, and 6 there. We doit for the sake of completeness and we would like to discuss some issues concerning[20, Theorem 6] as well.Suppose that K is a G -field, C = K G , and ϕ : G −→ Aut(
K/C )corresponds to the action of G on K . For a group G , we denote by b G the profinitecompletion of G . For a profinite group G , we denote by e G the universal Frattinicover of G (see [5, Chapter 22]). For a cardinal number κ and p ∈ P , we denote by b F κ ( p ) the free pro- p group of rank κ (see [5, Remark 17.4.7]).The lemma below originates from [20, Theorem 4]. It is also related to [6,Proposition 5.1], where a version of this lemma is proved in a more general context(see Remark 2.3(6)). Lemma 2.7.
There is a continuous epimorphism: α : b G −→ Gal (cid:0)(cid:0) C alg ∩ K (cid:1) /C (cid:1) such that the following diagram is commutative: b G α / / Gal( C alg ∩ K/C ) G ι O O ϕ / / Aut(
K/C ) . res O O Moreover, if ( K, G ) is p.e.c. and G is finitely generated, then the map α is anisomorphism.Proof. Let us consider a finite Galois extension C ⊆ C ′ such that C ′ ⊆ K . Weconsider the restriction map: r : G −→ Gal( C ′ /C ) . It is enough to show that the map r is onto. Let H := r ( G ) and C := ( C ′ ) H .Then G acts trivially on C , hence C ⊆ C and H = Gal( C ′ /C ), which we neededto show.For the moreover part, it is shown in the proof of [20, Theorem 4] (in the e.c.case) that G and Gal( C alg ∩ K/C ) have the same finite quotients, which is enough,since b G is small. We sketch below the argument given in [20]. Let π : G → H bean epimorphisms and ι : H → S m be an embedding, where S m is the symmetricgroup on m = | H | generators ( m is finite). Then the field of rational functions ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 7 K ′ := K ( t , . . . , t m ) has a natural structure of a G -extension of K (given by ι ◦ π ).It is easy to see that F ( t , . . . , t m ) H = ( K ′ ) G , hence the field of constants of K ′ has a Galois extension with Galois group isomor-phic to H . Since this last condition is first-order ( G is finitely generated), K isp.e.c., and the extension K ⊂ K ′ is regular, the result follows. (cid:3) We recall here a correspondence between Frattini covers and extensions of groupactions. It is just a reformulation of [8, Lemma 3.7].
Lemma 2.8.
We assume that • C ⊆ K is a finite Galois extension and G = Gal( K/C ) ; • G Gal( C ) is a closed subgroup; • C ′ := ( C sep ) G ; • K ′ := C ′ K = ( C sep ) G ∩ Gal( K ) .Then the following are equivalent: (1) K ⊆ K ′ is a G -field extension, where the G -field structure on K ′ is givenby Gal( K ′ /C ′ ) ; (2) res( G ) = G = Gal( K/C ) .Moreover, if any of the equivalent two conditions above holds, then we have: [ K ′ : K ] = [ G : G ] . Proof.
We just point out here that both the conditions are equivalent to the factthat the restriction map: res : Gal( K ′ /C ′ ) −→ Gal(
K/C )is an isomorphism, so K ′ gets the G -field structure (extending the one on K ) usingthis restriction isomorphism. (cid:3) We will use several times the following consequence of an implication fromLemma 2.8.
Proposition 2.9.
Suppose that we have a tower of fields K ⊆ K ⊆ K such that K/K and K/K are finite Galois extensions and we set: H := Gal( K/K ) , G := Gal( K/K ) . Assume that for i ∈ { , } , G i Gal( K i ) are closed subgroups such that: (1) for i ∈ { , } , we have res( G i ) = Gal( K/K i ) , (2) G ∩ Gal( K ) = G ∩ Gal( K ) .Let K ′ (resp. K ′′ ) be the H -field (resp. G -field) extension of K given by Lemma2.8. Then K ′ = K ′′ and the G -structure on K ′ expands the H -structure on K ′ .Proof. Let us define: C ′ := ( K sep ) G , C ′′ := ( K sep ) G . Then we have: K ′ = C ′ K, K ′′ = C ′′ K and since G ∩ Gal( K ) = G ∩ Gal( K ), we get that K ′ = K ′′ . ¨O. BEYARSLAN AND P. KOWALSKI We have the following commutative diagram:Gal( K ′ /C ′′ ) res / / ∼ = / / Gal(
K/K )Gal( K ′ /C ′ ) O O res / / ∼ = / / Gal(
K/K ) . O O By the description from Lemma 2.8 of both the G -field structure and the H -fieldstructure on K ′ , we see that the above diagram implies that the G -action expandsthe H -action. (cid:3) Remark 2.10.
After assuming Item (1) from Proposition 2.9, Item (2) there isequivalent to the following equality: G ∩ Gal( K ) = G . The next result generalizes an implication from Lemma 2.8 (a version of it, in amore general context, appeared as [7, Corollary 3.47]).
Proposition 2.11. If ( K, G ) is G -closed, then the restriction map Gal( C ) −→ Gal( C alg ∩ K/C ) is a Frattini cover. Hence, if C is PAC and ( K, G ) is G -closed (for example, when ( K, G ) is e.c. and G is finitely generated), then this restriction map is a universalFrattini cover.Proof. Let us consider the restriction map: α : Gal( C ) −→ Gal (cid:0) C alg ∩ K/C (cid:1) , and we take a closed subgroup G Gal( C ) such that α ( G ) = Gal (cid:0) C alg ∩ K/C (cid:1) . Itis enough to show that G = Gal( C ). Since ker( α ) G = Gal( C ), we get that (cid:0) C alg (cid:1) ker( α ) ∩ (cid:0) C alg (cid:1) G = C. Since the extension C ⊆ (cid:0) C alg (cid:1) ker( α ) is Galois (ker( α ) is clearly a normal subgroup),we get that ( C alg ) ker( α ) is linearly disjoint from ( C alg ) G over C using e.g. the remarkbelow the proof of Corollary 2.5.2 in [5] (the remark is for finite extensions, butsince both being Galois and being linearly disjoint are locally finite notions, it worksin general). Since we have C alg ∩ K ⊆ (cid:0) C alg (cid:1) ker( α ) , we get that C alg ∩ K is linearly disjoint from ( C alg ) G over C . Since K is G -closed,it is perfect. Hence the field C alg ∩ K is perfect as well and the field extension C alg ∩ K ⊆ K is regular. Therefore (by the definition of regularity), K is linearlydisjoint from C alg over C alg ∩ K . By the Tower Property for linear disjointness (see[5, Lemma 2.5.3]), we finally get that K is linearly disjoint from ( C alg ) G over C .Therefore, we have K ( C alg ) G ∼ = Frac (cid:0) K ⊗ C ( C alg ) G (cid:1) , hence the action of G on K extends to an action of G to K ( C alg ) G . Since ( K, G )is G -closed, we get that K ( C alg ) G = K , so ( C alg ) G ⊆ C alg ∩ K . By Galois theory,we get that G = Gal( C ), which finishes the proof. (cid:3) ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 9
Remark 2.12.
It is easy to see that the opposite implication to the one appearingin Proposition 2.11 is not true. It is enough to take an algebraically closed field C , K = C ( X ), and G = Z acting on K in the “classical difference way”, that is σ ( X ) = X + 1, where σ is a generator of the group Z .Our first corollary is exactly [20, Theorem 5] (it was also generalized to a moreabstract context in [6, Corollary 5.6]). Corollary 2.13. If ( K, G ) is e.c. and G is finitely generated, then we have: Gal( C ) ∼ = eb G. Proof.
It follows directly from Lemma 2.7 and Proposition 2.11. (cid:3)
The next corollary is much weaker than the statement in [20, Theorem 6], whichwill be discussed in Remark 2.18(2).
Corollary 2.14.
Suppose that ( K, G ) is e.c., then we have the following. (1) There is an epimorphism:
Gal( K ) −→ ker (cid:0) Gal( C ) → Gal (cid:0) C alg ∩ K/C (cid:1)(cid:1) . (2) If G is finitely generated, then ker (cid:18) eb G → b G (cid:19) ∼ = ker (cid:0) Gal( C ) → Gal (cid:0) C alg ∩ K/C (cid:1)(cid:1) and there is a monomorphism: ker (cid:18) eb G → b G (cid:19) −→ Gal( K ) . Proof.
Since the extension C alg ∩ K ⊆ K is regular (as in the proof of Proposition2.11), the restriction map r : Gal( K ) → Gal( C alg ∩ K )is onto. By Galois theory, we have the following isomorphism:ker (cid:0) Gal( C ) → Gal (cid:0) C alg ∩ K/C (cid:1)(cid:1) ∼ = Gal (cid:0) C alg ∩ K (cid:1) showing Item (1).For Item (2), by Corollary 2.13 we have:ker (cid:18) eb G → b G (cid:19) ∼ = ker (cid:0) Gal( C ) → Gal (cid:0) C alg ∩ K/C (cid:1)(cid:1) . Therefore, the profinite group Gal( C alg ∩ K ) is projective, hence the map r abovehas a section, which gives the result. (cid:3) We point out below that for a finite group G , the theory G − TCF is axiomatizedby Galois axioms, which was shown in [20] and [8]. We include here a version ofthe statement from [8], which is convenient for us to work with. For the proof ofthis version, we need the following result, which may be folklore. We recall that fora profinite group G , the rank of G , denoted rk( G ), is the minimal cardinality of aset of topological generators of G . Proposition 2.15.
Assume that H is a profinite group of finite rank. Then, anycontinuous epimorphism π : e H → H is a (universal) Frattini cover.
Proof.
The proof consists of two steps.
Step 1
The result holds if H is a pro- p group. Proof of Step 1.
Let r = rk( H ). By [5, Lemma 22.7.4], there is an epimorphism H → C rp (which is a Frattini cover). By [5, Lemma 22.5.4], we can assume that H = C rp .Let us take B ⊂ C rp such that | B | = r . Since r is finite, B generates the group C rp if and only if B is a basis of C rp considered as an F p -vector space. Therefore,the group Aut( C np ) = GL r ( F p ) acts transitively on the family of sets of generatorsof C rp of size r .By [5, Corollary 22.5.3], rk( e H ) = r and we fix e B , which is a set of generatorsof e H of size r . Let B be the image of e B by the universal Frattini cover map and B ′ := π ( e B ). Then, both B and B ′ have size r and generate C rp . Hence, there isan automorphism of C rp taking B to B ′ . Therefore, π is the composition of theuniversal Frattini cover map and this last automorphism, thus π is a Frattini coveritself. (cid:3) Let us take a closed subgroup G e H such that π | G : G −→ H is a Frattini cover (it exists by [5, Lemma 22.5.6]). We aim to show that G = e H .For necessary background regarding profinite Sylow theory, we refer the reader tothe beginning of Section 3. Step 2
For each p ∈ P , any p -Sylow subgroup of G is a p -Sylow subgroup of e H as well. Proof of Step 2.
By Step 1, for any p -Sylow subgroup P of e H , the map: π | P : P −→ π ( P )is a Frattini cover. Since π ( G ) = H , we get π ( P ∩ G ) = π ( P )and Step 2 follows. (cid:3) Step 2 implies that G = e H (it is enough to look at the finite quotients for whichit is clear), which finishes the proof. (cid:3) Remark 2.16. (1) The statement of Step 1 from the proof of Lemma 2.15 hasalready appeared in [1] as Lemma 4.4. Unfortunately, we gave an erroneousproof of [1, Lemma 4.4] (confusing two universal properties).(2) In the case of pro- p groups, Lemma 2.15 can be generalized to the followingstatement, which we will need in the sequel: Any continuous epimorphism of pro- p groups of the same finite rank is a Frattini cover .It can be proved in the same way as Step 1 in the proof of Lemma 2.15 wasshown. Namely, if π : G → H is a such an epimorphism and the rank is r ,then, by [5, Lemma 22.7.4], there are Frattini cover maps: p : G −→ C rp , p : H −→ C rp . ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 11
Hence, we can replace π with p ◦ π and conclude as in the proof of Step 1above.(3) It is easy to see that not every epimorphism of profinite (even finite) groupsof the same finite rank is a Frattini cover, consider for example the followingepimorphism: C −→ C × C . Proposition 2.17. If G is finite, then a G -field K is existentially closed if andonly if K is strict, perfect, the field of constants C := K G is PAC, and Gal( C ) ∼ = e G. Proof.
We use [8, Theorem 3.29], where the equivalence (1) ⇔ (4) says that K is ane.c. G -field if and only if it is strict, perfect, C is PAC and K is G -closed. Hence,we need to check the G -closedness condition only. By Lemma 2.8, we need to showthat the restriction map Gal( C ) −→ Gal(
K/C ) ∼ = G is a Frattini cover, which is given by our assumptions and Lemma 2.15. (cid:3) Remark 2.18.
We would like to point out several general observations concerningGalois axioms and absolute Galois groups.(1) The original theory ACFA(= Z − TCF) is not axiomatized by Galois ax-ioms. To see that, we notice first that the Galois axioms in the case of adifference field (
K, σ ) say that K is algebraically closed and C = Fix( σ ) ispseudofinite.By [9, Section 13.3], any model of ACFA of characteristic 0 has infinitetranscendence degree over Q . By [5, Theorem 18.5.6 and Theorem 18.6.1],for almost all (in the sense of the Haar measure) σ ∈ Gal( Q ), the fieldFix( σ ) is pseudofinite. Hence, such a difference field ( Q alg , σ ) satisfies theGalois axioms, but it is not existentially closed.(2) It is stated in [20, Theorem 6] that if G is finitely generated and finitelypresented and K is an e.c. G -field, then we haveGal( K ) ∼ = ker (cid:18) eb G −→ b G (cid:19) . The main part of the proof of [20, Theorem 6] is an argument, which issupposed to show that the monomorphism appearing in Corollary 2.14 isactually an isomorphism. We do not know how to make this argumentwork, we comment more on it below.(a) The monomorphism from Corollary 2.14 is an isomorphism in the caseof a finite group G , which was shown in [8, Theorem 3.40(2)].(b) In [1, Section 4], we use [20, Theorem 6] to show [1, Theorem 4.7]saying that if G is a finitely generated virtually free group, which isneither free nor finite, then the theory G -TCF is not simple, since theabsolute Galois group of underlying fields of models of G -TCF are notsmall. However, if a profinite groups is small, then its image by acontinuous epimorphism is also small. Therefore, in order to show [1,Theorem 4.7], it is enough to use just Corollary 2.14 instead of [20,Theorem 6]. (c) Nick Ramsey communicated to us a proof of the result saying that for G finitely generated and virtually free, the theory G -TCF is NSOP .However, this proof seems to be using the full version of [20, Theorem6].(d) Example 2.21(2) gives a counterexample for the isomorphism:Gal( K ) ∼ = ker (cid:18) eb G −→ b G (cid:19) in the case of G = C p ∞ , which is obviously not finitely generated.(e) Similar issues were discussed in a more general context in [6] (see [6,Conjecture 5.7] and [6, Remark 5.8]).2.3. Chains of theories.
In this subsection, we collect several well-known resultsabout chains of theories. They can be found e.g. in [15] or [17], but we includethem here for the sake of completeness.Let L be a language and T be an L -theory. It is easy to see that T is closedunder consequences (that is: T | = φ if and only if φ ∈ T ) if and only if T = \ M | = T Th( M ) . From now on, all the theories we consider are closed under consequences.Suppose that L ⊆ L ′ are languages, T is an L -theory, and T ′ is an L ′ -theory.We have the following obvious result. Fact 2.19.
The following are equivalent. (1) T ⊆ T ′ . (2) “ Mod( T ′ ) ⊆ Mod( T ) ”, i.e. for each M ′ | = T ′ , we have M ′ | = T . Let T be an inductive theory (the corresponding theories considered in this paperare even universal). If a model companion of T exists, then it is unique and wedenote it by T mc . The following result is crucial and appeared in [15] and [17]. Proposition 2.20.
Suppose we have an increasing sequence of languages ( L m ) m> and an increasing sequence of L m -theories ( T m ) m> . If the model companions ( T mc m ) m form an increasing sequence as well, then the model companion of T ∞ := S m T m exists and we have: T mc ∞ = ∞ [ m =1 T mc m . Moreover, if all the theories T mc m are simple, then the theory T mc ∞ is simple as well.Proof. Let us denote: L ∞ := ∞ [ m =1 L m , T ′∞ := ∞ [ m =1 T mc m . It is easy to see and it is pointed out e.g. in [15, Theorem 2] that T ′∞ inherits allthe “local” properties enjoyed by all the theories T mc m . In particular, T ′∞ is modelcomplete and if all the theories T mc m are simple, then the theory T ′∞ is simple aswell. Therefore, it is enough to show that each model of T ∞ embeds into a modelof T ′∞ . We will actually show that this last embedding property is also “local”. ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 13
Let us fix M | = T ∞ . We need to show that the theory diag + ( M ) ∪ T ′∞ isconsistent, where diag + ( M ) is the set of all atomic L ∞ -sentences with parametersfrom M which are true in M . Since we have:diag + ( M ) := ∞ [ m =1 diag + ( M | L m ) , the result follows. (cid:3) Example 2.21.
We give here an argument showing that for any p ∈ P , the theory C p ∞ − TCF exists, which may be considered as a “baby case” of the right-to-leftimplication in Theorem 1.1.(1) By Fact 2.19 and Proposition 2.20, it is enough to check that if (
K, σ ) | = C p m +1 − TCF, then (
K, σ p ) | = C p m − TCF. Let C = Fix( σ ) and C ′ =Fix( σ p ). Consider the following commutative diagram with exact rows:0 / / Gal( K ) < / / Gal( C ) = Z p res / / Gal(
K/C ) = C p m / / / / Gal( K ) = O O < / / Gal( C ′ ) < O O res / / Gal(
K/C ′ ) = pC p m < O O / / , where the description of the profinite group Gal( C ) comes from Proposition2.17. Hence, we have:Gal( C ′ ) = res − ( pC p m ) = p Z p ∼ = Z p . Since C ⊆ C ′ is a finite field extension and C is a perfect PAC field, then C ′ is perfect and PAC as well. Hence, ( K, σ p ) satisfies the Galois axiomsfor the theory C p m − TCF by Proposition 2.17.(2) By Item (1), it is easy to see that if K is an e.c. C p ∞ -field, then we have:Gal( K ) ∼ = Z p . (3) If A is any divisible group, K is an A -field, and C = K A , then C alg ∩ K = C ,since there are no non-trivial homomorphisms from a divisible group into aprofinite group. Therefore, for an A -closed field K , the extension C ⊆ K isregular, hence C is algebraically closed. In particular, if K | = C p ∞ − TCF,then Gal( C ) = 1, where C = K C p ∞ is the field of absolute constants. Remark 2.22.
We discuss here what may happen if the model companions fromProposition 2.20 exist, but they do not form an increasing chain.(1) The theories ( C p m − TCF) m> do not form an increasing chain. To see thatlet us take( K, σ, τ ) | = C p − TCF , C = Fix( σ, τ ) , C ′ = Fix( σ p , τ p ) . Then, we have:Gal( C ) / Gal( C ′ ) ∼ = Gal( C ′ /C ) ∼ = C p . By Proposition 2.17 and [5, Proposition 22.7.6] (this is a result of Tatesaying that projective pro- p groups are pro- p free), we get Gal( C ) ∼ = b F ( p ).Since [Gal( C ) : Gal( C ′ )] = p , we get by [5, Proposition 17.6.2] (the profi-nite Nielsen-Schreier formula) that H ∼ = b F p +1 ( p ) ≇ Gal( C ) . In particular, by Proposition 2.17, the C p -field ( C ′ , σ p , τ p ) is not existen-tially closed, so we get: C p − TCF * C p − TCF . This observation can not be immediately made into a proof of the non-existence of the theory C p ∞ − TCF, for which we will need the results ofSection 3.(2) As was noted in [17, Theorem 4], a model companion of the theory T ∞ (inthe notation from Proposition 2.20) may still exist, even when the theories( T mc m ) m do not form an increasing chain. We come across such a situationin the case of actions of torsion Abelian groups. Namely, let us define: C P := M p ∈ P C p ∼ = lim −→ m C p ...p m , where P = ( p i ) i> is an enumeration of the set of all primes. Then, onecan see (similarly as in Item (1) above) that: C − TCF * C − TCF , but (by Theorem 1.1), the theory C P -TCF still exists.3. Absolute Galois groups
In this section, we begin our proof of Theorem 1.1 (the main result of this paper)by describing the absolute Galois groups of certain fields of invariants.Firstly, we collect several notions from the theory of profinite groups, which wewill often use in the sequel without any references. Proofs of these results can befound in [5, Chapter 22.9]. The classical Sylow theory for finite groups generalizessmoothly to the profinite context after replacing the notion of a p -subgroup withthe notion of a closed pro- p subgroup. In particular, for p ∈ P and a profinitegroup G , p -Sylow subgroups of G exist and they are conjugate. We also have thecorresponding results about pronilpotent groups, that is: a profinite group G ispronilpotent if and only if it is the product of its unique p -Sylow subgroups. If G is a pronilpotent group and p ∈ P , then we denote by G ( p ) the unique p -Sylowsubgroup of G . Throughout this section, “cl” denotes the topological closure (in anambient profinite group).We will need a very general result about pronilpotent groups stated below. Itmay be folklore, but we were unable to find it in the literature. Proposition 3.1.
Let G be a profinite group, ( I, ) be a directed partially orderedset and ( P i ) i ∈ I be a direct system of closed pronilpotent subgroups of G (ordered byinclusion). Then, the subgroup P ∞ := cl [ i ∈ I P i ! is pronilpotent as well.Proof. For each i ∈ I , we have P i = Y p ∈ P ( P i ) ( p )ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 15 and for p ∈ P , we define: P ∞ ,p := cl [ i ∈ I ( P i ) ( p ) ! . Since the commutator map ( x, y ) [ x, y ] is continuous, we have: P ∞ ⊆ Y i ∈ P P ∞ ,p . Therefore, it is enough to show that each P ∞ ,p is a pro- p group. To ease thenotation, we assume that each P i is a pro- p group and we aim to show that P ∞ isa pro- p group as well.Let us fix a p -Sylow subgroup P G . For each i ∈ I , we define: X i := (cid:8) g ∈ G | P i ⊆ gP g − (cid:9) . Since for any fixed x ∈ G , the map g g − xg is continuous, it is easy to see thatfor each i ∈ I , the set X i is closed. Since all p -Sylow subgroups of G are conjugate,each X i is non-empty and ( X i ) i ∈ I a direct system ordered by the reversed inclusion.Since G is compact, we get that \ i ∈ I X i = ∅ . Let us take g ∈ T i ∈ I X i . Then for each i ∈ I , we have P i ⊆ gP g − . Since gP g − is pro- p , P ∞ is also pro- p , which finishes the proof. (cid:3) We state below a crucial result about absolute Galois groups of fixed subfieldsof A -closed fields, where A is a torsion Abelian group. Let us fix such a group A and we present it as: A = lim −→ i ∈ I A i for finite Abelian groups A i such that A = { } (0 appearing in the subscript is thesmallest element in ( I, )). For each p ∈ P , we denote by A ( p ) its p -power torsionsubgroup (which can be considered as its p -Sylow subgroup). Let us fix an A -field K . For each i ∈ I , we denote: K i := K A i and we have the following short exact sequence1 / / Gal( K ) < / / Gal( K i ) res i / / Gal(
K/K i ) / / , where res i is the appropriate restriction map. Theorem 3.2.
Suppose that K is A -closed and strict. Then we have the following. (1) For each i ∈ I , the profinite group Gal( K i ) is pronilpotent. (2) Suppose that for each p ∈ P , the group A ( p ) is finite. We enumerate theset of all primes P = ( p n ) n> and set: A n := A ( p ) ⊕ . . . ⊕ A ( p n ) . Let us take j, n ∈ N . If j n , then the restriction map: res n : Gal( K n ) ( p j ) −→ A ( p j ) = ( A n ) ( p j ) is a Frattini cover. Proof.
We proceed to show Item (1) and then we will notice that under the extraassumptions of Item (2), the proof of Item (1) gives the stronger conclusion fromItem (2). The following claim is crucial for our proof of Item (1) and the proof ofthis claim is rather long. Since K is a strict A -field, for each i ∈ I , we will identifyGal( K/K i ) with A i . Claim
For each i ∈ I , there is a closed subgroup W i Gal( K i ) such that:(1) the profinite group W i is pronilpotent;(2) we have: res i ( W i ) = A i ;(3) for each i, j ∈ I , if i j , then we have:Gal( K i ) ∩ W j = W i . Before proving the Claim, we will quickly see that it implies Item (1) from Theorem3.2. Let W be the common intersection of all W i ’s with Gal( K ) and K ′ := ( K alg ) W .By the Claim and Proposition 2.9, the action of A on K extends to K ′ . Since K is A -closed, we get K = K ′ . Therefore, Gal( K ) = W and for each i , we haveker(res i ) = Gal( K ) ⊆ W i . Since res i ( W i ) = A i , we get that W i = Gal( K i ), so all the profinite groups Gal( K i )are pronilpotent. Proof of Claim.
For each i ∈ I and p ∈ P , let n i,p be the cardinality of the finite p -group ( A i ) ( p ) . We define the following set of infinite tuples:Cl i := x ∈ Y p ∈ P Gal( K i ) n i,p | x satisfies the conditions (i)–(iii) below . Before stating the conditions (i)–(iii), we fix the obvious presentation of a tuple x ∈ Q p ∈ P Gal( K i ) n i,p : x = (cid:0) x ( p ) (cid:1) p ∈ P , x ( p ) ∈ Gal( K i ) n i,p . We give below the conditions defining the set Cl i .(i) For each p ∈ P , we have:res i (cid:0) x ( p ) (cid:1) = ( A i ) ( p ) . (ii) For each p ∈ P , the group cl( h x ( p ) i ) is a pro- p subgroup of Gal( K i ).(iii) For each p, q ∈ P , if p = q then[ x ( p ) , x ( q ) ] = 1 , i.e. the coordinates of the tuple x ( p ) commute with the coordinates of thetuple x ( q ) .For each x ∈ Cl i , we define: W xi := cl( h x i ) . Subclaim 1
For each i ∈ I , we have the following.(1) Cl i = ∅ .(2) Cl i is a closed subset of Q p ∈ P Gal( K i ) n i,p . ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 17 (3) For each x ∈ Cl i , the profinite group W xi is pronilpotent and res i ( W xi ) = A i . Proof of Subclaim 1.
For the proof of Item (1), we notice that by [19, Lemma 2.8.15]there is a closed subgroup W i Gal( K i ) such that res i | W i is a Frattini cover. Inparticular, by [5, Corollary 22.10.6(b)] the profinite group W i is pronilpotent andwe have the following decomposition: W i = Y p ∈ P ( W i ) ( p ) . For each p ∈ P , we have: res i (cid:0) ( W i ) ( p ) (cid:1) = ( A i ) ( p ) . Hence, there is x ( p ) ∈ ( W i ) n i,p ( p ) such that res i ( x ( p ) ) = ( A i ) ( p ) . Thus, we have: x := (cid:0) x ( p ) (cid:1) p ∈ P ∈ Cl i and Cl i is non-empty.For the proof of Item (2), it is clear that the conditions (i) and (iii) from thedefinition of Cl i are closed. Since any closed pro- p subgroup of Gal( K i ) is containedin a Sylow pro- p subgroup of Gal( K i ), x satisfies the condition (ii) for a prime p ifand only if the tuple x ( p ) is contained in a p -Sylow subgroup of Gal( K i ). Hence,it is enough to check that this last condition on the tuple x ( p ) is closed. Let us fix P , a p -Sylow subgroup of Gal( K i ). We set n := n i,p and consider the followingfunction:Ψ : Gal( K i ) × P n −→ P n , Ψ( g, ( x , . . . , x n )) = (cid:0) gx g − , . . . , gx n g − (cid:1) . Since all p -Sylow subgroups of Gal( K i ) are conjugate, the set of tuples x ( p ) satis-fying our last condition coincides with the image of the function Ψ. Since Ψ is acontinuous function between compact topological spaces, its image is closed.Item (3) is obvious from the definition of the group W xi . (cid:3) From Item (3) in Subclaim 1, we see that for each i ∈ I , there is a closed pronilpotentsubgroup W i Gal( K i ) such that res i ( W i ) = A i . To finish the proof of the Claim,we need to find such W i ’s satisfying the extra condition saying that for i j , wehave W j ∩ Gal( K i ) = W i . Firstly, we will find W i ’s satisfying the following weakercondition: W i ⊆ W j (for i j ).For each i, j ∈ I such that i j , we define the following coordinate projectionmap: π ji : Y p ∈ P Gal( K j ) r j,p −→ Y p ∈ P Gal( K j ) r i,p , where the projections are induced by the inclusions ( A i ) ( p ) ( A j ) ( p ) . We define:Cl ji := π ji (Cl j ) ∩ Y p ∈ P Gal( K i ) r i,p . Subclaim 2
For each i, j ∈ I such that i j we have the following.(1) Cl ji ⊆ Cl i .(2) Cl ji = ∅ .(3) Cl ji is closed. Proof of Subclaim 2.
To show Item (1), we consider the following commutative di-agram with exact rows:1 / / Gal( K ) < / / = (cid:15) (cid:15) Gal( K i ) (cid:15) (cid:15) res i / / A i (cid:15) (cid:15) / / / / Gal( K ) < / / Gal( K j ) res j / / A j / / . The right part of this diagram is a Cartesian square, that is res − j ( A i ) = Gal( K i ).Therefore, each x ∈ Cl ji satisfies Condition (i) from the definition of the set Cl i .Since Conditions (ii) and (iii) are clearly satisfied for any x ∈ Cl ji , Item (1) isproved.For Item (2), it is enough to notice (by the same Cartesian square argument asabove) that the condition res j ( W j ) = A j implies thatres i ( W j ∩ Gal( K i )) = A i . Item (3) is obvious, since π ji is a continuous map between compact topologicalspaces. (cid:3) The set Cl ji has the following interpretation: for any x ∈ Cl ji , there is y ∈ Cl j suchthat W xi ⊆ W yj , so W xi extends to W yj . We want to have this extension property“all the way along ( I, < )”: in particular for i i i . . . , we want to find x such that W x i extends to W x i which extends to W x i , etc. To this end, for any i i . . . i n from I , we defineCl i ,i i := π i i (cid:0) Cl i i (cid:1) , Cl i ,...,i n i := π i i (cid:16) Cl i ,...,i n i (cid:17) . To convey the main idea in a proper way, it is more convenient to continue in thespecial case when I = N and is the standard ordering on N . We will point outlater what one needs to do in the general case. We define:Cl ∞ := ∞ \ n =2 Cl ,...,n . As it was argued several times before in this proof, the compactness of Gal( K )implies that the set Cl ∞ is non-empty. Let us take x ∈ Cl ∞ . It follows from thedefinition of Cl ∞ that there is a sequence ( x i ∈ Cl i ) i> such that for each i > π i +1 i ( x i +1 ) = x i . Let us define: V i := W x i i Gal( K i ) . Then the profinite groups V i ’s are pronilpotent, they project onto the corresponding A i ’s and we have V i ⊆ V i +1 , so we have achieved the first step of approximatingthe conditions on W i from the statement of the Claim. We will correct these V i ’sto satisfy the conditions from the Claim fully. This is a very general procedure, westart from the following commutative diagram of inclusions, which summarizes our ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 19 situation: Gal( K ) < / / Gal( K ) < / / Gal( K ) < / / . . .V < / / < O O V < / / < O O V < / / < O O . . . . For each 0 < i < j , we define: V i,j := V j ∩ Gal( K i ) , V (1) i := cl [ j>i V i,j . It is finally the right moment to use Proposition 3.1 and thanks to this result eachprofinite group V (1) i is pronilpotent. Clearly, V (1) i ’s project again onto A i ’s and wehave V (1) i ⊆ V (1) i +1 . For each i, n > V ( n +1) i := (cid:16) V ( n ) i (cid:17) (1) , V ( ω ) i := cl ∞ [ n =1 V ( n ) i ! . Again, V ( ω ) i ’s are pronilpotent, they project onto A i ’s and we have V ( ω ) i ⊆ V ( ω ) i +1 .We can continue like this using transfinite induction as long as we wish. However,this procedure must finish after some (ordinal) number of steps — it is possible thatcountably many steps are enough, but for sure it is enough to take κ := | Aut( K ) | + of them. Then, for each i > W i := V ( κ ) i and, by the construction, these W i ’s satisfy the conditions of the Claim, whichfinishes the proof of the Claim in the case of ( I, ) = ( N , ).We sketch now how one can proceed in the case of an arbitrary directed poset( I, ). We choose a maximal antichain A in I . Without loss of generality, A isinfinite (otherwise, I can be taken to be N ). Then, we can assume that:( I, ) = (cid:0) [ A ] <ω , ⊆ (cid:1) , which is the set of finite subsets of A ordered by the inclusion relation. For any n >
0, let I n denote the subset of I consisting of subsets of A of cardinality n .Then we have: I = I ∪· I ∪· . . . and we can repeat the previous argument with taking an extra care about all theelements of I n at each level n . Namely, for any i ∈ I we defineCl i := \ a ∈A Cl iai , where ia := i ∪ { a } . The set Cl i is non-empty by the arguments as above. Then,we define Cl i := π iai (cid:0) Cl ia (cid:1) , Cl n +1 i := π i i (Cl nia )and we can continue as in the case of I = N above.Hence, we have obtained the subgroups W i Gal( K i ) for each i ∈ I whichsatisfy Items (1) and (2) from the Claim and such that for each i, j ∈ I , if i j ,then W i ⊆ W j . To correct these W i ’s to satisfy Item (3) from Claim, one can justrepeat the procedure described in the case of I = N . (cid:3) As it was explained immediately after the statement of the Claim, Item (1) (fromthe statement of Theorem 3.2, which we are still proving) directly follows from theClaim, whose proof was just finished above.We proceed now towards the proof of Item (2). Having the extra assumptionsfrom Item (2), we: • set r n,p as the rank (rather than just the cardinality) of ( A n ) ( p ) ; • replace Condition (iii) from the proof of the Claim with the following con-dition:(iii’) h res n (cid:0) x ( p ) (cid:1) i = A n .Condition (iii’) is still closed, since there is a fixed finite set of sequences of length r n,p generating the group ( A n ) ( p ) .We consider now the following commutative diagram:Gal( K j ) ( p j ) res j / / ( A j ) ( p j )= (cid:15) (cid:15) ( V j ) ( p j ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ O O (cid:15) (cid:15) ( V j +1 ) ( p j ) F . c . ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ (cid:15) (cid:15) Gal( K j +1 ) ( p j ) res j +1 / / ( A j +1 ) ( p j ) . By Remark 2.16(2), any continuous epimorphism of pro- p groups of the same finiterank is necessarily a Frattini cover. Hence, the map res j +1 restricted to ( V j +1 ) ( p j ) isa Frattini cover as indicated in the diagram above. Since res j (( V j ) ( p j ) ) = ( A j ) ( p j ) ,we get that ( V j +1 ) ( p j ) = ( V j ) ( p j ) . Hence, for each n > j , we have ( V n ) ( p j ) = ( V j ) ( p j ) .Therefore, if we repeat the process of getting W n ’s from V n ’s appearing at theend of the proof of Item (1), then for each n > j , we have( W n ) ( p j ) = ( V n ) ( p n ) . Since for each n > j , ( V n ) ( p j ) is a Frattini cover of ( A n ) ( p j ) , the proof is finished. (cid:3) Remark 3.3.
Item (1) in Theorem 3.2 cannot be improved towards the conclusionfrom the statement of Item (2). To see that, let us consider an existentially closed C p ∞ -field K and let A n := C p n . By the Claim from the beginning of Section 4, therestriction map: Gal( K n ) −→ A n = C p n is not a Frattini cover. 4. Negative results
This section is mostly about the proof of the left-to-right implication from Theo-rem 1.1. Using Remark 1.2(2), we assume that there is an infinite strictly increasingsequence P = C p < P < P < . . . ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 21 such that each P i is a finite p -subgroup of A . We aim to show that the theory A -TCF does not exist.Assume that the theory A -TCF exists and let K be an | A | + -saturated and ex-istentially closed A -field. We will reach a contradiction.By Theorem 3.2(1), for each i ∈ I , the profinite group Gal( K i ) is pronilpotent,hence it decomposes as: Gal( K i ) = Y p ∈ P Gal( K i ) ( p ) . By Corollary 2.5, for each i ∈ I , the field K i is PAC. Hence, the profinite groupGal( K i ) is projective and each pro- p group Gal( K i ) ( p ) is projective. As in Remark2.22(1), Gal( K i ) is pro- p free, so there is a cardinal κ i such that:Gal( K i ) ( p ) ∼ = b F κ i ( p ) . Claim
For each i ∈ I such that C p = P ⊆ A i , the restriction mapGal( K i ) ( p ) −→ ( A i ) ( p ) is not a Frattini cover. Proof of Claim.
Assume not and let us take i ∈ I as above such that the mapres i : Gal( K i ) ( p ) −→ ( A i ) ( p ) is a Frattini cover. Let r be the rank of ( A i ) ( p ) . By our assumption, r is finite and r >
2. Since the map res i is a Frattini cover, we get that r = rk(Gal( K i ) ( p ) ), hence r = κ i .For any j ∈ I such that i j , we have the following commutative diagram:1 / / Gal( K ) ( p )= (cid:15) (cid:15) < / / Gal( K i ) ( p ) ∼ = b F r ( p ) res i / / < (cid:15) (cid:15) ( A i ) ( p ) < (cid:15) (cid:15) / / / / Gal( K ) ( p ) < / / Gal( K j ) ( p ) ∼ = b F κ j ( p ) res j / / ( A j ) ( p ) / / . Hence, we have: t j := h b F κ j ( p ) : b F r ( p ) i = (cid:2) ( A j ) ( p ) : ( A i ) ( p ) (cid:3) . Therefore, by our main assumption on the sequence of groups ( A i ) i , the indices t j go to infinity when j → ∞ . This leads to a contradiction by the profinite versionof Nielsen-Schreier formula ([5, Proposition 17.6.2]), which we observe below.Since r is finite, κ j is finite as well for each j > i and we have r = 1 + t j ( κ j − . Since r >
2, then for each j > i we have κ j − >
0. Since t j ’s go to infinity with j ,then the constant r tends to infinity as well, which is obviously a contradiction. (cid:3) Using the Claim above, we obtain that for each i ∈ I such that C p ⊆ A i , themap: res i : Gal( K i ) −→ A i is not a Frattini cover, which is witnessed by a closed proper subgroup H i < Gal( K i )such that: res i ( H i ) = A i . Without loss of generality, H i is a maximal proper closed subgroup of Gal( K i ).Since the profinite group Gal( K i ) is pro- p , we get that[Gal( K i ) : H i ] = p. Hence, by Lemma 2.8, for each such i ∈ I , there is an A i -field extension K ⊂ L i ofdegree p . Since K is | A | + -saturated, there is an A -field extension K ⊂ L of degree p , which gives our final contradiction ( K is an existentially closed A -field, so it isalso A -closed) and finishes the proof of the left-to-right implication in Theorem 1.1.5. Positive results
This section is about the proof of the right-to-left implication in Theorem 1.1.Similarly as in Section 3, we start this section with a very general result, which willbe needed later.
Proposition 5.1.
Let K ⊇ K ⊇ K ⊇ . . . be a decreasing tower of fields and let K ∞ := ∞ \ n =1 K n . We assume that K ∞ is PAC. Let V be an algebraic variety over K ∞ such that forall n > , we have V ( K n ) = ∅ . Then V ( K ∞ ) = ∅ .Proof. Let V = V ∪ . . . ∪ V d be the decomposition of V into irreducible componentsover K alg ∞ . We assume that the result does not hold (for this fixed tower ( K n ) n )and take a counterexample V , which is minimal with respect to (dim( V ) , d ). Wewill reach a contradiction.Since K ∞ is PAC, we have d >
1. Without loss of generality, we may assumethat for each n , the set Q n := V ( K n ) ∩ V (cid:0) K alg ∞ (cid:1) is non-empty. Let W n be the Zariski closure of Q n for each n inside V (cid:0) K alg ∞ (cid:1) .Then, for each n we have the following: • W n is non-empty; • W n is defined over K n ; • W n ⊆ V ; • W n ( K n ) = ∅ ; • W n ’s form a descending chain.Hence, there is a variety W such that for n ≫
0, we have W = W n . Therefore, W is defined over K ∞ = T K n , and W satisfies the assumption of the statementwe are proving. But, since W ⊆ V and d >
1, either dim( W ) < dim( V ) or thenumber of irreducible components (over K alg ∞ ) of W is smaller than d (actually, ifdim( W ) = dim( V ), then W = V , so W is absolutely irreducible). By minimalityof V , we get W ( K ∞ ) = ∅ . But then V ( K ∞ ) = ∅ , a contradiction. (cid:3) Remark 5.2. (1) The conclusion of Lemma 5.1 can be easily strengthened byreplacing the variety V with a constructible set . By a constructible set de-fined over a field M , we just mean a quantifier-free formula in the languageof rings with parameters from M . If we evaluate this formula on M alg , ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 23 then we get a “classical” constructible set which is a Noetherian topologi-cal space with the induced Zariski topology and its irreducible componentsare constructible sets as well (defined over M alg ). To make the proof ofLemma 5.1 work in this context, one only needs to notice that since for anabsolutely irreducible variety W over a PAC field C , we have that W ( C ) isZariski dense in W ([5, Proposition 11.1.1]), then any absolutely irreducibleconstructible set over C has a C -rational point as well.(2) If we do not put assumptions on the intersection of the tower of fields, thenLemma 5.1 fails, we give an example below. For n >
0, let K n := (cid:16) F alg3 (cid:17) Z p × ... × Z pn (we enumerate the primes P = ( p n ) n> ). Each field extension F ⊂ K n is infinite algebraic, so, by [5, Corollary 11.2.4], each field K n is PAC.Therefore, it is enough to take an absolutely irreducible variety V over F such that V ( F ) = ∅ , for example: V = V ( Y − X + X + 1)(we could have taken any finite field in place of F , see [5, Example 11.2.9]).We proceed towards the right-to-left implication in Theorem 1.1. Let us fix firsta “good” torsion Abelian group B of a special kind, that is we assume that for all p ∈ P , the p -torsion subgroup B ( p ) is finite. We also fix an enumeration of theprimes P = ( p n ) n> and for each n ∈ N , we define the finite subgroup of B : B n := B ( p ) ⊕ . . . ⊕ B ( p n ) . Then ( B n ) n is an increasing sequence such that B = S n B n as in the assumptionsof Theorem 3.2(2). Let K be a B -field and, as usual, we define K n as K B n . Themain point is to show the following result below. We would like to point out thatthe conditions (1)–(3) below are exactly the Galois axioms from Definition 1.3.
Theorem 5.3.
The B -field K is existentially closed if and only if the followingconditions hold: (1) K is strict and perfect; (2) for each n ∈ N , K n is PAC; (3) we have Gal( K ) ∼ = ker (cid:18) eb B −→ b B (cid:19) = Y t> ker (cid:16) ] B ( p t ) −→ B ( p t ) (cid:17) , and for each n > , we have Gal( K n ) ∼ = f B n × Y t>n ker (cid:16) ] B ( p t ) −→ B ( p t ) (cid:17) . Proof.
For the implication “ ⇒ ”, we notice first that any e.c. B -field is strict andperfect (see e.g. [8, Lemma 3.1] and [8, Lemma 3.4], which hold for an arbitrarygroup). We proceed to show Items (2) and (3). Let us fix n ∈ N . By Theorem3.2(1), the profinite group Gal( K n ) is pronilpotent, hence we have:Gal( K n ) = Y p ∈ P Gal( K n ) ( p ) . We need to show that for any j >
0, we have: (i) if j < n , then Gal( K n ) ( p j ) ∼ = ] B ( p j ) ;(ii) if j > n , then Gal( K n ) ( p j ) ∼ = ker (cid:16) ] B ( p j ) −→ B ( p j ) (cid:17) . In the situation of Item (i), we get what we want directly from Theorem 3.2(2).In the situation of Item (ii), we consider the following short exact sequence:1 / / Gal( K n ) / / Gal( K j ) res / / Gal( K n /K j ) / / . Since K j = K B j and K n = K B n , we get (using that K is a strict B -field) thefollowing: Gal( K n /K j ) ∼ = B j /B n ∼ = B ( p n +1 ) ⊕ . . . ⊕ B ( p j ) . By the isomorphism above and Theorem 3.2(2), we get the following short exactsequence:1 / / Gal( K n ) ( p j ) / / Gal( K j ) ( p j ) ∼ = ] B ( p j ) res / / B ( p j ) / / , which gives the desired description of Gal( K n ) ( p j ) .For the implication “ ⇐ ”, let us assume that K is a B -field satisfying the con-ditions (1)–(3) above. We need the following conclusion of the Galois axioms inthis case. Claim 1 K is B -closed. Proof of Claim 1.
Let K ⊆ K ′ be an algebraic B -field extension. We aim to showthat K ′ = K . For each n > K ⊆ K ′ is an algebraic B n -field extension. Let K ′ n := ( K ′ ) B n and G n Gal( K n ) be a closed subgroup such that K ′ n = (cid:0) K alg (cid:1) G n . Let res n : Gal( K n ) −→ B n = Gal( K/K n )be the restriction map. By Lemma 2.8, we obtain thatres n ( G n ) = B n . By our assumption, the profinite group Gal( K n ) is pronilpotent and for each t n the map res n : Gal( K n ) ( p t ) −→ B ( p t ) = Gal( K/K n ) ( p t ) is a (necessarily universal) Frattini cover. Hence, for each t n , we obtain:( G n ) ( p t ) = ] B ( p t ) = Gal( K n ) ( p t ) . In particular, for any n > t = n ):( ∗ ) ( G n ∩ Gal( K )) ( p n ) = ^ B ( p n ) ∩ Gal( K n ) = ker (cid:16) ^ B ( p n ) −→ B ( p n ) (cid:17) = Gal( K ) ( p n ) . However, for each n >
0, we have (see Lemma 2.8): (cid:0) K alg (cid:1) G n ∩ Gal( K ) = K ′ . ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 25
Hence, there is a closed subgroup H Gal( K ) such that for every n > H = G n ∩ Gal( K ) , K ′ = (cid:0) K alg (cid:1) H . By ( ∗ ), we get that for each n > H ( p n ) = Gal( K ) ( p n ) . Therefore, H = Gal( K ) and K = K ′ , which we needed to show. (cid:3) The next claim is just a restatement of the Galois axioms.
Claim 2
For any n >
0, we have the following commutative diagram with exact rows (where,for clarity, we skip the trivial groups): Q t> ker (cid:16) ] B ( p t ) → B ( p t ) (cid:17) < / / f B n × Q t>n ker (cid:16) ] B ( p t ) → B ( p t ) (cid:17) / / B n Gal( K ) ∼ = O O < / / Gal( K n ) / / ∼ = O O Gal(
K/K n ) . ∼ = O O From now on, we identify all the isomorphic objects appearing in Claim 2. For n >
0, we define: K ( n ) := (cid:0) K alg (cid:1) f B n . Since we have: f B n · Y t> ker (cid:16) ] B ( p t ) → B ( p t ) (cid:17) = f B n × Y t>n ker (cid:16) ] B ( p t ) → B ( p t ) (cid:17) , we get by Claim 2 that: Gal (cid:0) K ( n ) (cid:1) · Gal( K ) = Gal( K n ) , hence we obtain: K ( n ) ∩ K = K n . Since K n ⊆ K is a finite Galois extension, we obtain that K ( n ) is linearly disjointfrom K over K n (see [5, Corollary 2.5.2] and the discussion below its proof). Wedefine now: K ′ ( n ) := K ( n ) K ∼ = K ( n ) ⊗ K n K. From the isomorphism above, K ′ ( n ) is naturally a B n -field extension of K . Claim 3
The fields K ′ ( n ) form a decreasing tower and we have the following: ∞ \ n =1 K ′ ( n ) = K. Proof of Claim 3.
From the definition of the field K ′ ( n ) and Claim 2, we obtain that:Gal (cid:16) K ′ ( n ) (cid:17) = Gal( K ) ∩ Gal( K ( n ) ) = ker (cid:16) f B n → B n (cid:17) . Hence, we get (“cl” below denotes the topological closure inside the profinite groupGal( K )): cl ∞ [ n =1 Gal (cid:16) K ′ ( n ) (cid:17)! = cl ∞ [ n =1 ker (cid:16) f B n → B n (cid:17)! = ker (cid:18) eb B → b B (cid:19) = Gal( K ) , which yields the claim by the Galois theory. (cid:3) By Claim 3, we see that the fields K ′ ( n ) approximate our field K . The next claimsays that they also “logically approximate” K , in the sense that these fields havebetter and better model-theoretic properties. Claim 4
For each n >
0, we have: K ′ ( n ) | = B n − TCF . Proof of Claim 4.
From the definition of the B n -field K ′ ( n ) , it follows that: (cid:16) K ′ ( n ) (cid:17) B n = K ( n ) . Since K n ⊆ K ( n ) is an algebraic field extension and K n is PAC, we get that K ( n ) is PAC as well. By the definition of K ( n ) , we get thatGal (cid:0) K ( n ) (cid:1) ∼ = f B n , hence, by Proposition 2.17, we obtain that the B n -field K ′ ( n ) is existentially closed. (cid:3) We are ready to show that K is an existentially closed B -field. Let us take aquantifier-free L B -formula ϕ ( x ) over K and a B -field extension K ⊆ K ′ such that: K ′ | = ∃ xϕ ( x ) . We aim to show that K | = ∃ xϕ ( x ).Let N > ϕ ( x ) ∈ L B N . Since K is B -closed (Claim 1), the fieldextension K ⊆ K ′ is regular. Let us take an arbitrary n > N . Since the fieldextension K ⊆ K ′ ( n ) is algebraic, K ′ is linearly disjoint from K ′ ( n ) over K (by thedefinition of regular extensions). Therefore, we have K ′ K ′ ( n ) ∼ = K ′ ⊗ K K ′ ( n ) and the field K ′ K ′ ( n ) has a natural B n -field structure extending those on K ′ and K ′ ( n ) over K . Since the formula ϕ ( x ) is quantifier-free, we have K ′ K ′ ( n ) | = ∃ xϕ ( x ).Since K ′ ( n ) is an existentially closed B n -field (Claim 4), we have K ′ ( n ) | = ∃ xϕ ( x ).For each n > N , the B N -field K ′ ( n ) is bi-interpretable with the pure field( K ′ ( n ) ) B N (see [8, Remark 2.3]). To proceed, we need the following claim. Thenotion of “uniform bi-interpretability” from this claim will be explained in the be-ginning of its proof. In this claim, we also set K ′ ( ∞ ) := K and K ( ∞ ) := K N . Claim 5
ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 27
The bi-interpretability between the B N -field K ′ ( n ) and the pure field ( K ′ ( n ) ) B N isuniform with respect to n ∈ { N, N + 1 , . . . , ∞} . In particular, there is a quantifier-free formula ψ ( y ) in the language of fields with parameters from K n such that forall n ∈ { N, N + 1 , . . . , ∞} we have: K ′ ( n ) | = ∃ xϕ ( x ) ⇔ (cid:16) K ′ ( n ) (cid:17) B N | = ∃ yψ ( y ) . Proof of Claim 5. If G is a finite group of order e , F is a G -field, and M := F G ,then (see [8, Remark 2.3]) there are M -bilinear maps m, a : M k × M k −→ M k such that ( M k , m, a ) is naturally bi-interpretable with the field F . Similarly, in thecase of a G -action, there are M -linear maps g , . . . , g e : M k −→ M k such that ( M k , m, a, g , . . . , g e ) is bi-interpretable with the G -field F .To prove our claim, it is enough to show that there are fixed K N -bilinear maps,which give K ′ ( n ) the B N -field structure for each n ∈ { N, N + 1 , . . . , ∞} . To thisend, it is enough to show that for all n ∈ { N, N + 1 , . . . , ∞} we have:( † ) K ′ ( n ) ∼ = (cid:16) K ′ ( n ) (cid:17) B N ⊗ K N K. We have the following commutative diagram of field extensions, where the arrowsare the inclusions and the Galois groups are indicated over some of the arrows: K ′ ( N ) K ′ ( n ) O O K Q t>n ker ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:16) K ′ ( n ) (cid:17) B N B N c c ❍❍❍❍❍❍❍❍ K N Q t>n ker < < ①①①①①①①①①① B N ^ ^ ❁❁❁❁❁❁❁❁❁❁❁ (cid:16) K ′ ( n ) (cid:17) B n = K ( n ) . f f ◆◆◆◆◆◆◆◆◆◆ Thanks to the diagram above, we see that: • the field K ′ ( n ) is the compositum of the fields (cid:16) K ′ ( n ) (cid:17) B N and K ; • the fields (cid:16) K ′ ( n ) (cid:17) B N and K are linearly disjoint over K N .Hence, we get the isomorphism from ( † ) above. (cid:3) Let us take the quantifier-free formula ψ ( y ) in the language of fields from Claim5. This formula corresponds to (or even: “this formula is ”, see Remark 5.2(1))a constructible set V defined over K N . By Claim 5, it is enough to show that We thank Junguk Lee for drawing a version of this diagram for us. V ( K N ) = ∅ . By Claim 5 again, we get that for each n > N , we have V ( K n ) = ∅ .By Claim 3, we obtain that: ∞ \ n = N K ( n ) = K N . Therefore, Remark 5.2(1) implies that V ( K N ) = ∅ , which finishes the proof thanksto Claim 5. (One could also arrange the original formula ϕ ( x ) in such a way thatthe resulting formula ψ ( x ) defines a variety. Then, using Proposition 5.1 (ratherthan Remark 5.2(1)) would be enough.) (cid:3) Remark 5.4.
As noted in the Introduction (below Definition 1.3), to see that theGalois axioms from Theorem 5.3 are first-order, it is enough to show that all theabsolute Galois groups Gal( K i ) appearing there are small. It is clear that if theprofinite group G is the product of its p -Sylow subgroups G ( p ) , then G is small ifand only if each G ( p ) is small. By Theorem 5.3, the profinite groups Gal( K i ) ( p ) aresmall, because they are topologically finitely generated.Therefore, for a torsion Abelian group B such that for all p ∈ P , the p -torsionsubgroup B ( p ) is finite, we get that the theory B -TCF exists and it is axiomatisedby Galois axioms from the statement of Theorem 5.3.We can conclude now the proof of our main result. Proof of Theorem 1.1.
Since the left-to-right implication was proved in Section 4,it is enough to show the right-to-left implication and the moreover part of Theorem1.1.For the right-to-left implication, let us assume that for each prime p , the p -primary part of A is either finite or it is the Pr¨ufer p -group. We decompose A as: A = A f ⊕ A ∞ , where for each p ∈ P , we have that ( A f ) ( p ) is finite, and ( A ∞ ) ( p ) = C p ∞ or ( A ∞ ) ( p ) is trivial. Let us set: P ∞ := { p ∈ P | ( A ∞ ) ( p ) = C p ∞ } = ( p i ) i> ,P := { q ∈ P | ( A f ) ( q ) = 0 } = ( q i ) i> , and for any m ∈ N and n > A ( m ) := A f ⊕ M p ∈ P ∞ C p m , ( A f ) n := n M k =1 ( A f ) ( q k ) , (cid:16) A ( m ) (cid:17) n := ( A f ) n ⊕ n M k =1 C p mk . Then A is the increasing union of the subgroups A ( m ) and each A ( m ) satisfies theassumptions on the group B in the statement of Theorem 5.3. By Theorem 5.3 andRemark 5.4, for each m ∈ N the theory A ( m ) -TCF exists and it is axiomatized bythe Galois axioms. By Fact 2.19 and Proposition 2.20, it is enough to show that if K | = A ( m +1) − TCF, then K | L A ( m ) | = A ( m ) − TCF. The proof of this last assertiondoes not differ much from the proof appearing in Example 2.21. Let us take n ∈ N and let us set P n := p . . . p n . Then, we have: (cid:16) A ( m ) (cid:17) n = P n (cid:16) A ( m +1) (cid:17) n . Let us also denote: K n := K ( A ( m +1) ) n , K ′ n := K ( A ( m ) ) n . ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 29
By Theorem 5.3(2), we get:Gal( K n ) ∼ = ^ ( A f ) n × Z P n × ∞ Y t = n +1 p t Z p t , where Z P n denotes Z p × . . . × Z p n . We have the following commutative diagramwith exact rows (generalizing the one from Example 2.21):1 / / Gal( K ) < / / Gal( K n ) res / / (cid:0) A ( m ) (cid:1) n / / / / Gal( K ) = O O < / / Gal( K ′ n ) < O O res / / P n (cid:0) A ( m +1) (cid:1) n< O O / / . Therefore, we obtain:Gal( K ′ n ) = res − (cid:16) P n (cid:16) A ( m +1) (cid:17) n (cid:17) ∼ = ^ ( A f ) n × P n Z P n × ∞ Y t = n +1 p t Z p t ∼ = ^ ( A f ) n × ∞ Y t =1 Z p t ∼ = ^ (cid:0) A ( m ) (cid:1) n × Y t>n ker (cid:18) ^ (cid:0) A ( m ) (cid:1) ( p t ) −→ (cid:16) A ( m ) (cid:17) ( p t ) (cid:19) . By Theorem 5.3(2), we get that K | L A ( m ) | = A ( m ) − TCF.For the moreover part, we need to show that the theory A − TCF is strictlysimple for A infinite. For simplicity, by Proposition 2.20 it is enough to showthat each theory A ( m ) -TCF is simple. We use [7, Corollary 4.31], which (veryconveniently for us) says that for any group G , if the theory G − TCF exists, thenit is simple if and only if the underlying fields of its models are bounded. Let ustake K | = A ( m ) − TCF. By Theorem 5.3, we have:Gal( K ) ∼ = Y p ∈ P ker (cid:18) ^ (cid:0) A ( m ) (cid:1) ( p ) −→ (cid:16) A ( m ) (cid:17) ( p ) (cid:19) . Each universal Frattini cover above is a small profinite group being finitely gener-ated. Hence each kernel above is small as well being an open subgroup of a smallprofinite group. Therefore, Gal( K ) is small, since all its pro- p components aresmall. As a result, the field K is bounded and the theory A − TCF is simple. Since K is PAC and not separably closed (if A = 0), by [13, Fact 2.6.7] the theory of thepure field K is not stable, so the theory A − TCF is also not stable. To see that A − TCF is not supersimple (if A is infinite), it is enough to look at any strictlyincreasing sequence of finite subgroups of A and consider the corresponding strictlydecreasing tower of definable subfields of invariants of K . (cid:3) Remark 5.5. (1) Example 2.21 can be generalized in the following way. Letus take A satisfying the equivalent conditions from Remark 1.2 and let K | = A − TCF. Then, we also have that K | = A (1) − TCF and the description of Gal( K ) comes from Theorem 5.3(3), that is:Gal( K ) ∼ = ker (cid:18) eb B −→ b B (cid:19) , where we have: B := A f ⊕ M p ∈ P ∞ C p for A f and P ∞ defined as in the beginning of the proof of Theorem 1.1 inthis section.(2) By expressing A as A = lim −→ m,n (cid:16) A ( m ) (cid:17) n , where again the finite subgroups (cid:0) A ( m ) (cid:1) n come from the beginning of theproof of Theorem 1.1 in this section, we see that the theory A -TCF isaxiomatised by Galois axioms in the sense of Definition 1.3. Example 5.6.
We can give now several examples of existentially closed A -fields.The ones from Items (2) and (3) below are in the spirit of (but, of course, mucheasier than) Hrushovski’s “non-standard Frobenius” from [9]. Let us define: C P := M p ∈ P C p , P = ( p i ) i> , P n := p . . . p n . (1) A C P -field is a field K with a collection of automorphisms ( σ p ) p ∈ P suchthat for all p, q ∈ P , we have σ p σ q = σ q σ p ; and for each p ∈ P , we have σ pp = id. A C P -field is strict if and only if for all p ∈ P , we have σ p = id.By Theorem 5.3, it is easy to see that if K is a strict and perfect C P -field,then K is e.c. if and only if all the fields of constants K n := Fix ( σ p ) ∩ . . . ∩ Fix ( σ p n )are pseudofinite.(2) Let q ∈ P and for each n >
0, we define the following C P -field: K q,n := (cid:16) F q Pn ; Fr P n /p , . . . , Fr P n /p n , id , id , . . . (cid:17) . Note that each C P -field K q,n is bi-interpretable with the difference field( F q Pn , Fr). Let U be a non-principal ultrafilter on the set of positive integersand we define the following C P -field: K q := Y n> K q,n , U . By Lo´s Theorem and Item (1), K q is an existentially closed C P -field ofcharacteristic q .(3) Let us define: K := Y n> K p n ,n , U , where each C P -field K p n ,n comes from Item (2) above. Similarly as in Item(2), K is an existentially closed C P -field of characteristic 0. ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 31 (4) Let us take q ∈ P and define: H := Y p ∈ P p Z p < b Z ∼ = Gal( F q ) , K := (cid:0) F alg q (cid:1) H . Then, we have: Gal( K/ F q ) ∼ = b Z / H ∼ = Y p ∈ P C p ∼ = c C P . Hence, K becomes naturally a C P -field. It is clear from Item (1) that K isan e.c. C P -field. Remark 5.7. (1) We note that if we consider in Example 5.6(2) the bi-interp-retable difference field ( F q Pn , Fr) and take the ultraproduct in the languageof difference fields, then the result is completely different: we would obtaina pseudofinite field of characteristic q with the Frobenius automorphism.(2) However, if we do the same in Example 5.6(3), then we get a pseudofinitefield of characteristic 0 with an automorphism of an infinite order, whichshould be generic in some sense. More precisely, we consider the followingdifference fields: K n := ( F q n , Fr) , q n := p P n n = p p ...p n n and their non-principal ultraproduct. Very similar difference fields wereconsidered in [21], that is: the difference fields from [21] are also ultra-products of finite Frobenius difference fields, but the order of growth of thecardinality of the finite fields in [21] seems to be much faster than in ourcase. 6. Miscellaneous results
In this section, we collect some results about model theory of group actions onfields, which did not fit to the course of the proof of the main result of this paper(Theorem 1.1). More precisely, we: • provide another argument for the non-existence of the theory A -TCF forcertain torsion Abelian groups A , • describe what we are able to prove for groups containing Z × Z , • discuss briefly the case of non-torsion Abelian groups.6.1. Another negative argument.
In this subsection, we present briefly a dif-ferent negative argument in the special case of A = C ( ω ) p . This was our originalargument and we think that it may have an independent interest. Intuitively, thecrucial property implying the non-existence of the theory A − TCF here is somekind of an auto-duplication of A inside A , i.e. A has a proper subgroup, which isisomorphic to A .We present C ( ω ) p as lim −→ C np and for a C ( ω ) p -field K , we set as usual K n := K C np .Let C = T n K n be the field of absolute constants of K . By a compactness argumentand the Galois theory, we obtain the following. Lemma 6.1.
Suppose that K is a strict C ( ω ) p -field, which is ℵ -saturated. Then,there are a , a , . . . ∈ K such that for each n , we have: • K = K n ( a , . . . , a n ) ; • [ C ( a n ) : C ] = p ; • the extension C ⊆ C ( a , . . . , a n ) is Galois and Gal( C ( a , . . . , a n ) /C ) ∼ = C np . Using Lemma 6.1, we can show the following improvement of Lemma 2.7 in thiscase (note that the profinite completion C × ωp = d C ( ω ) p is not small). Proposition 6.2.
Suppose that K is an ℵ -saturated and strict C ( ω ) p -field. Con-sider the following commutative diagram from Lemma 2.7: C × ωp α / / Gal( C alg ∩ K/C ) C ( ω ) p ι O O ϕ / / Aut(
K/C ) . res O O Then the map α is an isomorphism of profinite groups.Proof. We take a , a , . . . ∈ K given by Lemma 6.1. Then, for each n > C ⊂ C ( a , . . . , a n ) is Galois with Galois group being naturally isomorphicto C np . Hence, we obtain the following commutative diagram originating fromLemma 2.7: C × ωps (cid:1) (cid:1) ✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂ α / / Gal( C alg ∩ K/C ) res & & ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ C ( ω ) p ι O O ϕ / / Aut(
K/C ) res O O res * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ C np t n / / ⊂ qqqqqqqqqqqqq Gal( C ( a , . . . , a n ) /C ) , where s is the section of the inclusion map C np → C × ωp and t n is an isomorphism.Let ϕ n denote the following composition map:Gal( C alg ∩ K/C ) res / / Gal( C ( a , . . . , a n ) /C ) t − n / / C np . Then the map: ϕ := lim ←− n ( ϕ n ) : Gal( C alg ∩ K/C ) −→ C × ωp is the inverse map to the map α from Lemma 2.7. (cid:3) We need the following general result, which is rather obvious.
Lemma 6.3.
Let
Φ : G → H be an isomorphism of groups and assume that G − TCF exists. Then H − TCF exists and we have: H − TCF = L Φ ( G − TCF) . The next result uses the “auto-duplication” idea alluded to in the beginning ofthis subsection.
Proposition 6.4.
Suppose that C ( ω ) p − TCF exists. Then, for any existentiallyclosed C ( ω +1) p -field, its obvious reduct is an existentially closed C ( ω ) p -field. ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 33
Proof.
For each i >
0, let L i := L rings ∪ { σ , . . . , σ i } be the language of C ip -fields.Let L ω := S i L i be the language of C ( ω ) p -fields, and L ω +1 := L ω ∪ { σ } be thelanguage of C ( ω +1) p -fields.For each n >
0, we have the following group isomorphisms: s : C ( ω ) p −→ C ( ω ) p × C p , s ( σ i ) = σ i − ; c n : C ( ω ) p −→ C ( ω ) p , σ σ , σ σ , . . . , σ n σ n +1 , σ n +1 σ , σ >n +1 σ >n +1 . There are the corresponding bijections of languages: L s : L C ( ω ) p −→ L C ( ω ) p × C p , L c n : L C ( ω ) p −→ L C ( ω ) p . By Lemma 6.3 and our assumption, the theory C ⊕ ( ω +1) p − TCF exists. It isenough to show that: C ( ω ) p − TCF ⊂ (cid:16) C ( ω ) p × C p (cid:17) − TCF . Take ψ ∈ C ( ω ) p − TCF. Then, there is n > ψ ∈ L n . By Lemma6.3, L s ( ψ ) ∈ C ⊕ ( ω +1) p − TCF. By Lemma 6.3 again, L c n ( L s ( ψ )) ∈ C ⊕ ( ω +1) p − TCF.Since L c n ( L s ( ψ )) = ψ , the result follows. (cid:3) Theorem 6.5.
The theory C ( ω ) p − TCF does not exist.Proof.
Suppose that the theory C ( ω ) p − TCF exists and we will reach a contradiction.By Proposition 6.4, there is a C ( ω +1) p -field K such that K | = C ( ω +1) p − TCF , K | L ω | = C ( ω ) p − TCF . Let us set: C := K C ( ω ) p , C ′ = K C ( ω +1) p . Then, we have Gal(
C/C ′ ) = C p and:[Gal( C ′ ) : Gal( C )] = p = [Gal( C alg ∩ K/C ′ ) : Gal( C alg ∩ K/C )] . By Proposotion 6.2, we also have:Gal( C alg ∩ K/C ′ ) ∼ = C × ( ω +1) p . Hence Gal( C alg ∩ K/C ) is a codimension one F p -subspace of C × ( ω +1) p and it maybe identified with C × ωp .By Lemma 2.11, we have the following isomorphism:Ψ : Gal( C ′ ) −→ \ F ω +1 ( p ) , and Ψ (Gal( C )) = \ F ω ( p ), which is a contradiction, since the index [ \ F ω +1 ( p ) : \ F ω ( p )]is infinite. (cid:3) Groups containing Z × Z . Let G be an arbitrary group. The notion of ap.e.c. G -field, which we discussed in Section 2.1, was pointed out to us by UdiHrushovski in an attempt to show the following. Conjecture 6.6. If G has a subgroup isomorphic to Z × Z , then the theory G − TCF does not exist.
We discuss below a strategy for a proof of Conjecture 6.6, explain where is theproblem with this strategy, and give a weaker statement, which can be still provedusing this strategy.Firstly, Hrushovski’s proof of the non-existence of ( Z × Z )-TCF (see [11] and [1,Section 5.1]) gives a stronger result, which we formulate below. Theorem 6.7 (Hrushovski) . There is no ℵ -saturated and p.e.c ( Z × Z ) -field K such that: • the primitive third root of unity ζ belongs to K ; • we have: (1 , · ζ = ζ , (0 , · ζ = ζ . The hope was that if (in the situation of Conjecture 6.6) K is an ℵ -saturatedand e.c (even just p.e.c.) G -field, then its reduct to Z × Z would contradict Theorem6.7 using Proposition 2.2. However, the problem is that the induced action of Z × Z on K need not satisfy the conditions from Theorem 6.7. Actually, if G = Q × Q ,then this induced action never satisfies the conditions from Theorem 6.7.It is easy to give algebraic conditions on the group G yielding the conditionsfrom Theorem 6.7, which we do below. Proposition 6.8.
Suppose that: • there is H G such that H ∼ = Z × Z , • there is N < G of index such that H * N .Then, the theory G − TCF does not exist.Proof.
Assume that the theory G − TCF exists and we will reach a contradiction. Byour assumptions, there is a group homomorphism ϕ : G → C such that ϕ ( H ) = C .Since we have Gal( Q ( ζ ) / Q ) ∼ = C , the above homomorphism ϕ gives Q ( ζ ) the G -field structure such that the reductof this structure to H satisfies the conditions from Theorem 6.7. Since the theory G − TCF exists, we can extend the G -field Q ( ζ ) to an existentially closed and ℵ -saturated G -field K , and in this case the strategy described above works givinga contradiction with Theorem 6.7. (cid:3) We give below an explicit (although, looking a bit strange) statement, whicheasily follows from Proposition 6.8.
Corollary 6.9. If Z × Z embeds into G , then the theory ( G × C ) − TCF does notexist.Proof.
We take G to play the role of N from Proposition 6.8 and we also set: H := h ((1 , , a ) , ((0 , , a ) i , where we identify Z × Z with its image in G and take a as the generator of C . (cid:3) ODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS 35
Arbitrary Abelian groups.
It is tempting to extend Theorem 1.1 to thecase of an arbitrary Abelian group. However, there are the following problems.(1) On the negative argument side, it is not clear even how to show that thetheory ( Q × Q ) − TCF does not exist, as was pointed out in Section 6.2.(2) Regarding the positive argument side, it may be also unclear how to deale.g. with the case of C p ∞ × Q . We can present this group as: C p ∞ × Q = lim −→ n C p n × n ! Z , but, since the group C p n × Z is not finite, we can not use only the Galoisaxioms and we should also consider direct limits with respect to the Bass-Serre theory (see [1]). This may be doable, but such methods thematicallydo not fit to this paper and this circle of topics will be picked up elsewhere. Acknowledgements .We would like to thank Daniel Hoffmann, Udi Hrushovski, and Nick Ramseyfor their comments. The second author would like to thank the members of themodel theory group in Wroc law for their constructive remarks during model theoryseminars at Wroc law University.This work was partially done in the Nesin Mathematics Village (S¸irince, Izmir,Turkey), and we would like to thank the village for its hospitality.
References [1] ¨Ozlem Beyarslan and Piotr Kowalski. Model theory of fields with virtually free group actions.
Proc. London Math. Soc. , 118(2):221–256, 2019.[2] N. Bourbaki.
Algebra II: Chapters 4 - 7 . Elements of Mathematics. Springer Berlin Heidel-berg, 2003.[3] Zo´e Chatzidakis and Ehud Hrushovski. Model theory of difference fields.
Trans. AMS ,351(8):2997–3071, 2000.[4] Zo´e Chatzidakis, Ehud Hrushovski, and Ya’acov Peterzil. Model theory of difference fields.II. Periodic ideals and the trichotomy in all characteristics.
Proc. London Math. Soc. (3) ,85(2):257–311, 2002.[5] M.D. Fried and M. Jarden.
Field Arithmetic . Ergebnisse der Mathematik und ihrer Grenzge-biete. 3. Folge / A Series of Modern Surveys in Mathematics. 3rd Edition, Springer, 2008.[6] Daniel Hoffmann. On Galois groups and PAC substructures. Preprint, available on https://arxiv.org/abs/1805.11141 , accepted to
Fundamenta Mathematicae .[7] Daniel Hoffmann. Model theoretic dynamics in a Galois fashion.
Annals of Pure and AppliedLogic , 170(7):755–804, 2019.[8] Daniel Hoffmann and Piotr Kowalski. Existentially closed fields with finite group actions.
Journal of Mathematical Logic , 18(1):1850003, 2018.[9] Ehud Hrushovski. The Elementary Theory of the Frobenius Automorphisms. Preprint (24July 2012), available on .[10] Ehud Hrushovski. The Manin-Mumford conjecture and the model theory of difference fields.
Annals of Pure and Applied Logic , 112:43–115, 2001.[11] Hirotaka Kikyo. On generic predicates and automorphisms. Available on http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/25825/1/1390-1.pdf .[12] Hirotaka Kikyo and Anand Pillay. The definable multiplicity property and generic automor-phisms.
Annals of Pure and Applied Logic , 106:263–273, 2000.[13] B. Kim.
Simplicity Theory . Oxford Logic Guides. OUP Oxford, 2013.[14] Angus Macintyre. Generic automorphisms of fields.
Annals of Pure and Applied Logic ,88(2):165–180, 1997.[15] Alice Medvedev. Q ACFA. Available on http://arxiv.org/abs/1508.06007 .[16] Rahim Moosa and Thomas Scanlon. Model theory of fields with free operators in characteristiczero.
Journal of Mathematical Logic , 14(02):1450009, 2014. [17] ¨Ozcan Kasal and David Pierce. Chains of theories and companionability.
Proceedings of theAmerican Mathematical Society , 143(11):4937–4949, 2015.[18] Anand Pillay and Dominika Polkowska. On PAC and bounded substructures of a stablestructure.
The Journal of Symbolic Logic , 71(2):460–472, 2006.[19] Luis Ribes and Pavel Zalesskii.
Profinite groups . Springer New York, 2000.[20] Nils Sj¨ogren. The Model Theory of Fields with a Group Action.
Research Reportsin Mathematics, Department of Mathematics Stockholm University , 2005. Available on .[21] Tingxiang Zou. Pseudofinite difference fields. Available on https://arxiv.org/abs/1806.10026 . ♣ Boˇgazic¸i ¨Universitesi
E-mail address : [email protected] ♠ Instytut Matematyczny, Uniwersytet Wroc lawski, Wroc law, Poland
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