Amalgamation of types in pseudo-algebraically closed fields and applications

TL;DR

This paper explores the structure of unbounded PAC fields, demonstrating an amalgamation result for types over algebraically closed sets, and discusses applications including properties of omega-free PAC fields and imaginaries in PAC fields.

Contribution

It introduces a new amalgamation theorem for types in unbounded PAC fields and applies it to analyze properties like NSOP3 and imaginaries.

Findings

Omega-free PAC fields have property NSOP3

Description of imaginaries in PAC fields

Amalgamation result for types over algebraically closed sets

Abstract

This paper studies unbounded PAC fields and shows an amalgamation result for types over algebraically closed sets. It discusses various applications, for instance that omega-free PAC fields have the property NSOP3. It also contains a description of imaginaries in PAC fields.

Equations

$$\begin{array}{ccc}\mathcal{G}(E)& \stackrel{\Phi }{\leftarrow }& \mathcal{G}(F)\\ \mathrm{res}\downarrow & & \downarrow \mathrm{res}\\ \mathcal{G}(L)& \stackrel{{\Phi }_0}{\leftarrow }& \mathcal{G}(M)\end{array}$$

$$\begin{array}{ccc}\mathcal{G}al(E^{\prime }/E)& \stackrel{{\Phi }^{\prime }}{\leftarrow }& \mathcal{G}(F)\\ \mathrm{res}\downarrow & & \downarrow \mathrm{res}\\ \mathcal{G}(L)& \stackrel{{\Phi }_0}{\leftarrow }& \mathcal{G}(M)\end{array}$$

Full text

Introduction

Pseudo-algebraically closed fields (henceforth abbreviated by PAC) were introduced by Ax in his famous paper [Ax] on the theory of finite fields. The elementary theory of arbitrary PAC fields, studied among others by Cherlin-Van den Dries-Macintyre [[CDM]] and by Ershov [Er], puts in light an interesting dichotomy: definable sets are given, on the one hand by classical algebraic data, and on the other hand by elementary statements concerning the Galois group. Many of the properties of the theory of a PAC field thus reduce to the corresponding properties of its Galois group. For instance, if the subfield of algebraic numbers of the PAC field $F$ is decidable, then ${\rm Th}(F)$ will be decidable if and only if the “theory” of its absolute Galois group is decidable. One also knows that the structure of their models is complicated: a result of Duret ([Du]) asserts that a PAC field which is not separably closed has the independence property.

Interest for the model theory of PAC fields revived in the mid 90’s, when Hrushovski and Pillay ([HP]) were able to use stability theoretic techniques for groups definable in pseudo-finite fields, and more generally in bounded PAC fields (a field is bounded if for each $n>1$ it has only finitely many algebraic extensions of degree $n$). It was then observed that bounded PAC fields have a simple theory, because they satisfy the independence theorem (1991 result of Hrushovski, only published in 2005, [H]). Other results with a stability-theoretic flavour followed: in [[C1]], the author shows that a PAC field with a simple theory is necessarily bounded; a weak notion of independence is defined, and shown to be implied (in any field) by non-forking. In [[C2]], the study of unbounded PAC fields is continued, with emphasis on the theory of $\omega$-free PAC fields. The author shows that for these fields, forking is the transitive closure of weak independence, and shows versions of the independence theorem for various independence notions, the most difficult one being that $\omega$-free PAC fields of chararacteristic [math] satisfy the independence theorem with independence being the genuine non-forking. This last result is quite surprising, given that the theories of $\omega$-free PAC fields are not simple. This suggested that more can be done on unbounded PAC fields, and that their study might provide an insight of good behaviours of models of non-simple theories.

In this paper we continue the investigation of the behaviour of unbounded PAC fields. Our main result is an amalgamation result for types, similar to the (weak) independence theorem of [[C2]]. This result (Theorem LABEL:thm1) isolates the conditions under which amalgamation of types is possible. It has various consequences, notably a weak independence theorem over models for PAC fields $F$ such that $S{\mathcal{G}}(F)$ has a simple theory (Theorem LABEL:thm25), and the fact that Frobenius fields satisfy NSOP3 (see LABEL:thm41 and LABEL:cor42). It also appears as an ingredient in the description of imaginaries in PAC fields of finite degree of imperfection: an imaginary of the PAC field $F$ is equi-definable with a finite collection of pairs $(a,D)$, where $a$ is a tuple of elements of $F$ and $D$ is an imaginary of $S{\mathcal{G}}(F)$ (Theorem LABEL:thm31). We show by an example that this result is best possible.

The hope that PAC fields might provide good examples of things happening beyond simplicity was vindicated. Recent results of Chernikov and Ramsey ([CR], Theorem 6.2) show that the weak independence theorem proved in [[C2]] for Frobenius fields implies that the theory of a Frobenius field is NSOP1. Thus these fields provide a large family of new examples of structures with an NSOP1 theory. This is particularly useful as very few examples of theories with NSOP1 were known. It can be hoped that a further study of these PAC fields might lead to new insight on NSOP1 theories. The $\omega$-free PAC fields are particularly nice Frobenius fields, in which types and definable sets are well understood. As we show here, imaginaries are equally well understood.

Clearly, the connections between the neo-stability properties of the Galois group of a PAC field and those of the field also need to be explored further. Results of Nick Ramsey ([R]) suggest this is the case for the properties NSOP1 and NTP1.

The paper is organised as follows. In section 1, after setting up the notation, we recall or prove some technical results on fields and profinite groups. Section 2 contains the main result of this paper, Theorem LABEL:thm1, as well as various independence theorems and SOPn properties for $n\geq 3$. We conclude section 2 with some questions. Section 3 develops the part of the logic of complete systems which is interpretable in fields. In particular, it sets up the formalism which will enable us to deal with definable sets. This is applied in section 4 (Theorem LABEL:thm31) to give the description of imaginaries of PAC fields $F$ of finite degree of imperfection.

1 Notation and preliminary results

Recall first that a field $F$ is PAC if every absolutely irreducible variety defined over $F$ has an $F$-rational point. Equivalently, if $F$ is existentially closed in any regular extension. In this section we set up the notation, recall some classical results on PAC fields, and give two additional lemmas. We assume familiarity with elementary results on field extensions, see e.g. Chapter III of [[L]].

1.1**.**

Notation, conventions. We work in the usual language of rings ($\{+,-,\cdot,0,1\})$, sometimes expanded by adding constants for a $p$-basis. The separable closure of a field $K$ is denoted by $K^{s}$, and its absolute Galois group ${\mathcal{G}}al(K^{s}/K)$ by ${\mathcal{G}}(K)$. If $A\subseteq K$, then ${\rm acl}(A)$ denotes the model-theoretic closure of $A$ in the sense of ${\rm Th}(K)$. It is known that $K$ is a regular extension of ${\rm acl}(A)$.

We will often work inside the separable closure of a field $K$. In that case, we will denote by SCF the theory ${\rm Th}(K^{s})$, the notation $tp_{{\rm SCF}}(\ \ )$ will refer to the type in the field $K^{s}$. We use the notation ${\rm acl}_{K^{s}}(A)$ to denote the algebraic closure in the sense of ${\rm Th}(K^{s})$, i.e., the smallest subfield of $K^{s}$ containing $A$ and of which $K^{s}$ is a regular extension. We will say that two subsets of $K$ (or of $K^{s}$) are SCF-independent over some $E$ if they are independent in the sense of ${\rm Th}(K^{s})$.

In addition, unless otherwise specified, all fields will be subfields of some large algebraically closed field $\Omega$. If $A,B$ are two subfields, then $AB$ denotes the composite field.

An extremely useful and fundamental result on PAC fields is the so-called “embedding lemma” of Jarden and Kiehne:

Theorem 1.2**.**

(Lemma 20.2.2 in [[FJ]])* Let $E/L$ and $F/M$ be separable field extensions satisfying: $E$ is countable and $F$ is an $\aleph_{1}$-saturated PAC field; if ${\rm char}(F)=p>0$, assume in addition that $[E:E^{p}]\leq[F:F^{p}]$. Assume that there is an isomorphism $\varphi_{0}:L^{s}\to M^{s}$ such that $\varphi_{0}(L)=M$, and a commutative diagramme*

[TABLE]

where $\Phi_{0}:\sigma\mapsto\varphi_{0}^{-1}\sigma\varphi_{0}$, is the dual of $\varphi_{0}$, and $\Phi$ is a (continuous) homomorphism. Then $\varphi_{0}$ extends to an embedding $\varphi:E^{s}\to F^{s}$, with dual $\Phi$, and such that $F/\varphi(E)$ is separable.

Remarks 1.3**.**

We will use the following essentially immediate consequences of this result.

1. (1)

We may replace the countability hypothesis on $E$ by asking $F$ to be $|E|^{+}$-saturated. The proof is identical. 2. (2)

We will usually have that the extensions $E/L$ and $F/M$ are regular. This means that the restriction maps ${\mathcal{G}}(E)\to{\mathcal{G}}(L)$ and ${\mathcal{G}}(F)\to{\mathcal{G}}(M)$ are onto. Note that the conclusion will then be that $F/\varphi(L)$ is regular. Similarly, if $\Phi$ is onto, then the extension $F/\varphi(E)$ will be regular. 3. (3)

(Notation as above.) Let $E^{\prime}$ be a Galois extension of $E$ containing $L^{s}$, and $\Phi^{\prime}:{\mathcal{G}}(F)\to{\mathcal{G}}al(E^{\prime}/E)$ such that the following diagramme commutes:

[TABLE]

As ${\mathcal{G}}(F)$ is projective, the map $\Phi^{\prime}$ factors through a homomorphism $\Phi:{\mathcal{G}}(F)\to{\mathcal{G}}(E)$ (see Theorem 11.6.2 in [[FJ]]). Applying the embedding lemma therefore gives us an embedding $\varphi^{\prime}:E^{\prime}\to F^{s}$, with dual $\Phi^{\prime}$.

Complete systems associated to profinite groups

Cherlin, Van den Dries and Macintyre show in [[CDM]] how to associate to any profinite group $G$ a structure $SG$ in an $\omega$-sorted language ${\mathcal{L}}_{G}$, called the complete system of $G$, which encodes precisely the inverse system of all finite continuous quotients of $G$. The functor $G\mapsto SG$ is a contravariant functor, and defines a duality between the category of profinite groups with continuous epimorphisms and the category of complete systems with embeddings. The functor dual to $S$ is the functor $G$ which to a complete system $S$ associates the inverse limit of the inverse system of finite groups given by $S$. An important remark, which is at the core of the results of Cherlin Van den Dries and Macintyre, is that the functor $S{\mathcal{G}}$ commutes with ultraproducts and therefore with ultrapowers: If ${\mathcal{U}}$ is an ultrafilter on a set $I$ and $K$ is a field, then $S{\mathcal{G}}(K^{\mathcal{U}})\simeq(S{\mathcal{G}}(K))^{\mathcal{U}}$, where the second ultraproduct is taken in the $\omega$-sorted context (i.e., sort by sort). Hence, $K\equiv L$ implies $S{\mathcal{G}}(K)\equiv S{\mathcal{G}}(L)$. In an unpublished manuscript, Cherlin, Van den Dries and Macintyre also show that this $\omega$-sorted logic on $S{\mathcal{G}}(K)$ is in some sense the strongest logic of the Galois group ${\mathcal{G}}(K)$ which is interpretable in the field $K$. For more details on complete systems and their logic, see [[CDM]] or the Appendix of [[C2]].

We will first briefly recall the notation and definitions for arbitrary profinite groups, before going to the setting of Galois groups.