Model theory of endomorphisms of separably closed fields Zoé Chatzidakis – CNRS, Université Paris 7 Ehud Hrushovski ^{1}^{,}^{2} – Hebrew University

Introduction. The original motivation of this paper was to study the asymptotic theory of the difference fields (𝔽_{p}(t)^{s}, x ↦ x^{p}) as p goes to ∞, where 𝔽_{p}(t)^{s} denotes the separable closure of the field 𝔽_{p}(t). For each p, the field 𝔽_{p}(t)^{s} has a rich structure. For instance, the endomorphism x ↦ x^{p} is definable, as is the derivation D_{p} which satisfies D_{p}(t) = 1 (in the pure field language augmented by a constant symbol for t).

The asymptotic theory of the differential fields (𝔽_{p}(t)^{s}, D_{p}) was shown in [5] to be undecidable. Moreover, the map x ↦ x^{p} is definable, uniformly in p, in the differential field (𝔽_{p}(t)^{s}, D_{p}). In this paper we show that, a contrario, the asymptotic theory of the difference fields (𝔽_{p}(t)^{s}, x ↦ x^{p}) is decidable. It would be interesting to see where exactly the border between decidability and undecidability lies, among asymptotic theories of reducts of the differential fields (𝔽_{p}(t)^{s}, D_{p}). Other canonical theories in characteristic p are the theory of algebraically closed valued fields with distinguished Frobenius, and the theory of differentially closed fields with distinguished Frobenius. The asymptotic theory is decidable in both these cases. The differential case depends on the results of this paper. These facts, mentioned for the sake of comparison, will be shown elsewhere.

The study of the asymptotic theory of these difference fields led naturally to the concept of generic endomorphisms of fields, subject to certain constraints (K is a regular extension of the perfect closure of σ(K)). The theory of these difference fields, denoted SCFE_{e}, is then model complete in a language extending the language of difference fields: if K ⊂ L are models of SCFE_{e}, and K and σ(L) are linearly disjoint over σ(K), then K ≺ L. Many of the results obtained for the theory ACFA of generic difference fields (see (1.13) for a definition) generalise easily to this context, and we obtain for instance the decidability of the theories SCFE_{e}, a description of their completions, of the types, of independence (including the independence theorem over algebraically closed sets, which shows that their completions are simple). Much of this study depends on results obtained for the reducts to the language of pairs (namely, the pair (K, σ(K)), for (K, σ) a field with generic endomorphism σ). In addition, given a model (K, σ) of SCFE_{e}, we investigate a few natural difference fields associated to K: its inversive closure, its perfect closure and the field k = ⋂_{n∈ℕ} σ^{n}(K). We show that they are models respectively of ACFA, SCFE_{0} (provided K is ω-saturated), and ACFA. Furthermore, the only structure induced on k by K is the difference field structure, and k is stably embedded. Let us mention a result of independent interest involved in the proof of this last statement: Theorem (5.3). Let L be a model of ACFA, and let A ⊂ B be algebraically closed inversive difference subfields of L. Assume that if b ∈ B is transformally algebraic over A, then b ∈ A. Then tp(B/A) is stationary.

The machinery and results holding for models of ACFA also yield a characterisation of definable modular sets: they have finite SU-rank and are orthogonal to all fixed fields.

The paper is organised as follows. Section 1 contains a review of separably closed fields and difference fields, and sets up the notation. Section 2 studies pairs of separably closed fields (the inclusion not being an elementary one: the larger field contains the algebraic closure of the smaller field). Section 3 gives the axiomatisation of the theory of separably closed fields with a generic endomorphism, and shows that it is model complete in a natural language. It also shows that non-principal ultraproducts of the difference fields (𝔽_{p}(t)^{s}, x ↦ x^{q}) are models of this theory. Section 4 studies the elementary invariants, algebraic closure and independence, and gives a proof of the independence theorem. Section 5 studies the induced structure on the various difference fields which “live” in our models. Section 6 gives results on modular sets. Appendix A discusses the notion of “stationarity almost over a predicate”, and Appendix B gives a proof of Claim (5.3) not relying on the results of [13].

1Supported by the Israel Science Foundation

2Research carried out for the Clay Mathematics Institute as a Clay Prize research fellow

1 Preliminaries

In this section we review classical results on separably closed fields (and fields of characteristic p > 0), on difference fields and on models of ACFA. We assume familiarity with the basic notions of algebraic geometry: algebraic sets and varieties (or absolutely irreducible algebraic sets), generic points of varieties, dominant morphisms, linear disjointness, separable, primary and regular extensions, see e.g. Ch. I to III in [16]. For an introduction to classical model theory, one can consult [11].

(1.1). Setting and notation. We will always work inside a large algebraically closed field Ω, which will contain all fields considered. If K and L are subfields of Ω, we denote by KL the subfield of Ω composite of K and L, by K^{s} the separable closure of K inside Ω, i.e., the set of elements of Ω which are separably algebraic over K, and by K^{alg} the set of elements of Ω which are algebraic over K. Throughout the main body of the paper, L will denote the language of rings {+, −, ·, 0, 1}, L_{C} the language L ∪ {C}, where C is a unary relation symbol, and L_{σ} the language L ∪ {σ}, where σ is a unary function symbol.

Let p be the characteristic of Ω, and K a subfield of Ω, n ∈ ℕ. F rob will denote the identity map if p = 0, and the Frobenius automorphism x ↦ x^{p} if p > 0. For n ∈ ℤ, K^{p^{n}} denote the subfield of Ω image of K by F rob^{n}, and K^{p^{∞}} = ⋂_{n∈ℕ} K^{p^{n}}, and K^{p^{−∞}} = ⋃_{n∈ℕ} K^{p^{−n}} the perfect closure of K. Thus, if p = 0, all these fields are equal to K.

(1.2). p-bases. Up till (1.10), we assume p > 0. Details and proofs can be found in [1] §13. Let K be a subfield of Ω and k a subfield of K. Then kK^{p} is a subfield of K, and so K is a kK^{p}-vector space. We say that elements b_{1}, …, b_{n} ∈ K are p-independent over k if the set of p-monomials in b_{1}, …, b_{n}, i.e., monomials of the form b_{1}^{i(1)} ⋯ b_{n}^{i(n)} with 0 ≤ i(1), …, i(n) ≤ p − 1, is linearly independent in the kK^{p}-vector space K. Equivalently, if b_{i} ∉ kK^{p}(b_{1}, …, b_{i−1}) for i = 1, …, n.

A subset B of K is p-independent over k if every finite subset of B is p-independent over k. If B ⊂ K is not p-independent over k, then there is a finite subset B_{0} of B and b ∈ B \ B_{0} such that b ∈ kK^{p}[B_{0}]. A maximal p-independent over k subset of K is called a p-basis of K over k; if B is a p-basis of K over k, then K = kK^{p^{n}}[B] for any n ∈ ℕ. Any two p-bases of K over k have the same cardinality. A p-basis of K is a p-basis of K over 𝔽_{p}. We define the degree of imperfection of a field K as follows: if K has a finite p-basis B, then it is the size of B, and otherwise it is ∞. Observe that if B is a p-basis of K over k, then B is also a p-basis of K^{s} over k and over k^{s}, and the elements of B are algebraically independent over k.

(1.3). Recall that K is a separable extension of k if k and K^{p} are linearly disjoint over k^{p}. Equivalently, if any p-basis of k extends to a p-basis of K. If B ⊂ K is a transcendence basis of K over k such that K is separably algebraic over k(B), then B is called a separating transcendence basis of K over k. If K is finitely generated and separable over k then K has a separating transcendence basis over k. This result does not hold when K is infinitely generated over k: if t is transcendental over k, then the field ⋃_{n∈ℕ} k(t^{p^{−n}}) is a separable extension of k, but does not have a separating transcendence basis over k. If B ⊂ K is such that K is separably algebraic over k(B), then B will contain a p-basis of K over k, and therefore a separating transcendence basis of K over k is always a p-basis of K over k. The converse however only holds if K is finitely generated over k as a field.

(1.4). The λ-functions.

For each n fix an enumeration m_{i,n}(x̄) (0 ≤ i < p^{n}) of the p-monomials x_{1}^{i(1)} ⋯ x_{n}^{i(n)} with 0 ≤ i(1), …, i(n) ≤ p − 1, and define the (n + 1)-ary functions λ_{i,n} : K^{n} × K → K as follows: If the n-tuple b̄ is not p-independent, or if the (n+1)-tuple (b̄, a) is p-independent, then λ_{i,n}(b̄; a) = 0. Otherwise, the λ_{i,n}(b̄; a) satisfy

a = ∑_{i=0}^{p^{n}−1} λ_{i,n}(b̄; a)^{p}m_{i,n}(b̄).

Note that these functions depend on the field K, and that the above properties define them uniquely. These functions are definable in the pure ring K, and we call them the λ-functions of K. One checks easily that a subfield k of K is closed under the λ-functions of K if and only if K is a separable extension of k. Furthermore, if k is closed under the λ-functions of K and a ∈ k^{s}, then k(a) is also closed under the λ-functions of K, since k(a) = k(a^{p}).

(1.5). Separably closed fields. For each e ∈ ℕ ∪ {∞}, the theory expressing that K is a separably closed field of degree of imperfection e, is a complete theory (Ershov [10]), which we denote by SCF_{e}, and is stable (Wood [19]). If K is separably closed and {b_{1}, …, b_{e}} is a p-basis of K, then SCF_{e,b} = T h(K, b_{1}, …, b_{e}) is model complete in the language L(b_{1}, …, b_{e}) and eliminates imaginaries. By a result of Delon [8], in the language L_{λ} = {+, ·, 0, 1, λ_{i,n}, n ∈ ℕ, 0 ≤ i < p^{n}}, the theory of separably closed fields, expanded by axioms expressing the defining properties of the λ-functions, eliminates quantifiers; its completions are obtained by specifying the degree of imperfection. We will fix a bijection between the set of pairs (i, n) with n ∈ ℕ and 0 ≤ i < p^{n} and a set I.

(1.6). Algebraic and definable closures in separably closed fields.

We fix a degree of imperfection e ∈ ℕ ∪ {∞}, a separably closed field K of degree of imperfection e and characteristic p > 0. The model theoretic results on separably closed fields which appear below can be found in [8].

Let B be a subfield of K. Then dcl_{K}(B), the definable closure of B in the field K, is the field generated by closing B under the λ-functions of K. The algebraic closure of B, denoted by acl_{K}(B), is the separable closure of dcl_{K}(B).

(1.7). Generics in separably closed fields. Let K be as above, B = acl_{K}(B) ⊆ K, and let n ∈ ℕ. By stability of SCF_{e}, the generic n-type over B is unique. Let a = (a_{1}, …, a_{n}) ∈ K^{n}. The following are necessary and sufficient conditions for tp(a_{1}, …, a_{n}/B) to be generic:

(1) If e = 0, a_{1}, …, a_{n} are algebraically independent over B.

(2) If e = ∞, then a_{1}, …, a_{n} are p-independent over B in K.

(3) Assume e ∈ ℕ and B contains a p-basis {b_{1}, …, b_{e}} of K, and define A_{m} by induction on m as follows: A_{0} = {a_{1}, …, a_{n}}; A_{m+1} = {λ_{i,e}(b_{1}, …, b_{e}; b) | b ∈ A_{m}, 0 ≤ i < p^{e}}. Then for each m the elements of A_{m} are algebraically independent over B. Note also that dcl_{K}(B, a) = B(A_{m} | m ∈ ℕ).

(4) Assume e ∈ ℕ, {b_{1}, …, b_{f}} is a p-basis of B, and f < e. If n ≥ (e − f) and c is any (e − f)-sub-tuple of a, then c is p-independent over B, and tp(a \ c/B(c)^{s}) is the generic (n − e + f)-type over B(c)^{s}. If n ≤ e − f, then a is p-independent over B.

Observation. Assume that tp(a/E) is generic and that B is a subfield of K^{p^{∞}}. Then tp(a/EB) is also generic: this is because the generic type is orthogonal to all types realised in K^{p^{∞}}.

(1.8). Forking in separably closed fields. Let K be as above, and A = acl_{K}(A), B = acl_{K}(B) and C = acl_{K}(C) be subsets of K, with C ⊆ A ∩ B. We will describe necessary and sufficient conditions for tp(A/B) not to fork over C (in that case we also say that A and B are independent over C). Fix p-bases A_{0} ⊂ A of A over C and B_{0} ⊂ B of B over C.

(a) If e = ∞, then A and B are independent over C if and only if A_{0} ∪ B_{0} remain p-independent over C in K, and A and B are linearly disjoint over C.

(b) If e ∈ ℕ and one of A_{0}, B_{0} is empty, then A and B are independent over C if and only if they are linearly disjoint over C.

(c) Assume that e ∈ ℕ, and A_{0}, B_{0} ≠ ∅. Then tp(A/B) does not fork over C if and only if tp(A_{0}/B) does not fork over C, and tp(A/B, A_{0}) does not fork over (C, A_{0}). Clearly, tp(A_{0}/B) does not fork over C if and only if A_{0} realises the generic |A_{0}|-type over B. Hence, tp(A/B) does not fork over C if and only if A_{0} realises the generic |A_{0}|-type over B, and A and dcl_{K}(A_{0}, B) are linearly disjoint over dcl_{K}(C, A_{0}) = C(A_{0}).

Assume that A and B are independent over C. In cases (a) and (b), the composite field AB is closed under the λ-functions of K, and therefore dcl_{K}(A, B) = AB, acl_{K}(A, B) = (AB)^{s}. In case (c), we get dcl_{K}(A, B) = Adcl_{K}(A_{0}, B), and therefore acl_{K}(A, B) = (Adcl_{K}(A_{0}, B))^{s}. By (1.7)(3), dcl_{K}(A_{0}, B) is the union of purely transcendental extensions of B, and we get dcl_{K}(A, B) = ⋃_{m} AB(A_{m}) (in the notation of (1.7)(3)).

(1.9). Remarks. Let K ⊧ SCF_{e}.

(1) Let A = dcl_{K}(A) ⊆ K, and let B be a countable subset of K. Then dcl_{K}(A, B) is countably generated over A. If B ⊆ A^{alg}, then dcl_{K}(A, B) = A(B).

(2) Let A = dcl_{K}(A), B = dcl_{K}(B) and C = dcl_{K}(C) be subfields of K, with C ⊆ A∩B, and tp(A/B) not forking over C. The following conditions are equivalent:

(i) tp(A/C) has a unique non-forking extension to B.

(ii) A and B ∩ C^{s} are linearly disjoint over C.

Proof. (1) The second assertion is obvious. For the first assertion, if e ∈ ℕ, this is clear. If e ∈ ∞, see [8].

(2) That (i) implies (ii) is clear. Assume (ii) holds, and note that it implies that A ⊗_{C} B is an integral domain, with field of fractions the field AB. Hence, if AB is closed under the λ-functions of K, then dcl_{K}(AB) = AB, so that (i) is clear.

Let us therefore assume that AB is not closed under the λ-functions of K, and let A_{0} be a p-basis of A over C. Our assumption implies that tp(A_{0}/B) is the generic |A_{0}|-type over B, which is unique. To conclude, it will suffice to show that tp(A/A_{0}C) has a unique non-forking extension to dcl_{K}(B, A_{0}). Note that dcl_{K}(C, A_{0}) = C(A_{0}). The description of dcl_{K}(B, A_{0}) given in (1.7) shows that dcl_{K}(B, A_{0}) is a primary extension of B(A_{0}). Hence dcl_{K}(B, A_{0})∩C(A_{0})^{s} = B(A_{0})∩C(A_{0})^{s} = (B ∩C^{s})(A_{0}) = C(A_{0}). Then, Adcl_{K}(B, A_{0}) is closed under the λ-functions of K, and using the previous case, tp(A/C, A_{0}) has a unique non-forking extension to dcl_{K}(B, A_{0}).

(1.10). Difference rings and fields. Difference fields were first studied by Ritt in the 1930’s; we recall briefly definitions and some results, which can be found in Cohn’s book [7]. Unless otherwise indicated, the references are to [7].

A difference ring is a ring R with a distinguished injective endomorphism σ, and a difference field is a difference ring which is a field. A difference ring R is naturally an L_{σ}- structure, where L_{σ} = {+, ·, 0, 1, σ}. If σ(R) = R then R is called an inversive difference ring. If S is an inversive difference ring containing R such for every a ∈ S there is n ∈ ℕ such that σ^{n}(a) ∈ R, then S is called an inversive closure of R. Any two inversive closures of R are R-isomorphic; if R is a domain, so is its inversive closure ([2.5.2]).

(1.11). Difference polynomial rings. Let k be a difference field contained in an inversive difference field Ω, and let X_{1}, …, X_{n} be indeterminates. We define the difference polynomial ring k[X_{1}, …, X_{n}]_{σ} by taking the ring k[X_{1}, …, X_{n}]_{σ} to be the ordinary polynomial ring k[σ^{j}(X_{i}) | i = 1, …, n, j ∈ ℕ], and extending σ to k[X_{1}, …, X_{n}]_{σ} in the way suggested by the name of the generating elements. Note that σ is not onto. The order of a difference polynomial f is the largest m such that some indeterminate σ^{m}(X_{i}) appears in f.

Ideals I of k[X_{1}, …, X_{n}]_{σ} satisfying σ(I) ⊆ I are called σ-ideals. A perfect σ-ideal of k[X_{1}, …, X_{n}]_{σ} is a σ-ideal I satisfying moreover that aσ(a^{m}) ∈ I implies a ∈ I for all m ∈ ℕ. Thus a perfect σ-ideal is radical. A prime σ-ideal is a σ-ideal which is prime and perfect. Quotients of k[X_{1}, …, X_{n}]_{σ} by prime σ-ideals are difference domains, on which σ defines an embedding. While k[X_{1}, …, X_{n}]_{σ} has infinite ascending chains of σideals, it satisfies the ascending chain condition on perfect σ-ideals and on prime σ-ideals ([3.8.5]); in particular, every perfect σ-ideal of k[X_{1}, …, X_{n}]_{σ} is a finite intersection of prime σ-ideals.

Let a be a tuple of elements of Ω. We denote by I_{σ}(a/k) the ideal of k[X]_{σ} (X a tuple of indeterminates of the same length as a) of difference polynomials vanishing at a. Then I_{σ}(a/k) is a prime σ-ideal.

The difference field generated by a over k is denoted by k(a)_{ℕ}, and if k is inversive, then k(a)_{ℤ} denotes the inversive closure of k(a)_{ℕ}, i.e., the difference field k(σ^{i}(a) | i ∈ ℤ).

(1.12). Transformal transcendence bases. Let k ⊆ Ω be as above, and let a be a tuple of elements of Ω. If the transcendence degree tr.deg(k(a)_{ℕ}/k) of k(a)_{ℕ} over k is finite then we say that a is transformally algebraic over k. In that case, there is a non-negative integer m such that k(a)_{ℕ} ⊆ k(a, …, σ^{m}(a))^{alg}.

An element b ∈ Ω is transformally transcendental over k if the elements σ^{i}(b), i ∈ ℕ, are algebraically independent over k. Observe that a tuple a is either transformally algebraic over k, or contains an element which is transformally transcendental over k. We call a set B ⊆ Ω transformally independent over k if the elements σ^{j}(b), b ∈ B, j ∈ ℕ, are algebraically independent over k. If K is a difference subfield of Ω containing k, and B ⊂ K is transformally independent over k and maximal such, then B is called a transformal transcendence basis of K over k. Observe that K is then transformally algebraic over k(B)_{ℕ} ([5.5.1]). Any two transformal transcendence bases of K over k have the same cardinality, and this cardinality is called the transformal transcendence degree of K over k, and denoted by Δ(K/k) (see [5.5.2]). If a is a finite tuple, we also define Δ(a/k) = Δ(k(a)_{ℕ}/k); observe that Δ(a/k) ≤ tr.deg(k(a)/k) (the transcendence degree of k(a) over k).

(1.13). The theory ACFA

Recall that the model companion ACFA of the theory of inversive difference fields in the language L_{σ} is axiomatised by the scheme of axioms expressing the following properties of (K, σ):

(i) K is an algebraically closed field and σ is an automorphism of K.

(ii) If U and V are varieties defined over K and of the same dimension, such that V ⊆ U × σ(U) and the projections V → U and V → σ(U) are dominant, then there is a tuple a such that (a, σ(a)) ∈ V (here σ(U) denotes the variety image by σ of the variety U).

Models of ACFA are called generic difference fields.

(1.14). Conventions. Unless otherwise stated, we will always view difference fields as L_{σ}-structures. If K is a difference subfield of the difference field L, then K ⊆ L. Fix a sufficiently saturated model (Ω, σ) of ACFA. The uniqueness of the inversive closure of a difference field and the universal properties of Ω imply the following: If K ⊂ Ω is a separably closed difference field, and L is a difference field containing K, with |L| < |Ω|, then there is an L_{σ}(K)-embedding of L in Ω. From now on, all difference fields considered will be difference subfields of Ω, unless otherwise stated. Thus, if k is a difference subfield of Ω, then the inversive closure of k is simply ⋃_{n∈ℕ} σ^{−n}(k).

(1.15). Proposition ((2.11) in [4]). Let K be a difference subfield of Ω, and n > 0. If a and b are two n-tuples of transformally transcendental elements over K, then tp_{Ω}(a/K) = tp_{Ω}(b/K), i.e., the L_{σ}(K)-isomorphism which sends a to b extends to the algebraic closure of K(a)_{ℤ}.

(1.16). Algebraic closure, independence and SU-rank in models of ACFA. We work in (Ω, σ). Recall from [4] that the model-theoretic algebraic closure of a subset A of Ω is the smallest algebraically closed inversive difference subfield of Ω containing A, and we denote it by acl_{σ}(A). Hence, if k denotes the prime subfield of Ω, then acl_{σ}(A) = k(A)_{ℤ}^{alg}.

If A, B, C are subsets of Ω, we say that A and B are independent over C if acl_{σ}(AC) and acl_{σ}(BC) are linearly disjoint over acl_{σ}(C). This does correspond to non-forking in models of ACFA.

Let A = acl_{σ}(A) ⊂ Ω, and assume that |A| < |Ω|. Let a be a tuple of elements of Ω. The SU-rank of a over A, SU(a/A), is a rank based on forking, defined in the same way the U-rank is. The reader may consult [4] for details. Here we only need the dichotomy finite/infinite, which is easy to describe:

The tuple a has finite SU-rank over A iff a is transformally algebraic over A, and otherwise a has infinite SU-rank over A. If S ⊆ Ω^{n} is the set of realisations of some set of types over A, then S has finite SU-rank iff all its elements have finite SU-rank over A, and infinite SU-rank otherwise.

(1.17). Definition of modularity. Let T be a complete simple theory, M a sufficiently saturated model of T. Recall that M^{eq} is the multi-sorted structure obtained by adding to M all the imaginary elements of M (i.e., all equivalence classes of 0-definable equivalence relations), and that if A ⊂ M^{eq} then acl^{eq}(A) denotes the algebraic closure of A in the structure M^{eq}. Assume that T eliminates hyperimaginaries (i.e., if a ∈ M^{ω} and R is a type-definable equivalence relation on M^{ω}, then the R-equivalence class of a is equidefinable with some subset of M^{eq}), let D ⊆ M^{n} be invariant under Aut(M/E), for some E = acl(E) ⊂ M. We say that D is modular (also called one-based) if whenever a and b are tuples of elements of D, then a and b are independent over C = acl^{eq}(Ea) ∩ acl^{eq}(Eb).

As ACFA eliminates imaginaries (and hyperimaginaries) we have that acl^{eq}(E) = acl_{σ}(E). Hence, in models of ACFA, modularity translates as: D ⊆ Ω^{n} is modular if and only if whenever a and b are tuples of elements of D, then a and b are independent over C = acl_{σ}(Ea) ∩ acl_{σ}(Eb).

In the above condition, one may also replace b by an arbitrary tuple of Ω. For other properties of modularity in models of ACFA, see [4].

(1.18). Orthogonality. Let T be a complete (simple) theory, M a sufficiently saturated model of T, and A, B, subsets of M.

(1) Two complete types p over A and q over B are orthogonal if, for any set C containing A∪B, if a realises p and is independent from C over A, and b realises q and is independent from C over B, then a and b are independent over C.

(2) Let E = acl_{σ}(E) be a subset of Ω, and let S ⊆ Ω^{n} be a set of realisations of a set of types over E. We say that S is orthogonal to the fixed fields if over any set E′ containing E, every type realised in S is orthogonal to every type containing a formula of the form σ^{n}(x) = F rob^{m}(x) for some n ≠ 0 and m ∈ ℤ. The negation of “orthogonal to the fixed fields” is non-orthogonal to some fixed field.

(1.19). The dichotomy theorem for ACFA ([4], [6]). Let D ⊆ Ω^{n} be definable over an algebraically closed inversive difference field E. If D has infinite SU-rank, then D is not modular. Assume that D has finite SU-rank. If D is not modular then D is non-orthogonal to a fixed field.

If the characteristic of Ω is 0 and D is modular, then every type realised in D is stable, and D is stably embedded, i.e., every subset S of D definable in Ω is definable with parameters from D.

(1.20). Theorem ([4], [6]). Let G be an algebraic group defined over Ω, and let B be a definable modular subgroup of G(Ω). If X ⊆ G(Ω) is quantifier-free definable, then X ∩ B is a Boolean combination of cosets of definable subgroups of B. If char(Ω) = 0, then the same conclusion holds for arbitrary definable subsets X of G(Ω).

(1.21). Theorem. (Hrushovski [13], Macintyre [17]) Let Q be the set of prime powers, and for each q = p^{n}, let F_{q} be the difference field (𝔽_{p}^{alg}, x ↦ x^{q}). Let U be a non-principal ultrafilter on Q. Then F = ∏_{q∈Q} F_{q}/U is a model of ACFA.

2 Pairs of separably closed fields

We are interested in pairs of fields (K, C), where C and K are separably closed, and C^{alg} ⊆ K. Note that in particular C is usually not an elementary substructure of K. The examples we have in mind are:

(1) Take a separably closed field K of characteristic p, and consider a non-principal ultraproduct of the pairs (K, K^{p^{n}}), n ∈ ℕ.

(2) For each prime p, let K_{p} be a separably closed field of characteristic p, and q a power of p, and consider a non-principal ultraproduct of the pairs (K_{p}, K_{p}^{q}), p a prime.

(2.1). Setting, conventions and notations. We keep the notation and conventions of (1.1) and (1.14). Besides the characteristic p of Ω, we fix e_{1}, e_{2} ∈ ℕ ∪ {∞}, with e_{1} = e_{2} = 0 if p = 0.

We consider the languages L_{C} = L ∪ {C}, where C is a unary predicate, and L_{0} = L_{C} ∪ {R_{n} | n ∈ ℕ} ∪ {λ_{i}^{K}, λ_{i}^{C} | i ∈ I}, where each R_{n} is an n-ary relation, and I is the index set defined in (1.5) if p > 0, and I = ∅ if p = 0. Let T_{C} be the L_{C}-theory, whose models are structures K satisfying: K is a field of characteristic p and of degree of imperfection ≤ e_{1}, C is a subfield of K of degree of imperfection ≤ e_{2} and C^{p^{−∞}} ⊂ K, C^{s} ∩ K = C.

Consider now the L_{0}-theory T_{0} obtained by adding to T_{C} axioms expressing the following properties of the L_{0}-structure K:

– For each n ∈ ℕ, R_{n}(x_{1}, …, x_{n}) ⟺ the elements x_{1}, …, x_{n} are linearly independent in the C-vector space K.

– If p > 0, the λ_{i}^{K} are the λ-functions of K, and the λ_{i}^{C} are the λ-functions of C.

(2.2). Notation. We will use the notation of pairs, i.e., the notation (L, D) means that D is the interpretation of C in the model L of T_{C}. If (K, C) is a model of T_{C}, then (K, C) expands uniquely to an L_{0}-structure model of T_{0}, since the axioms of T_{0} uniquely define the interpretation of the symbols R_{n} and λ_{i}^{K}, λ_{i}^{C}.

Given a model (K, C) of T_{0}, we will often work in the L-structure K and in the Lstructure C. We will refer to these structures as the pure fields K and C. Given A ⊂ C and a tuple a in C, we will denote by tp_{C}(a/A) the type of a over A in the pure field C, and by dcl_{C}(A), acl_{C}(A) the definable and algebraic closure of A in the field C. Similarly, if A ⊂ K and a is a tuple from K, then tp_{K}(a/A) will denote the type of a over A in the pure field K, and dcl_{K}(A), acl_{K}(A) the definable and algebraic closures of the set A in the field K.

(2.3). Remarks. (1) Note that if (K_{1}, C_{1}) is an L_{0}-structure extending (K, C) and both are models of T_{0}, then:

– C_{1} and K are linearly disjoint over C.

– C_{1} is a separable extension of C and K_{1} is a separable extension of K.

(2) Assume that (K, C) and (K_{1}, C_{1}) are models of T_{0}, with K a subfield of K_{1}. Then the inclusion is an inclusion of L_{0}-structures if and only if the following conditions hold: K and C_{1} are linearly disjoint over C; C_{1} is a separable extension of C and K_{1} is a separable extension of K.

(3) Let (K, C) be a model of T_{0}, and let a be a tuple in K. Consider the variety V which is the locus of a over C (or rather over C^{alg}) and let k be its field of definition. Then k ⊆ dcl(a). Indeed, without loss of generality (replacing a by a^{p^{n}} for some integer n), we may assume that C(a) is a separable extension of C, and hence a regular extension of C. Consider the set M of monomials in the elements of the tuple a, and let M_{0} ⊂ M be maximal C-independent. Then all elements of M are in dcl(a), and if b ∈ M \ M_{0}, then b is a C-linear combination of elements of M_{0}, and the coefficients of this linear combination, being unique, are definable over a.

(4) Let a ∈ K. Then tp_{K}(a/C) is stationary (as C^{alg} = C^{p^{−∞}}), and we denote by Cb(tp_{K}(a/C)) the intersection of the canonical base Cb(tp_{K}(a/C^{alg})) of tp_{K}(a/C^{alg}) with C. We know that Cb(tp_{K}(a/C^{alg})) is the perfect closure of the field generated by the fields of definitions of all algebraic loci over C^{alg} of finite sub-tuples of dcl_{K}(a). Hence, by (3), Cb(tp_{K}(a/C)) is contained in dcl(a). Cb(tp_{K}(a/C)) can also be described as the smallest subfield A of C such that Adcl_{K}(a) and C are linearly disjoint over A.

(2.4). Definition. Consider the theory SCF_{C}(e_{1}, e_{2}) in the language L_{0} axiomatised by adding to T_{0} the following axioms:

– K and C are separably closed fields, of degree of imperfection e_{1} and e_{2} respectively.

– ∃x ¬C(x).

(2.5). Theorem. Let (K, C) ⊆ (K_{1}, C_{1}), (K_{2}, C_{2}), where (K_{1}, C_{1}) and (K_{2}, C_{2}) are models of SCF_{C}(e_{1}, e_{2}), and (K, C) ⊧ T_{0}. Then (K_{1}, C_{1}) ≡_{K} (K_{2}, C_{2}). Hence, the theory SCF_{C}(e_{1}, e_{2}) is model-complete and complete. It is the model companion of the theory T_{0}. Proof. Passing to elementary extensions, we may assume that (K_{1}, C_{1}) and (K_{2}, C_{2}) are sufficiently saturated. The proof is a standard back-and-forth argument. We consider the class I of partial L_{0}-isomorphisms φ satisfying:

– Dom(φ) = L_{1} contains K, is an L_{0}-substructure of K_{1} and a model of T_{0}.

– Im(φ) = L_{2} contains K, is an L_{0}-substructure of K_{1} and a model of T_{0}.

It is enough to show that given φ ∈ I and a ∈ K_{1}, there is ψ ∈ I extending φ and with a in its domain, and that given b ∈ K_{2}, there is ψ ∈ I extending φ and with b in its image.

Let a ∈ K_{1}, φ ∈ I with domain (L_{1}, D_{1}) and image (L_{2}, D_{2}). We will show that φ extends to an isomorphism ψ ∈ I, with domain (M_{1}, E_{1}) containing a and image (M_{2}, E_{2}). We know that L_{i} is closed under the λ-functions of K_{i}.

Consider A = Cb(tp_{K_{1}}(L_{1}, a/C_{1})); then D_{1} ⊆ A ⊆ dcl(L_{1}, a) by Remark (2.3)(3), and tp_{K_{1}}(L_{1}, a/A) has a unique non-forking extension to C_{1}. This implies that tp_{K_{1}}(a/L_{1}A) has a unique non-forking extension to L_{1}C_{1}, i.e., that (L_{1}A)^{s} ∩ dcl_{K}(L_{1}A, a) = L_{1}A.

Let B ⊆ C_{2} realise φ(tp_{C_{1}}(A/D_{1})). As C_{i} and L_{i} are linearly disjoint over D_{i} for i = 1, 2, φ extends to an isomorphism ψ : L_{1}dcl_{C_{1}}(A) → L_{2}dcl_{C_{2}}(B) sending A to B. Let b ∈ K_{2} realise the unique non-forking extension of ψ(tp_{K_{1}}(a/L_{1}A)) to L_{2}C_{2}; then ψ extends to an isomorphism θ : dcl_{K_{1}}(L_{1}dcl_{C_{1}}(A), a) → dcl_{K_{2}}(L_{2}dcl_{C_{2}}(B), b) sending a to b. By definition of A, we know that dcl_{K_{1}}(L_{1}dcl_{C_{1}}(A), a) and C_{1} are linearly disjoint over dcl_{C_{1}}(A). Our choice of b implies that dcl_{K_{2}}(L_{2}dcl_{C_{2}}(B), b) and C_{2} are also linearly disjoint over dcl_{C_{2}}(B). Hence θ ∈ I.

This shows one direction of the back-and-forth argument, and the other one follows by symmetry. Hence K_{1} ≡_{K} K_{2}. The model completeness of SCF_{C}(e_{1}, e_{2}) follows by taking K = K_{1}, and its completeness by taking K equal to the prime field. Clearly every model of T_{0} embeds in a model of SCF_{C}(e_{1}, e_{2}) (this is where we use the axiom C^{s} ∩ K = C).

(2.6). Corollary. SCF_{C}(e_{1}, e_{2}) is stable. Proof. It suffices to count the types. Let (K, C) be a model of SCF_{C}(e_{1}, e_{2}), and L ⊆ K. We may assume that L is a model of T_{0}. Let a ∈ K. By the proof of Theorem (2.5), tp(a/L) is described by the following data:

– The canonical base A of tp_{K}(dcl_{K}(L, a)/C) (see Remarks (2.3)(3) and (4)). Note that dcl_{K}(L, a) is countably generated over L. As L and C are linearly disjoint over D = L ∩ C, we obtain that A is countably generated over D. Note also that the elements of A are definable over L∪a. Then tp(a/L) contains the formulas defining the elements of A from L ∪ a, and also says that elements of dcl_{K}(L, a) which are A-linearly independent remain C-linearly independent.

– tp_{C}(A/D): by stability of the theory SCF_{e_{2}}, there are at most (|L|)^{ℵ_{0}} possibilities (since A is countably generated over D).

– tp_{K}(a/LA): again, there are at most |L|^{ℵ_{0}} possibilities.

(2.7). Corollary. Let (K, C) be a model of SCF_{C}(e_{1}, e_{2}), and let L be a substructure of K, which is a model of T_{0} and is relatively algebraically closed in K. Then L is algebraically closed (in the sense of (K, C)). Proof. This is immediate from the description of the types given in Corollary (2.6).

(2.8). Corollary. Let (K, C) ⊧ SCF_{C}(e_{1}, e_{2}). Then there is no induced structure on C, i.e., if S ⊆ K^{n} is definable, then S ∩ C^{n} is definable in C in the pure field language. Proof. By stability, S ∩ C^{n} is definable with parameters from C. The result follows from description of types given in Corollary (2.6).

(2.9). Proposition. Let (K, C) be a model of SCF_{C}(e_{1}, e_{2}), let (L, D) ⊆ (M, E) ⊆ K, with L and M algebraically closed in the pair (K, C), and let a ∈ K.

(1) Let A be the canonical base of tp_{K}(dcl_{K}(L, a)/C). The following are equivalent:

(i) tp(a/M) does not fork over L.

(ii) tp_{C}(A/E) does not fork over D, and tp_{K}(a/MC) does not fork over LA.

(2) tp(a/L) is stationary.

Proof. The fact that tp(a/L) has an extension to M satisfying the conditions of (1)(ii) is clear, and shows that (1)(i) implies (1)(ii). By stability, to prove (1) and (2) it therefore suffices to show that the conditions of (1)(ii) uniquely determine tp(a/M). Claim. MC is a primary extension of LA.

We know that M and C are linearly disjoint over E, and that each of the extensions M/(LE)^{s}, (LE)^{s}/E, and C/(EA)^{s}, (EA)^{s}/E is primary. Hence MC is a primary extension of (LE)^{s}(EA)^{s}.

From the non-forking of tp_{C}(A/E) over D and the linear disjointness of L and C over D, we deduce that the fields L, A and E are free over D (i.e., each field is free from the composite field of the other two over D). By Remark (1.9) in [4], this implies that (LE)^{alg}(EA)^{alg} ∩ (LA)^{alg} = L^{alg}A^{alg}, and therefore that

MC ∩ (LA)^{s} = (LE)^{s}(EA)^{s} ∩ (LA)^{s} = L^{alg}A^{alg} ∩ (LA)^{s} = LA.

By Remark (1.9)(2), tp_{K}(a/LA) has a unique non-forking extension to MC. Since tp_{K}(a/MC) describes in particular the isomorphism type over MC of the field dcl_{K}(M, a)C, this implies that A′ = Cb(dcl_{K}(M, a)/C) is contained in dcl_{K}(MA) ∩ C = EA, so that A′ = EA. The conditions of (1)(ii) uniquely determine tp_{C}(EA/E) and tp_{K}(a/MA), and by the discussion in (2.6) they uniquely determine tp(a/M).

(2.10). More on independence. Let L ⊆ M_{1}, M_{2} be algebraically closed subsets of (K, C) ⊧ SCF_{C}(e_{1}, e_{2}).

(1) M_{1} and M_{2} are independent over L if and only if tp_{C}(M_{1} ∩ C/M_{2} ∩ C) does not fork over L ∩ C, and tp_{K}(M_{1}/M_{2}C) does not fork over L(M_{1} ∩ C).

(2) The non-forking of tp(M_{1}/M_{2}) over L is equivalent to the following three conditions:

(a) tp_{K}(M_{1}/M_{2}) does not fork over L.

(b) tp_{K}(M_{1}M_{2}/C) does not fork over (M_{1} ∩ C)(M_{2} ∩ C).

(c) tp_{C}(M_{1} ∩ C/M_{2} ∩ C) does not fork over L ∩ C.

Proof. (1) is clear by Proposition (2.9). For (2), observe that (c) implies that tp_{K}(M_{1} ∩ C/M_{2} ∩ C) does not fork over L ∩ C. The result follows using forking calculus.

(2.11). Description of the algebraic and definable closure.

Let (K, C) ⊧ T_{0}, and let A ⊂ K. Let B = Cb(tp_{K}(A/C)). Then B ⊂ dcl(A), and hence dcl_{C}(B) ⊆ dcl(A). It follows that dcl_{K}(Adcl_{C}(B)) ⊆ dcl(A). The description of the types shows that dcl(A) = dcl_{K}(Adcl_{C}(B)), and therefore that acl(A) = dcl(A)^{s} = acl_{K}(Adcl_{C}(B)).

(2.12). Quantifier-elimination. For 1 ≤ n ∈ ℕ, consider the functions μ_{i,n} : K^{n+1} → C, i = 1, …, n, defined as follows: if the tuple ā = (a_{1}, …, a_{n}) is not linearly independent in the C-vector space K, or if b does not belong to the C-vector space generated by ā, then μ_{i,n}(ā; b) = 0 for i = 1, …, n. Otherwise the μ_{i,n}(ā; b) are uniquely defined by

b = ∑_{i=1}^{n} μ_{i,n}(ā; b)a_{i}.

One sees easily that in the language L_{0} expanded by function symbols for the μ_{i,n}, the theory SCF_{C}(e_{1}, e_{2}), together with the defining axioms for the functions μ_{i,n}, eliminates quantifiers. This follows from that fact that if A ⊂ (K, C) ⊧ SCF_{C}(e_{1}, e_{2}), then dcl(A) is the smallest field containing A and closed under the functions λ_{i,n}^{K}, λ_{i,n}^{C} and μ_{i,n}. Furthermore, note that for any n, R_{n}(x_{1}, …, x_{n}) is equivalent to x_{1} ≠ 0∧μ_{1,n}(x_{1}, …, x_{n}; x_{1}) = x_{1} so that the predicates R_{n} can be omitted from the language.

3 Fields with an endomorphism

(3.1). Conventions and setting. We keep the notation and conventions of (1.1) and (1.14). Besides the characteristic p of Ω, we fix some e ∈ ℕ ∪ {∞}, with e = 0 if p = 0.

Let T_{σ} be the L_{σ}-theory whose models are the L_{σ}-structures K satisfying: K is a field of characteristic p, σ is an endomorphism of K, and σ(K)^{p^{−∞}} ⊆ K, σ(K)^{s} ∩ K = σ(K).

Let L_{1} = L_{σ} ∪ {R_{n} | n ∈ ℕ}, and consider the L_{1}-theory T_{1} obtained by adding to T_{σ} the scheme of axioms expressing the following properties of the L_{1}-structure K:

– For each n ∈ ℕ, R_{n}(x_{1}, …, x_{n}) ⟺ the elements x_{1}, …, x_{n} are linearly independent in the σ(K)-vector space K.

(3.2). Remarks. Let K be a difference field, and L a model of T_{1}.

(1) Any difference field which is a model of T_{σ} expands uniquely to an L_{1}-structure which is a model of T_{1}. This is because the axioms added to T_{σ} define uniquely the predicates R_{n}.

(2) Assume that K is a model of T_{σ} and an L_{1}-substructure of L. Then K ⊧ T_{1} if and only if K and σ(L) are linearly disjoint over σ(K).

(3) If K ⊆ L are models of T_{1}, then, applying σ^{i}, we get that σ^{i}(K) and σ^{i+1}(L) are linearly disjoint over σ^{i+1}(K) for every i ∈ ℕ. Hence, by induction we obtain that K and σ^{i+1}(L) are linearly disjoint over σ^{i+1}(K).

(4) The universal part of T_{1} is much weaker than T_{1}, but is awkward to describe.

(3.3). Lemma. Let K ⊆ L be difference fields, and assume that σ(L) and K are linearly disjoint over σ(K).

(1) If a ∈ L is transformally algebraic over K, then a ∈ K(σ(a))_{ℕ}^{alg}.

(2) Assume that L = K(a)_{ℕ} for some finite tuple a. Let b be a subset of a forming a transcendence basis of L over Kσ(L). Then b is a transformal transcendence basis of L over K, and K(b)_{ℕ} and σ(L) are linearly disjoint over σ(K(b)_{ℕ}).

(3) If σ(L)^{p^{−∞}} ⊆ L, and the elements of the tuple b are p-independent over K in L, then they are algebraically independent over Kσ(L).

(4) If σ(K)^{1/p} ⊆ K, then L is a separable extension of K.

Proof. (1) Let m be minimal such that there is a non-zero polynomial F (X_{0}, …, X_{m}) ∈ K[X_{0}, …, X_{m}] with F (a, σ(a), …, σ^{m}(a)) = 0. If the variable X_{0} occurs in F, then a ∈ K(σ(a), …, σ^{m}(a))^{alg}. Assume that X_{0} does not occur in F. The linear disjointness of σ(L) and K over σ(K) then implies that F has its coefficients in σ(K). Hence, σ^{−1}(F) ∈ K[X_{1}, …, X_{m}], and vanishes at (a, …, σ^{m−1}(a)), contradicting the minimality of m.

(2) By (1), the elements of b are transformally independent over K. Moreover, as Kσ(L)(b) is a purely transcendental extension of Kσ(L), we get that K(b)_{ℕ} and σ(L) are linearly disjoint over σ(K(b)_{ℕ}).

It remains to show that b is indeed a transformal transcendence basis of L over K. Adjoining b to K, we may assume that b = ∅, so that a ∈ K(σ(a))_{ℕ}^{alg}. Because a is finite, there is some integer m such that a ∈ K(σ(a), …, σ^{m}(a))^{alg}. If i > 0, then applying σ^{i} and using induction, one then shows that

K(a, σ(a), …, σ^{i+m}(a)) ⊂ K(σ^{i+1}(a), …, σ^{i+m}(a))^{alg}.

This shows that tr.deg(K(a)_{ℕ}/K) is finite (≤ m tr.deg(K(a)/K)), i.e., that a is transformally algebraic over K.

(3) The assumption σ(L)^{p^{−∞}} ⊆ L implies that the elements of b are p-independent over Kσ(L) in L, and therefore they are algebraically independent over Kσ(L). Now apply (2).

(4) By hypothesis σ(K)^{1/p} and σ(L) are linearly disjoint over σ(K), and therefore σ(L) is a separable extension of σ(K). As σ is an isomorphism, L is a separable extension of K.

(3.4). Definition. Let SCFE_{e} be the L_{1}-theory axiomatised by the scheme of axioms expressing the following properties of the difference field (K, σ):

(i) T_{1}, ∃x R_{2}(1, x), K is a separably closed field of degree of imperfection e.

(ii) Assume that U and V are varieties defined over K, of the same dimension, and that V ⊆ U × σ(U) projects dominantly onto U and onto σ(U). If the characteristic is p > 0 and e > 0, assume moreover that if (a, b) is a generic of the variety V over K, then a ∈ K(b^{1/q})^{s} for some q = p^{n}, n ≥ 0. Then there is a ∈ K such that (a, σ(a)) ∈ V.

Let us explain why (ii) is indeed first order. Let c be a tuple of elements of K, and F (T, X), G(T, X, Y) tuples of polynomials with integer coefficients. It is then well-known that the following properties of the tuple c are elementary (see e.g. [9]):

F (c, X) = 0 defines an absolutely irreducible variety U, G(c, X, Y) = 0 defines an absolutely irreducible variety V contained in U × σ(U) and of the same dimension as U; the projections V → U and V → σ(U) are dominant.

If p > 0, e > 0, and V satisfies the last condition given in (ii), then there is a |Y |-tuple H(T′, X′, Y) of polynomials with integer coefficients such that H(c^{q}, X^{q}, Y) belongs to the ideal J generated by G(c, X, Y) in K[X, Y], and the matrix J(c, X, Y) = \frac{∂H}{∂X′} (c^{q}, X^{q}, Y) is non-singular (modulo the ideal J). This condition is also elementary in c, given F, G, H. Here we are using the well-known fact that an n-tuple a is separably algebraic over a field L if and only if there exists an n-tuple F (x) of polynomials over L which vanishes at a and is such that the determinant of the Jacobian matrix J_{F} (x) of F does not vanish at a.

(3.5). Theorem. The theory SCFE_{e} is the model companion of the theory obtained by adding to T_{1} an axiom saying that the degree of imperfection is ≤ e. Proof. We will first show that every model of T_{1} of degree of imperfection ≤ e embeds in a model of SCFE_{e}. Axiom (i) is no problem: σ extends to an endomorphism of K^{s}, and K is linearly disjoint from σ(K^{s}) = σ(K)^{s} over σ(K) because σ(K)^{s} ∩ K = σ(K). If (K, σ) is a model of T_{1} of characteristic p and degree of imperfection f < e, choose elements a_{1}, …, a_{e−f} ∈ Ω which are transformally independent over the inversive closure of K. Then the difference field generated over K by {a_{i}, σ(a_{i}^{1/p^{n}}) | n ∈ ℕ, i = 1, …, e−f} is a model of T_{1} of degree of imperfection e, and is an L_{1}-extension of K. If e ≠ 0, then σ(K) ≠ K. If e = 0 and σ(K) = K, let a ∈ Ω be transformally transcendental over K. Then the difference field K(a)_{ℕ}^{alg} is a model of T_{1} extending K, which is perfect, and on which the map σ is not onto.

We may therefore assume that K is separably closed and satisfies (i). Let U, V and q be as in (ii). Choose a ∈ Ω such that (a, σ(a)) is a generic of the variety V over the inversive closure of K. Since σ(a) is a generic of the variety σ(U) which is defined over σ(K),

the fields σ(K)(σ(a)) and σ^{−1}(K) are linearly disjoint over σ(K).  (1)

Observe that our conditions on U and V imply that for every m ≥ 0,

σ^{m+1}(a) ∈ σ^{m}(K(a))^{alg} and σ^{m}(a) ∈ σ^{m}(K(σ(a^{1/q}))^{s}.  (2)

Let M = σ(K(a))^{alg}; then σ(K(a)_{ℕ}) ⊆ M and a ∈ (KM)^{s} by (2). Also, M and σ^{−1}(K) are linearly disjoint over σ(K)^{alg}, by (1). Hence N = (KM)^{s} is a separable extension of K, is a separably closed difference field containing a and is a model of T_{σ}, and therefore expands uniquely to a model of T_{1}. In order to show that (K, σ) is an L_{1}-substructure of (N, σ), it suffices to show that σ(N) and K are linearly disjoint over σ(K), or equivalently, that N is linearly disjoint from σ^{−1}(K) over K.

The linear disjointness of M and σ^{−1}(K) over σ(K)^{alg} implies the linear disjointness of KM and σ^{−1}(K) over K; because K is separably closed, this implies the linear disjointness of N = (KM)^{s} and σ^{−1}(K) over K.

We now need to show that the models of SCFE_{e} are existentially closed. Let (K, σ) ⊧ SCFE_{e}, and let (L, σ) be a model of T_{1} containing (K, σ) and of degree of imperfection ≤ e. It is enough to show that there is an L_{1}(K)-embedding of L in some saturated elementary extension K^{∗} of K. Let us fix this K^{∗}. We may assume that L is separably closed, since σ(L)^{s} and K are linearly disjoint over σ(K).

It is enough to show the following:

(∗) There is an L_{σ}(K)-embedding φ of L into K^{∗} such that K^{∗} is a separable extension of φ(L), and σ(K^{∗}) is free from φ(L) over σ(φ(L)).

Indeed, the freeness condition will imply that σ(K^{∗})^{alg} = σ(K^{∗})^{p^{−∞}} and φ(L) are linearly disjoint over σ(φ(L))^{alg} = σ(φ(L))^{p^{−∞}}. The separability of K^{∗} over φ(L) will then imply the linear disjointness of σ(K^{∗}) and φ(L) over σφ(L). Thus the map φ will be an L_{1}(K)- morphism, and because L ⊧ T_{1}, will be an L_{1}-isomorphism.

Note that in order to show that K^{∗} is a separable extension of φ(L), it is enough to show that if a_{1} is a tuple of L which is p-independent over K (in L), then φ(a_{1}) is p-independent over K in K^{∗}. Similarly, to show that σ(K^{∗}) is free from φ(L) over σφ(L), it is enough to show that a tuple of elements of L which are algebraically independent over Kσ(L) is sent by φ to a tuple of elements which are algebraically independent over Kσ(K^{∗}). By compactness, it is therefore enough to show that given a finite tuple a in L and a sub-tuple a_{1} of a which is p-independent over K in L, there is an L_{σ}(K)-embedding φ of K(a)_{ℕ} in K^{∗} such that:

(a) φ(a_{1}) is p-independent over K in K^{∗}.

(b) If b ⊆ a is a transcendence basis of K(a)_{ℕ} over K(σ(a))_{ℕ}, then the elements of φ(b) are algebraically independent over Kσ(K^{∗}).

Consider M = σ(K(a)_{ℕ})^{alg}). Then KM(a) ⊆ L, and KM(a) is a finitely generated extension of KM, which contains K(a)_{ℕ}. We know that KM(a) is a separable extension of K, hence linearly disjoint from K^{p^{−∞}} over K; hence KM(a) is linearly disjoint from K^{p^{−∞}}M over KM; as M is perfect, this says that KM(a) is a separable extension of KM.

Let a_{1} ⊂ a be a tuple which is p-independent over K in L (a_{1} = ∅ if e ∈ ℕ), and let b ⊂ a be a separating transcendence basis of KM(a) over KM containing a_{1} (such a basis exists by the previous paragraph). Then the elements of b are transformally independent over K, form a transformal transcendence basis of K(a)_{ℕ} over K (because Δ(a/K) = tr.deg(K(a)_{ℕ}/K(σ(a))_{ℕ}), see lemma (3.3)), and K(a)_{ℕ} ⊆ KM(b)^{s}. Hence, for some q = p^{n}, we have that

a ⊂ (K(σ(a^{1/q}))_{ℕ}(b))^{s}.  (3)

From K ⊧ ∃x R_{2}(1, x), we deduce that σ(K) ≠ K. The saturation of K^{∗} then implies that the transcendence degree of K^{∗} over σ(K^{∗}) is infinite. If e = ∞, it also implies that any p-basis of K^{∗} over K is infinite. If e ∈ ℕ, choose c ∈ K^{∗} of the same length as b, whose elements are algebraically independent over Kσ(K^{∗}). If e = ∞, choose c ∈ K^{∗} of the same length as b, whose elements are p-independent over K (in K^{∗}, and hence algebraically independent over Kσ(K^{∗})).

By Proposition (1.15), there is a K-isomorphism of difference fields ψ : K(b)_{ℕ}^{alg} → K(c)_{ℕ}^{alg} which sends b to c. Letting M_{0} = σ(K(b)_{ℕ}^{alg}), this isomorphism restricts to a K-embedding ψ of the difference field k = (KM_{0}(b))^{s} into K^{∗}. Note that by our choice of b, we have that k is free from M over σ(k). Also, (kM)^{s} is a difference field and contains k(a)_{ℕ} (by (3)).

Consider I_{σ}(a/k). It is a prime σ-ideal; the ascending chain condition on perfect σ-ideals and the fact that a is transformally algebraic over k imply that there is an integer m such that if d ∈ Ω is such that there is a field isomorphism k(d, …, σ^{m}(d)) → k(a, …, σ^{m}(a)) which sends σ^{i}(a) to σ^{i}(d) for i = 0, …, m, and fixes k, then the difference fields k(a)_{ℕ} and k(d)_{ℕ} are k-isomorphic by an isomorphism sending a to d. Fix such an m, and let U be the algebraic locus of u = (a, …, σ^{m−1}(a)) over k, V the algebraic locus of (u, σ(u)) over k. Because a is transformally algebraic over k and by our choice of m, we know that σ(u) ∈ k(u)^{alg} and u ∈ k(σ(u))^{alg}. By equation (3), u ∈ k(σ(u^{1/q}))^{s}. Claim. σ(u) is a generic of the variety σ(U) over k.

From k(a)_{ℕ} ⊂ (kM)^{s}, we obtain σ(k(a)_{ℕ}) ⊆ σ(kM)^{s} ⊆ M. As k is free from M over σ(k), this implies that tr.deg(σ(u)/k) = tr.deg(σ(u)/σ(k)) = tr.deg(u/k) = dim(σ(U)), and this proves the claim.

Thus U, V, q satisfy the assumptions of axiom (ii) over the field k. Hence, if U′ and V′ are the images of U and V under the isomorphism ψ, axiom (ii) says that there is g ∈ K^{∗} such that (g, σ(g)) ∈ V′. The saturation of K^{∗} implies that we may assume that (g, σ(g)) is a generic of the variety V′ over ψ(k). Our choice of m implies that ψ extends to an isomorphism of difference fields φ : k(u)_{ℕ} → ψ(k)(g)_{ℕ} sending u to g. By our choice of b and c, the map φ satisfies conditions (a) and (b), and its domain contains K(a)_{ℕ}.

(3.6). Remark. Let K be a model of SCFE_{e}. Our proof shows that if L is a model of T_{1} containing K and of degree of imperfection e, then there is an elementary extension of K in which L K-embeds. This implies that in (ii) the dimension hypothesis on U, V can be dropped, i.e., that K satisfies the following scheme of axioms:

(ii’) Assume that U and V are varieties defined over K and that V ⊆ U × σ(U) projects dominantly onto U and onto σ(U). If the characteristic is p > 0 and e > 0, assume moreover that if b is a generic of the variety σ(U), then the field of definition of the irreducible components of the algebraic set V (b) = {a | (a, b) ∈ V } is contained in K(b^{1/q})^{s} for some q = p^{n}, n ≥ 0. Then there is a ∈ K such that (a, σ(a)) ∈ V.

(3.7). Ultraproducts of powers of Frobenius automorphisms. Consider the set Q of all prime powers, and for each q = p^{n} ∈ Q, choose a separably closed field K_{q} of characteristic p, which is not algebraically closed, and consider the difference field (K_{q}, σ_{q}), where σ_{q} : x ↦ x^{q}. Let U be a non-principal ultrafilter on Q, and consider K^{∗} = ∏_{q} K_{q}/U, with the distinguished endomorphism σ = (σ_{q})_{U}. Then K^{∗} is a model of T_{σ}, and therefore expands uniquely to an L_{1}-structure model of T_{1}.

Theorem. K^{∗} ⊧ SCFE_{e} for some e ∈ ℕ ∪ {∞}. Proof. Clearly K^{∗} is a separably closed field, of a certain degree of imperfection e, and σ is an endomorphism of K^{∗}, which is not onto as the fields K_{q} are not perfect. Note however that e > 0 if char(K^{∗}) = p > 0.

Let us consider an instance (U, V, q) of axiom (ii) (with q = 1 if the characteristic of K^{∗} is 0), and let c = (c_{r})_{U} ∈ K^{∗} and F (T, X), G(T, X, Y) be polynomials with integral coefficients such that the equations F (c, X) = 0 and G(c, X, Y) = 0 define the varieties U and V. If the characteristic of K^{∗} is 0, let J(T, X, Y) be the matrix \frac{∂G}{∂X} (T, X, Y), and if the characteristic is p > 0, let J(T^{q}, X^{q}, Y) be the matrix defined in (3.4).

For each r ∈ Q consider the difference field L_{r} = K_{r}^{alg}, with the automorphism σ_{r} : x ↦ x^{r}, and let L^{∗} = ∏_{r} L_{r}/U, and σ = (σ_{r})_{U}. Then (K^{∗}, σ) is a difference subfield of (L^{∗}, σ). There is A ∈ U such that if r ∈ A then r > q and the algebraic sets U_{r} and V_{r} defined by F (c_{r}, X) = 0 and G(c_{r}, X, Y) = 0 satisfy the following: U_{r} and V_{r} are varieties of the same dimension, V_{r} ⊆ U_{r} × σ_{r}(U_{r}), the projection maps V_{r} → U_{r} and V_{r} → σ_{r}(U_{r}) are dominant, and if (a_{r}, b_{r}) is a generic of the variety V_{r}, then J(c_{r}^{q}, a_{r}^{q}, b_{r}) has rank |Y |.

By Theorem (1.21), there is a set B ∈ U contained in A and such that for every r ∈ B, there is a tuple a_{r} ∈ L_{r} such that (a_{r}, σ_{r}(a_{r})) ∈ V_{r} and J(c_{r}^{q}, a_{r}^{q}, σ_{r}(a_{r})) has rank |Y |. If r ∈ B then a_{r}^{q} ∈ 𝔽_{p}(c_{r}^{q}, a_{r}^{r})^{s}; since q < r, a_{r} ∈ 𝔽_{p}(c_{r})^{s}, i.e., a_{r} ∈ K_{r}. This shows that a = (a_{r})_{U} ∈ K^{∗} and finishes the proof.

4 Elementary invariants, algebraic closure, independence, etc.

In this section we show how the proofs for models of ACFA generalise easily to our context. (4.1). Theorem. Let e ∈ ℕ ∪ {∞}, let (K_{1}, σ) and (K_{2}, σ) be models of SCFE_{e}, and assume that E is a separably closed model of T_{1} and is contained in K_{1} and in K_{2}. Then K_{1} ≡_{E} K_{2}. Proof. By (1.14), there is some E-embedding φ of K_{2} into Ω such that φ(K_{2}) is linearly disjoint from K_{1} over E. Then φ(K_{2}) ≡_{E} K_{2}, and we may therefore assume that K_{2} is linearly disjoint from K_{1} over E.

Let L = (K_{1}K_{2})^{s}. Then L is a difference subfield of Ω, as σ(K_{1}K_{2}) ⊆ K_{1}K_{2}. Moreover, as L contains the perfect closure of σ(K_{1}K_{2}), it is a model of T_{σ}. Expand L to an L_{1}- structure so that it is a model of T_{1}. Then K_{1} and K_{2} are difference subfields of L, and we want to show that they are L_{1}-substructures of L. Hence we need to show for i = 1, 2, that L is a separable extension of K_{i} and that σ(L) and K_{i} are linearly disjoint over σ(K_{i}).

Since K_{1} and K_{2} are linearly disjoint over E, L is a separable extension of K_{1} and of K_{2}. Since σ(K_{2}) is linearly disjoint from E over σ(E), and K_{2} is linearly disjoint from K_{1} over E, we deduce that σ(K_{2}) is linearly disjoint from K_{1} over σ(E). Hence σ(K_{1}K_{2}) and K_{1} are linearly disjoint over σ(K_{1}). Now σ(L) is the separable closure of the separable extension σ(K_{1}K_{2}) of the separably closed field σ(K_{1}), and σ(K_{1}K_{2}) is linearly disjoint from K_{1} over σ(K_{1}): this implies that σ(L) is linearly disjoint from K_{1} over σ(K_{1}) and shows that (K_{1}, σ) is an L_{1}-substructure of (L, σ). Similarly, (K_{2}, σ) is an L_{1}-substructure of (L, σ).

If e = ∞, or if E has degree of imperfection e, then L has degree of imperfection e, and therefore embeds in a model (M, σ) of SCFE_{e}. We then have (K_{i}, σ) ≺ (M, σ) by model-completeness of SCFE_{e}, and we get the result.

Assume now that p > 0, e ∈ ℕ, and let b_{1} be a p-basis of K_{1} over E, b_{2} a p-basis of K_{2} over E. Then b_{1} and b_{2} have the same size, and we fix a bijection f : b_{1} → b_{2}. Note that the elements of b_{1} and of b_{2} are transformally independent over E. Consider the field L_{1} = L((b − f(b))^{1/p^{n}} | b ∈ b_{1}, n ∈ ℕ). This is a purely inseparable extension of L, and σ(b − f(b))^{1/p^{n}} = σ(b)^{1/p^{n}} − σ(f(b))^{1/p^{n}} ∈ L for n ∈ ℕ. Moreover b_{1} and b_{2} are p-bases of L_{1} over E, and so L_{1} is a separable extension of K_{1} and of K_{2}, of degree of imperfection e. We now need to show that σ(L_{1}) and K_{i} are linearly disjoint over σ(K_{i}) for i = 1, 2, so that K_{1} and K_{2} will be L_{1}-substructures of L_{1}. Since σ(L_{1}) ⊆ σ(L)^{alg}, and σ(L)^{alg} is linearly disjoint from K_{1} over σ(K_{1})^{alg}, we obtain that σ(L_{1})σ(K_{1})^{alg} is linearly disjoint from K_{1} over σ(K_{1})^{alg}. Because L_{1} is a separable extension of K_{1}, the fields σ(L_{1}) and σ(K_{1})^{alg} are linearly disjoint over σ(K_{1}), and this implies that σ(L_{1}) and K_{1} are linearly disjoint over σ(K_{1}).

Similarly, σ(L_{1}) and K_{2} are linearly disjoint over σ(K_{2}). Hence L_{1} embeds in a model M of SCFE_{e}, and we conclude as in the previous case.

(4.2). Corollary. The completions of SCFE_{e} are obtained by describing the action of σ on the algebraic closure of the prime field. For each e, the theory SCFE_{e} is decidable, as well as the theory SCFE = ⋂_{e∈ℕ∪{∞}} SCFE_{e}. Proof. The first statement is immediate from Theorem (4.1). The others follow by standard arguments.

(4.3). Lemma. Let E = acl(E) ⊆ K ⊧ SCFE_{e}. Then E ⊧ T_{1}, and E is separably closed. Proof. Clearly E is a difference subfield of K, which is closed under the λ-functions of K, and is relatively algebraically closed in K, which implies that it is separably closed. Hence it is enough to show that E and σ(K) are linearly disjoint over σ(E). But this is clear by Remark (2.3)(3).

(4.4). If E = acl(E) ⊆ K ⊧ SCFE_{e}, then SCFE_{e} ∪ qftp(E) ⊢ tp(E). (Here, qftp(E) denotes the quantifier-free type of E). Proof. Immediate from Lemma (4.3) and Theorem (4.1).

(4.5). Let E ⊆ K ⊧ SCFE_{e}, and let a, b be tuples in K. Then tp(a/E) = tp(b/E) if and only if there is an E-isomorphism acl(Ea) → acl(Eb) which sends a to b. Proof. Clear by Theorem (4.1) and Corollary (4.4).

(4.6). Notation. We will work in the (pure) separably closed field K, and we need to introduce some notation: acl_{K}(−) and dcl_{K}(−) will denote the algebraic and definable closures in the field reduct K, and tp_{K}(−/−) will denote a type in the sense of K. Corollary. Let E be a difference subfield of the model K of SCFE_{e}, and assume that acl_{K}(E) = E, and that E and σ(K) are linearly disjoint over σ(E). Then acl(E) = E. Proof. Our assumption implies that E is separably closed, contains σ(E)^{alg}, and that K is a separable extension of E. Hence, if K_{1} is a difference field linearly disjoint from K over E and E-isomorphic to K, then K, K_{1} and E satisfy the hypotheses of Theorem (4.1). Hence any type realised in K \ E is realised anew in K_{1}, and this shows that E = acl(E) (by Corollary (4.5)).

(4.7). Description of the algebraic closure. Let K ⊧ SCFE_{e}, and let A ⊂ K. We build by induction on i ∈ ℕ a sequence of subsets of K which are contained in the definable closure of A as follows: Let B_{0} = dcl_{K}(A), and assume that we have defined a subset B_{i} of dcl(A). Let C_{i} be the canonical basis of tp_{K}(B_{i}/σ(K)), and let B_{i+1} = dcl_{K}(B_{i}, σ^{−1}(C_{i}), σ(B_{i}), C_{i}). Then B_{i+1} ⊂ dcl(B_{i}). Proposition. Let K, A, and the B_{n}, n ∈ ℕ, be defined as above. Then acl(A) = (⋃_{n∈ℕ} B_{n})^{s}. Proof. Let B = (⋃_{n∈ℕ} B_{n})^{s}. Then B is a separably closed difference subfield of K. Each B_{n} is closed under the λ-functions of K, and therefore K is a separable extension of B. Moreover, as B_{n} contains the perfect closure of σ(B_{n−1}), the field B contains σ(B)^{alg}. By definition, each B_{n} is linearly disjoint from σ(K) over C_{n} ⊂ σ(B_{n+1}), and therefore B is free from σ(K) over σ(B). As σ(K) is a regular extension of σ(B), this implies that B is linearly disjoint from σ(K) over σ(B). Hence B satisfies the hypotheses of Corollary (4.6), and B = acl(B).

(4.8). Corollary. Let K be a model of SCFE_{e}, and let D ⊆ K^{n} be 0-definable, defined by the formula φ(x).

(1) There is m and a set W ⊆ K^{n+m} defined by a positive quantifier-free formula such that if π : K^{n+m} → K^{n} is the natural projection, then π(W) = D.

(2) There is a partition D_{1}, …, D_{r} of D into definable sets, and for each i there is a subset W_{i} of K^{n+m_{i}} (for some m_{i}) such that the natural projection π_{i} : K^{n+m_{i}} → K^{n} restricts to a finite-to-one map from W_{i} onto D_{i}. Each W_{i} is defined by a formula ψ_{i}(x, y)∧ρ_{i}(x, y), where ψ_{i}(x, y) is a quantifier-free positive L_{1}-formula, and ρ_{i}(x, y) expresses that certain subtuples of x^{⌢}y are p-independent.

Proof. (1) By model-completeness, φ(x) is equivalent modulo SCFE_{e} to an existential formula. To conclude, note that modulo SCFE_{e} the formulas y ≠ 0 and ¬R_{m}(y_{1}, …, y_{m}) are equivalent to ∃z yz = 1 and ∃z_{1}, …, z_{m} ∑_{i=1}^{m} y_{i}σ(z_{i}) = 0 respectively.

(2) Let a be an n-tuple, and let A = a, and B and the B_{n}’s be defined as in (4.7). We know that modulo SCFE_{e}, qftp(B) ⊢ tp(a). As the field B is linearly disjoint from σ(K) over σ(B), the observation made in (1) shows that any formula of qftp(B) is implied by some positive quantifier-free formula of qftp(B) (modulo the theory SCFE_{e}).

Hence there is a tuple b in B, and a positive L_{1}-formula ψ_{a}(x, y) ∈ qftp(a, b) such that

SCFE_{e} ∪ ψ_{a}(x, y) ⊢ φ(x) or SCFE_{e} ∪ ψ_{a}(x, y) ⊢ ¬φ(x).

We will then show the following statement (∗): maybe enlarging b, there is a positive L_{1}-formula θ_{a}(x, y) ∈ qftp(a, b) and a formula ρ_{a}(x, y) ∈ tp(a, b) which expresses that certain subtuples of x^{⌢}y are p-independent, such that whenever a tuple (a′, b′) satisfies ρ_{a}(x, y) ∧ θ_{a}(x, y), then the set defined by θ_{a}(a′, y) ∧ ρ_{a}(a′, y) is finite or empty. This will show the result, since by compactness, finitely many of the formulas ∃y ψ_{a}(x, y) ∧ θ_{a}(x, y) ∧ ρ_{a}(x, y) cover φ(x).

The proof is done in two steps. We first show how to reduce to the case b ∈ B_{n} for some n, then show how the inductive definition of the B_{n}’s gives the result. Let k denote the prime field (𝔽_{p} or ℚ).

Choose a tuple c ∈ ⋃_{n} B_{n} such that b is integral algebraic over k[a, c]. Thus there is a quantifier-free positive L-formula θ(x, y, z) satisfied by (a, c, b), and such that for any (a′, c′) the set θ(a′, c′, z) is finite. Hence it suffices to show (∗) for tuples (a, b) where b ∈ ⋃_{n} B_{n}.

Observe that ⋃_{n} B_{n} is the smallest field containing a, which is closed under the functions σ, σ^{−1}|_{σ(K)}, the λ-functions of K and the functions μ_{i,n} of the pair (K, σ(K)) (see

(2.12)). If n > 0, then there is a positive quantifier-free L-definable set W ⊂ K^{n+1+p^{n}} such that for any c ∈ K^{n} which is p-independent, and d ∈ K, W (c, d) = {y ∈ K^{p^{n}} | (c, d, y) ∈ W } defines the tuple (λ_{i,n}(c; d))_{i<p^{n}} if d ∈ K^{p}(c), and is empty otherwise. Similarly, if n > 0, there is a positive quantifier-free L_{σ}-definable set W ⊂ K^{n+1+n} such that for any c ∈ K^{n} which is σ(K)-linearly independent, and d ∈ K, the set W (c, d) defines the tuple (σ^{−1}(μ_{i,n}(a, b)))_{1≤i≤n} if d belongs to the σ(K)-vector space generated by c, and is empty otherwise. This shows, using induction, that if b ∈ ⋃_{n} B_{n}, then (a, b) satisfies (∗), and finishes the proof of (2).

Remark. The sets W_{i} are almost quantifier-free definable. In fact, they are positively quantifier-free definable in the language L_{1} to which one has added symbols for the λfunctions of K. Similarly if e ∈ ℕ and one adds to L_{1} constant symbols for a p-basis of K.

(4.9). Independence. Definition. Let K ⊧ SCFE_{e}, and let A, B, E be subsets of K. We say that A and B are independent over E iff tp_{K}(acl(E, A)/acl(E, B)) does not fork over acl(E), and tp_{K}(acl(E, A)/acl(E, B)σ(K)) does not fork over acl(E)σ(acl(E, A)).

(4.10). Remarks and discussion. (1) The pair (K, σ(K)) is a reduct of K, and is a model of SCF_{C}(e, e). By (2.10), the independence of A and B over E is equivalent to the non-forking of tp_{(K,σ(K))}(acl(E, A)/acl(E, B)) over acl(E). Here tp_{(K,σ(K))}(−/−) denotes the type in the L_{0}-structure (K, σ(K)) (but acl is in the sense of the L_{1}-structure K).

(2) Hence independence is symmetric and transitive. We will show below that it corresponds to non-forking.

(4.11). Lemma. Let K ⊧ SCFE_{e}, E = acl(E) ⊆ B = acl(B) ⊆ K, and let a be a tuple of elements of K which are transformally algebraic over E.

(1) Then the elements of acl(Ea) are transformally algebraic over E, and acl(Ea) = E(a)_{ℕ}^{alg} ∩ K, and E and acl(Ea) have the same p-basis.

(2) a and B are independent over E if and only if acl(Ea) and B are linearly disjoint over E, if and only if E(a)_{ℕ} is free from B over E.

Proof. (1) Replacing a by a^{⌢}σ(a)^{⌢} ⋯^{⌢} σ^{m}(a) if necessary, we may assume that E(a)^{alg} = E(σ(a))^{alg} (by Lemma (3.3)). Claim. acl(Ea) = E(a)^{alg} ∩ K.

Because a is algebraic over E(σ(a)), E is closed under the λ-functions of K, and σ(a)^{1/p^{n}} ∈ K for every n, we get that A = E(a)^{alg} ∩ K is closed under the λ-functions of K, and has the same p-basis as E. Hence K is a regular extension of A. To finish the proof, we need to show that A and σ(K) are linearly disjoint over σ(A). The linear disjointness of E and σ(K) over σ(E) ⊂ σ(A) has two consequences:

– it is enough to show that A and Eσ(K) are linearly disjoint over Eσ(A),

– Eσ(K) is a regular extension of Eσ(A) (because σ(K) is a regular extension of σ(A)).

All elements of A being algebraic over Eσ(A), we obtain that A and Eσ(K) are linearly disjoint over Eσ(A).

(2) Clearly the independence of a and B over E implies the linear disjointness of acl(Ea) and B over E. Conversely, assume that A = acl(Ea) and B are linearly independent over E. We proved in (1) that a p-basis of E is also a p-basis of A, and this implies that tp_{K}(A/B) does not fork over E. It remains to show (see Proposition (2.9)) that tp_{K}(A/Bσ(K)) does not fork over Eσ(A): but this is clear, since A ⊂ (Eσ(A))^{alg}.

(4.12). Definition. Let K ⊧ SCFE_{e}, and let E = acl(E) ⊂ K. Let a be an n-tuple from K. We say that a is generic over E if tp_{K}(a/E) is the generic n-type over E of SCF_{e}. Note that then tp_{K}(a/Eσ(K)) is also generic, since σ(K) ⊆ K^{p^{∞}}.

(4.13). Proposition. Let K and E be as above, with K sufficiently saturated, and let n ∈ ℕ. Then K contains generic n-tuples. Moreover any two generic n-tuples realise the same type over E. Proof. By saturation of K, there is a ∈ K realising the generic n-type of SCF_{e} over E (or over (Eσ(K)^{alg})^{s}), and this shows the existence. If a is a generic n-tuple over E, then dcl_{K}(Eσ(K), a) is a union of purely transcendental extensions of Eσ(K)^{alg}, generated over Eσ(K)^{alg} by dcl_{K}(E, a). Let M_{1} be the difference subfield of K generated by σ(dcl_{K}(E, a)), and consider M = (M_{1}^{alg}dcl_{K}(E, a))^{s}. Then M is separably closed, M and σ(K) are linearly disjoint over σ(M) = M_{1} ∩ σ(K), M is closed under the λ-functions of K, and therefore M = acl(Ea).

If b is another generic n-tuple over E, then similarly acl(Eb) = (N_{1}^{alg}dcl_{K}(E, b))^{s} = N, where N_{1} is the difference subfield of K generated by σ(dcl_{K}(E, b)); moreover there is an E-isomorphism φ : dcl_{K}(E, a) → dcl_{K}(E, b) which sends a to b; because elements of dcl_{K}(E, a) which are algebraically independent over E are transformally independent over E, and similarly for elements of dcl_{K}(E, b), φ extends to a difference field isomorphism M → N. This shows the uniqueness. Remark. Let K and E be as above, and B an algebraically closed subset of K containing E. If a is generic over B, then a is also generic over E, and A = acl(Ea) and B are independent over E.

(4.14). Lemma. Let K and E be as above, with K sufficiently saturated, and let A, B be algebraically closed subsets of K. Then there is A′ realising tp(A/E) and independent from B over E. Proof. We will first assume that E contains a p-basis of A if e ∈ ℕ. Choose A′ realising the non-forking extension of tp_{(K,σ(K))}(A/E) to B, and let φ : A → A′ be an L_{0}(E)- isomorphism. Let τ = φσ|_{A}φ^{−1}. Then τ ∈ End(A′), and τ and σ agree on E. Our hypotheses imply that A′ and B are linearly disjoint over E, and that (A′B) has degree of imperfection ≤ e. Hence there is a unique ρ ∈ End(A′B) which extends σ on B and τ on A′: define ρ(a⊗b) = τ(a)⊗σ(b) for a ∈ A′ and b ∈ B, extend by linearity to A′ ⊗_{E} B, and then to its field of fractions A′B. As in Theorem (4.1), on shows that (A′B, ρ) expands to a model of T_{1} extending A′ and B. Let θ be an extension of ρ to (A′B)^{s}. Then, by the model completeness of T h(K), we may assume that (A′B)^{s} ⊆ K. This implies that (A′B)^{s} is algebraically closed in K, and proves the result.

Assume now that e ∈ ℕ, and that A_{0} ≠ ∅ is a p-basis of A over E. By Proposition (4.13), there is A′_{0} realising the |A_{0}|-generic over B, and A′_{0} and B are independent over E. Moving A by an E-automorphism, we may assume that A_{0} = A′_{0}. Then, by the first case, there is A′ realising tp(A/EA_{0}), independent from acl(BA_{0}) over acl(EA_{0}). Then A realises tp(A/E), and A is independent from B over E.

(4.15). Theorem. Let K ⊧ SCFE_{e} be sufficiently saturated, let E = acl(E) ⊆ K. Let a, b, c_{1}, c_{2} be tuples of elements in K satisfying:

(i) tp(c_{1}/E) = tp(c_{2}/E).

(ii) a and b are independent over E, a and c_{1} are independent over E, and b and c_{2} are independent over E.

Then there is c ∈ K realising tp(c_{1}/E, a) ∪ tp(c_{2}/E, b), such that c and (a, b) are independent over E. Proof. If e ∈ ℕ, we will first treat the case where E contains a p-basis of K. Let A = acl(E, a), B = acl(E, b), C = acl(E, c_{1}) and C_{2} = acl(E, c_{2}). Moving C by an A-automorphism, we may assume that C and acl(AB) are independent over E. Fix an L_{1}(E)-isomorphism f : C_{2} → C which sends c_{2} to c_{1}, and extend it to an L(B)- isomorphism g : (BC_{2})^{s} → (BC)^{s}. Consider the field L = (AB)^{s}(AC)^{s}(BC)^{s}. By Remark (1.9) in [4], we have (AB)^{s}(AC)^{s} ∩ (BC)^{s} ⊆ B^{alg}C^{alg}. As B^{alg}C^{alg} is a purely inseparable extension of BC, this implies that (BC)^{s} and (AB)^{s}(AC)^{s} are linearly disjoint over BC. By assumption, the endomorphism τ = gσg^{−1} of (BC)^{s} agrees with σ on B and on C; hence there is an automorphism ρ of L which agrees with σ on (AB)^{s}(AC)^{s} and with τ on (BC)^{s}, and we may extend ρ to L^{s}. Note that ρ(L^{s}) = ρ(ABC)^{s} = σ(L^{s}). Clearly also, L^{s} is a separable extension of (AB)^{s} and of C^{s}, and hence (AB)^{s} and C^{s} are L_{1}-substructures of (L^{s}, ρ). Hence, moving L by an (AB)^{s}-automorphism of K, we may assume that ρ = σ. Since acl(AC) = (AC)^{s} and acl(BC) = (BC)^{s}, c realises tp(c/A) ∪ tp(c_{2}/B). It remains to show that c is independent from AB over E. Clearly acl(AB) and acl(C) are linearly disjoint over E, by our choice of C and because acl(AB) = (AB)^{s}. Since L is an L_{1}- substructure of K and is model of T_{1}, we have that L and σ(K) are linearly disjoint over σ(L). This implies that Cacl(AB) and σ(K) are linearly disjoint over σ(Cacl(AB)), and therefore that C is independent from (AB) over E.

If e ∈ ℕ, we will show how to reduce to the case where E contains a p-basis of K. Let f be the degree of imperfection of E and assume that f < e. Choose an (e − f)-tuple d which is generic over acl(Eabc_{1}c_{2}), and let F = acl(Ed). By definition, (ii) holds over F. Moreover, the uniqueness of the generic type implies tp(c_{1}/F) = tp(c_{2}/F). Hence the assumptions of the theorem hold over F. Let c realise tp(c_{1}/F, a) ∪ tp(c_{2}/F, b), and independent from acl(F, a, b) over F. As c and F are independent over E, the transitivity of independence implies that c and (a, b) are independent over E.

(4.16). Corollary. Independence is non-forking, and all completions of SCFE_{e} are simple. Proof. Follows from [14].

(4.17). Remarks. (1) One cannot deduce, as in [3], that any of the completions of the theories SCFE eliminate imaginaries. One can however show, using techniques similar to those of [3], that the imaginaries of a model K of SCFE_{e} are inter-definable with the imaginaries of the pair (K, σ(K)).

(2) We will present an alternate proof of the independence theorem at the end of section 5.

(3) One can however deduce that any of the completions of SCFE eliminates hyperimaginaries. This follows immediately from the independence theorem: if (K, σ) is a model of some SCFE_{e}, E = acl(E) ⊂ K, and tp(a/E) = tp(b/E), then the Lascar strong types Lstp(a/E) and Lstp(b/E) are equal; this implies that Cb(Lstp(a/E)) ⊆ E and yields the result (see e.g. section 3.6 of [18]).

(4) If K is a model of SCFE_{e}, and a ∈ K is transformally algebraic over a difference subfield E of K, then tp(a/E) has finite SU-rank. This follows immediately from Corollary (4.16) and Lemma (4.11).

5 Various results

Notation and conventions are the same as in the previous sections.

(5.1). Theorem. Let K ⊧ SCFE_{e}, let n ≥ 1, and m ∈ ℕ if e > 0, m ∈ ℤ if e = 0. Let τ = σ^{n}F rob^{m}. Then (K, τ) ⊧ SCFE_{e}. Proof. The scheme of axioms (i) is clear: if e > 0, then m ≥ 0 and τ is an endomorphism of K; if e = 0, then K is perfect, and τ is an endomorphism of K. Case 1. n = 1 and m ≠ 0.

Then p > 0. Let U, V be varieties defined over K, with V ⊂ U × τ(U) projecting dominantly onto each of the factors, and such that if (a, b) is a generic of the variety V, then a ∈ K(b^{1/q})^{s} for some power q of p. We want to show that there is c ∈ K such that (c, σ(c^{p^{m}})) ∈ V. Consider the variety W with generic (a, b^{1/p^{m}}). Then b^{1/p^{m}} is a generic of the variety F rob^{−m}τ(U), i.e., of σ(U). Moreover, a ∈ K(b^{1/p^{m}q})^{s}. Hence, by axiom (ii), there is c ∈ K such that (c, σ(c)) ∈ W, i.e., (c, τ(c)) ∈ V. Case 2. n > 1, m = 0.

By model-completeness of SCFE_{e}, it suffices to show that if (L, τ) is a model of T_{1} extending (K, σ^{n}), there is a model (M, ρ) of T_{1} containing (K, σ) such that (L, τ) is an L_{1}-substructure of (M, ρ^{n}).

Without loss of generality, L is separably closed. For i = 0, …, n − 1, choose by induction on i isomorphisms f_{i} : L → L_{i} extending σ^{i} on K, such that: f_{0} = id_{L}, and L_{i} = f_{i}(L) is linearly disjoint from L_{0} ⋯ L_{i−1} over σ^{i}(K) for i > 0. Then L_{0} ⋯ L_{i−1} and L_{i} ⋯ L_{n−1} are linearly disjoint over σ^{i}(K) if i > 0, and M = L_{0}(L_{1} ⋯ L_{n−1})^{p^{−∞}} is a separable extension of L. Define σ_{i} : L_{i−1} → L_{i} for i = 1, …, n − 1 and σ_{n} : L_{n−1} → τ(L) by

σ_{i} = f_{i}f_{i−1}^{−1} for i = 1, …, n − 1, and σ_{n} = τf_{n−1}^{−1}.

For 1 < i ≤ n extend σ_{i} to L_{i−1}^{p^{−∞}}. Then σ_{n}σ_{n−1} ⋯ σ_{1} and τ agree on L_{0} = L. Moreover, each σ_{i} agrees with σ on σ^{i−1}(K); this, together with the linear disjointness assumptions on the L_{i}’s, shows that σ_{1} ∪ ⋯ ∪ σ_{n} = ρ extends uniquely to an endomorphism of M extending σ. Clearly (M, ρ) is a model of T_{σ}, and therefore expands to a model of T_{1}.

We need to show that ρ(M) and K are linearly disjoint over σ(K), and that ρ^{n}(M) and L are linearly disjoint over τ(L). Because M is a separable extension of L, letting M_{0} = L_{0} ⋯ L_{n−1}, it is enough to show that ρ(M_{0}) and K are linearly disjoint over σ(K), and that ρ^{n}(M_{0}) and L are linearly disjoint over τ(L).

By construction, L and L_{1} ⋯ L_{n−1} = ρ(L_{0} ⋯ L_{n−2}) are linearly disjoint over σ(K), and hence L and ρ(M_{0}) are linearly disjoint over σ(K)τ(L). We also know that τ(L) and K are linearly disjoint over σ^{n}(K), and hence σ(K)τ(L) and K are linearly disjoint over σ(K). As L ⊃ Kτ(L), this implies that ρ(M_{0}) and K are linearly disjoint over σ(K).

Since K and ρ(M_{0}) are linearly disjoint over σ(K), also K and ρ^{n}(M_{0}) are linearly disjoint over σ^{n}(K) (see Remark (3.2)(3)), and therefore K and ρ^{n}(KL_{1} ⋯ L_{n−1}) are linearly disjoint over σ^{n}(K). The linear disjointness of L and L_{1} ⋯ L_{n−1} over σ(K) implies the linear disjointness of L and ρ^{n}(KL_{1} ⋯ L_{n−1}) over σ^{n}(K). As τ(L) = ρ^{n}(L) ⊆ L, this gives that L and ρ^{n}(M_{0}) are linearly disjoint over τ(L). Case 3. n > 1, m ≠ 0.

Use Case 2 and Case 1.

(5.2). Proposition. Let K ⊧ SCFE_{e}, and consider L = ⋃_{n} σ^{−n}(K), the inversive closure of K. Then K^{alg} ⊧ SCFE_{0} and L ⊧ ACFA. Proof. If p = 0, then K = K^{alg}. Assume therefore that p > 0. Clearly K^{alg} is an algebraically closed field, and σ is an endomorphism of K which is not onto.

Let U and V be varieties of the same dimension, defined over K^{alg}, such that V ⊆ U × σ(U) and V projects dominantly onto U and onto σ(U). Let (a, b) be a generic of the variety V over K^{alg}. Our assumption on the dimensions of U and V implies that a ∈ K(b)^{alg} and b ∈ K(a)^{alg}. Choose powers q and r of p such that U and V are defined over K^{1/q}, and a ∈ K^{1/q}(b^{1/r})^{s}. The difference fields K and K^{1/q} are isomorphic, and therefore K^{1/q} is a model of SCFE_{e}. Hence K^{1/q} contains a tuple c such that (c, σ(c)) ∈ V, and this shows that K^{alg} ⊧ SCFE_{0}.

The second assertion follows immediately: observe that L = ⋃_{n} σ^{−n}(K^{alg}), where each σ^{−n}(K^{alg}) is a model of SCFE_{0}. As axiom (ii) for ACFA coincides with axiom (ii) for SCFE_{0}, and σ is an automorphism of the difference field L, it follows that L is a model of ACFA.

(5.3). Theorem. Let L be a model of ACFA, and let A = acl_{σ}(A) ⊆ B = acl_{σ}(B) ⊆ L. Assume that A contains all elements of B which are transformally algebraic over A. Then tp(B/A) is stationary (here tp is the type in the sense of the generic difference field L). Proof. We may assume that L is sufficiently saturated. Let C = acl_{σ}(C) containing A and independent from B over A. We want to show that tp(B/A) has a unique non-forking extension to C. We will first show that we may assume that C is transformally algebraic over A.

Step 1. tp(B/A) is orthogonal to all types of finite SU-rank over A, i.e.: if F = acl_{σ}(F) ⊃ A is linearly disjoint from B over A, and c ∈ acl_{σ}(F, B) is transformally algebraic over F, then c ∈ F (see (1.16)).

Indeed, let F, c be as above. By (2.13) in [4], the canonical base D of tp(F, c/B) is contained in acl_{σ}(A, F_{1}, c_{1}, …, F_{n}, c_{n}) for some n and independent realisations (F_{1}, c_{1}), …, (F_{n}, c_{n}) of tp(F, c/B). Hence D is transformally algebraic over acl_{σ}(F_{1}, …, F_{n}). As D ⊆ B, and B and acl_{σ}(F_{1}, …, F_{n}) are linearly disjoint over A, D must be transformally algebraic over A, whence D = A and c ∈ F. Step 2. We may assume that C = A(d)_{ℤ} where d is transformally algebraic over A.

First of all, we certainly may assume that C = A(d)_{ℤ} for some finite tuple d. Let c ⊂ d be a transformal transcendence basis of C over A, and let A′ = acl_{σ}(A, c). By (1.15), tp(A′/A) has a unique non-forking extension to B, or equivalently, tp(B/A) has a unique non-forking extension to A′. To show the result, it is therefore enough to show that tp(acl_{σ}(B, c)/A′) has a unique non-forking extension to A′(d)_{ℤ}. By step 1, A′ and acl_{σ}(B, c) satisfy the hypotheses satisfied by A and B. Step 3.

Let S be an A-definable set of finite SU-rank containing d, defined by the formula θ(a, x). If tp(B/A) has several non-forking extensions to A(d)_{ℤ}, then there is a B-definable subset X of S, such that d ∈ X and some realisation of a non-forking extension of tp(d/A) to B is in S \ X. By elimination of imaginaries, X has a code b ∈ B. Thus if an Aautomorphism of some elementary extension of L fixes all elements satisfying θ(a, x) then necessarily it also fixes b. We will assume by way of contradiction that b contains an element b_{1} ∉ A. Claim. There is an elementary extension K of L, and an automorphism of K which is the identity on all elements of K which are transformally algebraic over A, and which moves b_{1}. Proof. By Theorem (1.21), there is an ultrafilter U on the set Q of prime powers, and a sequence of algebraically closed fields F_{q}, q ∈ Q, such that the difference field A embeds in ∏_{q∈Q}(F_{q}, F rob_{q})/U, where F rob_{q} : x ↦ x^{q}. Let K_{q} be an algebraically closed field properly containing F_{q}, and fix c_{q} ∈ K_{q} \ F_{q}, and h_{q} ∈ Aut(K_{q}/F_{q}) such that h_{q}(c_{q}) ≠ c_{q}. Consider the structure M_{q} = (K_{q}, F_{q}, F rob_{q}, h_{q}, c_{q}), and let M = (K, F, σ, h, c) be a highly saturated elementary extension of ∏_{q∈Q} M_{q}/U. Then there is an embedding φ of the difference field A in F, and h ∈ Aut(K/F), h(c) ≠ c. All elements of K_{q} are transformally transcendental over F_{q} (because σ_{q} is algebraic), and therefore all elements of K are transformally transcendental over F.

Since b_{1} ∉ A, b_{1} is transformally transcendental over A. Then tp(c/φ(A)) = φ(tp(b_{1}/A)) (by Proposition (1.15)). Using the saturation of M, there is a difference field embedding ψ of L in K which extends φ and such that ψ(b_{1}) = c. Then ψ(L) ≺ K.

The claim allows us to conclude that b ∈ A: the elements of K satisfying θ(φ(a), x) are transformally algebraic over A, and therefore are in F. Hence they are fixed by h and this implies that h fixes ψ(b); but this contradicts the claim.

Remark. Another way of stating Theorem (5.3) is to say, that over an algebraically closed set, all types of SU-rank ωn for some n, are stationary. In Appendix B, we give an alternate proof of the claim, which does not rely on the results of [13].

(5.4). Corollary. Let K be a model of SCFE_{e}, and E = acl(E) ⊂ A = acl(A) subfields of K. Assume that all elements of A \ E are transformally transcendental over E. Then tp(A/E) is stationary. Proof. We will first assume that E contains a p-basis of K if e ∈ ℕ. Let B = acl(B) ⊂ K containing E, and let A_{1}, A_{2} realise non-forking extensions of tp(A/E) to B. Let L, E′, A′_{1}, A′_{2} and B′ denote the inversive closures of K, E, A_{1}, A_{2}, B respectively. Then all elements of A′_{1} are transformally transcendental over E′, and therefore tp_{L}(A′_{1}/B′) = tp_{L}(A′_{2}/B′) by Theorem (5.3) and Proposition (5.2) (tp_{L} denotes the type in the generic difference field L). Hence there is an L_{σ}(B′)-isomorphism φ : acl_{σ}(B′, A′_{1}) → acl_{σ}(B′, A′_{2}) which sends A_{1} to A_{2}. Since E contains a p-basis of K if e ∈ ℕ, we get that acl(B, A_{1}) = (BA_{1})^{s}, acl(B, A_{2}) = (BA_{2})^{s}, so that φ restricts to an L_{1}(B)-isomorphism acl(B, A_{1}) → acl(B, A_{2}). This shows that tp(A_{1}/B) = tp(A_{2}/B).

Assume now that e ∈ ℕ, and that [E : E^{p}] = p^{f}, with f < e. Let B = acl(B) contain E, independent from A over E, and let B_{0} realise the generic (e − f)-type over (A, B). Then tp(A/E) has a unique non-forking extension to acl(E, B_{0}) by Proposition (4.13), and it suffices to show that tp(acl(A, B_{0})/acl(E, B_{0})) has a unique non-forking extension to acl(B, B_{0}). The discussion in (1.8) gives that dcl_{K}(A, B_{0}) is contained in A(B_{0}, C)^{alg}, for some set C of elements which are algebraically independent over AK^{p^{∞}}(B_{0}), and therefore over Aσ(K)(B_{0}). We then have acl(A, B_{0}) ⊂ A(B_{0}, C)_{ℕ}^{alg} ∩ K, and the elements of C are transformally independent over A(B_{0}). By step 1 of the proof of Theorem (5.3), the elements of acl(A, B_{0}) which are transformally algebraic over acl(E, B_{0}) are already in acl(E, B_{0}). The previous case gives the result.

(5.5). Proposition. Let K be a model of SCFE_{e}, and let k = ⋂_{n} σ^{n}(K). Then k ⊧ ACFA, and if D ⊂ K^{n} is definable, then D ∩ k^{n} is definable in the difference field (k, σ). Thus there is no induced structure on the L_{σ}-structure k, and k is stably embedded. Proof. Let L be the inversive closure of K. Then L ⊧ ACFA by Proposition (5.2). Note that σ(k) = k and k = ⋂_{n} σ^{n+1}(K)^{alg} is algebraically closed.

If a ∈ K is transformally algebraic over k, then k(a)_{ℤ} ⊂ k(σ(a), …, σ^{m}(a))^{alg} for some m, so that a ∈ k. By Remark (1.1)(1) in [4], k is a model of ACFA.

Let B = acl(B) ⊂ K, and let A = B ∩ k. Then B and k are linearly disjoint over A = acl(A), and all elements of B \ A are transformally transcendental over A. By Corollary (5.4), tp(B/A) is stationary, and in particular has a unique non-forking extension to k.

This implies (see Lemma 1 in the Appendix of [4]) that k is stably embedded, and that any elementary automorphism of k lifts to an automorphism of K. But, as k = acl(k) is perfect, any L_{σ}-automorphism of k is elementary in K, and this gives the result.

(5.6). Theorem. Let K be a model of SCFE_{e}, let n ≥ 1 and m ∈ ℤ, and let F be the subfield of K consisting of the elements satisfying σ^{n}(x) = F rob^{m}(x). Then F is a pseudo-finite field, and every subset D of F^{ℓ} definable in K is definable using parameters from F. If n = 1, then D is definable in the pure field F. Proof. All elements of F are transformally algebraic, and therefore F ⊆ ⋂_{n∈ℕ} σ^{n}(K) = k. Let D ⊆ K^{ℓ} be definable. By Proposition (5.5), D ∩ k^{ℓ} is definable in the difference field (k, σ), and k is a model of ACFA. The result follows by (7.1)(5) in [6].

(5.7). Proposition. Let K be an ω-saturated model of SCFE_{e}. Then K^{p^{∞}} ⊧ SCFE_{0}. Proof. Clearly K^{p^{∞}} is algebraically closed, and σ restricts to an endomorphism of K^{p^{∞}}. Because K is ω-saturated, there is a ∈ K^{p^{∞}}, a ∉ σ(K), and therefore σ is not onto.

Let U and V be varieties defined over K^{p^{∞}}, of the same dimension, and such that V ⊂ U × σ(U), and V projects dominantly onto U and onto σ(U). Fix a generic (a, b) of the variety V over K^{p^{∞}}, and for each n ≥ 1, let U_{n}, V_{n} be the varieties defined over K of which (a, a^{1/p^{n}}) and (a, a^{1/p^{n}}, b, b^{1/p^{n}}) are generics. Then each pair (U_{n}, V_{n}) satisfies the assumptions of (3.4)(ii). By ω-saturation of K, there are c, c_{n}, n ∈ ℕ in K, such that (c, c_{n}, σ(c), σ(c_{n})) ∈ V_{n} for all n. Then (c, σ(c)) ∈ V, and c ∈ K^{p^{∞}}.

(5.8). Sketch of an alternate proof of the independence theorem. Let K, E, a, b, c_{1} and c_{2} be as in Theorem (4.15), A = acl(Ea), B = acl(Eb), C_{1} = acl(Ec_{1}) and C_{2} = acl(Ec_{2}). Let A^{0} be the difference subfield of A consisting of the elements of A which are transformally algebraic over E, and define similarly B^{0}, C_{1}^{0} and C_{2}^{0}. By Lemma (4.11), we know that A^{0} ⊆ (Eσ(A^{0}))^{alg}, and similarly for B^{0} and the C_{i}^{0}. Proceed as in the first part of the proof of Theorem (4.15) to show that there is C^{0}, independent from (A, B) over E, and realising tp(C_{1}^{0}/A^{0}) ∪ tp(C_{2}^{0}/B^{0}). Then consider isomorphisms ψ_{1} : acl(A^{0}, C_{1}^{0}) → acl(A^{0}, C^{0}) and ψ_{2} : acl(B^{0}, C_{2}^{0}) → acl(B^{0}, C^{0}), fixing A^{0} and B^{0} respectively, and which witness the fact that C^{0} realises tp(C_{1}^{0}/A^{0}) ∪ tp(C_{2}^{0}/B^{0}). Let C realise a non-forking extension of ψ_{1}(tp(C_{1}/C_{1}^{0})) to (A, B). Then C is independent from (A, B) over E, because C^{0} is independent from (A, B) over E. By Corollary (5.4), tp(C_{1}/C_{1}^{0}) is stationary. Hence, C realises ψ_{1}(tp(C_{1}/A^{0}, C_{1}^{0})) and ψ_{2}(tp(C_{2}/B^{0}, C_{2}^{0})). This implies that C realises tp(C_{1}/A) ∪ tp(C_{2}/B).

6 Study of modularity

(6.1). Proposition. Let E = acl(E) ⊂ K ⊧ SCFE_{0}, and assume that acl(Ea) contains an element b which is transformally transcendental over E. Then the set S of realisations of tp(a/E) is not modular. Proof. Since acl(Ea) = dcl(Ea)^{s}, we may assume that b ∈ dcl(Ea). Choose c ∈ F ix(σ) independent from a over E, and let d = b · c. Then tp(d/E) = tp(b/E), and the set of realisations of tp(d/Eb) is in definable bijection with F ix(σ). Hence there is a definable subset S′ of S which projects onto F ix(σ). By Theorem (5.6), F ix(σ) has no induced structure, and every infinite ∞-definable subset of F ix(σ) is non-modular.

(6.2). Remarks. If e ≠ 0 and we do not assume that b ∈ K^{p^{∞}}, then the result does not necessarily hold. Let a ∈ K and assume that (the pure field type) tp_{K}(a/E) is minimal and orthogonal to the generic type of K^{p^{∞}}. Using Lemmas (3.3) and (4.11) one can then show that tp(a/E) is minimal. By (1.15), one also obtains that tp_{K}(a/E) uniquely determines tp(a/E). Furthermore, if tp_{K}(a/E) is modular [resp. non-trivial], so is tp(a/E). These observations show that there are many non-trivial minimal modular types which are realised by non-transformally algebraic elements, using e.g. any of the types described in [12] or in [2].

In the case of a definable set S however, the result extends to the non-perfect case.

(6.3). Proposition. Let K be a model of SCFE_{e}, and let S ⊂ K^{n} be definable over E = acl(E). If S contains elements which are not transformally algebraic over E, then S is non-modular. Proof. Enlarging E, we will assume that it contains a p-basis of K if e ∈ ℕ. We also assume that K is sufficiently saturated. By Corollary (4.8), using the fact that modularity and non-modularity are preserved under finite covers, we may assume that S is defined by a formula φ(x) ∧ θ(x) ∧ ρ(x), where φ(x) is a quantifier-free L_{σ}(E)-formula, θ(x) is a conjunction of formulas of the form R_{n}(x_{1}, e), where e is a tuple of elements of E and x_{1} ⊆ x, and ρ(x) is an L(E)-formula expressing that certain tuples in x ∪ E are p-independent in K.

First of all, assume that there is some a ∈ S such that some sub-tuple of a is pindependent over E in K. This means that a contains a realisation a_{1} of the generic 1-type over E, and implies that tp(a/E) is non-modular: if c ∈ F ix(σ) is independent from a_{1} over E, then tp(a_{1}c/E) also realises the generic 1-type over E. As in (6.1), this implies that S is non-modular.

Hence we may assume that S is defined by φ(x) ∧ θ(x). Let a ∈ S, not transformally algebraic over E. We will show that there is b in S such that the difference fields E(a)_{ℕ} and E(b)_{ℕ} are isomorphic, some element b_{1} of b realises the generic type over E. Reasoning as in the previous paragraph will then show that non-modularity of S.

We know that E(a)_{ℕ} is a separable extension of E. We proceed exactly as in the proof in Theorem (3.5) that the models of SCFE_{e} are existentially closed: we let M = σ(E(a)_{ℕ})^{alg}, and select a_{1} ⊂ a, a separating transcendence basis of EM(a) over EM. As a is not transformally algebraic over E, we know that a_{1} is non-empty. We then choose b_{1} realising the generic |a_{1}|-type, and, as in Theorem (3.5), find b extending b_{1}, such that the difference fields E(a)_{ℕ} and E(b)_{ℕ} are E-isomorphic, by an isomorphism f sending a to b. By construction, the elements of b_{1} are algebraically independent over Eσ(K). This implies that any formula of the form R_{n}(y, e) satisfied by a sub-tuple of a will also be satisfied by the corresponding sub-tuple of b. Hence, b ∈ S.

(6.4). Theorem. Let K ⊧ SCFE_{e} be sufficiently saturated, and let E = acl(E) ⊂ K. Let S ⊂ K^{n} be a subset which is invariant under Aut(K/E), and such that the elements of S are transformally algebraic over E. The following conditions are equivalent:

(1) S is non-modular.

(2) S is non-orthogonal to some fixed field, i.e., there is a tuple a ∈ S, and a set F = acl(F) containing E, such that acl(F a) \ F contains an element b satisfying σ^{n}(b) = F rob^{m}(b) for some n ≥ 1 and m ∈ ℤ.

Proof. We will work in the inversive closure L of K, and will denote by tp_{L} the types in the generic difference field L. Without loss of generality, if e ∈ ℕ, then E contains a p-basis of K.

If S is not modular, there are tuples a and b, with a a tuple of elements of S, such that a and b are not independent over acl^{eq}(Ea) ∩ acl^{eq}(Eb). Let A = acl(Ea), B = acl(Eb) and C = A ∩ B. Then C = acl(C), and therefore acl_{σ}(A) ∩ acl_{σ}(B) = acl_{σ}(C). As C ⊂ acl^{eq}(Ea) ∩ acl^{eq}(Eb), we certainly have that A and B are not independent over C.

Our assumption on the elements of S and Lemma (4.11)(2) imply that A and B are not linearly disjoint over C. Let d ∈ A be such that the field C(d)_{ℕ} and B are not linearly disjoint over C. Then acl_{σ}(C)(d)_{ℤ} and acl_{σ}(B) are not linearly disjoint over acl_{σ}(C). By Proposition (1.19), tp_{L}(d/acl_{σ}(C)) is non-orthogonal to the formula σ^{n} = x^{p^{m}} for some n ≥ 1 and m ∈ ℤ (where p is the characteristic of K if it is positive, and 1 if it is 0). Let F ix(τ) be the subfield of L defined by σ^{n} = x^{p^{m}}. Then F ix(τ) ⊆ K. Claim. There are independent realisations d_{1}, …, d_{k} of tp(d/C) in K, such that F ix(τ)∩ C(d_{1}, …, d_{k})_{ℕ} contains an element b not in C. Proof. Indeed, since tp_{L}(d/acl_{σ}(C)) is non-orthogonal to σ^{n}(x) = x^{p^{m}}, there is an integer k and independent realisations d_{1}, …, d_{k} of tp_{L}(d/acl_{σ}(C)), and an element b ∈ F ix(τ) ∩ acl_{σ}(C)(d_{1}, …, d_{k})_{ℤ}, b ∉ acl_{σ}(C). Note that σ(F ix(τ)) = F ix(τ). Hence, replacing b by σ^{ℓ}(b) for a suitable ℓ, we may assume that b ∈ C(d_{1}, …, d_{k})_{ℕ}.

This only depends on the isomorphism type of the difference field C(d_{1}, …, d_{k})_{ℕ}, and therefore we may assume that d_{1}, …, d_{k} are independent realisations of tp(d/C).

Since d_{1}, …, d_{k} are independent realisations of tp(d/C), we may choose a_{1}, …, a_{k} in K such that tp(a_{i}, d_{i}/C) = tp(a, d/C). Then b ∈ C(a_{1}, …, a_{k})_{ℕ}.

The other direction is easy: F ix(τ) ⊆ k = ⋂_{ℓ∈ℕ} σ^{ℓ}(K), and the only induced structure on k is that of an L_{σ}-structure model of ACFA (by (5.5)). By the main result of [6], every non-algebraic type which is non-orthogonal to a fixed field is non-modular.

(6.5). Concluding remarks.

Let K be a model of SCFE_{e}, E = acl(E) ⊂ K. If D ⊂ K^{n} is definable and consists of elements transformally algebraic over E, then we know by Remark (4.17) that D has finite SU-rank. Moreover forking in K coincides with forking in the sense of the L_{σ}-structure L which is the inversive closure of K. Hence, if D is modular and the characteristic of K is 0, then we know that all types realised in D are stable, and that D is stably embedded over E, see (1.19).

Let G be a modular group definable over E in K. We know by Proposition (6.3) that if a ∈ G then a is tranformally algebraic over E, so that by Lemma (4.11), acl(Ea) is contained in E(a)_{ℕ}^{alg}. By the results of [15] and this description of algebraic closure, there is a definable map f : G_{1} → H(K), where H is an algebraic group, G_{1} is a definable subgroup of G of finite index, and Ker(f) is finite. Theorem (1.23) then yields a nice description of quantifier-free definable subsets of f(G_{1}).

Appendix A - Stationarity almost over a predicate

In this appendix, we mention a general result that (using Theorem (5.3)) gives another proof of the validity of the independence theorem in SCFE over bases contained in k.

A.1 Setting. Let T be a complete theory in a language L. We assume that we have a notion of independence in U^{eq} which is well-behaved, i.e., it is Aut(U)-invariant and, for all A ⊆ B ⊆ C ⊆ U, and tuple a in U we have:

(i) (Extension property) There is a′ realising tp(a/A) and which is independent from B over A.

(ii) (Transitivity) a is independent from C over A if and only if it is independent from B over A and from C over B.

(iii) (Finite character) For any D ⊆ U, D is independent from B over A if and only if every finite tuple of elements of D is independent from B over A.

(iv) (Symmetry) a is independent from B over A if and only if B is independent from a over A.

(v) (Local character) There is A_{0} ⊂ A, with |A_{0}| ≤ |L| + ℵ_{0}, such that a is independent from A over A_{0}.

A.2 Definition. Let U be a sufficiently saturated model of a theory T in a language L. Let S ⊂ U^{eq} be preserved under all automorphisms of U. We say that U is stationary almost over S if for all subsets A ⊂ B of U, tp(acl^{eq}(B)/acl^{eq}(B)∩acl^{eq}(S, A)) is stationary.

Here stationarity is meant with respect to our independence notion: if C contains acl^{eq}(B)∩acl^{eq}(S, A), then there is a unique type over C extending tp(acl^{eq}(B)/acl^{eq}(B)∩ acl^{eq}(S, A)) and whose realisations are independent from C over acl^{eq}(B) ∩ acl^{eq}(S, A).

A.3 Remark. If U is stationary almost over S, then S is stably embedded. Proof. See Lemma 1 in the Appendix of [4].

A.4 Proposition. Assume that U is stationary almost over S. Let A, B, C_{1}, C_{2}, E, be algebraically closed subsets of U^{eq} satisfying the following conditions:

(1) A is independent from B over E, C_{1} is independent from A over E and C_{2} is independent from B over E.

(2) tp(C_{1}/E) = tp(C_{2}/E).

(3) There is C̃ ⊂ acl^{eq}(S) realising tp(C_{1} ∩ acl^{eq}(S)/A ∩ acl^{eq}(S)) ∪ tp(C_{2} ∩ acl^{eq}(S)/B ∩ acl^{eq}(S)), which is independent from (A ∩ acl^{eq}(S), B ∩ acl^{eq}(S)) over Ẽ = E ∩ acl^{eq}(S).

Then there is C ⊆ U^{eq} realising tp(C_{1}/A) ∪ tp(C_{2}/B) and which is independent from (A, B) over E. Proof. Let C̃ be as given by (3), and independent from (A, B) over Ẽ.

Let A′ = acl^{eq}(E, A∩acl^{eq}(S)), B′ = acl^{eq}(E, B∩acl^{eq}(S)). By stationarity almost over S, tp(A′/A∩acl^{eq}(S)) has a unique non-forking extension to (A∩acl^{eq}(S), C̃), and therefore tp(C̃/A′) = tp(C_{1} ∩acl^{eq}(S)/A′). Hence C′ = acl^{eq}(E, C̃) and C′_{1} = acl^{eq}(E, C_{1} ∩acl^{eq}(S)) have the same type over A′. Similarly, C′ and C′_{2} = acl^{eq}(E, C_{2} ∩ acl^{eq}(S)) have the same type over B′.

Choose C independent from (A, B), and such that tp(C, C′/E) = tp(C_{1}, C′_{1}/E). We know that tp(C_{1}/C′_{1}) is stationary, and therefore tp(C/A) = tp(C_{1}/A) because C and C_{1} are independent from A over C′, C′_{1}, respectively. Similarly, tp(C/B) = tp(C_{2}/B).

A.5 Corollary. Let E = acl^{eq}(E) ⊆ U^{eq}, and assume that in acl^{eq}(S), the independence theorem holds over Ẽ = E ∩ acl^{eq}(S). Then it holds in U over E.

A.6 Corollary. If T h(acl^{eq}(S)) is simple, and the restriction of the independence notion to acl^{eq}(S) coincides with non-forking, then T is also simple, and our notion of independence coincides with non-forking. Proof. By results of Kim-Pillay [14], the independence theorem holds in acl^{eq}(S) over submodels, and our results imply that it holds in U over submodels. Moreover, the properties satisfied by our independence notion imply that it coincides with non-forking.

Appendix B

In this appendix we provide a proof of Claim (5.3) not relying on the results of [13].

B.1 Proposition. Let T be a first-order theory, with saturated model U. Assume given a function rk defined on consistent formulas over U, into ℕ∪{∞}. Assume rk(φ) ≤ rk(ψ) if φ implies ψ. Define rk(a/B) = inf{rk(φ) | φ ∈ tp(a/B)}.

Assume

(i) Definability: if rk(φ(x, a)) = m ∈ ℕ, then there exists ψ ∈ tp(a) such that ψ(a′) implies rk(φ(x, a′)) = m.

(ii) Density: over any set A, any consistent formula over A is implied by some consistent formula of finite rank over A.

(iii) Additivity: rk(ab/C) = rk(b/C) + rk(a/Cb).

(iv) Zero: rk(φ) = 0 iff φ has finitely many solutions.

(v) For any set A, there exists c ∉ A such that if rk(d/A) < ∞ then tp(d/A) implies tp(d/Ac).

Then there exists an automorphism of U fixing the finite rank part (i.e. fixing pointwise every 0-definable set of finite rank), and which is not the identity.

B.2 Definition. tp(a/B) is j-isolated if for some φ(x) ∈ tp(a/B), whenever φ(a′) holds, then rk(a′/B) = rk(a/B).

B.3 Remark. In the definition of j-isolated, rk(a/B) is necessarily finite (using density.)

B.4 Lemma. tp(b_{1}b_{2}/C) is j-isolated iff both tp(b_{1}/C) and tp(b_{2}/Cb_{1}) are. Proof. First suppose tp(b_{1}b_{2}/C) is j-isolated, say by φ(x_{1}, x_{2}). Then φ(b_{1}, x_{2}) j-isolates tp(b_{2}/Cb_{1}), by additivity of rk. Find ψ(x_{1}) ∈ tp(b_{1}/C) such that ψ(b′_{1}) implies rk(φ(b′_{1}, x_{2})) = rk(φ(b_{1}, x_{2})). Then ψ(x_{1}) ∧ (∃x_{2}) φ(x_{1}, x_{2}) j-isolates tp(b_{1}/C). The converse is similar.

B.5 Definition. (B/C) is j-atomic if for any finite tuple b from B, tp(b/C) is j-isolated.

B.6 Lemma. Assume (B/C) is j-atomic, and (a/BC) is j-isolated. Then (aB/C) is j-atomic. Proof. Similar to the above.

B.7 Lemma. Let A ⊂ U. Then there exists a model M, A ⊂ M, with (M/A) j-atomic. Proof. Let B ⊂ U be maximal j-atomic over A, and assume by way of contradiction that B is not an elementary substructure of U. Then there is a formula φ(x) ∈ L(B) which is realised in U and not in B. Take such a formula of minimal rank m, which by density is in ℕ. If m = 0, then any realisation of φ is in acl(B) and its type over B is isolated. Assume m > 0, and take b ∈ U satisfying φ. Then rk(b/B) = m (by the minimality of rk(φ)), and so tp(b/B) is j-isolated. By (B.6), (Bb/A) is j-atomic, and this gives us the required contradiction. B.8 Proof of Proposition B.1. Now let M_{0} be a small model (size of language.) Using (v), construct a large set I such that rk(d/M_{0}) < ∞ implies tp(d/M_{0}) ⊢ tp(d/M_{0}I).

Let M be a j-atomic model over M_{0}I. If a ∈ M with rk(a/M_{0}) < ∞, then tp(a/M_{0}I) is j-isolated; since tp(a/M_{0}) ⊢ tp(a/M_{0}I), tp(a/M_{0}) is also j-isolated. But this implies that a ∈ M_{0}. Thus the finite rank part of M is contained in M_{0}. By Morley’s two cardinal theorem (since I is large compared to M_{0}), if we choose a Skolemization L′, T′ of the language, there exists a model M′ of T′ with an indiscernible sequence over M_{0} and which does not realise any finite rank non-algebraic type over M_{0}. This model clearly has automorphisms fixing the finite rank part (which is contained in M_{0}). Hence so does U. B.9 Remarks. (1) Note that Proposition B.1 also holds if (v) is weakened to: For any set A, there exists c ∉ A such that if rk(d/A) < ∞ and tp(d/Ac) is j-isolated then so is tp(d/A), and they have the same rank.

Indeed, one constructs a large set I such that if rk(d/M_{0}) < ∞ and tp(d/M_{0}I) is j-isolated, then so is tp(d/M_{0}) and they have the same rank. Thus all elements of M not in M_{0} have infinite rank over M_{0}.

(2) If T satisfies (i) – (v), and A ⊂ U, so does T (A) (= T h(U, a)_{a∈A})). B.10 Proposition. The hypotheses hold for ACFA: take rk to be transformal order, i.e., rk(φ(x, a)) = sup{tr.deg(A(b)_{ℤ}/A)| ⊧ φ(b, a)}, where A is the difference field generated by a; in (v), take c transformally transcendental over A (see (1.15)). B.11 Corollary (Claim (5.3)). Let A ⊂ B be algebraically closed difference subfields of a model L of ACFA, and assume that all elements of B which are transformally algebraic over A are already in A. Let b ∈ B \ A. Then in some elementary extension U of L, there is an automorphism h of U which is the identity on the set of elements of U which are transformally algebraic over A, and which moves b. Proof. By Propositions B.1, B.10 and Remark B.9 (2), there is a saturated model U of ACFA containing A, and with an automorphism h which is the identity on the set of elements transformally algebraic over A, and moves some element c. Then c is transformally transcendental over A, as is b. By (1.15), tp(b/A) = tp(c/A), so that there is an A-automorphism φ of U which sends b to c. Then hφ(b) ≠ φ(b), so that ψ = φ^{−1}hφ is the identity on the set of elements transformally algebraic over A and moves b.

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Current addresses: UFR de Mathématiques Université Paris 7, Case 7012 2, place Jussieu 75251 Paris Cedex 05 France e-mail: zoe@logique.jussieu.fr

Institute of Mathematics The Hebrew University Givat Ram 91904 Jerusalem Israel e-mail: ehud@math.huji.ac.il