Generic automorphisms of separably closed fields Zoé Chatzidakis, CNRS - Université Paris 7

Introduction. Generic automorphisms of a saturated structure M appear in a paper by Lascar [L], where they are used towards the study of the automorphism group Aut(M). In many cases, if σ is a generic automorphism of M, then the structure (M, σ) is existentially closed in the class of models of T h(M) with a distinguished automorphism σ. A natural question arises: given a model-complete theory T, does the theory of models of T with a distinguished automorphism σ have a model companion T_{A}? In [CP], Pillay and I showed that if this model-companion T_{A} exists, then it is well-behaved, and we obtained a description of the completions of T_{A} and of the types. In case T is moreover stable, then we showed that T_{A} is simple (unstable in most cases), and got a description of the imaginaries in terms of the imaginaries of the models of T.

At the moment there is no general criterion for existence of the theory T_{A}. We know of its existence when T is the theory of algebraically closed fields, see [M] and [CH], or when T is the theory of differentially closed fields of characteristic 0 (unpublished result of Hrushovski). There are also some criteria of existence when T is ω-stable of finite rank, more details are given in [CP], see also [KkP]. The theory T_{A} is known not to exist for various unstable theories T, see Kikyo’s paper [K].

Consider the theory T = SCF_{e} of separably closed fields of degree of imperfection e (e ∈ ℕ ∪ {∞}). Does T_{A} exist? Recall that a field with a distinguished automorphism σ is called a difference field, so our question is equivalent to: does the theory of separably closed difference fields of a fixed degree of imperfection have a model-companion?

As stated, the question has an immediate negative answer, since SCF_{e} is not model complete in the language L of rings. Hence we first need to enrich the language L, and expand the theory SCF_{e} accordingly, so that in the new language, if K ⊆ L are fields then L is a separable extension of K. This is done by adding to L new function symbols, called the λ-functions (see §1), getting a language L_{λ}, and adding to T the axioms defining these functions to obtain the theory SCF_{e,λ}. As these functions are definable within the field language, we are not adding any definable sets, but only strengthening the notion of substructure. If e is finite, one can instead add constant symbols b_{1}, …, b_{e} for the elements of a p-basis. Then SCF_{e,b} = SCF ∪ {b_{1}, …, b_{e} form a p-basis} is complete and model-complete in L(b_{1}, …, b_{e}), but does not eliminate quantifiers.

Even given this first adjustment, the problem is slightly delicate, because separably closed fields have a very rich and complicated structure. For instance, observe that if σ ∈ Aut(K) and a ∈ K satisfies the equation σ(x) = x^{p}, then necessarily a ∈ ⋂_{n∈ℕ} K^{p^{n}}. Thus the formula σ(x) = x^{p} implies an infinite conjunction of L-formulas (modulo T ∪ {σ is an automorphism}).

The main result of this paper is, that if T is the appropriate extension of the theory of separably closed fields in L_{λ} (or possibly in L(b_{1}, …, b_{e}) if we only consider fields of finite degree of imperfection), then T_{A} exists. We then derive some easy consequences along the lines of [CP] and [CH].

This paper is organised as follows. In §1, we set up the notation and conventions, and recall the basic facts on difference fields and separably closed fields needed in the paper. In §2 we show the existence and give axiomatisations of the model companion SCFA_{e,λ} of the theory of difference fields models of T_{λ} and of degree of imperfection ≤ e, see (2.6) and (2.12); the cases of finite and infinite degree of imperfection need to be treated separately. We deduce the existence and an axiomatisation SCFA_{e,b} of the model companion SCFA_{e,b} of the theory (in the language L_{σ}(b̄)) of difference fields with fixed p-basis b̄ = {b_{1}, …, b_{e}}, see (2.7). In §3, we describe the completions of SCFA_{e,λ}, its types, the formulas, and prove the independence theorem and elimination of imaginaries of the completions of SCFA_{e,b}. These results are slightly stronger than those obtained by applying the results of [CP], but their proofs follow the same lines. In §4, we extend various results obtained in [CH] for the model companion ACFA of the theory of difference fields to SCFA_{e,λ}: let (K, σ) ⊧ SCFA_{e,λ}; then F ix(σ) is a PAC-field of degree of imperfection e, with absolute Galois group ℤ̂; if n ≥ 1 then (K, σ^{n}) ⊧ SCFA_{e,λ}; if K is ω-saturated, then ⋂_{n} K^{p^{n}} ⊧ ACFA. We also show that if K is a separable extension of the difference field k and is a model of SCFA_{e,λ}, then the subfield of K consisting of elements which are transformally algebraic over k, is an elementary substructure of K. In §5 we study modular sets.

1. Preliminaries on difference fields and fields of positive characteristic

(1.1) Setting and notation. We will always work inside a large algebraically closed field Ω of characteristic p > 0, which will contain all fields considered. The language L_{σ} is obtained by adjoining to the language L = {+, −, ·, 0, 1} of rings a unary function symbol for σ. A difference field is a field K with a distinguished automorphism σ, and is naturally an L_{σ}-structure. I should mention that our definition of difference fields slightly differs from the usual definition which only requires σ to be a field embedding, i.e., not necessarily onto. Our difference fields are called inversive by Cohn. All the basic algebraic results on difference fields can be found in the first few chapters of Cohn’s book [C].

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. Since we are in characteristic p, the map F rob : x ↦ x^{p} defines a monomorphism on K, called the Frobenius automorphism, and the image K^{p} of K by this map is a subfield of K. Similarly, we denote by K^{1/p} the image of K by the inverse of the Frobenius automorphism. We denote by K^{p^{∞}} the field ⋂_{n∈ℕ} K^{p^{n}}, and by K^{p^{−∞}} the field ⋃_{n∈ℕ} K^{1/p^{n}}. 𝔸^{k} denotes the k-dimensional affine space, and 𝔸^{k}(K) its K-rational points. We also view 𝔸^{k}(K) as a k-dimensional K-vector space.

We assume familiarity with the basic notions of algebraic geometry: algebraic sets and varieties (or absolutely irreducible algebraic sets), generic points, linear disjointness, separable and regular extensions, see e.g. Ch. I to III in [L2].

(1.2) The theory ACFA

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

— K is an algebraically closed field and σ is an automorphism of K.

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

(1.3) Difference polynomial rings

Let k ⊆ K be difference fields, with K a sufficiently saturated model of ACFA. 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 domains, on which σ defines an embedding. Thus they embed uniquely in a smallest difference field. If ā is an n-tuple of K, we define I_{σ}(ā/k) = {f(X̄) ∈ k[X̄]_{σ} | f(ā) = 0}, where X̄ = (X_{1}, …, X_{n}). Then I_{σ}(ā/k) is a prime σ-ideal of k[X_{1}, …, X_{n}]_{σ}.

While k[X_{1}, …, X_{n}]_{σ} has infinite ascending chains of σ-ideals, it satisfies the ascending chain condition on perfect σ-ideals and on prime σ-ideals. A σ-equation (over k) is an equation of the form f(x_{1}, …, x_{n}) = 0 where f(X_{1}, …, X_{n}) ∈ k[X_{1}, …, X_{n}]_{σ}. The set of solutions (in K^{n}) of a set of σ-equations is called a σ-closed set; it can be defined by a finite set of σ-equations. Thus the topology on K^{n} whose basic closed sets are the σ-closed sets is Noetherian. A σ-closed set is called irreducible if it is not the union of two proper σ-closed subsets. Every σ-closed set of K^{n} is the union of finitely many irreducible σ-closed sets, which are called its irreducible components. If the irreducible σ-closed set V is defined by σ-equations over k, then V is defined over k, and the set of difference polynomials over k vanishing on V is a prime σ-ideal of k[X_{1}, …, X_{n}]_{σ}, denoted by I(V). If ā ∈ K^{n} and I(V) = I_{σ}(ā/k), then a is called a generic of V over k.

(1.4) Transformal transcendence bases. Let k ⊆ K be as above, and let ā be a tuple of elements of K. We denote by k(ā)_{σ} the difference field generated by ā over k, i.e., the difference subfield k(σ^{i}(ā) | i ∈ ℤ) of K. If the transcendence degree tr.deg(k(ā)_{σ}/k) of k(ā)_{σ} over k is finite then we say that ā is transformally algebraic over k. In that case, there is a non-negative integer m such that k(ā)_{σ} ⊆ k(ā, …, σ^{m}(ā))^{alg}. Observe that since σ and σ^{−1} are automorphisms of k(ā)_{σ}, we then have k(ā)_{σ} ⊆ k(σ^{j}(ā), …, σ^{j+m}(ā))^{alg} for every j ∈ ℤ.

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

(1.5) p-bases.

Details and proofs can be found in [B] §13. Let K be a field of characteristic p > 0 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. 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 a p-basis of K over k is also a p-basis of K^{s} over k and over k^{s}.

(1.6) Derivations. Let R ⊂ S be commutative rings. Recall that an R-derivation on S is an additive map δ : S → S which vanishes on R and satisfies δ(xy) = δ(x)y + xδ(y). Let A ⊆ Ω be a finite set, and assume that every k-derivation on k(A) vanishes on k(A). Then k(A) is separably algebraic over k, see [L1] Proposition X.7.2. We will use the following easy consequence of this: Lemma. Let A, B be finite subsets of Ω.

(1) A ⊆ k(A^{p}) if and only if A is separably algebraic over k.

(2) Assume that A is separably algebraic over k(B^{p}) and that B is separably algebraic over k(A). Then (A, B) is separably algebraic over k.

Proof. (1) If a is separably algebraic over k then a ∈ k(a^{p}), and this gives the right-to-left implication. Conversely, assume that A ⊂ k(A^{p}) and let D be a k-derivation on k(A). Then D vanishes on k(A^{p}), and therefore vanishes on A. Hence A is separably algebraic over k.

(2) Let D be a k-derivation on k(A, B). Then D vanishes on k(B^{p}) and therefore vanishes on A since A is separably algebraic over k(B^{p}). This implies that D vanishes on B, and therefore that (A, B) is separably algebraic over k.

(1.7) 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^{1/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. Observe however that if B is a p-basis of K over k, then the elements of B are algebraically independent over k.

(1.8) Separably closed fields and the λ-functions.

For each e ∈ ℕ ∪ {∞}, the theory expressing that K is a separably closed field of degree of imperfection e, is a complete theory (Ershov [E]), which we denote by SCF_{e}, and is stable (Wood [W]). 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}).

For each n fix an enumeration m_{i,n}(x̄) (1 ≤ 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=1}^{p^{n}} 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 language, and we call them the λ-functions of K.

Consider the language L_{λ} = L ∪ {λ_{i,n} | n ∈ ℕ, 1 ≤ i ≤ p^{n}}, and let T_{λ} be the L_{λ}-theory obtained by adjoining to the theory of fields axioms expressing the defining properties of the functions λ_{i,n}. Let SCF_{e,λ} = SCF_{e} ∪ T_{λ}. Then SCF_{e,λ} is complete and eliminates quantifiers but does not eliminate imaginaries (Delon [D]). Fix a separably closed field K. When the degree of imperfection of K is finite, one may fix a p-basis {b_{1}, …, b_{e}} of K, and only consider the unary functions λ_{i,e}(b_{1}, …, b_{e}; x). Then T h(K) eliminates quantifiers and imaginaries in the language L_{λ}(b_{1}, …, b_{e}) = L ∪ {b_{1}, …, b_{e}, λ_{i,e}(b_{1}, …, b_{e}; −) : 1 ≤ i ≤ p^{e}} (Delon [D]).

(1.9) Lemma. Let L be a subfield of K. Then K is a separable extension of L if and only if L is closed under the λ-functions of K. Proof. This follows easily from the definition of the λ-functions of K, but we will give the proof. Assume that K is not a separable extension of L. Choose a tuple (a_{1}, …, a_{n}) ∈ L^{n} which is p-independent in L but not p-independent in K, with n minimal such. Then a_{1}, …, a_{n−1} are p-independent in K, and a_{n} ∈ K^{p}[a_{1}, …, a_{n−1}], a_{n} ∉ L^{p}[a_{1}, …, a_{n−1}]. Hence λ_{i,n−1}(a_{1}, …, a_{n−1}; a_{n}) ∈ K \L for some i. This shows one direction. For the other assume that for some a_{1}, …, a_{n} ∈ L and some i the element λ_{i,n−1}(a_{1}, …, a_{n−1}; a_{n}) is not in L. Note that the λ_{j}(a_{1}, …, a_{n−1}; a_{n}) are (the unique) solutions of the equation (∗) : ∑_{j} X_{j}^{p}m_{j,n−1}(a_{1}, …, a_{n−1}) = a_{n}. From λ_{i,n−1}(a_{1}, …, a_{n−1}; a_{n}) ∉ L, we obtain first that {a_{1}, …, a_{n−1}} is p-independent in K, and then that {a_{1}, …, a_{n}} is p-independent in L: if not, there would be elements b_{j} in L satisfying (∗) in L, and we would have b_{j} = λ_{j,n−1}(a_{1}, …, a_{n−1}; a_{n}) ∈ K for all j. Thus a_{1}, …, a_{n} is p-independent in L but not in K.

(1.10) Basic λ-terms. We work in a large separably closed field K, with finite p-basis B = {b_{1}, …, b_{e}}. We let I = I(B) = {1, …, p^{e}}, λ_{i}(B; −) = λ_{i,e}(b_{1}, …, b_{e}; −) and m_{i}(B) = m_{i,e}(b_{1}, …, b_{e}). If μ ∈ I^{n}, we define by induction on n the function λ_{μ}(B; −) and the p^{n}-monomial m_{μ}(B) as follows: if n = 0, then λ_{μ}(B; x) = x, m_{μ}(B) = 1; if n ≥ 1, write μ = ν^{⌢}i, where ν ∈ I^{n−1}, i ∈ I, and define

λ_{μ}(B; x) = λ_{i}(B; λ_{ν}(B; x)), m_{μ}(B) = m_{i}(B)^{p^{n−1}}m_{ν}(B).

Then we have, for any a ∈ K and n ∈ ℕ:

a = ∑_{μ∈I^{n}} λ_{μ}(B; a)^{p^{n}}m_{μ}(B).

If μ ∈ I^{n}, we call λ_{μ} a λ-term of level n. Note that all basic λ-terms are terms of the language L_{λ}(b_{1}, …, b_{e}). We let I^{<ω} = ⋃_{n∈ℕ} I^{n}.

If B ⊆ K is infinite, then by abuse of notation, we will denote by (λ_{μ}(B; x) | μ ∈ I^{n}) the set of all terms λ_{μ}(b_{1}, …, b_{e}; x) where e ∈ ℕ, b_{1}, …, b_{e} ∈ B, and μ ∈ I({b_{1}, …, b_{e}})^{n}. Note that for a given element c ∈ K, the set {λ_{μ}(B; c) | μ ∈ I^{n}} is finite, and that c ∈ 𝔽_{p}[λ_{μ}(B; c) | μ ∈ I^{n}]^{p^{n}}[B]. Lemma. Let k ⊆ L ⊆ K, assume that L is a separable extension of k, and that B is a p-basis of k and of K. Let C be a p-basis of L over k, and let k⟨C⟩ = k(λ_{μ}(B; c) | c ∈ C, μ ∈ I^{<ω}), the λ-functions being those of K.

(1) If c ∈ L^{p^{n}}(B), then λ_{μ}(B; c) ∈ L for every μ ∈ I^{n}.

(2) Lk⟨C⟩ is closed under the λ-functions of K. If a ∈ L and ν ∈ I^{n}, then λ_{ν}(B; a) ∈ L[λ_{μ}(B; c) | c ∈ C, μ ∈ I^{n}].

(3) If D is another p-basis of K and a ∈ L, ν ∈ I^{n}, then λ_{ν}(B; a) ∈ 𝔽_{p}[λ_{μ}(D; a), λ_{μ}(B; d) | d ∈ D, μ ∈ I^{n}]. If D ⊂ k, then λ_{ν}(B; a) ∈ k[λ_{μ}(D; a) | μ ∈ I^{n}].

Proof. (1) As B ⊂ L, we have L^{p^{n}}(B) = L^{p^{n}}[B], and c ∈ L^{p^{n}}[B_{0}] for some finite subset B_{0} of B, so that we may assume that B is finite. Then c = ∑_{j∈I} c_{j}^{p}m_{j}(B) for some c_{j} ∈ L^{p^{n−1}}. Thus c_{j} = λ_{j}(B; c) ∈ L^{p^{n−1}}. Since λ_{μ} = λ_{ν} ∘ λ_{j} for some ν ∈ I^{n−1} and j ∈ I, an induction on n gives us that λ_{ν}(B; c_{j}) = λ_{μ}(B; c) ∈ L.

(2) By assumption, B ∪ C is a p-basis of L; hence L = L^{p^{n}}[B, C] for every n ≥ 0. By definition of the λ-functions, we also have that c ∈ (𝔽_{p}[λ_{μ}(B; c) | μ ∈ I^{n}])^{p^{n}}[B] for any c ∈ K. Hence L ⊆ (L[λ_{μ}(B; c) | c ∈ C, μ ∈ I^{n}])^{p^{n}}[B], and (1) gives that λ_{ν}(B; a) ∈ L[λ_{μ}(B; c) | c ∈ C, μ ∈ I^{n}]. Hence Lk⟨C⟩ is closed under the λ-functions of K.

(3) By assumption, K = K^{p^{n}}[D] = K^{p^{n}}[B]. If a ∈ L, then a ∈ (𝔽_{p}[λ_{μ}(D; a) | μ ∈ I^{n}])^{p^{n}}[D], and if d ∈ D then d ∈ (𝔽_{p}[λ_{μ}(B; d) | μ ∈ I^{n}])^{p^{n}}[B]. Thus a ∈ 𝔽_{p}[λ_{μ}(D; a), λ_{μ}(B; d) | μ ∈ I^{n}, d ∈ D]^{p^{n}}[B], and (1) gives the first assertion. The second assertion follows from the fact that D ⊂ k and that k is closed under the λ-functions of K.

(1.11) λ-polynomial rings over k when k has finite degree of imperfection.

Let B be a p-basis of k of finite size e. Then the set I defined above has size p^{e}. We define k⟨X⟩_{≤n,B} to be the quotient of the polynomial ring k[X_{μ} | μ ∈ I^{≤n}] by the ideal generated by the polynomials

X_{μ} − ∑_{i∈I} X_{μ^{⌢}i}^{p}m_{i}(B)

for μ ∈ I^{≤n−1}, and we let k⟨X⟩_{B} = ⋃_{n} k⟨X⟩_{≤n,B}. If we consider only one p-basis B, then we will omit B from the notation. We define k⟨X_{1}, …, X_{n}⟩_{≤n,B} and k⟨X_{1}, …, X_{n}⟩_{B} analogously.

Then B is a p-basis of (the field of fractions of) k⟨X_{1}, …, X_{n}⟩_{B}. If B is a p-basis of an extension K of k and a_{1}, …, a_{n} ∈ K, then there is a unique k-morphism k⟨X_{1}, …, X_{n}⟩_{B} → K which sends X_{i} to a_{i}. Note also that k⟨X_{1}, …, X_{n}⟩_{B} is generated as a ring by the elements λ_{μ}(B; X_{i}), i = 1, …, n, μ ∈ I(B)^{<ω}.

If C is another p-basis of k, then there is a natural k[X_{1}, …, X_{n}]-isomorphism : k⟨X_{1}, …, X_{n}⟩_{B} → k⟨X_{1}, …, X_{n}⟩_{C}. This follows from the previous observation, and the fact that by Lemma 1.10(3), we have k⟨X_{1}, …, X_{n}⟩_{≤m,C} ⊆ k⟨X_{1}, …, X_{n}⟩_{≤m,B} and k⟨X_{1}, …, X_{n}⟩_{≤m,B} ⊆ k⟨X_{1}, …, X_{n}⟩_{≤m,C} for every m ∈ ℕ.

(1.12) Lemma. Let K be a separable extension of k, with p-basis B over k, and let C ⊂ k be p-independent. Consider the fields k_{1} = k(c^{1/p} | c ∈ C) and k_{2} = k(c^{1/p^{n}} | c ∈ C, n ∈ ℕ). Then B is a p-basis of k_{1}K over k_{1} and of k_{2}K over k_{2}. Proof. Clearly k_{1}K = k_{1}K^{p}[B] and k_{2}K = k_{2}K^{p}[B], so we only need to show that B is p-independent over k_{1} in k_{1}K and over k_{2} in k_{2}K. Since K is a separable extension of k, the fields k^{p^{−∞}} and K are linearly disjoint over k. Hence K is linearly disjoint from k_{1} and from k_{2} over k. This implies that k_{1}K^{p} and K are linearly disjoint over kK^{p}: hence linearly independent elements of the kK^{p}-vector space K stay linearly independent in the k_{1}K^{p}-vector space k_{1}K. This shows that B stays p-independent over k_{1} in k_{1}K. The proof for k_{2}K is similar.

2. The main result Let e ∈ ℕ ∪ {∞}, and L_{λ,σ} = L_{λ} ∪ {σ}. In this section we prove that the L_{λ,σ}-theory

of difference fields models of T_{λ} and of degree of imperfection ≤ e has a model companion, and that the L_{σ}(b_{1}, …, b_{e})-theory of difference fields with p-basis {b_{1}, …, b_{e}} has a model companion. Before embarking into preparatory lemmas, let me explain the strategy of the proof.

Clearly existentially closed difference fields (in L_{λ,σ} or in L_{σ}(b_{1}, …, b_{e})) need to be separably closed, since any automorphism of a field extends to an automorphism of its separable closure. By a linearisation argument already used for the axiomatisation of the theory ACFA of existentially closed difference fields, one reduces the problem of solving systems of σ-equations to the problem of finding a point ā in some variety U defined over K, such that (ā, σ(ā)) ∈ V for some variety V contained in U × σ(U) and projecting generically onto U and σ(U). While in the theory of difference fields, we know that such a problem always has a solution in some extension (cf. the axiomatisation of ACFA, see [CH] or [M], or (1.2)), here we need to find conditions on V, elementary in the parameters of K used to define V, and which ensure that there is a solution ā in some L_{λ,σ}-extension L of K. In particular, such an ā must generate a difference field which is separable over K.

It turns out that the main difficulty is to find elementary conditions which ensure that K(ā)_{σ} is a separable extension of K: if we manage to do that, then Lemma (2.2) allows us to conclude. We then reduce the problem to the case where dim(U) = dim(V), see Lemma (2.1). To show that K(ā)_{σ} is a separable extension of K, it is enough to show that K(ā, σ(ā), …, σ^{m}(ā)) is a separable extension of K for every m ∈ ℕ. We solve this problem in (2.4) and (2.5).

Let me mention a case where the solution of the problem is easy: let K(V) = K(x̄, ȳ), and assume that ȳ is separably algebraic over K(x̄). This condition is certainly elementary in the parameters defining V. Let (ā, ā_{1}) be a generic over K of the algebraic set V, and let σ be an extension of σ to Ω which sends ā to ā_{1}. Then σ(ā) ∈ K(ā)^{s}, and therefore K(ā, …, σ^{m}(ā)) ⊆ K(ā)^{s} for every m ≥ 0, which implies that K(ā)_{σ} is a separable extension of K. Similarly, if x̄ is separably algebraic over K(ȳ) in K(V). However there are examples of equations solvable in a separable extension of K which are not of this form, e.g., the equation σ^{2}(x)^{p} + bσ(x) + x^{p} = 0 has a solution in a separable extension of K (Note also that if a is a solution of this equation which is generic over K, then the degree of transcendence of K(a)_{σ} over K is 2, and K(a)_{σ} has same p-basis as K).

(2.1) Lemma. Let k be a difference field, and k(ā)_{σ} a separable extension of k. Then there is a transformal transcendence basis b̄ ⊆ ā of k(ā)_{σ} over k, such that k(ā)_{σ} is a separable extension of k(b̄)_{σ}. Proof. The proof is by induction on the transformal transcendence degree Δ(ā/k) of ā over k. If it equals 0, there is nothing to prove, so we will assume that it is positive. It suffices to find an element b ∈ ā such that b is transformally transcendental over k and k(ā)_{σ} is a separable extension of k(b)_{σ}: we will then use induction applied to the extension k(ā)_{σ} of k(b)_{σ} to conclude.

For n ∈ ℕ consider the field K_{n} = k(ā, σ(ā), …, σ^{n}(ā)). Replacing ā by ā^{⌢} ⋯^{⌢} σ^{m}(ā) for some m, we may assume that tr.deg(K_{n}/k) = d + ne where d = tr.deg(ā/k) and e = Δ(ā/k). Each K_{n} is a finitely generated separable extension of k, and therefore has a separating transcendence basis over k of size d+ne. In other words: (1) [K_{n} : k(K_{n}^{p})] = p^{d+ne}. Claim. There is b ∈ ā such that for any n ≥ 0 the elements b, σ(b), …, σ^{n}(b) are pindependent over k in K_{n}.

Otherwise, for each b ∈ ā, there is an n = n(b) such that the elements b, σ(b), …, σ^{n}(b) are not p-independent over k in K_{n}. Take such an n, minimal; then for some i ≤ n we have σ^{i}(b) ∈ k(K_{n}^{p})(b, σ(b), …, σ^{i−1}(b)). As σ(K_{n}) ⊆ K_{n+1}, we get that σ^{i+1}(b) ∈ k(K_{n+1}^{p})(b, …, σ^{i−1}(b)), …, σ^{i+m}(b) ∈ k(K_{n+m}^{p})(b, …, σ^{i−1}(b)) for m ≥ 0, so that any subset of {b, …, σ^{m+n}(b)} which is p-independent in K_{m+n} has size ≤ n.

Let N = sup{n(b) | b ∈ ā}, and let m ≥ N. By the above, any subset of {ā, …, σ^{m}(ā)} which is p-independent in K_{m} has size at most |ā|N. Then [K_{m} : k(K_{m}^{p})] ≤ p^{|ā|N}, which contradicts the unboundedness of [K_{m} : k(K_{m}^{p})] given by equation (1) and proves the claim.

Take an element b satisfying the conclusion of the claim. Then the elements σ^{i}(b), i ∈ ℕ, are p-independent over k in the field k(σ^{i}(ā) | i ∈ ℕ). Using the fact that σ^{−1} is an automorphism, we deduce that the elements σ^{i}(b), i ∈ ℤ, are p-independent over k in k(σ^{i}(ā) | i ∈ ℤ) = k(ā)_{σ}. Hence k(ā)_{σ} is a separable extension of k(b)_{σ}.

(2.2) Lemma. Let k be a difference field with finite p-basis B, and let k(ā)_{σ} be a finitely generated difference field extending k which is separable over k. Then k(ā)_{σ} embeds into a difference field M with p-basis B. Proof. By (2.1), it suffices to show the result in the following two cases:

(a) ā is transformally independent over k.

(b) ā is transformally algebraic over k. The separable closure of k has also p-basis B, so we may assume that k is separably

closed.

(a) Note that k(ā)_{σ} is a purely transcendental extension of k, with transcendence basis {σ^{i}(d) | d ∈ ā, i ∈ ℤ} over k. As σ induces a permutation of this transcendence basis, the automorphism σ of k(ā)_{σ} extends uniquely to an automorphism of M = k(σ^{i}(d)^{1/p^{n}} | d ∈ ā, i ∈ ℤ, n ∈ ℕ). Then M is a separable extension of k, with p-basis B.

(b) let C be a p-basis of k(ā)_{σ} over k. Then k(ā)_{σ} = k(ā^{p})_{σ}(C). If C = ∅, then we are done: B is a p-basis of k(ā)_{σ}. Assume therefore that C ≠ ∅. Consider the quotient of the polynomial ring k(C)⟨X_{c} | c ∈ C⟩_{B} by the ideal generated by the polynomials X_{c} − c for c ∈ C, and let M_{0} be its field of fractions. Then M_{0} is a separable extension of k, with p-basis B, and we may choose M_{0} free from k(ā)_{σ} over k(C), because the elements of C are p-independent over k in k(ā)_{σ}. Hence M = M_{0}k(ā)_{σ} is a separable extension of k, with p-basis B (by (1.10)(1) and because k(ā)_{σ} = k(ā^{p})_{σ}(C)). We consider the λ-functions of M indexed by I = I(B), and define σ(λ_{μ}(B; c)) = λ_{μ}(σ(B); σ(c)) for μ ∈ I^{<ω}. We need to show that σ defines an automorphism of M. For n ∈ ℕ let M_{n} = k(ā)_{σ}(λ_{μ}(B; c) | c ∈ C, μ ∈ I^{n}), let 0_{n} be the element of I^{n} corresponding to the p^{n}-monomial 1, and let J_{n} = I^{n} \ {0_{n}}. Then the elements λ_{μ}(B; c), c ∈ C, μ ∈ J_{n}, are algebraically independent over k(ā)_{σ} and generate M_{n} over k(ā)_{σ}. As σ(B) is also a pbasis of M, we know that for c ∈ C we have σ(c) = ∑_{μ∈I^{n}} λ_{μ}(σ(B); σ(c))^{p^{n}}m_{μ}(σ(B)), and by definition of σ this coincides with ∑_{μ∈I^{n}} σ(λ_{μ}(B; c)^{p^{n}})σ(m_{μ}(B)). Hence σ extends uniquely to a ring homomorphism M_{n} → M_{n}. We have:

\begin{aligned}k(ā)_{σ}(λ_{μ}(B; c) | c ∈ C, μ ∈ I^{n}) & = k(ā)_{σ}(λ_{μ}(B, σ(c)) | c ∈ C, μ ∈ I^{n}) (by (1.10)(2)) \\ & = k(ā)_{σ}(λ_{μ}(σ(B), σ(c)) | c ∈ C, μ ∈ I^{n}) (by (1.10)(3)) \end{aligned}

This shows that σ is onto, and therefore is also injective as the elements λ_{μ}(σ(B), σ(c)), c ∈ C, μ ∈ J_{n}, must be algebraically independent over k(ā)_{σ}. Hence σ defines an automorphism of M_{n} for every n, and therefore defines an automorphism of M.

(2.3) An example. The following example shows that Lemma (2.2) does not generalise to the case where B is infinite.

Let b be an element transformally transcendental over 𝔽_{p}, k = 𝔽_{p}(b)_{σ}^{s}, and let a be a solution of σ(x) = x + b. If L is a difference field containing k(a)_{σ} which is separable over k, then {σ^{i}(b) | i ∈ ℤ} ∪ {a} is p-independent in L.

Let B = {σ^{i}(b) | i ∈ ℤ}, let L be a difference field containing k(a)_{σ}, separable over k, and assume that a ∈ L^{p}[B]. Let m < n ∈ ℤ be such that a ∈ L^{p}[σ^{m}(b), …, σ^{n}(b)] and n − m is minimal. Let B_{0} = {σ^{m}(b), …, σ^{n}(b)}, and write a = ∑_{i} λ_{i}(B_{0}; a)^{p}m_{i}(B_{0}). By minimality of n − m, we get that σ^{m}(b) ∈ L^{p}[a, σ^{m+1}(b), …, σ^{n}(b)] and σ^{n}(b) ∈ L^{p}[σ^{m}(b), …, σ^{n−1}(b), a]. This implies that there are indices i and j such that λ_{i}(B_{0}; a) ≠ 0, λ_{j}(B; a) ≠ 0, and the exponents of σ^{m}(b) in m_{i}(B_{0}) and of σ^{n}(b) in m_{j}(B_{0}) are positive. Then σ(a)−a = ∑_{k} λ_{k}(σ(B_{0}; σ(a))^{p}m_{k}(B_{0})−λ_{k}(B_{0}; a)^{p}m_{k}(B_{0}), and the coefficients of the monomials m_{i}(B_{0}) and σ(m_{j}(B_{0})) in this expression are non-zero, because σ^{m}(b) ∉ σ(B_{0}), σ^{n+1}(b) ∉ B_{0}. But this contradicts the p-independence of B and the fact that σ(a)−a = b.

(2.4) Lemma. Let k(ā)_{σ} be a separable extension of k, with σ(ā) ∈ k(ā)^{alg}, and let d = tr.deg(k(ā)/k).

Then there is a separating transcendence basis c̄ ∪ d̄ of k(ā, σ(ā), …, σ^{d}(ā)) over k, such that c̄ ∪ σ^{−1}(d̄) ⊆ ā ∪ σ(ā) ∪ ⋯ ∪ σ^{d−1}(ā) and σ^{−1}(d̄) is separably algebraic over k(c̄^{p}, d̄). Proof. Let 0 ≤ i ≤ d + 1, and consider the field

K_{d,i} = k(ā^{p}, …, σ^{i−1}(ā^{p}), σ^{i}(ā), …, σ^{d}(ā)).

Then K_{d,0} = k(ā, …, σ^{d}(ā)), K_{d,d+1} = k(ā^{p}, …, σ^{d}(ā^{p})) = k(K_{d,0}^{p}), and K_{d,i} is a purely inseparable extension of K_{d,i+1} for every i ≤ d. As K_{d,0} is a separable, finitely generated extension of k, of transcendence degree d over k, we have: [K_{d,0} : K_{d,d+1}] = p^{d}. Hence, there is 0 ≤ i ≤ d such that K_{d,i} = K_{d,i+1}, and we fix such an i. Choose d̄ ⊆ σ^{i+1}(ā) ∪ ⋯ ∪ σ^{d}(ā) maximal p-independent over k in K_{d,0}. Then K_{d,i} = K_{d,i+1} = k(K_{d,0}^{p})[d̄], and K_{d,0} = K_{d,i+1}(ā, …, σ^{i−1}(ā)). Choose c̄ ⊆ ā ∪ ⋯ ∪ σ^{i−1}(ā) such that c̄ ∪ d̄ is a p-basis of K_{d,0} over k. Then c̄, d̄ satisfy the conclusion: since σ^{−1}(d̄) ∈ K_{d,i}, we have that σ^{−1}(d̄) ∈ k(K_{d,0}^{p})[d̄] ⊆ k(c̄^{p}, d̄^{p})^{s}[d̄] ⊆ k(c̄^{p}, d̄)^{s}.

(2.5) Lemma. Let U and V be varieties of dimension d defined over the difference field k, and assume that V ⊆ U × σ(U), and that V projects generically onto U and onto σ(U). Write k(V) = k(x_{1}, …, x_{n}, y_{1}, …, y_{n}) and assume that there is 1 ≤ r ≤ d such that (x_{1}, …, x_{r}, y_{r+1}, …, y_{d}) is a separating transcendence basis of k(V) over k, and that x_{r+1}, …, x_{d} are separably algebraic over k(x_{1}^{p}, …, x_{r}^{p}, y_{r+1}, …, y_{d}). Then there is a tuple ā such that k(ā)_{σ} is a separable extension of k, and (ā, σ(ā)) is a generic of V over k. Proof. Choose in some difference field (K, σ) containing k (e.g., in a model of ACFA), a tuple ā = (a_{1}, …, a_{n}) such that (ā, σ(ā)) is a generic of the algebraic set V. Our assumption on the dimensions of U and V implies that σ(ā) is algebraic over k(ā), so that k(ā)_{σ} ⊆ k(ā)^{alg}. Let c̄ = (a_{1}, …, a_{r}), and d̄ = (σ(a_{r+1}), …, σ(a_{d})). Then c̄ ∪ d̄ is a separating transcendence basis of k(ā, σ(ā)) over k, and σ^{−1}(d̄) is separably algebraic over k(c̄^{p}, d̄). We will show by induction on m ≥ 1 that

(i) σ^{−1}(d̄), …, σ^{m−2}(d̄) are separably algebraic over k(c̄^{p}, σ^{m−1}(d̄)), and (ii) c̄ ∪ σ^{m−1}(d̄) is a separating transcendence basis of k(ā, …, σ^{m}(ā)) over k. For m = 1, this is our assumption. Assume the result proved for m. Then σ(c̄) ∈ k(c̄, d̄)^{s} and by applying σ to (i), we get that d̄, …, σ^{m−1}(d̄) ∈ k(σ(c̄)^{p}, σ^{m}(d̄))^{s}. By Lemma (1.6)(2) applied to k(c̄, σ^{m}(d̄)), d̄ and σ(c̄), we obtain that σ(c̄), d̄ ∈ k(c̄, σ^{m}(d̄))^{s}, and therefore that d̄, …, σ^{m−1}(d̄) ∈ k(c̄^{p}, σ^{m}(d̄))^{s} (as σ(c̄)^{p} ∈ k(c̄^{p}, σ^{m}(d̄^{p}))^{s}). From σ^{−1}(d̄) ∈ k(c̄^{p}, d̄)^{s} we deduce (i).

We know that ā ∈ k(c̄, d̄)^{s} ⊆ k(c̄, σ^{m}(d̄))^{s}, and by induction hypothesis that σ(ā), …, σ^{m+1}(ā) ∈ k(σ(c̄), σ^{m}(d̄))^{s}. As σ(c̄) ∈ k(c̄, d̄)^{s}, we get (ii).

Hence, k(ā, …, σ^{m}(ā)) is a separable extension of k for every m ≥ 1. This implies that k(ā)_{σ} is separable over k.

(2.6) Theorem. Let e ∈ ℕ. The theory of difference fields of degree of imperfection ≤ e which are models of T_{λ}, has a model companion SCFA_{e,λ}. The theory SCFA_{e,λ} is axiomatised by formulas expressing the following properties of the L_{λ,σ}-structure (K, σ):

(i) SCF_{e,λ}, (ii) σ is an automorphism of K, (iii) If U and V are varieties defined over K and satisfy the hypotheses of (2.5), then there

is a tuple ā such that (ā, σ(ā)) ∈ V.

Proof. We will first show that (iii) is expressible by an infinite collection of first-order statements. Let U and V be varieties defined over K. Then there is a finite tuple c̄ of elements of K, and finite tuples F (X̄, T̄) and G(X̄, Ȳ , T̄) of polynomials over 𝔽_{p}, such that the ideal of K[X̄] of polynomials vanishing on U is generated by the tuple F (X̄, c̄), and the ideal of K[X̄, Ȳ] of polynomials vanishing on V is generated by G(X̄, Ȳ , c̄). Write K[V] = K[x̄, ȳ], x̄ = (x_{1}, …, x_{n}), ȳ = (y_{1}, …, y_{n}), and let r ≤ d be such that K(x_{1}, …, x_{r}, y_{r+1}, …, y_{d}) is a separating transcendence basis of K(V) over K, and x_{r+1}, …, x_{d} are separably algebraic over K(x_{1}^{p}, …, x_{r}^{p}, y_{r+1}, …, y_{d}). Since the algebraic varieties U and V are defined over 𝔽_{p}(c̄), this implies that 𝔽_{p}(c̄)(V) is separably algebraic over 𝔽_{p}(c̄)(x_{1}, …, x_{r}, y_{r+1}, …, y_{d}), and x_{r+1}, …, x_{d} are separably algebraic over 𝔽_{p}(c̄)(x_{1}^{p}, …, x_{r}^{p}, y_{r+1}, …, y_{d}).

If d̄ is a finite tuple of K of the same length as c̄, define U(d̄) to be the algebraic set defined by the equations F (X̄, d̄) = 0, and V (d̄) to be the algebraic set defined by G(X̄, Ȳ , d̄) = 0. We claim that there is a quantifier-free formula φ of the language L of rings satisfied by (c̄, σ(c̄)), and such that if (d̄, σ(d̄)) satisfies φ, then the algebraic sets U(d̄) and V (d̄) have the following properties:

— U(d̄) and V (d̄) are (absolutely irreducible) varieties of dimension d.

— V (d̄) ⊆ U(d̄) × U(σ(d̄)), and the projections V (d̄) → U(d̄) and V (d̄) → U(σ(d̄)) are generically onto.

— 𝔽_{p}(d̄)(V (d̄)) is separably algebraic over 𝔽_{p}(d̄)(x_{1}, …, x_{r}, y_{r+1}, …, y_{d}), and the elements x_{r+1}, …, x_{d} of 𝔽_{p}(d̄)(V (d̄)) are separably algebraic over 𝔽_{p}(d̄)(x_{1}^{p}, …, x_{r}^{p}, y_{r+1}, …, y_{d}).

Since the theory of algebraically closed fields is strongly minimal and eliminates the quantifier ∃^{∞}, the property of the Zariski closure of a definable set of being of a given dimension, is elementary in the parameters defining this set. Then, standard results on polynomial rings over fields (see e.g. [DS]) and elimination of quantifiers of the theory of algebraically closed fields give that the first two items are elementary properties of d̄.

The conditions of separability given in the third item are easily expressible using the polynomials G(x̄, ȳ, d̄), since they correspond to some sub-determinants of the Jacobian of G(x̄, ȳ, d̄) being non-zero, see [L1] Proposition X.7.3. Note that as d = dim(V (d̄)), the elements x_{1}, …, x_{r}, y_{r+1}, …, y_{d} of 𝔽_{p}(d̄)(V (d̄)) are necessarily algebraically independent over 𝔽_{p}(d̄).

Using compactness, we deduce that there is a scheme of axioms expressing the properties (i), (ii) and (iii). It now remains to prove that SCFA_{e,λ} is indeed the model-companion of the theory of difference fields of degree of imperfection ≤ e which are models of T_{λ}. We will first show that every difference field (K, σ) of degree of imperfection ≤ e which is a model of T_{λ} embeds in a model of SCFA_{e,λ}.

First note that if K is a model of T_{λ}, then any field automorphism σ of K is an L_{λ}-automorphism, since the λ-functions of K are first-order definable in K. Also, any difference field of degree of imperfection f < e embeds in one of degree of imperfection e: let t̄ be an (e − f)-tuple of elements algebraically independent over K, and extend σ to K(t̄) so that it fixes the elements of t̄. Then the degree of imperfection of K(t̄) is e. Also, any automorphism of a field K extends to an automorphism of the separable closure K^{s} of K, and we may therefore assume that K satisfies (i) and (ii).

Let U and V be varieties defined over K and satisfying the hypotheses of (iii). By (2.5), there is a separable extension K(ā)_{σ} of K such that (ā, σ(ā)) is a generic of V. By Lemma (2.2) there is a difference field M containing K(ā)_{σ}, separable over K and with the same p-basis as K. Thus we have shown that every occurrence of axiom (iii) can be satisfied in some separable extension M of K with the same p-basis as K. A standard limit argument then shows that K embeds in a model of SCFA_{e,λ}. It remains to show that SCFA_{e,λ} is model-complete, i.e., that if K ⊆ L are models of SCFA_{e,λ} then K is existentially closed in L.

Let (K, σ) ⊆ (L, σ) models of SCFA_{e,λ}. As e is finite, if B is a p-basis of K, then B is also a p-basis of L, and therefore L is an elementary L-extension of K. Any quantifierfree formula of L_{λ} is equivalent, modulo the L_{λ}(B)-theory of (K, B), to an existential L(B)-formula. Hence, it is enough to show that any quantifier-free L_{σ}-formula φ(x̄) with parameters in K which is satisfied in L is already satisfied in K. Using the usual trick of replacing the formula (x ≠ 0) by (∃y xy = 1), we may furthermore assume that φ(x̄) is positive, i.e., that φ(x̄) is a finite conjunction of difference equations with coefficients in K. Let ā be a tuple of L satisfying φ(x̄). Because φ is a finite conjunction of difference equations, any tuple d̄ with I_{σ}(d̄/K) ⊇ I_{σ}(ā/K) will satisfy φ. We will show that such a tuple exists in an elementary extension K^{∗} of K, and therefore also in K.

By Lemma (2.1), there is a subtuple b̄ of ā such that b̄ is transformally independent over K and K(ā)_{σ} is a separable transformally algebraic extension of k = K(b̄)_{σ}. Choose m large enough such that k(ā)_{σ} ⊆ k(ā, …, σ^{m}(ā))^{alg}, and such that if ā′ is a tuple (in some model of ACFA containing K) such that there is a k-isomorphism sending (ā′, …, σ^{m}(ā′)) to (ā, …, σ^{m}(ā)), then this k-isomorphism extends to a k-isomorphism of difference fields k(ā′)_{σ} → k(ā)_{σ} (see 8.20.13 in [C] for the existence of such an m). By (2.4) applied to k = K(b̄)_{σ}, there is an integer n > m and a separating transcendence basis c̄ ∪ d̄ of k(ā, …, σ^{n}(ā)) over k with c̄∪σ^{−1}(d̄) ⊆ ā∪⋯∪σ^{n−1}(ā), and such that σ^{−1}(d̄) is separably algebraic over k(c̄^{p}, d̄).

Let U be the algebraic locus of (ā, …, σ^{n−1}(ā)) over k^{s} (i.e., the smallest algebraic set defined over k^{s} and containing (ā, …, σ^{n−1}(ā)), and V the algebraic locus of (ā, …, σ^{n−1}(ā), σ(ā), …, σ^{n}(ā)) over k^{s}. As K(ā)_{σ} is a separable extension of k = K(b̄)_{σ}, the algebraic sets U and V are varieties defined over k^{s}, of the same dimension (because k(ā)_{σ} ⊆ k(ā, …, σ^{n−1}(ā))^{alg}), and they satisfy the hypotheses of (2.5) over k^{s} after some renaming of the variables.

Let K^{∗} be a sufficiently saturated elementary extension of K. It then suffices to show that φ(x̄) is satisfied in K^{∗}. We first claim that K^{∗} contains arbitrarily large finite tuples of elements which are transformally transcendental over K. First observe that any instance of (iii) with the varieties U and V defined over K, has a realisation ē in K^{∗} with (ē, σ(ē)) a generic of the algebraic variety V over K: this follows from the saturation of K^{∗} and the fact that using (iii), any formula (x̄, σ(x̄)) ∈ V ∧ zg(x̄, σ(x̄)) = 1, where g ∈ K[V] \ {0}, has a solution in K. Thus for any n, m, there is an m-tuple ē of elements of K^{∗} satisfying σ^{n}(x) = x and tr.deg(k(ē)_{σ}/k) = nm. By saturation, there is ē ∈ (K^{∗})^{m} such that the elements σ^{i}(g), g ∈ ē, i ∈ ℕ, are algebraically independent over K.

Choose a tuple c̄ of K^{∗}, of the same length as b̄, and which is transformally independent over K. Then the difference fields k = K(b̄)_{σ} and K(c̄)_{σ} are K-isomorphic, by an isomorphism f sending b̄ to c̄. By Proposition 2.10 of [CH], if τ is an automorphism of K(c̄)_{σ}^{s} which agrees with σ on K(c̄)_{σ}, then τ is conjugate to σ by an element of Gal(K(c̄)_{σ}^{s}/K(c̄)_{σ}). Hence, the isomorphism f extends to a difference field embedding f : k^{s} → K^{∗}. By axiom (iii) and saturation of K^{∗}, there is a tuple ē in K^{∗}, such that (ē, σ(ē)) is a generic of f(V) over K(c̄)_{σ}^{s}. Then ē = (d̄, …, σ^{n−1}(d̄)) for some tuple d̄ of the same length as ā, and I_{σ}(ā/K) = I_{σ}(d̄/K) because n > m, so that d̄ satisfies φ.

(2.7) We only added the λ-functions symbols to the language to ensure that if (L, σ) is an extension of (K, σ) then L is separable over K. In case the degree of imperfection e is finite, this could have been achieved as well by adding constant symbols for the elements of a p-basis, together with the axioms expressing that these elements form a p-basis. Theorem. Consider the language L_{σ}(b_{1}, …, b_{e}), and the class of difference fields (K, σ) with p-basis {b_{1}, …, b_{e}}. The existentially closed members of this class form an elementary class, axiomatised by (ii), (iii) above, together with (i’): SCF_{e,b}. Proof. This follows easily from (2.6) and the following observations: a field expands uniquely to a model of T_{λ}; if K ⊆ L are two difference fields with the same p-basis, then K is an L_{λ,σ}-substructure of L.

(2.8) Let us denote this theory by SCFA_{e,b}. Note that this gives a slightly less general result than (2.6), as the p-basis is fixed by σ. However, the advantage is that the theory SCF_{e,b} of separably closed fields with p-basis {b_{1}, …, b_{e}} eliminates imaginaries. It is also model-complete but doesn’t eliminate quantifiers as the λ-functions are only existentially definable. We will see that it implies that any completion of SCFA_{e,b} eliminates imaginaries (in the language L_{σ}(b̄)).

(2.9) The case e = ∞ necessitates a few more lemmas. The problem comes from the fact that quantifier-free L_{λ}-formulas are NOT equivalent modulo SCF_{∞,λ} to existential L-formulas, because the λ-functions allow us to say that elements are p-independent. Indeed, the formula ⋀_{i}(λ_{i,n}(x_{1}, …, x_{n}; y) = 0) ∧ y ≠ 0 says that either (x_{1}, …, x_{n}) are p-dependent, or that (x_{1}, …, x_{n}, y) are p-independent. We have a first reduction: Lemma. Let φ(x̄) be a quantifier-free L_{λ}-formula. Then, modulo SCF_{∞,λ}, φ(x̄) is equivalent to a finite disjunction of L-formulas of the form

∃ȳ ψ(x̄, ȳ) ∧ θ(x̄, ȳ),

where ψ(x̄, ȳ) is quantifier-free, and θ(x̄, ȳ) is a conjunction of universal formulas expressing that some subtuples of x̄^{⌢}ȳ are p-independent. Proof. Unraveling the λ-functions appearing in φ, it is easy to see that there are: an integer m, a tuple ȳ = (y_{1}, …, y_{m}), a quantifier-free L-formula ψ_{0}(x̄, ȳ), and a formula θ_{0}(x̄, ȳ), which is a conjunction of formulas of the form y_{i} = λ_{j,k}(z̄; t) with (z̄, t) some (k + 1)-subtuple of (x̄, y_{1}, …, y_{i−1}), such that φ(x̄) is equivalent to the formula

∃ȳ ψ_{0}(x̄, ȳ) ∧ θ_{0}(x̄, ȳ).

Modulo SCF_{∞,λ}, the formula y = λ_{j,k}(z̄; t) is equivalent to a formula (∃ū ψ_{j}(z̄, t, y, ū)) ∨ (y = 0 ∧ θ_{j}(z̄, t)), where ψ_{j}(z̄, t, y, ū) is the quantifier-free L-formula

(t = ∑_{i=1}^{p^{k}} u_{i}^{p}m_{i,k}(z̄) ∧ u_{j} = y) ∨ (y = 0 ∧ 0 = ∑_{i=1}^{p^{k}} u_{i}^{p}m_{i,k}(z̄) ∧ ⋁_{i=1}^{p^{k}} u_{i} ≠ 0),

and θ_{j}(z̄, t) is the universal L-formula expressing that (z̄, t) is p-independent. The result follows.

(2.10) Lemma. Let ā be transformally algebraic over the difference field k. Assume that σ(ā) ∈ k(ā)^{alg}, and that k(ā)_{σ} is separable over k. For some m, there are tuples b̄ ⊂ {ā, …, σ^{m−1}(ā)}, c̄ ⊆ {σ(ā), …, σ^{m−1}(ā)} and d̄⊂ {σ(ā), …, σ^{m}(ā)}, such that

(a) b̄, c̄, d̄ is a separating transcendence basis of k(ā, …, σ^{m}(ā)).

(b) c̄ ∈ k(b̄^{p}, d̄^{p}, σ(c̄))^{s}, and σ(c̄) ∈ k(b̄^{p}, d̄^{p}, c̄)^{s}.
(c) σ^{−1}(d̄) ∈ k(b̄^{p}, c̄, d̄)^{s}.

Proof. Let c̄ ⊆ {σ^{i}(ā) | i ∈ ℤ} be a p-basis of k(ā)_{σ} over k. Replacing c̄ by σ^{i}(c̄) for some i ∈ ℕ and ā by ā^{⌢} ⋯^{⌢} σ^{j}(ā) for some j ∈ ℕ, we may assume that ā contains c̄ ∪ σ^{−1}(c̄), and that c̄ ∈ k(ā)^{p}(σ^{−1}(c̄)), σ^{−1}(c̄) ∈ k(ā)^{p}(c̄). Let d = tr.deg(k(ā)_{σ}/k), and consider the fields K_{d,i}, i = 0, …, d + 1, defined in (2.4). Let i be such that K_{d,i+1} = K_{d,i}; then K_{d,i+1} contains the tuple σ^{d}(c̄), which is p-independent over k. Let d̄ ⊂ {σ^{i+1}(ā), …, σ^{d}(ā)} be such that (σ^{d}(c̄), d̄) is maximal p-independent over k in K_{d,0}. Let b̄ ⊆ {ā, …, σ^{i−1}(ā)} be such that b̄, c̄, d̄ is a p-basis of K_{d,0} over k. Then σ^{−1}(d̄) ∈ k(b̄^{p}, c̄, d̄)^{s} (see (2.4)), and so (b̄, c̄, d̄) is our desired tuple.

(2.11) Lemma. Let U and V be varieties of dimension d, defined over a difference field k. We assume that V ⊆ U × σ(U), and that V projects generically onto U and σ(U). Let us write k(V) = k(x_{1}, …, x_{n}, y_{1}, …, y_{n}). We also assume that there are integers 1 ≤ r ≤ s ≤ d such that

— (x_{1}, …, x_{r}, y_{r+1}, …, y_{d}) is a separating transcendence basis of k(V) over k.

— x_{r+1}, …, x_{s} are separably algebraic over k(x_{1}^{p}, …, x_{r}^{p}, y_{r+1}, …, y_{s}, y_{s+1}^{p}, …, y_{d}^{p}), and y_{r+1}, …, y_{s} are separably algebraic over k(x_{1}^{p}, …, x_{r}^{p}, x_{r+1}, …, x_{s}, y_{s+1}^{p}, …, y_{d}^{p}).

— x_{s+1}, …, x_{d} are separably algebraic over k(x_{1}^{p}, …, x_{r}^{p}, y_{r+1}, …, y_{d}). Then in some difference field extending k, there is a tuple ā such that k(ā)_{σ} is separable over k, (ā, σ(ā)) ∈ V and (a_{r+1}, …, a_{s}) is p-independent over k in k(ā)_{σ}. Proof. Choose in some difference field (K, σ) containing k (e.g., in a model of ACFA), a tuple ā = (a_{1}, …, a_{n}) such that (ā, σ(ā)) is a generic of the algebraic set V. Our assumption on the dimensions of U and V implies that σ(ā) is algebraic over k(ā), so that k(ā)_{σ} ⊆ k(ā)^{alg}. Let b̄ = (a_{1}, …, a_{r}), c̄ = (σ(a_{r+1}), …, σ(a_{s})), and d̄= (σ(a_{s+1}), …, σ(a_{d})). Then b̄ ∪ c̄ ∪ d̄ is a separating transcendence basis of k(ā, σ(ā)) over k, and σ^{−1}(d̄) is separably algebraic over k(b̄^{p}, c̄, d̄). Moreover, c̄ is separably algebraic over k(b̄^{p}, σ^{−1}(c̄), d̄^{p}) and σ^{−1}(c̄) is separably algebraic over k(b̄^{p}, c̄, d̄^{p}), and therefore, for every M ≥ 0, we have

(∗)  k(ā^{p}, …, σ^{M+m}(ā)^{p})(c̄) = k(ā^{p}, …, σ^{M+m}(ā)^{p})(σ^{M−1}(c̄)).

As in (2.5), one shows that for all M:

(a) σ^{−1}(d̄), …, σ^{M−2}(d̄) ∈ k(b̄^{p}, σ^{M−1}(c̄), σ^{M−1}(d̄))^{s}.

(b) b̄ ∪ σ^{M−1}(c̄, d̄) is a separating transcendence basis of k(ā, …, σ^{m+M}(ā)) over k. By (∗), if −1 ≤ j ≤ M, then b̄, σ^{j}(c̄), σ^{M−1}(d̄) is also a separating transcendence basis of k(ā, …, σ^{m+M}(ā)) over k. This implies that c̄ is p-independent over k in all the fields k(σ^{i}(ā) | −n ≤ i ≤ n), and therefore is p-independent over k in k(ā)_{σ}.

(2.12) Theorem. The theory of difference fields models of T_{λ} has a model companion SCFA_{∞,λ}, which is axiomatised by expressing the following properties of the L_{λ,σ}-structure (K, σ):

(i) SCF_{∞,λ}, (ii) σ is an automorphism of K, (iv) For all m, if U and V are varieties defined over K satisfying the hypotheses of (2.11)

(over K), then K satisfies:

∀z_{1}, …, z_{m}, ∃x_{1}, …, x_{n}, (x̄, σ(x̄)) ∈ V
∧({z_{1}, …, z_{m}} p-independent → {z_{1}, …, z_{m}, x_{r+1}, …, x_{s}} p-independent).

Proof. As in Theorem (2.6), one shows that these axioms are indeed first-order. We will first show that every difference field model of T_{λ} embeds in a model of SCFA_{∞,λ}. Let K be a difference field, and let U, V be varieties satisfying the hypothesis of (2.11), let b̄ be a tuple of elements of K, which we may assume p-independent in K. Consider the separable extension L = K(ā)_{σ} of K given by Lemma (2.11). Then a_{r+1}, …, a_{s} are p-independent over K, and therefore also over 𝔽_{p}(b̄). We conclude as in (2.6) that every difference field embeds in a model of SCFA_{∞,λ}.

It remains to show that SCFA_{∞,λ} is model complete. Let K be a model of SCFA_{∞,λ}, and L a difference field containing K and separable over K. Let φ(x̄) be a quantifier-free L_{λ,σ}-formula with parameters in K, and assume that it is satisfied in L by a tuple ā. By (2.9), enlarging ā if necessary, we may replace φ(x̄) by a formula ψ(x̄) ∧ θ(x̄), where ψ(x̄) is a quantifier-free positive L_{σ}(K)-formula, and θ(x̄) is an L(K)-formula expressing that certain subtuples of (x̄, ē) are p-independent, for some ē in K.

We will show that there is d̄ in some elementary extension K^{∗} of K, such that K^{∗} is a separable extension of K(d̄)_{σ}, and such that the difference fields K(ā)_{σ} and K(d̄)_{σ} are isomorphic by a K-isomorphism f sending ā to d̄. Let ē ∈ K; if a subtuple of ā is p-independent over 𝔽_{p}(ē) in L, then it is also p-independent over 𝔽_{p}(ē) in K(ā)_{σ}, and therefore its image by f is p-independent over 𝔽_{p}(ē) in K(d̄)_{σ}, and also in K^{∗} as K^{∗} is separable over K(d̄)_{σ}. Finding such a d̄ will then finish the proof.

By (2.1), there is a tuple b̄ ⊆ ā such that b̄ is transformally transcendental over K, K(ā)_{σ} is separable and transformally algebraic over K(b̄)_{σ}. The saturation of K^{∗} implies that K^{∗} contains a tuple c̄ of the same length as b̄, of elements transformally transcendental over K, and such that K^{∗} is a separable extension of K(c̄)_{σ}. Indeed, let n = |b̄|; then for any m, the partial type saying that the elements σ^{i}(x_{j}), i = 0, …, m − 1, j = 1, …, n, are distinct and p-independent over K (in K^{∗}), is finitely consistent, and therefore realised in K^{∗}. Choose a tuple c̄ = (c_{1}, …, c_{n}) such that the elements σ^{i}(c_{i}), i ∈ ℤ, 1 ≤ j ≤ n, are p-independent over K. Then the tuple c̄ is transformally transcendental over K, and K^{∗} is a separable extension of K(c̄)_{σ}.

As in the proof of (2.6), we show that there is a K-isomorphism of difference fields f : K(b̄)_{σ}^{s} → K(c̄)_{σ}^{s} ⊆ K^{∗}. We use the saturation of K^{∗}, (2.10) and (2.11), axiom (iv), and reason as in the proof of (2.6) to get a tuple d̄ in K^{∗}, and a difference field isomorphism extending f : K(b̄, ā)_{σ} → K^{∗} and sending ā to d̄. By axiom (iv) and (2.11), we may assume that the p-basis of K(d̄)_{σ} over K(c̄)_{σ} stays p-independent in K^{∗}, and this implies that K^{∗} is a separable extension of K(d̄)_{σ}. This finishes the proof.

(2.13) At first sight, the axiomatisations of SCFA_{e,λ} are quite different for e ∈ ℕ and for e = ∞. We will show that SCFA_{∞,λ} is nevertheless the limit of the theories SCFA_{e,λ}, e ∈ ℕ.

Proposition. Let θ be an L_{λ,σ}-sentence, and assume that SCFA_{∞,λ} ⊧ θ. There is e_{0} such that for all e ≥ e_{0}, SCFA_{e,λ} ⊧ θ. Proof. By compactness, it suffices to show the assertion for all sentences occuring in the scheme of axioms given in (2.12). This is clear for axioms of type (i) or (ii). Let K ⊧ SCFA_{e,λ}, let b̄ be an m-tuple of p-independent elements of K, and U, V varieties defined over K and satisfying the hypotheses of (2.11). Assume moreover that m + (s − r) ≤ e (in the notation of (2.11)). We will show that there is a tuple ā in K such that (ā, σ(ā)) ∈ V and {b̄, a_{r+1}, …, a_{s}} is p-independent.

Let θ(x̄, ȳ) be the L_{λ,σ}(K)-formula expressing this property of the tuple (ā, b̄), and let K(ā)_{σ} be the extension of K constructed in Lemma (2.11). Then {a_{r+1}, …, a_{s}} is contained in a p-basis C of K(ā)_{σ} over K. Fix a p-basis B of K containing the m-tuple b̄, and consider the extension M of K(ā)_{σ} constructed in Lemma 2.2(b). Then M is a separably algebraic extension of a field M_{0} which is isomorphic to the field of fractions of K⟨X_{c} | c ∈ C⟩_{B}. Hence, it is enough to show that the elements of {b̄, X_{a_{i}} | i = r+1, …, s} are p-independent in the field of fractions of K⟨X_{c} | c ∈ C⟩_{B}. This will immediately follow from the following: Claim. Let c_{1}, …, c_{n} be distinct elements of B. Then B \ {c_{1}, …, c_{n}} ∪ {X_{1}, …, X_{n}} is a p-basis of the field of fractions K_{n} of K⟨X_{1}, …, X_{n}⟩_{B}. Proof. This is proved by induction on n. Assume first that n = 1. We identify I(B) with I(B \ {c_{1}}) × {0, …, p − 1}, by viewing a p-monomial in B as the product of a p-monomial in B \ {c_{1}} with a p-monomial in c_{1}. This allows us to write

X = ∑_{i=0}^{p−1}( ∑_{j∈I(B\{c_{1}})} X_{i,j}^{p}m_{j}(B \ {c_{1}}))c_{1}^{i}.

Hence c_{1} satisfies an equation of degree p − 1 over K_{1}^{p}[B \ {c_{1}}, X]; c_{1} is also purely inseparable over this field, and hence is an element of this field. As B is a p-basis of K_{1}, so is B \ {c_{1}} ∪ {X}, which proves the claim when n = 1.

Assume the result proved for n and view K_{n+1} as the field of fractions of K_{n}⟨X_{n+1}⟩_{B}. By induction, B_{n} = B \ {c_{1}, …, c_{n}} ∪ {X_{1}, …, X_{n}} is a p-basis of K_{n}, and by (1.11), K_{n}⟨X_{n+1}⟩_{B} is naturally isomorphic to K_{n}⟨X_{n+1}⟩_{B_{n}} by a K_{n}[X_{n+1}]-isomorphism. Hence the case n = 1 gives us that B_{n} \ {c_{n+1}} ∪ {X_{n+1}} = B \ {c_{1}, …, c_{n+1}} ∪ {X_{1}, …, X_{n+1}} is a p-basis of K_{n+1}.

By the claim, M is a separable extension of K, of the same degree of imperfection as K, and which satisfies ∃x̄ θ(x̄, b̄). By model completeness of SCFA_{e,λ}, K also satisfies this formula. Note that the integer e_{0} does not depend on the parameters defining U and V but only on m + s − r. This proves the proposition.

3. Elementary invariants and types Recall that if K is separably closed and A ⊆ K, then the model-theoretic algebraic

closure of A inside the L-structure K is obtained by taking the separable closure of the L_{λ}-substructure of K generated by A (see [D]). Let K be a model of SCFA_{e,λ} for some e. If A ⊆ K, we define acl_{σ}(A) (or acl_{σ,K}(A) if we need to specify in which field we take the λ-closure) to be the smallest separably closed subfield of K, closed under σ, σ^{−1} and the λ-functions of K, and which contains A. We know by results in [CP] that any completion of SCFA_{e,λ} is simple. We also know that if K ⊧ SCFA_{e,λ}, then the theory of K is completely determined by the isomorphism type of any difference separably closed subfield K_{0} of K of degree of imperfection e. Thus, a priori, the theory of K might depend on the action of σ on a given p-basis. We will show that this is not the case, and that T h(K) is completely determined by the isomorphism type of its difference subfield 𝔽_{p}^{s}. We will also obtain a slight generalisation of the independence theorem, and show that types over algebraically closed sets coincide with Lascar strong types. We start with a description of p-bases when e ∈ ℕ.

(3.1) Lemma. Let K ⊧ SCFA_{e,λ}, and assume that B is a finite p-independent subset of K such that σ(B) ⊂ K^{p}[B]. Then there is a finite set D of elements fixed by σ, such that K^{p}[D] = K^{p}[B].

Proof. Let e = |B|, fix an enumeration of the p-monomials m_{i}(B) in B, i = 1, …, p^{e}, and let m_{i}(σ(B)) = σ(m_{i}(B)). As σ(B) is also p-independent in K, we have that K^{p}[σ(B)] = K^{p}[B]. Let A = (a_{j,i}) be the (p^{e} × p^{e})-matrix with entries in K defined by m_{i}(σ(B)) = ∑_{j} a_{j,i}^{p}m_{j}(B). Then A is an invertible matrix. Consider the following system of equations, in the unknown X̄ = (X_{1}, …, X_{p^{e}}):

(1)  ( X_{1} ; ⋮ ; X_{p^{e}} ) = A ( σ(X_{1}) ; ⋮ ; σ(X_{p^{e}}) ) .

If (c_{1}, …, c_{p^{e}}) is a solution of (1) in 𝔸^{p^{e}}(K), then

σ(∑_{i=1}^{p^{e}} c_{i}^{p}m_{i}(B)) = ∑_{i=1}^{p^{e}} σ(c_{i})^{p}m_{i}(σ(B))

= ∑_{i=1}^{p^{e}} σ(c_{i})^{p} ∑_{j=1}^{p^{e}} a_{j,i}^{p}m_{j}(B)

= ∑_{j=1}^{p^{e}} ∑_{i=1}^{p^{e}}(σ(c_{i})a_{j,i})^{p}m_{j}(B)

= ∑_{j=1}^{p^{e}} c_{j}^{p}m_{j}(B),

so that ∑_{i=1}^{p^{e}} c_{i}^{p}m_{i}(B) is fixed by σ and belongs to K^{p}[B]. Let F = F ix(σ). The set of solutions of (1) in 𝔸^{p^{e}}(K) forms an F -vector space, and we will first show that it generates a K-vector space of dimension p^{e}. We prove by induction on i ≤ p^{e} that there are p^{e}-tuples c̄_{1}, …, c̄_{i} which are solutions of (1) and are (K-)linearly independent. Assume that we have already found these elements c̄_{1}, …, c̄_{i}, and that i < p^{e}. Consider the (p^{e} × (i + 1))-matrix with columns X̄, c̄_{1}, …, c̄_{i}. Since the vectors c̄_{1}, …, c̄_{i} are independent in 𝔸^{p^{e}}(K), this matrix has a submatrix of size (i + 1) × (i + 1) with non-zero determinant g(X̄) ∈ K[X̄]. Let U ⊆ 𝔸^{p^{e}+1} be the variety defined by g(X̄)X′ = 1, and let V ⊆ U × σ(U) be defined by X̄ = AȲ. Then U and V satisfy the hypotheses of axiom (iii) of SCFA_{e,λ} (note that k(U) = k(V)), and therefore there is (c̄, c′) in K such that c̄ = Aσ(c̄) and g(c̄)c′ = 1. Then c̄_{1}, …, c̄_{i}, c̄ are linearly independent solutions of (1).

Fix linearly independent solutions c̄_{1} = (c_{1,j}), …, c̄_{p^{e}} = (c_{p^{e},j}) of (1), and let d_{i} = ∑_{j=1}^{p^{e}} c_{i,j}^{p}m_{j}(B) for i = 1, …, p^{e}. Then the elements d_{1}, …, d_{p^{e}}, are linearly independent in the K^{p}-vector space K. We claim that {d_{1}, …, d_{p^{e}}} contains a p-independent subset of size e. This is again proved by induction on i ≤ p^{e}: assume that j(1), …, j(i) are such that d_{j(1)}, …, d_{j(i)} are p-independent in K. Then K^{p}[d_{j(1)}, …, d_{j(i)}] is a K^{p}-vector space of dimension p^{i}. If i < e then there is an n such that d_{n} ∉ K^{p}[d_{j(1)}, …, d_{j(i)}], i.e., such that d_{j(1)}, …, d_{j(i)}, d_{n} are p-independent in K, and we let j(i + 1) = n. Hence, K^{p}[B] ∩ F contains a p-independent set D of size e. This implies that K^{p}[B] = K^{p}[D].

(3.2) Corollary. Let e ∈ ℕ and let K be a model of SCFA_{e,λ}. Then K has a p-basis consisting of elements fixed by σ.

(3.3) Corollary. Let K ⊧ SCFA_{∞,λ}, and assume that K is transformally algebraic (over ∅). Then K has a p-basis contained in F ix(σ).

Proof. If ā is a tuple of elements of K, then the field 𝔽_{p}(ā)_{σ} has finite transcendence degree, and therefore K^{p}(ā)_{σ} is a finite extension of K^{p}. By (3.1), there is a finite tuple b̄ of elements of F ix(σ) such that K^{p}(ā)_{σ} = K^{p}(b̄). Hence, we get that K ⊆ K^{p}(F ix(σ)), and therefore F ix(σ) contains a p-basis of K.

(3.4) Lemma. Let k be a separably closed difference field, let K_{1} and K_{2} be difference fields containing k, which are linearly disjoint and separable over k. Let B_{1} be a p-basis of K_{1} over k and B_{2} a p-basis of K_{2} over k, and assume that there is an injection f : B_{1} → B_{2}, and that the elements of B_{1} ∪ f(B_{1}) are fixed by σ. Then K_{1} and K_{2} are contained in a common difference field L, which is separable over K_{1} and over K_{2}, and in which B_{2} is a p-basis of L over k. If f is onto, then B_{1} is also a p-basis of L over k. Proof. As K_{1} and K_{2} are linearly disjoint over k, the field K_{1}K_{2} has a unique structure of difference field extending those of K_{1} and K_{2}; this result is well-known but we will repeat the proof. To avoid confusion, let us denote by σ_{i} the automorphism of K_{i} for i = 1, 2. By assumption σ_{1} and σ_{2} agree on k. Since K_{1} and K_{2} are linearly disjoint over k, their composite K_{1}K_{2} is separable over K_{1} and over K_{2}, isomorphic to the field of fractions of K_{1} ⊗_{k} K_{2}. Define σ on K_{1}K_{2} by setting σ(a ⊗ b) = σ_{1}(a) ⊗ σ_{2}(b) for a ∈ K_{1} and b ∈ K_{2}, and extending to K_{1}K_{2}. This is well-defined as K_{1} and K_{2} are linearly disjoint over k and σ_{1} and σ_{2} agree on k.

Let C = {b − f(b) | b ∈ B_{1}} and L = K_{1}K_{2}[c^{1/p^{n}} | c ∈ C, n ∈ ℕ]. By assumption, the elements of C are fixed by σ, which implies that the automorphism σ of K_{1}K_{2} extends uniquely to an automorphism of L, with σ(c^{1/p^{n}}) = c^{1/p^{n}} for c ∈ C, n ∈ ℕ. Clearly we have L = kL^{p}[B_{2}] so it is enough to show that B_{1} and B_{2} stay p-independent in L: this will ensure that L is separable over K_{1} and over K_{2}.

As K_{1} and K_{2} are linearly disjoint over k, we know that B_{1} ∪ B_{2} is a p-basis of K_{1}K_{2} over k. Then B_{1} ∪ C ∪ (B_{2} \ f(B_{1})) is also a p-basis of K_{1}K_{2} over k. Thus K_{1}K_{2} is a separable extension of k(C), with p-basis B_{1} ∪ (B_{2} \ f(B_{1})) over k(C). Lemma (1.12) tells us that B_{1} ∪ (B_{2} \ f(B_{1})) remains p-independent over k(c^{1/p^{n}} | c ∈ C, n ∈ ℕ) in L. This proves the result for B_{1}; the proof for B_{2} is similar, using the fact that B_{2} ∪ C is a p-basis of K_{1}K_{2} over k.

(3.5) Theorem. Let (K_{1}, σ_{1}) and (K_{2}, σ_{2}) be models of SCFA_{e,λ}, with a common difference subfield (A, σ), and let A_{i} =_{def} acl_{σ_{i},K_{i}}(A) for i = 1, 2. Then

(K_{1}, σ_{1}) ≡_{A} (K_{2}, σ_{2}) ⟺ (A_{1}, σ_{1}) ≃_{A} (A_{2}, σ_{2}).

Proof. The left-to-right implication is clear. For the reverse implication, let f : (A_{1}, σ_{1}) → (A_{2}, σ_{2}) be an A-isomorphism of difference fields. Let g be an A-automorphism of the field Ω which extends f^{−1}. Because A_{1} is separably closed and K_{1}, g(K_{2}) are separable over A_{1}, we may choose g such that the field g(K_{2}) is linearly disjoint from K_{1} over A_{1}; then g(A_{2}) = A_{1} is a substructure of the difference field (g(K_{2}), gσ_{2}g^{−1}), and it is enough to show that (g(K_{2}), gσ_{2}g^{−1}) ≡_{A_{1}} (K_{1}, σ_{1}). Hence we may assume that A = A_{1} = A_{2}, and that K_{1} and K_{2} are linearly disjoint over the separably closed field A. Then there is a unique automorphism σ of K_{1}K_{2} which extends σ_{1} and σ_{2} (see the proof of (3.4)).

Our assumption then implies that K_{1} and K_{2} are separable extensions of A, and that they have the same degree of imperfection e ∈ ℕ ∪ {∞}. If e = ∞, then by (2.12) the L_{λ,σ}-structure (K_{1}K_{2}, σ) embeds in a model L of SCFA_{∞,λ}, separable over K_{1} and over K_{2}.

If e ∈ ℕ, then by (3.1) K_{1} and K_{2} have p-bases which are fixed by σ_{1}, σ_{2}, respectively. Let C_{i} ⊂ K_{i} ∩ F ix(σ_{i}) be a p-basis of K_{i} over A for i = 1, 2. If B is a p-basis of A, then |B| + |C_{i}| = e, and therefore |C_{1}| = |C_{2}|. By (3.4), there is a difference field L_{0} containing K_{1}K_{2}, which is separable over A, and in which C_{1} and C_{2} are p-bases over A. Thus L_{0} is a separable extension of K_{1} and of K_{2}, of degree of imperfection e. By (2.6), L_{0} embeds in a model L of SCFA_{e,λ}, separable over K_{1} and over K_{2}.

In both cases, the model completeness of SCFA_{e,λ} gives us that (K_{i}, σ_{i}) ≺ (L, σ) for i = 1, 2, and therefore that (K_{1}, σ_{1}) ≡_{A} (K_{2}, σ_{2}).

(3.6) This result has numerous consequences, which we now list. Most of them already appear in [CP].

Corollaries. Let K be a model of SCFA_{e,λ}.

(1) Let A = acl_{σ}(A) and B = acl_{σ}(B) be subfields of K. Any difference field isomorphism f between A and B is elementary.

(2) Let K ⊧ SCFA_{e,λ} and let A ⊆ K, ā and b̄ tuples from K. Then tp(ā/A) = tp(b̄/A) if and only if there is a difference field isomorphism f : acl_{σ}(A, ā) → acl_{σ}(A, b̄) which fixes A and sends ā to b̄.

(3) The completions of SCFA_{e,λ} are obtained by describing the isomorphism type of the difference field 𝔽_{p}^{s}.

(4) Let e ∈ ℕ, b̄ = {b_{1}, …, b_{e}}. The completions of SCFA_{e,b} are obtained by describing the isomorphism type of the L_{σ}(b̄)-structure (𝔽_{p}(b_{1}, …, b_{e})^{s}, b_{1}, …, b_{e}).

(5) Let A ⊆ K. Then acl_{σ}(A) is the model theoretic algebraic closure of A in K.

(6) Let φ(x̄) be a formula of L_{σ}. Then φ(x̄) is equivalent modulo SCFA_{e,λ} to a disjunction of formulas of the form

(∗)  ∃ȳ ψ(x̄, ȳ)

where ψ(x̄, ȳ) is a positive quantifier-free L_{λ,σ}-formula. Moreover there is a finite tuple F (x̄) of terms of the language L_{λ,σ} such that in any difference field K, if (ā, b̄) is a tuple of K satisfying ψ, then b̄ is separably algebraic over 𝔽_{p}(F (ā)).

(7) If b̄ = {b_{1}, …, b_{e}}, then any completion of SCFA_{e,b} eliminates imaginaries. Proof. (1) Extend f to an automorphism of Ω, and consider the difference field (L, τ) = (f(K), fσf^{−1}). Then acl_{τ,L}(B) = B; by (3.5) this implies that (L, τ) ≡_{B} (K, σ). As f defines an isomorphism between the difference fields (K, σ) and (L, τ), this implies that f is elementary.

(2) – (4) are clear by (3.5).

(5) As acl_{σ}(A) is separably closed, and K is a separable extension of acl_{σ}(A), there is an acl_{σ}(A)-automorphism g of Ω such that the field g(K) is linearly disjoint from K over acl_{σ}(A). As in the proof of (3.5), there is a model M of SCFA_{e,λ}, containing the difference fields (K, σ) and (g(K), gσg^{−1}), and separable over K and g(K). Then K ≺ M, g(K) ≺ M, and every type over A realised in K \ acl_{σ}(A) is realised anew in g(K), and is therefore non-algebraic.

(6) Let Σ be the set of formulas of the form given by (∗). Then Σ is closed under finite conjunctions.

Let K be a model of SCFA_{e,λ}, and let ā be a tuple of elements of K. By (3.5), the type of ā (over ∅) is obtained by describing the isomorphism type of the difference field acl_{σ}(ā) with distinguished constants the elements of ā. Hence it is axiomatised by an infinite conjunction of formulas of Σ (note that for any formula θ(ȳ) we always have θ(ȳ) ⟺ θ(σ(ȳ)); this allows us to describe the isomorphism type of acl_{σ}(ā) without using σ^{−1}). Use compactness to deduce that each formula is equivalent modulo SCFA_{e,λ} to a disjunction of formulas of Σ.

(7) Let K ⊧ SCFA_{e,b}. Then any algebraically closed substructure of K is a separably closed field of degree of imperfection e, because the language contains symbols for elements of a p-basis. Thus Theorem (3.7) of [CP] (the independence theorem) holds over any algebraically closed subset of (K, b̄). The proof of (2.9) in [CP] then gives us the result.

(3.7) Definition of independence, and the independence theorem. Let K ⊧ SCFA_{e,λ}, let A, B, C ⊆ K. We say that A and B are independent over C iff the fields acl_{σ}(C, A) and acl_{σ}(C, B) are linearly disjoint over acl_{σ}(C), and if moreover K is a separable extension of acl_{σ}(C, A)acl_{σ}(C, B) when e = ∞.

The following result is a simple generalisation of (3.7) in [CP], and implies that any completion of SCFA_{e,λ} is simple, and that independence as defined above is non-forking, provided acl_{σ}(C) contains a p-basis of K if e is finite (see [KP]). It also gives a description of the Lascar strong types: Lstp(ā/A) ≃ tp(ā/acl_{σ}(A)), provided acl_{σ}(A) contains a p-basis of K when e ∈ ℕ.

Theorem. Let K ⊧ SCFA_{e,λ}, let E = acl_{σ}(E) ⊆ K. Let ā, b̄, c̄_{1} and c̄_{2} be tuples from K which satisfy:

(i) ā and b̄ are independent over E, ā and c̄_{1} are independent over E, b̄ and c̄_{2} are independent over E.

(ii) tp(c̄_{1}/E) = tp(c̄_{2}/E). (iii) If e ∈ ℕ, then E contains a p-basis of K. Then in some elementary extension of K there is c̄ realising tp(c̄_{1}/E ∪ ā) ∪ tp(c̄_{2}/E ∪ b̄), which is independent from (ā, b̄) over E. Proof. The proof is absolutely standard, and follows the lines of (3.7) in [CP]. Below we will indicate some changes or remarks, which will allow the reader to modify suitably (3.7) of [CP]. Let A = acl_{σ}(E, ā), B = acl_{σ}(E, b̄), C = acl_{σ}(E, c̄_{1}), C_{2} = acl_{σ}(E, c̄_{2}), and T = SCF_{e}.

(1) Note that our independence assumptions on A, B, C and C_{2} simply mean: tp_{T} (A/B), tp_{T} (A/C) and tp_{T} (B/C_{2}) do not fork over E, see [D].

(2) Moving c̄_{1} by an A-automorphism, we may assume that C is independent (in the sense of T) from (A, B) over E: if e = ∞, this is immediate; if e is finite, this is because E contains a p-basis of K.

(3) dcl_{T} (−) will denote the λ-closure, and acl_{T} (−) the separable closure of dcl_{T} (−) (i.e., we are not working in T^{eq} as in [CP]). In our case, the independence assumptions give: dcl_{T} (A, B) = AB, acl_{T} (A, B) = (AB)^{s}, etc.

(4) The claim is replaced by the following: Claim. (AB)^{s}(AC)^{s} and (BC)^{s} are linearly disjoint over BC.

By Remark (1.9) in [CH], we know that (AB)^{s}(AC)^{s}∩(BC)^{s} ⊆ B^{alg}C^{alg}. As B^{alg}C^{alg} is purely inseparable over BC, this implies that (AB)^{s}(AC)^{s} ∩ (BC)^{s} = BC, which gives the claim because (BC)^{s} is a Galois extension of BC.

(5) From this new claim, we deduce that σ ∪ σ_{2} defines an automorphism of L = (AB)^{s}(AC)^{s}(BC)^{s}. Since the degree of imperfection of L is ≤ e, we can embed L into a model K^{∗} of SCFA_{e,λ}. The result then follows from (3.6)(2).

(3.8) Corollary. Independence as defined in (3.7) coincides with non-forking.

(3.9) Corollary. Let A = acl_{σ}(A) ⊆ K ⊧ SCFA_{e,λ}, and assume that A contains a p-basis of K if e is finite. Then the Lascar strong type over A of a tuple ā in K is implied by tp(ā/A).

Proof. We first recall the definition of the Lascar strong type. Let K^{∗} be a large saturated elementary extension of K, and consider the group H of automorphisms of K^{∗} generated by all groups Aut(K^{∗}/M) where A ⊆ M ≺ K^{∗}. Then two elements of K^{∗} have the same Lascar strong type over A if they are conjugate by some element of H.

It is enough to show that if b̄ realises tp(ā/A) and is independent from ā over A, then there is a difference field M containing A, such that M ≡_{A} K and tp(ā/M) = tp(b̄/M). Since A is separably closed, there is an A-automorphism f of Ω such that f(K) is free from acl_{σ}(A, ā, b̄) over A. Then (K, fσf^{−1}) ≡_{A} (K, σ). Since ā and b̄ realise the same type over A, there is an A-isomorphism g : acl_{σ}(A, ā) → acl_{σ}(A, b̄) which sends ā to b̄. By the independence theorem (3.7), there is M realising tp(f(K)/acl_{σ}(A, ā)) ∪ g(tp(f(K)/acl_{σ}(A, ā)), and independent from acl_{σ}(A, ā, b̄) over A. Then we have M ≡_{A} K, and tp(ā/M) = tp(b̄/M).

4. Properties of models of SCFA_{e,λ}.

In this section, we show how some of the results of [CH] generalise to the theory SCFA_{e,λ}. If K ⊧ SCFA_{e,λ}, we study F ix(σ), the difference fields (K, σ^{n}), and the difference field K^{p^{∞}} when K is ω-saturated.

(4.1) Proposition. Let (K, σ) be a model of SCFA_{e,λ}, and let n > 0. Then (K, σ^{n}) ⊧ SCFA_{e,λ}. If (K, σ) is κ-saturated, so is (K, σ^{n}).

Proof. The result will follow from the following claim:

Claim. Let (k, σ) be a separably closed difference field, let (K, σ^{n}) be a separably closed difference field extending (k, σ^{n}), separable over k, and of the same degree of imperfection as k if e ∈ ℕ. Then there is a difference field (L, σ) containing (k, σ) and such that (L, σ^{n}) extends (K, σ^{n}) and is a separable extension of K.

The proof of this claim is identical to the proof of (1.12) of [CH], but we will repeat it. For i = 1, …, n − 1, choose a field extension K_{i} of k, which is isomorphic to K by an isomorphism f_{i} : K → K_{i} which agrees with σ^{i} on k, and let K_{0} = K, f_{0} = id. Since k is separably closed and the K_{i}’s are separable extensions of k, we may assume that the K_{i}’s are linearly disjoint over k, so that the composite L of K_{0} ⋯ K_{n−1} is the field of fractions of the domain K_{0} ⊗_{k} K_{1} ⊗_{k} ⋯ ⊗_{k} K_{n−1}. Note that L is a separable extension of K_{0}. For i = 0, …, n − 2, let σ_{i} = f_{i+1}f_{i}^{−1} : K_{i} → K_{i+1}; then σ_{i} is an isomorphism which agrees with σ on k. Let σ_{n−1} = σ^{n}f_{n−1}^{−1} : K_{n−1} → K; then σ_{n−1} is an isomorphism which agrees with σ on k. Thus the isomorphisms σ_{i}, i = 0, …, n−1, extend to a unique automorphism σ′ of L, which agrees with σ on k. It remains to show that σ′^{n} agrees with σ^{n} on K. Let a ∈ K; then σ′^{n}(a) = σ_{n−1} ⋯ σ_{1}σ_{0}(a) = (σ^{n}f_{n−1}^{−1})(f_{n−1} ⋯ f_{1}^{−1})(f_{1}f_{0}^{−1})(a) = σ^{n}(a).

Thus any instance of axiom (iii) or (iv) for (K, σ^{n}) has a solution in a difference field (L, σ) containing (K, σ), separable over K and of the same degree of imperfection. This shows that (K, σ^{n}) is a model of the scheme of axioms (iii) and (iv) if e = ∞. It is also clearly a separably closed σ^{n}-difference field of degree of imperfection e, and is therefore a model of SCFA_{e,λ}. The statement about saturation is proved similarly.

(4.2) Proposition. Let (K, σ) be a model of SCFA_{e,λ}, let τ = σ^{n}F rob^{m} for some n ≥ 1 and m ∈ ℤ, and let F = F ix(τ), the subfield of K fixed by τ. Then F is a PAC field (i.e., every absolutely irreducible variety defined over F has an F -rational point), and Gal(F^{s}/F) ≃ ℤ̂. Furthermore, K is a separable extension of F, the degree of imperfection of F is e if m = 0, and 0 otherwise.

Proof. Let us first do the case m ≠ 0. Then F ⊆ k = K^{p^{∞}}, so that F is perfect. Let V be a variety defined over F. Let (L, σ) be a model of ACFA containing k, linearly disjoint from K over k and let F′ be the subfield of L fixed by σ^{n}Frob^{m}. The composite field KL is then a difference field, separable over K and of the same degree of imperfection, so that it embeds in some elementary extension K^{∗} of K. Since V is defined over F ⊂ k, and L is a model of ACFA the variety V has an F′-rational point, and Gal(F′^{s}/F′) ≃ ℤ̂. Hence, in K^{∗}, if F^{∗} = F ix(σ^{n}Frob^{m}), then the same statements are true of F^{∗}.

If m = 0, then by (4.1) we may assume that τ = σ. Let V ⊆ Ω^{n} be an absolutely irreducible variety defined over F, and consider the subvariety W of V × V, intersection of V × V with the diagonal of Ω^{n} × Ω^{n}. Then W satisfies the assumptions of axiom (iii) or (iv), and so (K, σ) contains a tuple ā with (ā, σ(ā)) ∈ W. Then ā ∈ V (F) by definition of W.

Fix n > 1, let t_{1}, …, t_{n} be algebraically independent over K, and extend σ to K(t_{1}, …, t_{n}) by setting σ(t_{i}) = t_{i+1} for i < n, and σ(t_{n}) = t_{1}. Then the difference field K(t_{1}, …, t_{n}) is a separable extension of K, and satisfies the sentence ∃x σ^{n}(x) = x ∧ ⋀_{i=1}^{n−1} σ^{i}(x) ≠ x. As K is existentially closed, K also satisfies that sentence, and therefore F has at least one extension of degree n for every n > 1. On the other hand, we know that Gal(F^{s}/F) is generated by the restriction of σ to F^{s}, which implies that F has at most one extension of degree n for each n. Hence Gal(F^{s}/F) ≃ ℤ̂.

As the λ-functions of K are definable, F is closed under the λ-functions of K, and therefore K is a separable extension of F.

If e ∈ ℕ, then (3.2) shows that F contains a p-basis of K. If e = ∞, then axiom (iv) ensures that F contains infinitely many elements which are p-independent in K.

Remark. The conclusion that F ix(σ) is PAC with absolute Galois group isomorphic to ℤ̂ is also proved by Kudaibergenov, see [Ku].

(4.3) Proposition. Let (K, σ) be a model of SCFA_{e,λ} for some e ∈ ℕ, let τ = σ^{n}Frob^{m} for some n ≥ 1 and m ∈ ℤ, and let F = F ix(τ). Then every definable subset D of F^{ℓ} is definable using only parameters from F. If n = 1, then D is definable in the pure field F.

Proof. Let D ⊆ F^{ℓ} be definable in K. The automorphism σ fixes D, and therefore, by elimination of imaginaries of SCFA_{e,b} ((3.6)(7)), the set D is definable by a formula of L_{λ,σ} with parameters in F.

If n = 1, then any field automorphism of F is an L_{λ,σ}-automorphism (because σ|_{F} =

x^{p^{−m}} is definable in the pure field language, and because K is a separable extension of F). Fix a finite set of parameters c̄ in F over which D is defined, and let F_{0} be a countable elementary substructure of F containing c̄. By compactness, it suffices to show that SCFA_{e,b} ∪ tp_{F} (ā/F_{0}) ⊢ tp_{K}(ā/F_{0}) for any ā ∈ D (here, tp_{F} (ā/F_{0}) denotes the type of the tuple ā over F_{0} in the pure field F, and tp_{K}(ā/F_{0}) the type of ā over F_{0} in the difference field K). Without loss of generality, we may assume that F is sufficiently saturated.

Let d̄∈ F realise tp_{F} (ā/F_{0}). Then there is a field automorphism f of F which leaves F_{0} fixed and sends ā to d̄. Since F_{0} is an elementary substructure of F, F and F_{0}^{s} are linearly disjoint over F_{0}, and F_{0}^{s}F = F^{s}. Hence f extends to an automorphism of F^{s} which is the identity on F_{0}^{s}. This implies that f commutes with σ because Gal(F^{s}/F) ≃ Gal(F_{0}^{s}/F_{0}). By (3.6)(1), f is an elementary partial map of the difference field K, which gives the conclusion.

(4.4) Lemma. Let k ⊆ K be difference fields, and assume that K is separable over k. Let L be the subfield of K consisting of elements transformally algebraic over k. Then K is a separable extension of L. Proof. Let ā be a finite tuple of elements of L. By definition of L, the transcendence degree of k(ā)_{σ} over k is finite. Hence there is a finite tuple b̄ ⊆ {σ^{i}(ā) | i ∈ ℕ}, maximal p-independent over k in K. Then k(ā)_{σ} ⊆ kK^{p}[b̄]. As σ is an automorphism of K, we also have k(ā)_{σ} ⊆ kK^{p}[σ^{n}(b̄)] for any n. Also, there is a positive integer m such that k(ā)_{σ} = k(ā^{p})_{σ}[ā, …, σ^{m}(ā)].

We will show that there is a finite tuple ū of elements of K, such that σ^{n}(ā) ⊆ k(ā^{p})_{σ}(b̄, ū^{p}) for every n ∈ ℤ. Indeed, choose ū such that ā, …, σ^{m}(ā) ∈ k(b̄, ū^{p}); then k(ā)_{σ} ⊆ k(ā^{p})_{σ}[ā, …, σ^{m}(ā)] ⊆ k(ā^{p})_{σ}(b̄, ū^{p}). As σ is an automorphism of K, each σ^{n}(b̄) is also p-independent in K. Consider the k(ā^{p})_{σ}(ū^{p})-vector space V generated by the p-monomials in b̄. It has dimension p^{|b̄|}. The p-monomials in σ^{n}(b̄) are also linearly independent in V because the elements of σ^{n}(b̄) are p-independent in K, and this implies that k(ā^{p})_{σ}(b̄, ū^{p}) = k(ā^{p})_{σ}(σ^{n}(b̄), ū^{p}) for every n ∈ ℤ. Fix a p-basis B of k. Then σ^{n}(B) is also a p-basis of k and k = k^{p}[B] = k^{p}[σ^{n}(B)]. Hence for every n ∈ ℤ we have:

(1)  k(ā)_{σ} ⊆ (k(ā)_{σ}(ū))^{p}[B, b̄] = (k(ā)_{σ}(ū))^{p}[σ^{n}(B), σ^{n}(b̄)].

Fix c ∈ k(ā)_{σ} and a finite subset B_{0} of B such that c ∈ K^{p}[B_{0}, b̄]. From equation (1), we get that λ_{i}(B_{0} ∪ {b̄}; c) and λ_{i}(σ^{n}(B_{0}) ∪ {σ^{n}(b̄)}; σ^{n}(c)) ∈ k(ā)_{σ}(ū) for every n ∈ ℤ (the λ-functions are those of K). As the latter is the image of the former by σ^{n}, we obtain that the difference field generated by the elements λ_{i}(B_{0} ∪ {b̄}; c), c ∈ k(ā)_{σ}, B_{0} a finite subset of B, has finite transcendence degree over k(ā)_{σ}, and therefore is contained in L.

This shows that L is closed under the λ-functions of K, and therefore that K is a separable extension of L.

(4.5) Proposition. Let e ∈ ℕ ∪ {∞}, and let k be a difference field of degree of imperfection ≤ e. Let (K, σ) be a model of SCFA_{e,λ}, separable over k. Let L be the set of elements of K which are transformally algebraic over k. Then L ≺ K. Proof. The difference field L is certainly separably closed, hence L satisfies (i) and (ii). Any instance of axiom (iii) or (iv) is satisfied in a transformally algebraic extension of L contained in K, hence in L by definition of L. If e ∈ ℕ, then K has a p-basis consisting of elements fixed by σ (by (3.1)), and these elements are therefore in L. If e = ∞, then F ix(σ) ⊆ L has infinite degree of imperfection by (4.2). By Lemma (4.4), K is a separable extension of L. The model-completeness of SCFA_{e,λ} gives the result.

(4.6) Proposition. Let K be an ω-saturated model of SCFA_{e,λ}, and let k = K^{p^{∞}}. Then k ⊧ ACFA.

Proof. Clearly k is a difference subfield of K, and is algebraically closed. We need to show that if U and V are varieties defined over k, with V ⊆ U × σ(U), and such that V projects generically onto U and onto σ(U), then there is a tuple ā in k such that (ā, σ(ā)) ∈ V. Let k_{0} be a countable algebraically closed difference subfield of k over which U and V are defined. Choose, in some saturated model of ACFA containing k_{0}, an element ā satisfying (ā, σ(ā)) ∈ V, and such that k_{0}(ā)_{σ} and K are linearly disjoint over k_{0}. Consider k_{1} = k_{0}(ā)_{σ}^{alg}. As k_{0} is algebraically closed, k_{1} is a separable extension of k_{0}, and therefore k_{1}K is a separable extension of K, of the same degree of imperfection. Hence there is an elementary extension L of K containing k_{1}K, and ā ∈ L^{p^{∞}}. By ω-saturation of K, tp(k_{1}/k_{0}) is realised in K.

(4.7) Proposition. Let k and K be as above. If D is a quantifier-free definable subset of K^{n}, then D ∩ k^{n} is quantifier-free definable in the difference field (k, σ).

Proof. We will first show that we may assume that D is quantifier-free definable by an L_{σ}-formula, and then that we can assume that its defining parameters are in k. The first assertion follows from the fact that the value of the λ-term λ_{i,m}(x_{1}, …, x_{m}; x_{m+1}) at any tuple (a_{1}, …, a_{m+1}) with a_{j} ∈ K^{p} is a_{j}^{1/p} if j = m + 1, a_{1}, …, a_{m} are p-independent, and the p-monomial corresponding to λ_{i,m} is the p-monomial 1, and is 0 otherwise. Hence there is an integer ℓ such that a tuple (a_{1}, …, a_{n}) is in D ∩ k^{n} if and only if (a_{1}^{1/p^{ℓ}}, …, a_{n}^{1/p^{ℓ}}) satisfies some L_{σ}(K)-quantifier-free formula, if and only if (a_{1}, …, a_{n}) satisfies an L_{σ}(K)- quantifier-free formula. Hence we may assume that D is a Boolean combination of σ-closed subsets of K^{n}.

For the second assertion we may therefore assume that D is a σ-closed subset of K^{n}. Then D′ = D ∩ k^{n} is a σ-closed subset of k^{n}. If ā_{i}, i ∈ ℕ, is an infinite indiscernible sequence of tuples in D′, then D′ is defined over acl_{σ}(ā_{i}, i ∈ ℕ), see (2.13) in [CH]. As acl_{σ}(ā_{i}, i ∈ ℕ) ⊆ k, we get the result.

Remark. There are however definable subsets of k^{n} which need parameters from K. E.g., if the characteristic is ≠ 2, let b ∈ K \ k be such that σ(b) = b, and consider the set D defined by:

σ(x) = x ∧ ∃y σ(y) = y ∧ y^{2} = x − b.

Then D ∩ k is not definable in the difference field (k, σ).

5. Modularity In this section we investigate modularity of definable subsets of models of SCFA_{e,b}

(for a fixed e ∈ ℕ). A word of warning on the terminology: our notion of modularity does not coincide with the notion of modularity defined on minimal sets; in the case of a stable theory, our notion agrees with one-basedness. Recall that in models of ACFA we have the following results: ([CHP]) Let S be the set of realisations in a model of ACFA of a set of types over some set E. If S is modular, then all elements of S are transformally algebraic over E. Assume that all elements of S are transformally algebraic over E. Then S is modular if and only if S is orthogonal to all formulas σ^{n}(x) = x^{p^{m}}, with n ≥ 1, m ∈ ℤ.

We investigate how these results extend to models of SCFA_{e,b}. We are not able to show the first assertion in our case. Assume now that S is such that any tuple of S is transformally algebraic over E. We are then able to show that non-modularity of S implies that S is non-orthogonal to some formula σ^{n}(x) = x^{p^{m}} as above; the converse only holds if S is definable, and we give examples of modular ∞-definable subsets of F ix(σ^{n}F rob^{−m}). It is also likely that there are ∞-definable modular sets which contain elements not transformally algebraic over E. We derive some consequences of modularity for quantifier-free definable subsets of modular subgroups of an algebraic group. We end the chapter with some questions and an example.

(5.1) Definition of modularity. Let T be a simple theory, M a large saturated model of T and E = acl(E) a small subset of M. Let S ⊆ M^{n} be a set which is invariant under Aut(M/E). So, S is either a definable set, or is the union of realisations of a set of types over E.

We say that S is modular if whenever ā and b̄ are finite tuples of members of S and C = acl^{eq}(E, ā) ∩ acl^{eq}(E, b̄), then tp(ā/C, b̄) does not fork over C.

(5.2) We will consider the theories T = ACFA and T = SCFA_{e,b}. Note that any completion of either theory is simple and eliminates imaginaries. Recall that in case T = ACFA, the algebraic closure of a subset of a model of ACFA is simply the smallest algebraically closed difference field containing that set (see (1.7) in [CH]). In case T = SCFA_{e,b}, algebraic closure coincides with acl_{σ}, see (3.6)(5).

In both theories, independence of sets A, B over C is defined as linear disjointness of acl(C, A) and acl(C, B) over acl(C), and the independence theorem holds, thus showing that our notion of independence coincides with non-forking. It follows that the modularity of S can be expressed as follows: if ā and b̄ are finite tuples of members of S, then the fields acl(E, ā) and acl(E, b̄) are linearly disjoint over acl(E, ā) ∩ acl(E, b̄).

The following lemma is routine. Note that we are using here that if A = acl(A) and ā, b̄ are independent over A, then acl(A, ā) ∩ acl(A, b̄) = A. The corresponding statement with acl^{eq} may be false in an arbitrary simple theory.

Lemma. Let K ⊧ SCFA_{e,b} be sufficiently saturated, and let S be a set of realisations of a set of types over some E = acl_{σ}(E) ⊆ K. Assume that S is modular. Let φ(x̄, ȳ) be a formula over E, and assume that for every ā ∈ S, the set defined by φ(ā, ȳ) is finite. Then the set U of tuples b̄ such that there is ā ∈ S satisfying φ(x̄, b̄) is modular. Proof. Assume that U is not modular, and let c̄ = (c̄_{1}, …, c̄_{n}) and d̄ = (d̄_{1}, …, d̄_{m}) be tuples of elements of U such that c̄ and d̄ are not independent over C = acl_{σ}(E, c̄) ∩ acl_{σ}(E, d̄). Choose tuples ā = (ā_{1}, …, ā_{n}), b̄ = (b̄_{1}, …, b̄_{m}) in S such that φ(ā_{i}, c̄_{i}) and φ(b̄_{j}, d̄_{j}) hold for all i, j, and such that ā and d̄are independent over E ∪c̄, and b̄ and ā are independent over E ∪ d̄. Then acl_{σ}(E, ā) ∩ acl_{σ}(E, b̄) = C: our independence assumptions imply that acl_{σ}(E, ā) ∩ acl_{σ}(E, b̄) ⊆ acl_{σ}(E, d̄), and acl_{σ}(E, ā) ∩ acl_{σ}(E, c̄, d̄) ⊆ acl_{σ}(E, c̄). This contradicts the modularity assumption on S.

(5.3) Lemma. Let K ⊧ SCFA_{e,b}, and S ⊆ K^{n} a set of realisations of a set of types over E = acl_{σ}(E) ⊆ K. If S is modular, then for any tuple ā of elements of S and any tuple b̄ of elements of K, ā and b̄ are independent over acl_{σ}(E, ā) ∩ acl_{σ}(E, b̄). Proof. Let ā be a tuple of elements of S and b̄ a tuple of elements of K and assume that ā and b̄ are not independent over C = acl_{σ}(E, ā) ∩ acl_{σ}(E, b̄). Choose c̄ in acl_{σ}(E, ā) and d̄ in acl_{σ}(E, b̄) such that the fields C(c̄)_{σ} and C(d̄)_{σ} are not linearly disjoint over C. Thus there is some algebraic set V defined over C(d̄)_{σ} and not over C, such that (c̄, …, σ^{m}(c̄)) is a generic of the variety V over C(d̄)_{σ}. Let (ā_{n}, c̄_{n}), n ∈ ℕ, be an infinite independent over acl_{σ}(E, b̄) sequence of realisations of tp(ā, c̄/acl_{σ}(E, b̄)), with (ā_{1}, c̄_{1}) = (ā, c̄). Then acl_{σ}(E, ā_{1}) ∩ acl_{σ}(E, ā_{i} | i > 1) = C, and the tuples (c̄_{i}, …, σ^{m}(c̄_{i})), i > 1, are generics of V, algebraically independent over acl_{σ}(E, b̄). By a classical result in algebraic geometry, a variety is defined over the algebraic closure of a finite set of independent generic solutions, and this implies that V is defined over the algebraic closure of 𝔽_{p}(c̄_{2}, …, c̄_{n})_{σ} for some n. This shows that ā_{1} and (ā_{2}, …, ā_{n}) are not independent over C, and therefore shows that S is not modular.

(5.4) Definitions of orthogonality. We work in a model K of the theory T, where T = ACFA or T = SCFA_{e,b}.

(1) Recall that two complete types p over A and q over B are orthogonal if, for any set C containing A ∪ B, if ā realises p and is independent from C over A, and b̄ realises q and is independent from C over B, then ā and b̄ are independent over C.

(2) Let p be a complete type over A, S a set of realisations of a set of types over E = acl(E). Then p is orthogonal to S if for any F = acl(F) containing E and ā ∈ S, tp(ā/F) is orthogonal to p.

(3) If S and T are two sets of realisations of sets of types over some E = acl(E), then S and T are orthogonal if for any F = acl(F) containg E, and ā ∈ S, b̄ ∈ T, the tuples ā and b̄ are independent over F.

(4) Let E = acl(E) be a subset of K, and let S ⊆ K^{n} be a set of realisations of a set of types over E. We say that S is orthogonal to the fixed fields if S is orthogonal to all formulas σ^{n}(x) = x^{p^{m}} where n ≥ 1, m ∈ ℤ. Similarly, we say that a type p over E is orthogonal to the fixed fields if it is orthogonal to all formulas σ^{n}(x) = x^{p^{m}} where n ≥ 1, m ∈ ℤ.

Remark. If tp(ā/E) is orthogonal to the fixed fields, and b ∈ acl(E, ā), then tp(b/E) is orthogonal to the fixed fields.

(5.5) The following result should have been proved in [CHP], but was somehow overlooked: Proposition. Let L be a model of ACFA, let E = acl(E) ⊆ B = acl(B) ⊆ L, let ā ∈ L be transformally algebraic over E, and assume that acl(E, ā) ∩ B = E. If ā and B are not independent over E, then tp(ā/E) is non-orthogonal to a fixed field. Proof. Assume that B and ā are not independent over E, but that tp(ā/E) is orthogonal to all fixed fields. Then E(ā)_{σ} and B are not linearly disjoint over E. By [CH] (2.13)(1), Cb(ā/B) is contained in the algebraic closure of the difference field generated by finitely many realisations of tp(ā/E), and therefore is transformally algebraic over E. Hence there is b̄ ∈ B, with tr.deg(E(b̄)_{σ}/E) < ∞, such that E(ā)_{σ} and E(b̄)_{σ} are not linearly disjoint over E. We may therefore assume that B = acl(E, b̄). The proof is by induction on SU(b̄/F), for all F and all ā′ such that tp(ā′/F) is orthogonal to all fixed fields, and acl(F, ā′) ∩ acl(F, b̄) = F.

Since b̄ has finite SU-rank over E, by (3.4)(1) of [CH], there is F = acl(F) containing E and independent from b̄ over E, and c ∈ acl(F, b̄) such that SU(c/F) = 1. We may choose this F independent from (ā, b̄) over E, and this implies that acl(F, ā)∩acl(F, b̄) = F. Note that c is independent from ā over F because SU(c/F) = 1 and c ∉ acl(F, ā). Hence tp(ā/F, c) is orthogonal to all fixed fields and SU(b̄/F, c) < SU(b̄/E). To use the induction hypothesis and show the result, it is therefore enough to show that acl(F, ā, c)∩acl(F, b̄) = acl(F, c).

Assume that this is not the case and let d ∈ acl(F, ā, c) ∩ acl(F, b̄), d ∉ acl(F, c). Using the semi-minimal analysis of tp(b̄/acl(F, c)) (see [CHP] (7.3) and [CH] (5.4)), we may assume that either tp(d/F, c) is modular of rank 1, or that tp(d/F, c) is qf-internal to some fixed field defined by an equation σ^{n}(x) = x^{p^{m}} for some n ≥ 1 and m ∈ ℤ. Since tp(ā/F, c) is orthogonal to all fixed fields, it will be orthogonal to anything which is qf-internal to a fixed field. It follows that tp(d/F, c) is modular and of rank 1.

If tp(c/F) is modular, then so is tp(c, d/F) (see (7.4) in [CHP]), which gives us a contradiction, as acl(F, c, d) ∩ acl(F, ā) = F and d ∈ acl(F, ā, c). If SU(d/F) = 1 and tp(c/F) is non-modular, we also reach a contradiction: d ∈ acl(F, ā, c) implies d ∈ acl(F, ā). Hence we are left with the case where SU(d/F) = 2 and tp(c/F) is nonmodular. Note that c ∈ acl(F, d). Let (c, d) = (c_{1}, d_{1}), …, (c_{n}, d_{n}), …, be an infinite sequence of independent realisations of tp(c, d/acl(F, ā)); because ā and (c, d) are not independent over F, the sequence (c_{n}, d_{n}) is not independent over F, and there is a largest i such that SU(c_{1}, d_{1}, …, c_{i}, d_{i}/F) = 2i. Then d_{1}, …, d_{i} are independent over F, and SU(d_{i+1}/F, d_{1}, …, d_{i}) ≤ 1. By choice of our sequence, c_{i+1} ∉ acl(F, d_{1}, …, d_{i}), and d_{i+1} ∈ acl(F, d_{1}, …, d_{i}, c_{i+1}). Hence, tp(d_{1}/acl(F, d_{2}, …, d_{i}, c_{i+1})) does not fork over F, and is not almost orthogonal to the modular rank-1-type tp(d_{i+1}/acl(F, c_{i+1})). This means that tp(d/F) is non-orthogonal to a modular type of rank 1. Hence, there is F′ = acl(F′), independent from (ā, b̄) over F, such that acl(F′, d) contains an element e realising a modular type of SU-rank 1 over F′. By the first case, we have that acl(F′, ā, e) ∩ acl(F′, b̄) = acl(F′, e). By induction hypothesis on SU(b̄/F′, e) < SU(b̄/F′), we get that (ā, e) and b̄ are independent over acl(F′, e), and therefore that ā and b̄ are independent over F′, which gives us the desired contradiction. Hence this case cannot happen, which means that acl(F, ā, c) ∩ acl(F, b̄) = acl(F, c), and we are done.

(5.6) Proposition. Let K ⊧ SCFA_{e,b}, E = acl_{σ}(E) ⊆ K, S ⊆ K^{n} the set of realisations of a set of types over E, and assume that all elements of S are transformally algebraic over E. If S is non-modular, then S is non-orthogonal to a fixed field. Proof. Let L be a model of ACFA containing K, and assume that S is not modular. Then there are ā_{1}, …, ā_{m} ∈ S, and B = acl_{σ}(B) ⊇ E such that (ā_{1}, …, ā_{m}) and B are not independent over C = acl_{σ}(E, ā_{1}, …, ā_{m}) ∩ B. Let d̄ ∈ A = acl_{σ}(E, ā_{1}, …, ā_{m}) be such that the fields C(d̄)_{σ} and B are not linearly disjoint over C. We may assume that d̄ contains ā_{1}^{⌢} ⋯^{⌢} ā_{m}. By Lemma 4.4, d̄ is transformally algebraic over E, and therefore also over C.

We will denote by acl_{L}(−) the algebraic closure in the sense of L, and by tp_{L} types in the sense of T h(L). Recall that A = acl_{L}(A) if and only if A is an algebraically closed difference field. Hence, if A = acl_{σ}(A), then acl_{L}(A) = A^{alg} = A^{p^{−∞}}.

Our assumption on A, B, C implies that acl_{L}(A) ∩ acl_{L}(B) = acl_{L}(C), and that d̄ is not independent from B over C (in the sense of L). By (5.5), this implies that tp_{L}(d̄/C) is non-orthogonal to a formula σ^{n}(x) = x^{p^{m}} for some n ≥ 1 and m ∈ ℤ, and we may assume that (n, m) = 1. Let F ix(τ) be the subfield of L fixed by τ = σ^{n}F rob^{−m}.

Claim. There are independent realisations d̄_{1}, …, d̄_{k} of tp(d̄/C) in K such that C(d̄_{1}, …, d̄_{k})_{σ}∩ F ix(τ) contains an element not in C.

Indeed, since tp_{L}(d̄/C) is non-orthogonal to σ^{n}(x) = x^{p^{m}}, there are integers k, ℓ, and independent realisations d̄_{1}, …, d̄_{k} of tp_{L}(d̄/C^{alg}) in L, elements b_{1}, …, b_{ℓ} of F ix(τ) independent over C and such that the fields acl_{L}(C, d̄_{1}, …, d̄_{k}) and acl_{L}(C, b_{1}, …, b_{ℓ}) are not linearly disjoint over C^{alg} (see Remark (3.1)(1) in [CH]). acl_{L}(C, b_{1}, …, b_{ℓ}) is algebraic over C(b_{1}, …, b_{ℓ}, …, σ^{n−1}(b_{ℓ})), and hence we obtain that (b_{1}, …, σ^{n−1}(b_{ℓ})) are not algebraically independent over C(d̄_{1}, …, d̄_{k})_{σ}. Let V be the algebraic locus of (b_{1}, …, σ^{n−1}(b_{ℓ})) over C(d̄_{1}, …, d̄_{k})_{σ}; then (b_{1}, …, σ^{n−1}(b_{ℓ})) is a generic of V, is fixed by τ, and therefore is also a generic of τ(V). Hence τ(V) = V, the field of definition of V is fixed by τ, contained in the perfect hull of C(d̄_{1}, …, d̄_{k})_{σ}, and not contained in C^{alg}. This implies that C(d̄_{1}, …, d̄_{k})_{σ} contains an element of F ix(τ) not in C.

The above result depends only on the isomorphism type of the difference field C(d̄_{1}, …, d̄_{k})_{σ}, and so one may furthermore impose that the tuples d̄_{1}, …, d̄_{k} realise independent realisations of tp(d̄/C) in K. This proves the claim.

Hence we have ā_{1}, …, ā_{mk} in S, such that acl_{σ}(E, ā_{1}, …, ā_{mk}) ∩ F ix(τ) contains an

element b not in C, and hence not in E. Thus there is i such that b ∈ acl_{σ}(E, ā_{1}, …, ā_{i}), b ∉ acl_{σ}(E, ā_{1}, …, ā_{i−1}), and we have shown that S is non-orthogonal to a fixed field.

(5.7) Proposition. Let K ⊧ SCFA_{e,b}, let S be a definable modular subset of K^{n}, defined over E = acl_{σ}(E). Then S is orthogonal to the fixed fields.

Proof. By (5.2) it suffices to show that any infinite definable subset S of a fixed field k = {x | σ^{n}(x) = x^{p^{m}}}, n ≥ 1, m ∈ ℤ, (n, m) = 1 if m ≠ 0, is non-modular.

We will first assume that m = 0. By (4.3), we know that S is definable in the pure field language within k. Let B be the p-basis of k (which is also a p-basis of K). By Theorem 3.5 and Proposition 4.1, the type of a tuple ā of k is determined by the isomorphism type of its algebraic closure, which is the relative separable closure in k of the closure of 𝔽_{p}(B, ā) under the λ-functions λ_{i}(B; −), i ∈ I(B). Hence, using compactness, it follows that S = π(W (k)) for some algebraic set W and algebraic morphism π, which is finite-to-one on W (k). By (5.2), we may therefore replace S by W (k), where W is a (absolutely irreducible) variety defined over k. Then the function field k(W) is a regular extension of k. Let x ∈ k(W) be an element of a separating transcendence basis of k(W) over k, and let y ∈ k(W) be such that k(W) ∩ k(x)^{s} = k(x, y). Then k(W) is a regular extension of k(x, y). If f(x, Y) ∈ k(x)[Y] is the minimal polynomial of y over k(x), then the equation f(X, Y) = 0 defines an absolutely irreducible curve C, and we have a rational map g : W → C, defined over k; then, for all but finitely many (a, b) ∈ C(k^{alg}), g^{−1}(a, b) is absolutely irreducible. Hence, since k is PAC, g(W (k)) is a cofinite subset of the infinite set C(k). Thus we have reduced the problem to showing that C(k) is not modular.

The proof now uses a trick already used for pseudo-finite fields. Observe first that k(x^{1/p^{n}}, y | n ∈ ℕ) is a separable extension of k, because k(x, y) is separably algebraic over k(x). Let k_{1} be a saturated extension of k, and let (a_{1}, a_{2}), (b_{1}, b_{2}) ∈ C(k_{1}), with a_{1}, b_{1} ∈ k_{1}^{p^{∞}}, transcendental and algebraically independent over k. Consider (k_{1})^{alg}(x, y). By 11.7 and 12.9 of [FJ], there is a Zariski open subset U of 𝔸^{2} such that if a, b ∈ U, then the polynomial f(a + bx, Y) is irreducible over (k_{1})^{alg}(x, y); since f has its coefficients in k, the Zariski open set U is defined over k, and therefore contains a_{1} and b_{1}.

This shows that the algebraic set defined by f(x, y) = 0 = f(a_{1} + b_{1}x, y′) is absolutely irreducible, and defined over k_{1}. Hence, it has a solution (c_{1}, c_{2}, d_{2}) in k_{1}, and we may assume that c_{1} belongs to k_{1}^{p^{∞}} and is transcendental over k(a_{1}, b_{1}). Let d_{1} = a_{1} + b_{1}c_{1}. Then (c_{1}, c_{2}), (d_{1}, d_{2}) ∈ C(k_{1}). Since a_{1}, b_{1}, c_{1}, d_{1} ∈ k_{1}^{p^{∞}}, acl_{σ}(k, a_{1}, b_{1}) ⊆ k(a_{1}, b_{1})^{alg} and acl_{σ}(k, c_{1}, d_{1}) ⊆ k(c_{1}, d_{1})^{alg}, and therefore acl_{σ}(k, a_{1}, b_{1}) ∩ acl_{σ}(k, c_{1}, d_{1}) ⊆ k^{alg} ∩ k_{1} = k. However, (a_{1}, b_{1}) and (c_{1}, d_{1}) are not independent over k, and this shows that C(k) is not modular.

If m ≠ 0, then k ⊆ K^{p^{∞}}. Proposition (7.1)(1) of [CHP] gives that if E is a difference field, and a ∈ k, a ∉ E, then tr.deg(a/E) = n. This implies that in K, any element of k is either in E or independent from E, so that SU(k) = 1. Hence SU(S) = SU(k) = 1, and the non-modularity of the field k implies the non-modularity of S. Observe that this also proves the result for an ∞-definable S.

(5.8) Proposition. Let K ⊧ SCFA_{e,b}, let S be a definable modular subset of K^{n}, defined over E = acl_{σ}(E). Then every tuple ā ∈ S is transformally algebraic over E. Proof. By (5.2) and (3.6)(6), we may assume that S is defined by a quantifier-free L_{σ}- formula. Assume by way of contradiction that there is ā = (a_{1}, …, a_{n}) ∈ S which is not transformally algebraic over E. By Lemma 2.1, ā contains an element which is transformally transcendental over E, say a_{1}, and such that E(ā)_{σ} is a separable extension of E(a_{1})_{σ}. If S_{1} is the projection of S on the first coordinate, then S_{1} is also modular by (5.2), and because S is defined by a quantifier-free formula of L_{σ}, S_{1} contains all elements of K which are transformally transcendental over E. Then E(ā, a_{1}^{p^{n}} | n ∈ ℕ)_{σ} is a separable extension of E, and we may therefore assume that a_{1} ∈ K^{p^{∞}}. Let b = σ(a_{1}) − a_{1}; then a_{1} ∉ acl_{σ}(E, b), and acl_{σ}(E, b) ⊆ E(b)_{σ}^{alg} because b ∈ K^{p^{∞}}. Hence the set S_{2} defined by x ∈ S_{1} ∧ σ(x) − x = b contains a_{1}, and is infinite, hence is modular and orthogonal to the fixed fields by (5.2) and (5.7). But the set of solutions of σ(x) − x = b equals a_{1} + F ix(σ), and we get a contradiction.

(5.9) Remark. Proposition 5.7 does not generalise to ∞-definable subsets of K^{n} if e > 0. Indeed, let k = F ix(σ). Hrushovski has shown the following result (see 2.15, 2.16 and 5.6 in [H]): Let L be a separably closed field, let G be a simple abelian variety defined over L but not isomorphic to one defined over L^{p^{∞}}. Then Γ = ⋂_{n}[p^{n}]G(L) is modular.

Assume that L = k^{s}, and that G is defined over k. Observe the following: if A ⊆ k, then acl_{L}(A) (the model-theoretic algebraic closure of A in the sense of the separably closed field L) equals acl_{σ}(A): indeed, acl_{L}(A) is obtained by taking first the λ-closure of the field generated by A under the λ-functions of L, and then its separable closure. On k, the λ-functions of K, k and L agree (as they have a common p-basis B), and this shows the assertion.

This implies that if A, B ⊆ k, then A and B are independent over C = acl_{σ}(A) ∩ acl_{σ}(B) if and only if they are independent over C in the separably closed field L. Hence Γ ∩ G(k) is a modular subgroup of G(K).

(5.10) Two questions. (1) Let S be an ∞-definable subset of K^{n}, and assume that S is modular. Does it imply that every element of S is transformally algebraic over the set over which S is defined?

(2) Is there a criterion analogous to (5.6) for modular subsets of models of SCFA_{∞,λ}? Recall that by (3.7), forking may arise at the level of p-independence and so we may have acl_{σ}(E, ā) and acl_{σ}(E, b̄) linearly disjoint over E = acl_{σ}(E), but ā and b̄ not independent over E. We also do not have a good description of imaginaries in SCF_{∞,λ}.

(5.11). Modularity is an important tool in the study of definable sets, and in our context, modular subgroups of algebraic groups have good properties. Theorem. Let K ⊧ SCFA_{e,λ}, let G be an algebraic group definable over K and let S be a subgroup of G(K) which is ∞-definable by quantifier-free L_{λ,σ}-formulas and which is modular. Let U ⊆ G(K) be quantifier-free definable. Then U ∩S is a Boolean combination of cosets of definable subgroups of S. If S is defined over E = acl_{σ}(E), then so are these subgroups. Proof. We assume K sufficiently saturated, and let B be the p-basis of K (which is fixed by σ and contained in E because the elements of B are in the language). Consider the set Δ of quantifier-free L_{λ,σ}-formulas. Then over any subset E = acl_{σ}(E) of K, there are at most |E|^{ℵ_{0}} Δ-types: indeed, the Δ-type of a tuple ā over E is determined by the L_{λ}-type of the countable tuple (σ^{i}(ā))_{i∈ℤ} over E. Since the theory SCF_{e,b} is stable, there are at most |E|^{ℵ_{0}} L_{λ}-types of countable tuples over E, and hence at most |E|^{ℵ_{0}} Δ-types over E. The result now follows by a result of Pillay [P], because the formulas of Δ are stable and witness forking. Since Pillay’s result is unpublished, we will give an alternate proof of the result in our particular case, using the underlying topology on cartesian powers of K. Since we will be dealing with integers which are powers of p, we will use the notation 𝔸^{m}(K) to mean the cartesian product of m copies of K.

We put two topologies on 𝔸^{m}(K): the σ-topology and the λσ-topology. The σtopology is the topology whose basic closed sets are defined by positive quantifier-free L_{σ}-formulas, and we call them σ-closed. This topology is Noetherian (see [C]), and therefore any σ-closed set X is the union of finitely many irreducible σ-closed sets, called the irreducible components of X. Note that 𝔸^{m}(K) is compact and Hausdorff.

The λσ-topology on 𝔸^{m}(K) is generated by the closed sets X ⊆ 𝔸^{m}(K) defined by positive Δ-formulas; we call these sets basic λσ-closed. The λσ-tology is not Noetherian, but every λσ-closed set is the intersection of countably many basic λσ-closed sets. Again, each 𝔸^{m}(K) is compact and Hausdorff.

For each n ∈ ℕ, we identify K with 𝔸^{p^{en}}(K) via the map ψ_{n} : x ↦ (λ_{μ}(B; x))_{μ∈I(B)^{n}}. Then the algebraic group G gives rise to an algebraic group G_{n} = ψ_{n}(G) defined over K, of dimension p^{en}dim(G).

If X is a basic λσ-closed subset of 𝔸^{m}(K), there is an integer n such that X_{n} = ψ_{n}(X) is σ-closed in 𝔸^{mp^{ne}}(K), and then X = ψ_{n}^{−1}(X_{n}). Thus the λσ-topology on 𝔸^{m}(K) is the smallest topology such that all maps ψ_{n} : 𝔸^{m}(K) → 𝔸^{mp^{en}}(K) are continuous, where 𝔸^{mp^{en}}(K) is equipped with the σ-topology.

We call a λσ-closed set X irreducible if it is not the proper union of two closed subsets. We say that a λσ-closed set X is defined over C if it is defined by positive Δ-formulas with parameters in C. If X is defined over C = acl_{σ}(C), and a ∈ X, we say that a is a generic of X over C if X is the smallest λσ-closed set defined over C and containing a. By saturation of K, any (non-empty) irreducible λσ-closed set has a generic.

If H is an ∞-Δ-definable subgroup of G(K), then H is λσ-closed, and H is the intersection of definable λσ-closed subgroups of G(K): indeed, H = ⋂_{n} ψ_{n}^{−1}(H̃_{n}), where H̃_{n} is the σ-closure of ψ_{n}(H) in G_{n}(K). We define the connected component H^{0} of H as the intersection of all λσ-closed subgroups of H of finite index in H. Then H^{0} is irreducible. Indeed, if H^{0} is not irreducible, then for some n the σ-closure H̃_{n}^{0} of ψ_{n}(H^{0}) is not irreducible in the σ-topology. Since the σ-topology is Noetherian, H̃_{n}^{0} has a σ-closed proper subgroup of finite index H′, and ψ_{n}^{−1}(H′)∩H^{0} is then a proper λσ-closed subgroup of H^{0} of finite index in H^{0}.

Note that if H is defined over A = acl_{σ}(A) ⊇ E, then so are the groups H̃_{n} and H̃_{n}^{0}, and therefore so is H^{0}. Moreover, [H̃_{n} : H̃_{n}^{0}] < ∞, and by elimination of imaginaries, this implies that all cosets of H̃_{n}^{0} in H̃_{n} are defined over A. Hence, all cosets of H^{0} in H are defined over A.

The proof of the theorem follows the line of the classical proof, using the λσ-topology. Let a be a tuple in S^{0}, let E ⊆ A = acl_{σ}(A) ⊆ K, and let X be the λσ-closure of the set of realisations of the quantifier-free type of a over A. We will show that X = Ha, where H is an ∞-definable subgroup of S, and that H is defined over E.

By definition, X is λσ-closed, defined over A, and is an irreducible λσ-closed set, since A = acl_{σ}(A). Let H = {g ∈ S | gX = X}. Then H is a λσ-closed subgroup of S, and Ha ⊆ X by definition of H.

Let g ∈ S be a generic of S^{0} over A ∪ a. Then b = ga is independent from a over A [Proof: if V is a basic λσ-closed set defined over acl_{σ}(A, a) and containing b, then V a^{−1} contains g, and therefore also S^{0}, and is defined over A. Hence V contains S^{0}a, and this shows that b is a generic of S^{0}a over A∪a; as S^{0}a is defined over A = acl_{σ}(A), this implies that b is independent from a over A.] Consider the λσ-closed set Y = gX. Then b is a generic of Y over A ∪ g. Claim. The fields of definition of Y and of gH have the same algebraic closure over A.

Let τ be an automorphism of (K, σ) which fixes A. Then τ(X) = X and τ(H) = H, as X is defined over A; hence τ(Y) = Y if and only if τ(gX) = gX if and only if τ(g)X = gX if and only if g^{−1}τ(g) ∈ H if and only if τ(gH) = gH. Let C_{1} = acl_{σ}(C_{1}), resp. C_{2} = acl_{σ}(C_{2}), be the smallest algebraically closed subsets of K containing A over which Y, resp. gH, are defined. Assume C_{1} ⊈ C_{2}. By (3.6)(5), there is some A-automorphism τ of (K, σ) which fixes C_{2} and moves C_{1}; then τ(Y) ≠ Y and τ(gH) = gH, which gives us a contradiction. Similarly, we cannot have C_{2} ⊈ C_{1}, and therefore C_{1} = C_{2} = C.

b is a generic of Y over A ∪ g, and Y is defined over C and over A ∪ g. This implies that b and g are independent over C, and that C ⊆ acl_{σ}(A, b) ∩ acl_{σ}(A, g) (here we are using the fact that modularity is preserved under addition of constants).

From the independence of a and b over A, we deduce that a is independent from b over C, and therefore the set of realisations of qftp(a/C, b) is dense in X. Note that if a′ realises qftp(a/C, b), then ba′^{−1} realises qftp(g/C, b), and is therefore an element of gH. Since gH is λσ-closed, this implies that bX^{−1} ⊆ gH. From Ha ⊆ X we deduce that

b(a^{−1}H) ⊆ bX^{−1} ⊆ gH,

and therefore that bX^{−1} = gH, and X = Ha.

The proof of the theorem now follows by compactness. We may assume that U is defined by positive quantifier-free formulas. If a ∈ U, then Ha ⊆ U, and this implies that H′a ⊆ U for some quantifier-free definable subgroup H′ of G(K) containing H. Thus U ∩ S is a finite union of translates of quantifier-free definable subgroups of S. Remark. In the particular case that the subgroup S is contained in G(K^{p^{∞}}), there is a direct proof of this result. By (4.7), S = G(K^{p^{∞}}) ∩ S′, where S′ is ∞-definable in the language L_{σ}. As any descending sequence of σ-closed subsets of a difference field stabilises, this implies that S is a quantifier-free definable subgroup of the difference field (K^{p^{∞}}, σ). Hence, by (4.6) the results of [CHP] apply and give us the desired conclusion.

(5.12) An example and some questions. The criterion for modularity given in (5.6) is very unpractical. Even in simple cases, it is quite difficult to determine whether a set is modular or not. We will illustrate this with an example: assume that p ≠ 2, and consider the subgroup S of 𝔾_{m}(K) defined by σ(x) = x^{2}. We do not know whether this group is modular (but conjecture that it should be).

The “generic” of S is the type specifying that σ(x) = x^{2} and that for each n ∈ ℕ, the elements λ_{μ}(B; x), μ ∈ I(B)^{n}, are algebraically independent. At the other extreme, we have the type σ(x) = x^{2} ∧ x ∈ K^{p^{∞}}: the set of its realisations is the group 𝔾_{m}(K^{p^{∞}}) ∩ S, which is modular by (7.4) in [CHP] (since the multiplicity of x over σ^{n}(x) is 2^{n}, and hence unbounded).

In order to prove that S is modular, one needs to solve the following problem: let a ∈ S, let E = acl_{σ}(E) contain a p-basis, and show that there is no element b ∈ acl_{σ}(E, a) \ E satisfying σ^{n}(x) = x^{p^{m}} for some n ≥ 1. In fact (see [CHP] (7.1)(2)), it suffices to show that there is no such b in the λ-closure of the field E(a). From the equation σ(a) = a^{2}, we deduce equations relating the elements λ_{i}(B; a), i ∈ I(B), and the elements σ(λ_{i}(B; a)) = λ_{i}(σ(B); a^{2}), i ∈ I(B). However, determining which extra algebraic relations are allowed is quite complicated.

Of independent interest would be to describe the quantifier-free definable subgroups of S. The first difficulty here is to determine the quantifier-free L_{λ}-definable subgroups of 𝔾_{m}(K). Note that these are λ-closed (i.e., defined by L_{λ}-equations). Examples of such groups are as follows: let n be an integer, and c_{1}, …, c_{m} ∈ K; then K^{p^{n}}[c_{1}, …, c_{m}] is a subfield of K definable in K, and all definable subfields of K are of this form (Messmer [Me]). Then 𝔾_{m}(K^{p^{n}}[c_{1}, …, c_{m}]) is a λ-closed definable subgroup of 𝔾_{m}(K). The first question to answer: is every λ-closed subgroup of 𝔾_{m}(K) of this form? Then one needs to investigate the λ-closed subgroups of 𝔾_{m}(K)^{n} for n ∈ ℕ in order to describe the quantifierfree definable subgroups of S: such a subgroup is defined by (g, σ(g), …, σ^{n−1}(g)) ∈ U for some n and λ-closed subgroup U of 𝔾_{m}(K)^{n}.

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