MODEL THEORY OF DIFFERENCE FIELDS LECTURE NOTES NOTRE DAME, SEP 2000

ZOÉ CHATZIDAKIS

Introduction. This is the set of notes of a special topics course I gave at Notre Dame in the Fall 2000 (September 20-October 13). I develop here some of the model theory of difference fields (a difference field is simply a field with a distinguished automorphism σ), at a fairly elementary level. I reproduce many of the classical proofs of stability theory in the particular context we are working in, my feeling being that a proof in a concrete situation is much easier to understand than in a more general context. I tried to avoid using results from stability theory, and succeeded except at one or two places (where the neophyte is asked to just accept the result). I also inserted some comments for people with a working knowledge of stability theory, and these are enclosed by the symbols □□□. These comments can simply be skipped.

The notes are organized as follows. Chapter 1 gives some preliminary algebraic results and definitions (difference fields, varieties, Zariski topology, etc.). Chapter 2 introduces the theory ACFA of generic difference fields, and proves elementary results about it. Chapter 3 introduces the notions of independence and SU-rank, and shows various results about them. In chapter 4 we study the fixed field, and in chapter 5 the notions of orthogonality and modularity. Chapter 6 introduces generics and stabilizers of groups. Finally, chapter 7 contains some of the hard results in the area, and the applications by Hrushovski to problems in number theory. At the end of the notes, you will find some references and “further reading” on difference fields, with comments.

I would like to thank A. Berenstein for his careful reading of the notes and many helpful suggestions.

Notation

ℕ, ℤ, ℚ, ℂ  the natural numbers, the integers, the rational

and complex numbers

𝔽_{q}  the field with q elements
L(E)  language obtained by adjoining to L constant symbols

for elements of E

AB  subfield of Ω generated by A and B
A[B]  subring of Ω generated by A and B
A^{alg}  algebraic closure of the field A
K[X_{1}, …, X_{n}]_{σ} = K[X_{1}, …, X_{n}, X_{1}^{σ}, …, X_{n}^{σ}, …, X_{i}^{σ^{j}}, …]

= difference polynomial ring in X_{1}, …, X_{n}

I(S)  ideal of polynomials vanishing on the set S
I_{σ}(S)  ideal of difference polynomials vanishing on the set S
I(ā/K)  ideal of polynomials over K vanishing at ā
I_{σ}(ā/K)  ideal of difference polynomials over K vanishing at ā
cl_{σ}(A)  smallest difference field containing A

= field generated by {σ^{i}(A) | i ∈ ℤ}, with the action of σ

acl_{σ}(A)  cl_{σ}(A)^{alg}
U^{σ}  variety conjugate of U under σ
U(K)  points of the algebraic set U with their coordinates in K
qfdiag(E)  set of quantifier-free L(E)-sentences which hold in some

L-structure containing E

qftp  quantifier-free type
tp_{ACF}  type in the reduct to the language of fields {+, −, ·, 0, 1}
tr.deg  transcendence degree
dim  dimension of an algebraic set
⊥  orthogonal
T or(A)  torsion points of the group A
T or_{p′}(A)  torsion points of the group A of order prime to p
□□□  The text enclosed by these symbols is meant

as comments for those with a working knowledge of stability theory §1. Definitions, preliminary results.

1.1. Definition. A difference field is a field K with a distinguished endomorphism σ. An inversive difference field is a difference field where the endomorphism is onto (and therefore an automorphism). Given a difference field K, there is a “smallest” inversive difference field extending K, which is unique up to K-isomorphism. Therefore we will adopt the following CONVENTION: In what follows, all difference fields will be assumed to be inversive

Consider the language L = {+, −, ·, 0, 1, σ}, where +, −, · are binary operations, 0, 1 are constants and σ is a unary function symbol. Then every difference field is naturally an L-structure. Inclusion of difference fields corresponds to inclusion of L-structures, and similarly for morphisms of difference fields. A good reference for basic results on difference fields is R.M. Cohn’s book, Difference algebra [1].

1.2. Difference polynomials, difference equations. Let (K, σ) be a difference field, and X̄ = (X_{1}, …, X_{n}) a tuple of indeterminates. The difference polynomial ring in X̄ over K, denoted K[X̄]_{σ}, is defined as follows:

As a ring, K[X̄]_{σ} is simply the polynomial ring in the indeterminates X_{1}, …, X_{n}, X_{1}^{σ}, …, X_{n}^{σ}, …, X_{1}^{σ^{m}}, . . ..

The action of σ on K[X̄]_{σ} is the one suggested by the names of the indeterminates. Then K[X̄]_{σ} is also an L-structure.

Remarks. (1) A difference ring is a ring with a distinguished injective endomorphism. So, K[X̄]_{σ} is a difference ring.

(2) This map σ is injective, but is not surjective. One could instead take K[X_{j}^{σ^{i}}]_{j=1,... ,n,i∈ℤ}.

The elements of K[X̄]_{σ} are called difference polynomials. If f(X̄) ∈ K[X̄]_{σ}, then f(X̄) = 0 is called a σ-equation, and the set {ā ∈ K^{n} | f(ā) = 0} is called a σ-closed set.

1.3. Definitions. (1) Let T be a theory, M a model of T. Then M is existentially closed (among models of T) if whenever N is a model of T containing M, then every existential sentence with parameters in M which is satisfied in N is already satisfied in M.

(2) A generic difference field is a difference field K such that every finite system of σ-equations (over K) which has a solution in some difference field extending K, has already a solution in K.

1.4. Exercise 1. Show that if K is a generic difference field, then K is an existentially closed difference field. [Hint: recall that in a field x ≠ 0 ⟺ ∃y xy = 1].

1.5. Our first goal is to show that the generic difference fields form an elementary class (with axiomatization called ACFA). We will then study some easy properties of this theory. We first need however to recall some definitions of basic algebraic geometry and of field theory. Proofs and details can be found in S. Lang’s book, Introduction to algebraic geometry [2].

1.6. Algebraic sets, varieties, etc.. Here and in what follows we fix a large algebraically closed field Ω. All fields considered will be subfields of Ω. If K is a subfield of Ω, we denote by K^{alg} the algebraic closure of K, i.e., the set of elements of Ω which are algebraic over K. We first define affine algebraic sets. Affine n-space, 𝔸^{n}(Ω), is simply Ω^{n} (for n ∈ ℕ). An algebraic subset of Ω^{n} is a subset defined by polynomial equations f_{1}(X̄) = ⋯ = f_{m}(X̄) = 0, where the f_{i}(X̄) are polynomials in X̄ = (X_{1}, …, X_{n}), with coefficients in Ω.

These sets are also called Zariski closed. The Zariski closed subsets of Ω^{n} generate a topology on Ω^{n}, called the Zariski topology. This topology is Noetherian (the ascending chain condition on ideals of Ω[X̄] implies the descending chain condition on Zariski closed sets).

1.7. Let V be an algebraic set. We define

I(V) = {f(X̄) ∈ Ω[X̄] | f(ā) = 0 for all ā ∈ V }.

The coordinate ring of V, Ω[V], is defined to be Ω[X̄]/I(V).

We say that V is defined over the subfield K of Ω, if I(V) is generated by I(V) ∩ K[X̄]. In this case we can also form the coordinate ring of V over K, K[V] = K[X̄]/I(V) ∩ K[X̄].

A closed subset V of Ω^{n} is irreducible if whenever V = V_{1} ∪ V_{2} with V_{1}, V_{2} closed subsets of Ω^{n}, then V = V_{1} or V = V_{2}. An irreducible closed set is also called a variety.

If V is a variety, we define the field of rational functions of V to be the field of fractions of Ω[V]; we denote it by Ω(V).

Because the Zariski topology is Noetherian, it follows that every algebraic set V can be written (uniquely up to permutation) as

V = V_{1} ∪ ⋯ ∪ V_{m}

for some m and varieties V_{1}, …, V_{m} such that V_{i} ⊈ V_{j} for i ≠ j. The V_{i} are called the irreducible components of V.

1.8. Some classical results.

1 - An (affine) algebraic set V is a variety if and only I(V) is prime.

2 - If V ⊆ W are algebraic sets, then I(V) ⊇ I(W).

3 - (Nullstellensatz) Let I be an ideal of Ω[X̄], and let V (I) = {ā ∈ Ω^{n} | f(ā) = 0 for all f(X̄) ∈ I}. Then

V (I) ≠ ∅ ⟺ I ≠ (1).

4 - If I ⊆ J are ideals in Ω[X̄] then V (I) ⊇ V (J).

5 - If I is a radical ideal of Ω[X̄] (i.e., a^{n} ∈ I implies a ∈ I), then I(V (I)) = I. 6 - If V is an algebraic subset of Ω^{n} then V (I(V)) = V. 7 - If V_{1}, …, V_{m} are the irreducible components of V, then I(V_{1}), …, I(V_{m})

are the minimal prime ideals containing I(V), and I(V) = ⋂_{i=1,... ,m} I(V_{i}).

1.9. Remarks. (1) If K is a subfield of Ω, one can also define the notion of K-irreducible subsets of Ω^{n}: an algebraic set V is K-irreducible if it is defined over K and cannot be written as the union of two proper algebraic subsets which are defined over K. This corresponds to the ideal I(V) ∩ K[X̄] being prime.

(2) One can show that if the algebraic set V is defined over K, then V is a variety if and only if V is K^{alg}-irreducible (if and only if I(V) ∩ K[X̄] generates a prime ideal in K^{alg}[X̄]).

(3) It may be that an algebraic set is defined by polynomial equations with their coefficients in K, but is not defined over K. This can happen in positive characteristic. Indeed, assume that K is a field of characteristic p > 0, and that a ∈ K does not have a p-th root in K. Then the set V = {a^{1/p}} is defined by the equation X^{p} − a = 0 (with coefficients in K), but I(V) is generated by X − a^{1/p}, and V is therefore not defined over K as a^{1/p} ∉ K.

(4) If V is an algebraic set, there is a smallest field K over which V is defined, and this field is called the field of definition of V. If V is defined by polynomial equations with coefficients in K, then the field of definition of V is contained in

— K if char(K) = 0,

— the perfect hull of K (= closure of K under p-th roots) if char(K) = p > 0. [Recall that in characteristic p > 0, the Frobenius map x ↦ x^{p} defines an injective endomorphism of the field K, as (x + y)^{p} = x^{p} + y^{p}. Hence, K^{1/p^{n}} = {a ∈ Ω | a^{p^{n}} ∈ K} is a subfield of Ω, and so is the perfect hull K^{1/p^{∞}} of K: K^{1/p^{∞}} = ⋃_{n∈ℕ} K^{1/p^{n}}.]

1.10. Dimension, generics. Let V ⊆ Ω^{n} be a variety defined over K. We define the dimension of V, dim(V), to be tr.deg(Ω[V]/Ω). Note that if V ⊆ Ω^{n}, then dim(V) ≤ n. The tuple ā = (a_{1}, …, a_{n}) is a generic of V over K if the K-morphism K[X̄] → K[ā] which sends X_{i} to a_{i} for i = 1, …, n, has kernel I(V) ∩ K[X̄].

If V is an algebraic set, then dim(V) is the maximum of the dimensions of the irreducible components of V.

Remark. If V is K-irreducible, then the irreducible components of V are conjugate under Aut(K^{alg}/K), and therefore have the same dimension. Thus one can also define dim(V) as tr.deg(K[V]/K), and define a notion of generic over K.

1.11. Locus of a point. Let K be a subfield of Ω, and ā = (a_{1}, …, a_{n}). Define

I(ā/K) = {f(X̄) ∈ K[X̄] | f(ā) = 0},

where X = (X_{1}, …, X_{n}). Then I(ā/K) is a prime ideal of K[X̄]. The set V of points of Ω^{n} at which all elements of I(ā/K) vanish, is therefore K-irreducible, and is called the locus of ā over K. Then

— ā is a generic of V over K,

— Let b̄ ∈ Ω^{n}. Then b̄ ∈ V ⟺ I(b̄/K) ⊇ I(ā/K).

— Assume that b̄ = (b_{1}, …, b_{n}) ∈ V. Then there is a unique K-morphism φ : K[ā] → K[b̄] which sends a_{i} to b_{i} for i = 1, …, n. This morphism is an isomorphism if and only if b̄ is a generic of V.

— Let V_{1} be an irreducible component of V, let L = K(ā) ∩ K^{alg}, and let L̂ be the normal closure of L over K (= the field generated by all conjugates of L under the action of Aut(K^{alg}/K)). Then the set {V_{1}^{τ} | τ ∈ Aut(L̂/K)} is precisely the set of irreducible components of V.

Remark - warning. If the characteristic is p > 0, the locus V of ā over K is not necessarily defined over K. One has: V is defined over K if and only if K(ā) is a separable extension of K, i.e.: K(ā) and K^{1/p} are linearly disjoint over K.

1.12. Morphisms.

Let V ⊆ Ω^{n}, W ⊆ Ω^{m} be algebraic sets. Let us write Ω[V] = Ω[x̄], where x̄ = (x_{1}, …, x_{n}), and each x_{i} is the image of X_{i} in Ω[V].

One can think of an element of Ω[V] as a function V → Ω. Indeed, clearly an element f(X̄) of Ω[X̄] defines a function : Ω^{n} → Ω. This function restricts to a function f̂ defined on V. One then has: f̂ = ĝ if and only if f − g vanishes on V, if and only if f − g ∈ I(V).

A morphism f : V → W is given by a tuple (f_{1}(x̄), …, f_{m}(x̄)) of elements of Ω[V], such that for all ā ∈ V one has (f_{1}(ā), …, f_{m}(ā)) ∈ W. Note that it suffices to check this for generics of the irreducible components of V.

Given f : V → W as above, we obtain a dual morphism of Ω-algebras f^{∗} : Ω[W] → Ω[V], given by f^{∗}(g) = g ∘ f, i.e., f^{∗}(g)(x̄) = g(f_{1}(x̄), …, f_{m}(x̄)) ∈ Ω[V].

Assume that f, V and W are all defined over K, and let ā be a generic of V over K. One has that f^{∗} is injective if and only if f(V) is Zariski dense in W, if and only f(ā) is a generic of W over K. (The proof is an exercise). In this case, one says that f is generically onto.

§2. The theory ACFA. Notation is as in the previous chapter.

2.1. Consider the theory, called ACFA, whose models are the L-structures K satisfying:

(1) K is an algebraically closed field.

(2) σ is an automorphism of K.

(3) If U and V are varieties defined over K, with V ⊆ U × U^{σ}, such that the projections of V to U and to U^{σ} are generically onto, then there is a tuple ā in K such that (ā, σ(ā)) ∈ V.

Explanation of the notation

— σ extends to an automorphism of K[X̄] which leaves the elements of X̄ fixed. Then U^{σ} = V (σ(I(U)), i.e., U^{σ} ∩ K^{n} = σ(U ∩ K^{n}).

— The projection maps are induced by π_{1} : U × U^{σ} → U and π_{2} : U × U^{σ} → U^{σ}. Our hypothesis simply says that π_{1}(V) is Zariski dense in U, and π_{2}(V) is Zariski dense in U^{σ}. Equivalently, if whenever (ā, b̄) is a generic of V over K, then ā is a generic of U over K, and b̄ a generic of U^{σ} over K. Why (3) is first-order. Write I(U) = (f_{1}(X̄), …, f_{m}(X̄)), and I(V) = (g_{1}(X̄, Ȳ), …, g_{s}(X̄, Ȳ)). Choose a tuple ū, and polynomials F_{i}(Ū, X̄) ∈ ℤ[Ū, X̄], G_{j}(Ū, X̄, Ȳ) ∈ ℤ[Ū, X̄, Ȳ], such that f_{i}(X̄) = F_{i}(ū, X̄) for i = 1, …, m and g_{j}(X̄) = G_{j}(ū, X̄, Ȳ) for j = 1, …, s.

Fact: The following properties of the tuple ū are expressible by a first-order formula:

— F_{1}(ū, X̄), …, F_{m}(ū, X̄) generate a prime ideal I in K[X̄],

— G_{1}(ū, X̄, Ȳ), …, G_{s}(ū, X̄, Ȳ) generate a prime ideal J in K[X̄, Ȳ], which intersects K[X̄] in I.

The second property tells us that the dual of the projection map gives an inclusion K[X̄]/I ⊆ K[X̄, Ȳ]/J, which exactly says that V (J) projects generically onto V (I). From the fact, we deduce that each instance of axiom (3) is elementary. (Note that (3) is in fact a scheme of axioms: one for each triple (n, m, d), where n is an upper bound on the number of variables of U, m an upper bound on the number of polynomials defining the varieties U and V, and d an upper bound on the degree of these polynomials.)

Remarks. The above fact holds for an arbitrary field K, and the formulas expressing the required properties of the tuple ū do not depend on the field K. For a proof, see e.g. the paper by Van den Dries and Schmidt [13]. In this paper there are other very nice (and useful) definability results for ideals in polynomial rings over fields. E.g., define uniformly the minimal prime ideals containing an ideal I, which when dualised, corresponds to finding the K-irreducible components of an algebraic set.

Recall that every formula of the field language is equivalent modulo the theory of algebraically closed fields, to a quantifier-free formula. Hence for instance the property of ū, that F_{1}(ū, X̄), …, F_{m}(ū, X̄) generate a prime ideal in K^{alg}[X̄], is an elementary property of the tuple ū in K.

2.2. Theorem. Every difference field embeds in a model of ACFA. The models of ACFA are exactly the generic difference fields.

Proof. Let (K, σ) be a difference field. Then σ lifts to an automorphism of K^{alg}, and so axioms (1) and (2) are no problem. So, let K be an algebraically closed difference field, let U and V be varieties satisfying the hypotheses of (3). We want to find a difference field L extending K, and containing a tuple ā with (ā, σ(ā)) ∈ V.

Let (ā, b̄) be a generic of V over K (recall, we work in Ω). Then ā is a generic of U over K, and b̄ is a generic of U^{σ} over K. This exactly says that I(b̄/K) = σ(I(ā/K)), so that σ extends uniquely to a morphism τ : K(ā) → K(b̄) sending ā to b̄. Let L = K(ā, b̄)^{alg}. By properties of algebraically closed fields, τ lifts to an automorphism ρ of L. Hence (L, ρ) is a difference field extending (K, σ) and contains a solution to our equation.

A standard chain argument now shows that every difference field embeds in a model of ACFA (Exercise). We now need to show that the models of ACFA are generic. Let K be a model of ACFA, f_{1}(X̄), …, f_{m}(X̄) ∈ K[X̄]_{σ}, and assume that there is a difference field L containing K, and a tuple ā in L such that f_{1}(ā) = ⋯ = f_{m}(ā) = 0. We want to show that there is such an ā in K.

Let ℓ ∈ ℕ be such that f_{1}(X̄), …, f_{m}(X̄) ∈ K[X̄, …, σ^{ℓ}(X̄)]. Consider the varieties

— U with generic over K the tuple b̄ = (ā, σ(ā), …, σ^{ℓ−1}(ā)),

— V with generic over K the tuple (b̄, σ(b̄)).

Then σ(b̄) = (σ(ā), …, σ^{ℓ}(ā)) is a generic of U^{σ}. Thus V ⊆ U × U^{σ}, and projects generically onto U and onto U^{σ}. By axiom (3), there is c̄ ∈ K^{nℓ} such that (c̄, σ(c̄)) ∈ V. Then c̄ can be written (d̄, σ(d̄), …, σ^{ℓ−1}(d̄)). Since I(c̄, σ(c̄)/K) contains I(b̄, σ(b̄)/K), we get that I(d̄, σ(d̄), …, σ^{ℓ}(d̄)/K) contains I(ā, σ(ā), …, σ^{ℓ}(ā)/K), and therefore that f_{1}(d̄) = ⋯ = f_{m}(d̄) = 0.

2.3. Corollaries. (1) ACFA is model complete, i.e., if K_{1} ⊆ K_{2} are models of ACFA, then K_{1} ≺ K_{2}.

(2) Every formula is equivalent, modulo ACFA, to an existential formula.

2.4. Exercise 2. Show that if all models of a theory T are existentially closed (among models of T), then T is model complete. [Hint of proof: show first that if A ⊆ B and A is existentially closed in B, then there is an elementary extension C of A containing B. Let now A ⊆ B be models of T. Using the first step, construct elementary chains (A_{i})_{i∈ω} and (B_{i})_{i∈ω}, with A_{0} = A, B_{0} = B, and A_{i} ⊆ B_{i} ⊆ A_{i+1} ⊆ B_{i+1}. Then ⋃_{i} A_{i} = ⋃_{i} B_{i} is an elementary extension of both A and B.]

2.5. Definition. Let E ⊆ K_{1}, K_{2} be subfields of Ω. One says that K_{1} and K_{2} are algebraically independent over E, or free over E, if for every n ∈ ℕ, whenever a_{1}, …, a_{n} ∈ K_{1} are algebraically independent over E, then they remain algebraically independent over K_{2}.

This notion corresponds to independence in the theory of algebraically closed fields, and is a symmetrical notion: K_{1} is free from K_{2} over E if and only if K_{2} is free from K_{1} over E (see Lang’s book [2]). It is also transitive: if K_{2} ⊆ K_{3}, then K_{1} and K_{3} are free over E if and only if K_{1} and K_{2} are free over E, and K_{1}K_{2} and K_{3} are free over E.

2.6. Definition. Let E ⊆ K_{1}, K_{2} be subfields of Ω. One says that K_{1} and K_{2} are linearly disjoint over E if for every n ∈ ℕ, whenever a_{1}, …, a_{n} ∈ K_{1} and b_{1}, …, b_{n} ∈ K_{2} are such that ∑_{i=1}^{n} a_{i}b_{i} = 0 and not all b_{i}’s are 0, then there are c_{1}, …, c_{n} ∈ E such that ∑_{i=1}^{n} a_{i}c_{i} = 0 and not all c_{i}’s are 0.

Recall also that the tensor product K_{1} ⊗_{E} K_{2} is defined as follows: Fix a basis B_{i} of the E-vector space K_{i}, with 1 ∈ B_{i}, i = 1, 2. Then, as an E-vector space, K_{1} ⊗_{E} K_{2} has basis

{a ⊗ b | a ∈ B_{1}, b ∈ B_{2}}.

If c = ∑_{a∈B_{1}} c_{a}a and d = ∑_{b∈B_{2}} d_{b}b (with the c_{a} and d_{b} in E; all but finitely many of the c_{a}’s and d_{b}’s are 0), then we write c ⊗ d for the element ∑_{a∈B_{1},b∈B_{2}} c_{a}d_{b}(a ⊗ b). Multiplication on the elements of the basis is given by

(a ⊗ b) · (c ⊗ d) = (ac) ⊗ (bd),

and extended by linearity to the whole space. Note that K_{1} and K_{2} embed in K_{1} ⊗_{E} K_{2}, via a ↦ a ⊗ 1 and b ↦ 1 ⊗ b. If e ∈ E, a ∈ K_{1} and b ∈ K_{2}, then we have ae ⊗ b = e(a ⊗ b) = a ⊗ eb.

Fact. K_{1} and K_{2} are linearly disjoint over E if and only if K_{1}⊗_{E} K_{2} is a domain, equal to K_{1}[K_{2}], the subring of Ω generated by K_{1} and K_{2}.

For details and proofs, see Lang’s book [2]. The proof of the fact is not difficult. One should note that linear disjointness is a symmetrical notion, and implies algebraic independence. It is also transitive. Here are some special cases:

— If E is an algebraically closed field, then K_{1} and K_{2} are algebraically independent over E if and only if they are linearly disjoint over E.

— If one of K_{1} or K_{2} is a Galois extension of E, then K_{1} and K_{2} are linearly disjoint over E if and only if K_{1} ∩ K_{2} = E.

2.7. If E is a subset of K, one denotes by L(E) the language obtained by adjoining to L constant symbols for the elements of E. Then K expands naturally to an L(E)-structure, by interpreting the constant symbol corresponding to e ∈ E by the element e itself. The set of quantifier-free L(E)-sentences true in K is denoted by qfdiag(E), and it describes the isomorphism type of the Lsubstructure of K generated by E. So, a model of qfdiag(E) will be an Lstructure containing an isomorphic copy of the L-substructure of K generated by E. Elementary equivalence in the language L(E) is denoted by ≡_{E}. Theorem. Let K_{1} and K_{2} be models of ACFA, containing a common algebraically closed difference subfield E. Then K_{1} ≡_{E} K_{2}. Proof. To avoid confusion we will denote by σ_{i} the automorphism of K_{i} that we are considering. Step 1. Choose an E-isomorphic copy K′_{2} of the field K_{2} (by an E-isomorphism φ), which is free from K_{1} over E, and let σ′_{2} = φσ_{2}φ^{−1}. Because E is algebraically closed, K_{1} and K′_{2} are then linearly disjoint over E. Then the difference fields (K_{2}, σ_{2}) and (K′_{2}, σ′_{2}) are E-isomorphic via φ, and therefore elementarily equivalent over E. Hence, replacing K_{2} by K′_{2}, we may assume that K_{1} and K_{2} are linearly disjoint over E. Step 2. We will now show that σ_{1} and σ_{2} have a common extension to K_{1}K_{2} (the subfield of Ω generated by K_{1} and K_{2}). We first define τ(a ⊗ b) = σ_{1}(a) ⊗ σ_{2}(b) for a ∈ K_{1}, and b ∈ K_{2}. Since σ_{1} and σ_{2} agree on E, and K_{1} and K_{2} are linearly disjoint over E, this is well-defined. This extends (by linearity) to K_{1} ⊗_{E} K_{2}, which we identify with K_{1}[K_{2}], and we then extend τ to the field of fractions of K_{1}[K_{2}], i.e., to K_{1}K_{2}. Step 3. The difference field (K_{1}K_{2}, τ) embeds in a model L of ACFA. Because ACFA is model complete (Corollary 2.3(1)) we then have K_{i} ≺ L for i = 1, 2, which implies that K_{1} and K_{2} satisfy the same L(E)-sentences, and shows the result.

2.8. Corollary. Let E be an algebraically closed difference field. Then ACFA∪qfdiag(E) is complete. Proof. If K_{1} and K_{2} are models of ACFA∪qfdiag(E), then K_{1} and K_{2} contain difference subfields E_{1} and E_{2} respectively, which are isomorphic to E. Moving K_{2} by an isomorphism, we may assume that E_{1} = E_{2}. The result then follows by Theorem 2.7.

2.9. Corollary. The completions of ACFA are obtained by adjoining to ACFA a description of the isomorphism type of the difference field consisting of elements algebraic over the prime field. Proof. If T = T h(K) is a complete theory containing ACFA, then T will specify the characteristic, and therefore the isomorphism type of the prime field k. Note that the elements of k are in fact (interpretations of) terms of the language L. Let L be a finite Galois extension of k of degree n over k, α a generator of L over k, and p(X) ∈ k[X] its minimal (monic) polynomial. Since L is Galois over k, and k is fixed by σ, σ(L) will contain all the roots of p(X), and therefore will equal L. Hence, σ(α) = ∑_{i=0}^{n−1} a_{i}α^{i} for some elements a_{i} ∈ k. The sentence

∃x p(x) = 0 ∧ σ(x) = ∑_{i=0}^{n−1} a_{i}x^{i}

will therefore belong to T. The set of all such sentences will describe the isomorphism type of the difference subfield k^{alg} of K.

The converse follows from Theorem 2.7.

2.10. Notation. If A is a subset of an algebraically closed difference field K, we denote by cl_{σ}(A), the difference field generated by A, and by acl_{σ}(A) the smallest algebraically closed difference field containing A. Note that cl_{σ}(A) is the field generated by the sets σ^{i}(A), i ∈ ℤ, and that acl_{σ}(A) is simply the algebraic (field-theoretic) closure of cl_{σ}(A).

Recall that the model-theoretic definable and algebraic closure are defined as follows: let M be a model of a theory, and A ⊂ M. The definable closure of A in M, denoted dcl(A), is the set of elements a ∈ M, such that there is some formula φ(x) ∈ L(A), which is satisfied by a in M and by no other element of M. We will then say that the formula φ(x) defines a. So, dcl(A) will in particular contain all elements of the substructure of M generated by A. The algebraic closure of A in M, denoted acl(A), is the set of elements a ∈ M which satisfy some L(A)-formula which is satisfied by only finitely many elements of M.

Clearly dcl(A) ⊆ acl(A), and A ⊆ B implies dcl(A) ⊆ dcl(B) and acl(A) ⊆ acl(B). Moreover, one can show that acl(acl(A)) = acl(A) (and of course dcl(dcl(A)) = dcl(A)). In the particular case of difference fields, clearly cl(A) ⊇ cl_{σ}(A), and acl(A) ⊇ acl_{σ}(A).

2.11. Corollary. Let (K_{1}, σ_{1}) and (K_{2}, σ_{2}) be models of ACFA, containing a common difference subfield (E, σ). Then

K_{1} ≡_{E} K_{2} ⟺ (E^{alg}, σ_{1}|_{E^{alg}}) ≃_{E} (E^{alg}, σ_{2}|_{E^{alg}}).

Proof. The left to right implication is clear, as elements of E^{alg} are algebraic over E. For the converse, let φ : E^{alg} → E^{alg} be an E-isomorphism such that φσ_{1} = σ_{2}φ. Extend φ to an automorphism ψ of Ω, and let (K′_{1}, σ′_{1}) = (ψ(K_{1}), ψσ_{1}ψ^{−1}) be the difference field image of K_{1} by ψ. Then K_{1} ≡_{E} K′_{1}, and σ′_{1} and σ_{2} agree on E^{alg}. So we may apply 2.7

2.12. Corollary. Let φ(x̄) be a formula. Then, modulo ACFA, φ(x̄) is equivalent to a disjunction of formulas of the form ∃y ψ(x̄, y), where ψ(x̄, y) is quantifier-free, and for every difference field K and (ā, b) in K satisfying ψ, we have that b is algebraic over (ā, σ(ā), …).

2.13. Exercise 3. Give a proof of 2.12. [Hint: First note that if E is a separably closed difference field, then qfdiag(E) ⊢ qfdiag(E^{alg}) modulo the theory of difference fields. Then, for every model K of ACFA, and tuple ā satisfying φ, find a formula ψ_{a} implying φ and of the required form. Use compactness to conclude.]

2.14. Definition of types and saturated models. Let M be an L-structure, A a subset of M, and ā an n-tuple of elements of M. The type of ā over A in M, denoted tp(ā/A) or sometimes tp_{M}(ā/A), is the set of L(A)-formulas φ(x̄) satisfied by ā in M, where x̄ is a fixed n-tuple of variables. A partial n-type over A is a set Γ(x̄) of L(A)-formulas in the variables x̄, such that every finite conjunction of elements of Γ(x̄) is satisfiable by some n-tuple of M. By compactness, a partial n-type over A will be realised in some elementary extension M^{∗} of M, i.e., M^{∗} will contain some tuple ā which satisfies all formulas of Γ(x̄).

If κ is an infinite cardinal, an L-structure M is κ-saturated if for every A ⊆ M with |A| < κ, every (partial) type over A is realised in M. The structure M is saturated iff it is |M|-saturated. Saturated models are quite useful, as they realise many types and have many automorphisms: if M is saturated and f : A → B is an elementary map between two subsets of M of size < |M|, then f extends to an automorphism of M.

2.15. Corollary. Let E be a difference subfield of a model K of ACFA, and let ā and b̄ be tuples in K of the same length. Then tp(ā/E) = tp(b̄/E) if and only if there is an E-isomorphism (of difference fields) acl_{σ}(Eā) → acl_{σ}(Eb̄) sending ā to b̄.

Proof. Extend the E-isomorphism acl_{σ}(Eā) → acl_{σ}(Eb̄) to an E-isomorphism φ : K → K_{1} (for some difference field K_{1}). Then tp_{K}(ā/E) = tp_{K_{1}}(b̄/E) (since φ(ā) = b̄). By Theorem 2.7, K_{1} ≡_{acl_{σ}(Eb̄)} K, which implies that tp(ā/E) = tp(b̄/E).

2.16. Corollary. Let K be a model of ACFA, and A a subset of K. Then acl_{σ}(A) equals the model-theoretic algebraic closure of A, acl(A).

Proof. Clearly acl_{σ}(A) ⊆ acl(A). Choose an acl_{σ}(A)-isomorphic copy K_{1} of K, which is linearly disjoint from K over acl_{σ}(A). As in 2.7, KK_{1} embeds in a model L of ACFA. By 2.15, if a ∈ K \ acl_{σ}(A), then there is b ∈ K_{1} \ acl_{σ}(A) realising the same type over acl_{σ}(A) as a. Hence no type realised in K \ acl_{σ}(A) is algebraic.

2.17. Saturated models of ACFA. Let K be a saturated model of ACFA. Then K has the following property: if E is an algebraically closed difference subfield of K, and F is a difference field extending E, and with |F | < |K|, then there is an E-embedding of F into E.

Thus saturated models of ACFA, of large enough cardinality, play the role of universal models of algebraic geometry.

§3. σ-closed sets, independence and SU-rank. We keep the notation and conventions introduced before. We work in a (sufficiently saturated) model K of ACFA.

3.1. σ-closed sets, and the topology it generates. We work in a (sufficiently saturated) model K of ACFA. In analogy with the Zariski topology, we define the σ-topology on K^{n}. Given B ⊂ K[X̄]_{σ}, X̄ = (X_{1}, …, X_{n}), we set

V (B) = {ā ∈ K^{n} | f(ā) = 0 for all f(X̄) ∈ B}.

Dually, if S ⊆ K^{n}, we define

I_{σ}(S) = {f(X̄) ∈ K[X̄]_{σ} | f(ā) = 0 for all ā ∈ S}.

If E is a difference subfield of K and ā ∈ K^{n}, we define

I_{σ}(ā/E) = {f(X̄) ∈ E[X̄]_{σ} | f(ā) = 0}.

We call the sets of the form V (B) the σ-closed subsets of K^{n}. One checks easily that V (I_{σ}(B)) = B. Some of the following results are theorems (see Cohn’s book): Facts and remarks. Note that I_{σ}(S) is an ideal I of K[X̄] with the following properties:

(i) f ∈ I ⟺ σ(f) ∈ I. (ii) If f^{m}σ(f)^{n} ∈ I, then f ∈ I. Ideals satisfying (i) are called reflexive σ-ideals (and σ will then induce an injective endomorphism on the quotient of K[X̄] by such an ideal). Ideals satisfying in addition condition (ii) are called perfect σ-ideals. Prime ideals satisfying (i) and (ii) are called prime σ-ideals. Perfect σ-ideals are the analogues of radical ideals, and are intersections of prime σ-ideals.

3.2. Fact. Even though K[X̄]_{σ} does not satisfy the a.c.c. on σ-ideals, it satisfies it on perfect σ-ideals, and this implies that the σ-topology on K^{n} is Noetherian.

3.3. Exercise 4. Show that if I is a prime σ-ideal then I_{σ}(V (I)) = I. [Warning: your proof should use the fact that K is a model of ACFA. You may use the fact that a field with a distinguished endomorphism embeds in an inversive difference field].

3.4. Definition. Let E be a difference subfield of K, and a ∈ K. We say that a is transformally transcendental over E if I_{σ}(a/E) = 0. Otherwise, we say that a is transformally algebraic over E. A tuple is transformally algebraic over E if all its elements are.

If a is transformally transcendental over E, then the elements σ(a), i ∈ ℕ, are algebraically independent over E. Hence, applying σ^{−1}, so are the elements σ(a), i ∈ ℤ. Thus, the difference field generated by a over E is isomorphic to E(X^{σ^{i}} | i ∈ ℤ) (with the obvious action of σ).

Similarly, one says that n-tuple ā is it transformally independent over E, if I_{σ}(ā/E) = (0). There are notions of transformal transcendental bases, transformal transcendental degree of an extension, etc.

Assume now that a is transformally algebraic over E, and let m be least such that some f(X) = F (X, X^{σ}, …, X^{σ^{m}}) ∈ I_{σ}(a/E). Choose such an f(X) of lowest degree when viewed as a polynomial in X^{σ^{m}}. Then F (a, …, σ^{m−1}(a), Y) is irreducible over E(a, …, σ^{m−1}(a)) because I_{σ}(a/E)∩E[X, …, X^{σ^{m}}] is prime, and is the minimal polynomial of σ^{m}(a) over E(a, …, σ^{m−1}(a)).

From F (a, …, σ^{m}(a)) = 0, we deduce that σ(F)(σ(a), …, σ^{m+1}(a)) = 0, so that the minimal polynomial of σ^{m+1}(a) over E(a, …, σ^{m}(a)) divides σ(F)(σ(a), …, σ^{m}(a), Y), and therefore has degree bounded above by the degree of F (a, …, σ^{m−1}(a), Y). It follows that I_{σ}(a/E), as a σ-ideal, is finitely generated, since from some point on, the degree of the minimal polynomial of σ^{n}(a) over E(a, …, σ^{n−1}(a)) must stabilize.

Note that we also have that

acl_{σ}(Ea) = E(a, …, σ^{m−1}(a))^{alg}.

Similarly, if ā is transformally algebraic over E, then there is an n such that acl_{σ}(Eā) = E(ā, …, σ^{n}(ā))^{alg}.

3.5. Exercise 5.

(1) Use the preceding remarks to show that every prime σ-ideal of E[X̄]_{σ}, is finitely generated (as a σ-ideal).

(2) (harder) Let I be a perfect σ-ideal of K[X̄], X̄ = (X_{1}, …, X_{n}), and assume that I ∩K[X_{i}]_{σ} ≠ 0 for all i = 1, …, n. Show that any descending sequence of σ-closed subsets F_{j} of V (I) is finite [Hint: Look at the Zariski closures of {(ā, σ(ā), …, σ^{m}(ā)) | ā ∈ F_{j}} for m large enough, use induction on the dimension of the algebraic sets considered.]

(3) Using (2), show that the σ-topology is Noetherian.

3.6. Definition. Let A, B, C be subsets of a model K of ACFA. We say that A and B are independent over C if the fields acl_{σ}(CA) and acl_{σ}(CB) are free (or equivalently, linearly disjoint) over acl_{σ}(C). We denote it by A | ⌣_{C}B.

Remarks. Independence is clearly a symmetrical notion, and is transitive, i.e.: let B′ ⊂ B. Then

A | ⌣_{C}B ⟺ A | ⌣_{C}B′ and A | ⌣_{C∪B′}B.

Note also that by definition

A | ⌣_{C}B ⟺ acl_{σ}(C, A) | ⌣_{acl_{σ}(C)}acl_{σ}(C, B)

and that

acl_{σ}(C, A) = (acl_{σ}(C)acl_{σ}(A))^{alg}.

Assume for simplicity, that C ⊆ B are algebraically closed difference fields. Then A and B are independent over C if and only if, for every tuple ā ∈ A, the ideal I_{σ}(ā/B) is generated by its intersection with C[X̄]_{σ}.

Moreover, independence satisfies the extension property: given A, B and C, there is A′ realising tp(A/C) in some elementary extension of K such that A′ and B are independent over C. Indeed, without loss of generality, we may assume that C, A and B are algebraically closed difference fields, with C ⊂ A ∩ B. Let A′ be a C-isomorphic copy of A which is free from B over C. Then A′ is linearly disjoint from B over C, reasoning as in Step 2 of Theorem 2.7, we get that there is an elementary extension L of K which contains the difference fields A′ and B. Then A′ and B are independent over C, and tp_{L}(A′/C) = tp_{L}(A/C).

3.7. Exercise 6. Let K be a model of ACFA, and A, B, C, D algebraically closed difference subfields of K. Give a proof, or convince yourself, of the following facts:

(1) (Symmetry) If A | ⌣_{C}B then B | ⌣_{C}A.

(2) (Transitivity). If C ⊆ A ∩ B and B ⊆ D,

A | ⌣_{C}D ⟺ A | ⌣_{C}B and A | ⌣_{B}D.

(3) A | ⌣_{C}B if and only if for every finite tuple b̄ from B, A | ⌣_{C}b̄.

(4) (Extension) In some elementary extension of K there is A′ realising tp(A/C) with A′ | ⌣_{C}B. (This was already explained above)

(5) There is a finite subset E of C such that A | ⌣_{E}C.

3.8. The independence theorem. Let K be a model of ACFA (sufficiently saturated), let E = acl_{σ}(E) ⊆ K, and ā, b̄, c̄_{1} and c̄_{2} tuples in K, such that (i) tp(c̄_{1}/E) = tp(c̄_{2}/E), (ii) ā and c̄_{1} are independent over E, ā and b̄ are independent over E and b̄

and c̄_{2} are independent over E. Then there is c̄ realising tp(c̄_{1}/E ∪ā)∪tp(c̄_{2}/E ∪b̄), independent from (ā, b̄) over E. Proof. Let c̄ realise tp(c̄_{1}/E), independent from (ā, b̄) over E. Let A = acl_{σ}(Eā), B = acl_{σ}(Eb̄), C = acl_{σ}(Ec̄), and fix E-isomorphisms (of difference fields) φ_{1} : acl_{σ}(Ec̄_{1}) → C and φ_{2} : acl_{σ}(Ec̄_{2}) → C, with φ_{i}(c̄_{i}) = c̄.

Let σ_{0} be the restriction of σ to (AB)^{alg}C. Since A is linearly disjoint from acl_{σ}(E, c̄_{1}) and from C over E, we may extend φ_{1} to a field-isomorphism ψ_{1} : acl_{σ}(Ac̄_{1}) → (AC)^{alg}, which is the identity on A. Then σ_{1} = ψ_{1}σψ_{1}^{−1} is an automorphism of (AC)^{alg} which agrees with σ on A and on C. Indeed, σ_{1} agrees with σ on A because ψ_{1} is the identity on A, and on C because ψ_{1} extends the difference field isomorphism φ_{1} : acl_{σ}(Ec̄_{1}) → C. Note also that by definition of σ_{1}, the isomorphism ψ_{1} is an isomorphism between the difference field (acl_{σ}(Ac̄_{1}), σ) and the difference field ((AC)^{alg}, σ_{1})

Similarly, we may extend φ_{2} to a field-isomorphism ψ_{2} : acl_{σ}(B, c̄_{2}) → (BC)^{alg} which is the identity on B. The automorphism σ_{2} = ψ_{2}σψ_{2}^{−1} agrees with σ on B and C.

Assume that there is an automorphism τ of L = (AB)^{alg}(AC)^{alg}(BC)^{alg} which extends σ_{0}, σ_{1} and σ_{2}. Then we can find some model M of ACFA containing (L, τ). As τ extends σ_{0}, we then have that tp_{M}(AB/E) = tp_{K}(AB/E) (by 2.7). By 2.15, we also have

tp_{M}(c̄/A) = tp_{K}(c̄_{1}/A), and tp_{M}(c̄/B) = tp_{K}(c̄_{2}/B)

because τ extends σ_{1} and σ_{2}, and the ψ_{i} are difference fields isomorphisms fixing A and B respectively. Clearly, c̄ is independent from (A, B) over E, and this will have finished the proof.

It remains to show that there is such an automorphism τ of L. To do that, it suffices to show that σ_{0} and σ_{1} have a common (and necessarily unique) extension τ_{1} to (AB)^{alg}(AC)^{alg}, and that τ_{1} and σ_{2} have a common extension to L.

To show that σ_{0} and σ_{1} have a common extension τ_{1} to (AB)^{alg}(AC)^{alg}, it is enough to show that their domains are linearly disjoint over their intersection, and that σ_{0} and σ_{1} agree on this intersection. Similarly for τ_{1} and σ_{2}.

The domain of σ_{0} is (AB)^{alg}C, the domain of σ_{1} is (AC)^{alg}, which is a Galois extension of AC. By definition, σ_{0} is the restriction of σ (∈ Aut(K)), and we know that σ_{1} and σ agree on AC. It follows that it suffices to show that

(AB)^{alg}C ∩ (AC)^{alg} = AC.  (1)

(Here we are using the fact that (AC)^{alg} is a Galois extension of AC to reduce the linear disjointness over AC to intersecting in AC). Similarly, to show that τ_{1} and σ_{2} have a common extension to L, it will be enough to show that

(AB)^{alg}(AC)^{alg} ∩ (BC)^{alg} = BC.  (2)

Unfortunately the proof of either of these equations uses tools slightly beyond the scope of this course, since I had chosen not to assume anything known in stability theory. For sake of completeness I will give the proof using stability results. The reader unfamiliar with stable theories may just skip this part and admit the equations (1) and (2), and therefore the result. Let us prove (2) first. Algebraists would prove it via specialisations, the ideas are essentially the same (but not the way it is said).

Let α ∈ (BC)^{alg} ∩ (AB)^{alg}(AC)^{alg}, and write α = ∑_{i=1}^{n} β_{i}γ_{i}, where β_{i} ∈ (AB)^{alg}, γ_{i} ∈ (AC)^{alg}. Let ā′, b̄′ and c̄′ be tuple of elements from A, B and C respectively, and f_{i}(X̄, Ȳ , U), g_{i}(X̄, Z̄, V) be polynomials over E, such that f_{i}(ā′, b̄, U) is the minimal polynomial of β_{i} over AB, and g_{i}(ā′, c̄′, V) is the minimal polynomial of γ_{i} over AC for i = 1, …, n. Then

K ⊧ ∃u_{1}, …, u_{n}, v_{1}, …, v_{n} ⋀_{i}(f_{i}(ā′, b̄′, u_{i}) = 0∧g_{i}(ā′, c̄′, v_{i}) = 0)∧α = ∑_{i} u_{i}v_{i}.

Note that this is a formula of the field language, satisfied by (ā′, b̄′, c̄′, α). By assumption, ā′ is independent from (BC)^{alg} over E, in the sense of the theory of algebraically closed fields. Hence every formula represented in tp_{ACF} (b̄′, c̄′, α/A) is already represented in tp_{ACF} (b̄′, c̄′, α/E) (here tp_{ACF} denotes the type in the theory of algebraically closed fields). This precisely means that there is a tuple ē in E such that

K ⊧ ∃u_{1}, …, u_{n}, v_{1}, …, v_{n} ⋀_{i}(f_{i}(ē, b̄′, u_{i}) = 0 ∧ g_{i}(ē, c̄′, v_{i}) = 0) ∧ α = ∑_{i} u_{i}v_{i}.

If β′_{1}, …, β′_{n}, γ′_{1}, …, γ′_{n} satisfy ⋀_{i}(f_{i}(ē, b̄′, β′_{i}) = 0 ∧ g_{i}(ē, c̄′, γ′_{i}) = 0) ∧ α = ∑_{i} β′_{i}γ′_{i}, then β′_{i} ∈ B and γ′_{i} ∈ C, which shows that α ∈ BC, and proves (2). Permuting A, B, C, we obtain that (AC)^{alg} ∩(AB)^{alg}(BC)^{alg} = AC, from which we get (1).

3.9. □□□ Corollary. All completions of ACFA are supersimple, and independence corresponds to non-forking. Proof. See the paper by Kim and Pillay [15]. They show that if you have an independence notion, which is

(i) symmetric, (ii) transitive, (iii) has the extension property, (iv) is such that, given a finite tuple ā and a set A, there is A_{0} ⊆ acl^{eq}(A) such

that ā and A are independent over A_{0}, and |A_{0}| ≤ |L| + ℵ_{0},

(v) satisfies the independence theorem.

Then this notion of independence coincides with non-forking, and the theory is simple. If in (iv), the set A_{0} can always be taken finite, then the theory is supersimple.

In our case, given the complete theory of a model K of ACFA, items (i) – (iii) are immediate because similar statements hold for non-forking in algebraically closed fields, (v) is 3.8. From our definition of independence and the descending chain condition on σ-closed sets (3.2), it follows that in (iv) we can always take A_{0} to be finite. □□□

3.10. Definitions of forking and of the SU-rank. The SU-rank is a rank on types, based on non-independence in the same way the U-rank is. Let K be a model of ACFA, sufficiently saturated, E ⊆ F algebraically closed subsets of K, and ā a tuple of elements, p = tp(ā/E), q = tp(ā/F). We say that q forks over E, or that q is a forking extension of p, iff ā | ⌣ / _{E}F. Otherwise, we say that q does not fork over E, or that q is a non-forking extension of p to F. We define SU(p) = SU(ā/E) ≥ α by induction on the ordinal α:

— SU(p) ≥ 0

— SU(p) ≥ α + 1 iff p has a forking extension q such that SU(q) ≥ α, iff there is B = acl_{σ}(B) containing E such that acl_{σ}(Eā) and B are not linearly disjoint over acl_{σ}(E), and SU(ā/B) ≥ α.

— If α is a limit ordinal, then SU(p) ≥ α iff SU(p) ≥ β for all β < α.

We then define SU(p) to be the least α such that SU(p) ≱ α + 1 if it exists, and ∞ otherwise.

3.11. Exercise 7. Let ā, E ⊆ F, K be as above. Give a proof, or convince yourself of the following facts:

(1) SU(ā/F) ≤ SU(ā/E). [Hint: show by induction on α that SU(ā/F) ≥ α implies SU(ā/E) ≥ α.]

(2) If ā | ⌣_{E}F, then SU(ā/E) = SU(ā/F). [Again, use induction on α. The independence theorem intervenes in the proof.]

(3) Deduce from Remark 3.6 and the noetherianity of the σ-topology that SU(ā/E) < ∞.

3.12. Remark. As explained in the exercise, Fact 3.2 yields that the SUrank of a type (of a finite tuple) exists. It is however much stronger: You could imagine that there could exist an infinite descending sequence of σ-closed sets all defined over acl_{σ}(∅).

3.13. Natural sum on ordinals. Every ordinal can be written uniquely as

α = ω^{α_{1}}a_{1} + ⋯ + ω^{α_{n}}a_{n},

where α_{1} > ⋯ > α_{n} ≥ 0 are ordinals, and a_{1}, …, a_{n} are positive integers. If β = ω^{β_{1}}b_{1} + ⋯ + ω^{β_{m}}b_{m}, then we define α ⊕ β as follows. First, relaxing the condition on a_{1}, …, a_{n}, b_{1}, …, b_{m} to be positive and allowing them to be 0, we may assume that m = n and α_{i} = β_{i} for i = 1, …, n. Then one sets

α ⊕ β = ω^{α_{1}}(a_{1} + b_{1}) + ⋯ + ω^{α_{n}}(a_{n} + b_{n}).

One verifies that ⊕ is commutative and transitive.

While ⊕ coincides with the usual ordinal addition on finite ordinals, it definitely does not on infinite ones. For instance 1 ⊕ ω = ω ⊕ 1 = ω + 1, but 1 + ω = ω.

3.14. Properties of the SU-rank. One can show that the SU-rank satisfies the so-called Lascar inequality: given another tuple b̄,

SU(ā/Eb̄) + SU(b̄/E) ≤ SU(ā, b̄/E) ≤ SU(ā/Ab̄) ⊕ SU(b̄/A).

This is shown by induction. For the first inequality, one shows that SU(b̄/E) ≥ α implies SU(ā/Eb̄)+α ≥ SU(ā, b̄/E). For the second, that SU(ā, b̄/E) ≥ α implies SU(ā/Eb̄) ⊕ SU(ā/E) ≥ α. The proof is not difficult, but one can also consult e.g. Wagner’s book on simple theories [19].

3.15. SU-rank of definable sets, or of formulas. Let S ⊆ K^{n} be a definable set, defined by a formula φ(x̄) over some E, and assume that K is sufficiently saturated (otherwise our definition will not make sense). We define SU(φ) = SU(S) = sup{SU(ā/A) | ā ∈ S}. One can show that this sup is attained, i.e., that there is some ā satisfying φ and such that SU(ā/A) = SU(φ).

3.16. Remarks. Note the following special cases:

SU(ā/E) = 0 ⟺ ā ∈ acl_{σ}(E)
SU(ā/E) = 1 ⟺ ā ∉ acl_{σ}(E) and for all F ⊃ E,

either ā is independent from F over E, or ā ∈ acl_{σ}(F).

To simplify, let us assume that E is an algebraically closed difference field (since SU(ā/acl_{σ}(E)) = SU(ā/E).) Let us also assume for the moment, that the tuple ā is transformally algebraic over E, that is, that tr.deg(cl_{σ}(Eā)/E) = n is finite. Then, tp(ā/F) forks over E if and only if

tr.deg(cl_{σ}(F ā)/F) < tr.deg(cl_{σ}(Eā)/E).

This implies that SU(ā/E) ≤ n. Equality however does not always hold. For instance, one can show that any non-realised type containing the formula σ^{2}(x) = x^{2}, or the formula σ^{2}(x) = x^{2} + 1, has SU-rank 1.

3.17. The type of SU-rank ω. Let E = acl_{σ}(E), and consider an element a ∈ K which is transformally transcendental over E, that is, the elements σ^{i}(a), i ∈ ℤ, are algebraically independent over E. Let b_{0} = a, and b_{n} = σ(b_{n−1}) − b_{n−1} for n ≥ 1. Then, letting L_{n} = cl_{σ}(Eb_{n}), we get that each L_{n} contains L_{n+1}, and has transcendence degree 1 over L_{n+1}. Hence, SU(b_{n}/Eb_{n+1}) = 1 (since it is not 0, and is at most 1). Hence, SU(a/Eb_{n}) = n, which implies that SU(a/E) ≥ ω. On the other hand, any forking extension of tp(a/E) has finite SU-rank (since if a is not independent from F over E, then a satisfies some σ-equation, i.e., tr.deg(cl_{σ}(F a)/F) < ∞). Hence, SU(a/E) = ω. Note that

ω = SU(a, b_{1}/E) = SU(a/Eb_{1}) + SU(b_{1}/E) < SU(a/Eb_{1}) ⊕ SU(b_{1}/E) = ω + 1.

3.18. Other examples. Let V be a variety of dimension d defined over some E = acl_{σ}(E) ⊆ K. Assume that if ā = (a_{1}, …, a_{n}) is a tuple of V (K) which is generic over E, then the elements a_{1}, …, a_{d} are algebraically independent over E. Consider the type p_{V} (x̄) over E which says that x_{1}, …, x_{d} are transformally independent over E (i.e., satisfy non non-trivial difference equation over E), and that (x_{1}, …, x_{n}) ∈ V. According to the additivity rule (one can also see it directly), if ā realises p_{V}, then SU(ā/E) = ω_{d}. The type p_{V} (x̄) is in a sense a generic type of the variety V. One can show that this type is complete.

Finally, one shows easily that if ā is any tuple of K, then SU(ā/E) is of the form ωn + m for some non-negative integers n, m.

3.19. Definition of imaginaries, etc. Let M be a model of a complete theory T in a language L. We assume M to be sufficiently saturated. Let S ⊆ M^{n} be a 0-definable set, and let E ⊆ S^{2} be a 0-definable equivalence relation. Then each E-equivalence class ā/E, with ā ∈ S, is called an imaginary element.

Given S and E as above, the set S/E is interpretable in M. To each such pair, we associate a new sort, and let M^{eq} be the many-sorted structure with sorts indexed by the pairs (S, E) as above, the “real sort” being the L-structure M, and the sort indexed by (S, E) being the set S/E; there is also a projection map S → S/E for each (S, E). The structure M^{eq} is then interpretable in M. The difference between many-sorted logic and 1-sorted logic, is that all quantifiers, variables and constants have a sort attached to them, so that you will have things like ∀x ∈ M, ∀y ∈ S/E. Note also that M^{n} becomes a sort, so that an n-tuple of M can be thought of as an element of M^{eq}.

An important example of imaginary is the following. Let D ⊆ M^{k} be a definable set, defined by some formula φ(x̄, b̄), where φ(x̄, ȳ) ∈ L and b̄ is an n-tuple from M. Define an equivalence relation E on M^{n} by:

E(ȳ, z̄) : ∀x̄ (φ(x̄, ȳ) ↔ φ(x̄, z̄)).

Then the equivalence class b̄/E has the following property: for all ρ ∈ Aut(M), ρ(D) = D if and only if ρ fixes b̄/E.

The imaginary element b̄/E will be called a code for D. We say that T eliminates imaginaries if whenever D ⊆ M^{k} is definable (with parameters), then there is tuple d̄ in M, such that for any ρ ∈ Aut(M),

ρ(D) = D ⟺ ρ fixes the elements of the tuple d̄.

If the language L has at least two terms t_{0}, t_{1} and T ⊧ t_{0} ≠ t_{1}, then T eliminates imaginaries if and only if whenever S ⊆ M^{n} is 0-definable, and E ⊆ S^{2} is a 0definable equivalence relation, then there is a 0-definable function f : S → M^{k} for some k, such that for every ȳ, z̄ ∈ S, we have

M ⊧ f(ȳ) = f(z̄) ⟺ E(ȳ, z̄).

3.20. Some facts. (1) If T eliminates imaginaries, and A ⊂ M, then T (A) eliminates imaginaries as well (T (A) is the set of sentences in the language L(A) obtained by adjoining to L a constant symbol for each element of A, which hold in the L(A)-structure (M, a)_{a∈A}.)

(2) □□□ Assume that M is stable. Working in M^{eq} instead of M, one has the following: if a is independent from b over cd and from c over bd, then a is independent from (bc) over acl^{eq}(bd) ∩ acl^{eq}(cd). Note that a tuple of elements of M can be thought of as an imaginary element. The proof uses the following ingredients. By elimination of imaginaries and stability, we know that every type p over an algebraically closed set E is stationary (i.e., has a unique non-forking extension to any superset of E), and is definable over E, i.e., given a formula φ(x̄, ȳ) ∈ L, there is a formula d_{φ}(ȳ) ∈ L(E) such that for every tuple b̄ in E, we have that

φ(x̄, b̄) ∈ p ⟺ M ⊧ f_{φ}(b̄).

The non-forking extension p′ of p to a set F containing E will then be defined analogously: for every b̄ ∈ F, φ(x̄, b̄) ∈ p′ ⟺ M ⊧ d_{φ}(b̄). One defines the canonical base of p, denoted by Cb(p), to be the set of codes of the formulas d_{φ}(ȳ) (and by elimination of imaginaries, this is a subset of E). By definition we have that p does not fork over Cb(p), and that the restriction of p to Cb(p) is stationary. Moreover, if E_{0} ⊂ E is such that p does not fork over E_{0}, then acl(E_{0}) contains Cb(p).

Hence, we look at the canonical base of tp(a/acl(bcd)) and from the independence relations, deduce that it is contained in acl(bd) ∩ acl(cd).

We will use the fact that every completion of the theory ACF of algebraically closed fields eliminates imaginaries and is stable. Note that the above proof extends to the case of M simple with stable forking (for those of you who know what this means). □□□

(2’) Let us rephrase (2) in a more algebraic language. We work within a sufficiently large algebraically closed field Ω. It is known that any algebraic set V defined over Ω has a smallest field of definition. If one looks at what it means in terms of independence, it translates as follows: let E and F be algebraically closed subfields of Ω, let ā be a tuple of elements of Ω, and assume that

ā | ⌣_{E}F and ā | ⌣_{F} E.

This means that if V is the locus of ā over (EF)^{alg}, then V is defined over E and V is defined over F. Indeed, let V_{1} be the locus of ā over E. As E ⊂ EF, we certainly have V_{1} ⊇ V. From the fact that ā is free from F over E, we know that the dimensions of V_{1} and V (which equal respectively tr.deg(ā/E) and tr.deg(ā/EF)) are equal. As V_{1} is irreducible, this implies that V = V_{1}, so that V is defined over E. Reasoning similarly with the locus of ā over F, one obtains that V is defined over F.

The uniqueness of the smallest field of definition of V then implies that V is defined over E ∩ F. Hence, all equations over EF satisfied by ā are implied by equations over E ∩ F. This implies that tr.deg(ā/E ∩ F) = dim(V), so that

ā | ⌣_{E∩F} EF.

(3) In the case of fields, any finite set has a code. Let us show how it works for elements of M: we want to code the definable set {a_{1}, …, a_{n}}. Consider the polynomial f(X) = ∏_{i=1}^{n}(X − a_{i}), and let b_{1}, …, b_{n} be its coefficients. Then any permutation of {a_{1}, …, a_{n}} fixes b_{1}, …, b_{n}, and conversely. Hence the tuple (b_{1}, …, b_{n}) is a code for the set {a_{1}, …, a_{n}}. There is a similar trick for finite sets of tuples.

(4) The uniqueness of the field of definition of algebraic sets also gives easily that any completion of the theory of algebraically closed fields eliminates imaginaries. Indeed, let K be an algebraically closed field. By elimination of quantifiers, we know that any definable subset D of K^{n} is a Boolean combination of algebraic sets. Then D can be written V \W, where V is the Zariski closure of D, and W is some definable subset of V, with the property that every irreducible component of its Zariski closure W̄ is strictly contained in some irreducible component of V. In particular, dim(W̄) < dim(V). Clearly any automorphism ρ of K which leaves D invariant will also leave V and W invariant. The automorphism ρ leaves V invariant if and only if it fixes the field of definition of V. The result follows by induction on the dimension.

This type of proof generalises to other theories of fields which eliminate quantifiers: the theory of differentially closed fields of characteristic 0, and also the theory of separably closed fields of finite degree of imperfection (in that case, one needs however to add some constant symbols to the language).

3.21. Theorem Any completion of ACFA eliminates imaginaries. Sketch of the proof. □□□ We work in a saturated model K of ACFA. We are given a 0-definable function f, and a tuple ā, and we look at the equivalence class e of ā for the equivalence relation E(x̄, ȳ) ⟺ f(x̄) = f(ȳ). We want to show that there is a real tuple which is equi-definable with e.

Let E = acl^{eq}(e) ∩ K. If e is definable over E, then we are done: choose a tuple b̄ ∈ E over which e is definable. Then b̄ ∈ acl^{eq}(e). Since we are in a field, there is a tuple c̄ which codes the finite set of conjugates of b̄ over e. Then c̄ and e are equi-definable. Hence, we may assume that e is not definable over E, and in particular that ā ∉ E. Our aim is to show that there is a tuple b̄ realising tp(ā/e) which is independent from ā over E.

Let p = tp(ā/e). Since p is non-algebraic, there is b̄ realising p and such that acl^{eq}(eā) ∩ acl^{eq}(eb̄) = acl^{eq}(e), and therefore

acl_{σ}(Eā) ∩ acl_{σ}(Eb̄) = E.  (∗)

Choose b̄ realising p, satisfying (∗) and of maximal SU-rank over Eā (note that f(b̄) = e). Let c̄ realise tp(b̄/Eā), independent from b̄ over Eā. Then c realises p, as e ∈ dcl^{eq}(ā). Moreover, acl_{σ}(Ec̄) ∩ acl_{σ}(Eb̄) ⊆ acl_{σ}(Eb̄) ∩ acl_{σ}(Eā) (because acl_{σ}(Eāb̄) ∩ acl_{σ}(Eāc̄) = acl_{σ}(Eā)), and therefore the tuple (b̄, c̄) also satisfies (∗). By maximality of the SU-rank of b̄ over Eā and because tp(a/e) = tp(b/e), we get that

SU(c̄/Eb̄) ≤ SU(b̄/Eā).

We also know that

SU(c̄/Eāb̄) = SU(c̄/Eā) = SU(b̄/Eā).

As SU(c̄/Eāb̄) ≤ SU(c̄/Eb̄), we obtain that

SU(c/Eāb̄) = SU(b̄/Eā) = SU(c̄/Eb̄),

so that cl_{σ}(c̄) is independent from acl_{σ}(Eāb̄) over acl_{σ}(Eā) and over acl_{σ}(Eb̄) (in the sense of the theory of algebraically closed fields). By elimination of imaginaries of the theory of algebraically closed fields, this implies that cl_{σ}(c̄) is (ACF-)independent from cl_{σ}(ā, b̄) over acl_{σ}(Eā) ∩ acl_{σ}(Eb̄) = E, i.e., that c̄ is (ACFA-) independent from (ā, b̄) over E. Hence SU(b̄/Eā) = SU(c̄/Eā) = SU(c̄/E), so that ā and b̄ were independent over E.

Hence, we have shown that there is a realisation b̄ of p, which is independent from ā over E. Since e is not definable over E, there is ā′ realising tp(ā/E) and with f(ā′) ≠ f(ā). Since tp(ā′/E) = tp(ā/E), there is c̄′ realising tp(ā′/E), with f(ā′) = f(c̄′), and independent from ā′ over E and we may assume that this c̄′ is also independent from b̄ over E. Apply the independence theorem to tp(ā/Eb̄) ∪ tp(ā′/Ec̄′) to derive a contradiction. □□□

3.22. Corollary. Let ā, b̄, c̄, d̄ be tuples in K, and assume that ā is independent from b̄ over (c̄d̄) and from c̄ over (b̄d̄). Then ā is independent from (b̄c̄) over acl_{σ}(b̄c̄) ∩ acl_{σ}(b̄d̄).

Proof. Either use the remark about simple theories with stable forking given in 3.20(2), or equivalently, the fact that independence in models of ACFA corresponds to independence of algebraically closed sets for the theory of algebraically closed fields. One uses also the fact that acl_{σ}(AB) = (acl_{σ}(A)acl_{σ}(B))^{alg}.

3.23. A useful remark. Let K be a model of ACFA, E an algebraically closed subset of K and ā a tuple of elements of K. As in the case of algebraically closed fields, one can show that I_{σ}(ā/E) has a smallest field of definition (as a σ-ideal). Or more simply, this follows from the elimination of imaginaries. So, let us suppose that E is the algebraic closure of this smallest field of definition, so that in particular it is the algebraic closure of a finite tuple, and therefore is ranked by the SU-rank.

Claim. For n sufficiently large, if ā_{0}, …, ā_{n} are independent realisations of tp(ā/E), then E ⊆ acl_{σ}(ā_{0}, …, ā_{n}).

Proof. We assume K sufficiently saturated. Construct by induction on n ∈ ℕ a sequence ā_{n}, n ∈ ℕ, of realisations of tp(ā/E), with ā_{n} | ⌣_{E}ā_{0}, …, ā_{n−1} for every n. Then

SU(E/ā_{0}, …, ā_{n}) ≤ SU(E/ā_{0}, …, ā_{n−1})

for every n, so that there is some index m such that SU(E/ā_{0}, …, ā_{m+1}) = SU(E/ā_{0}, …, ā_{m}). Take the smallest such m. Then

E | ⌣_{ā_{0},... ,ā_{m}} ā_{m+1},

and by assumption

ā_{m+1} | ⌣_{E}ā_{0}, …, ā_{m}.

Corollary 3.22 then gives

ā_{m+1} | ⌣_{E∩acl_{σ}(ā_{0},... ,ā_{m})}E, ā_{0}, …, ā_{m},

and therefore E ⊆ acl_{σ}(ā_{0}, …, ā_{m}).

3.24. Exercise 8. Show that the formula σ^{2}(x) = x^{2} has SU-rank 1. I.e., you need to show that given any algebraically closed difference field E and element a satisfying σ^{2}(x) = x^{2}, one cannot have tr.deg(cl_{σ}(Ea)/E) = 1.

§4. Study of the fixed field. A particularly important definable subset of a model K of ACFA is the fixed field, F ix(σ) = {x ∈ K | σ(x) = x}. It turns out that this subfield is responsible for much of the bad behaviour of models of ACFA. We will show that F ix(σ) is a pseudo-finite field, i.e., is elementary equivalent to an ultraproduct of finite fields (or equivalently, is an infinite model of the theory of finite fields). Pseudo-finite fields were first studied by Ax, see [9]. Some very nice results on pseudo-finite fields were also obtained in two papers by E. Hrushovski and A. Pillay ([11] and [12])

4.1. Theorem. Let K be a model of ACFA, and F = F ix(σ). Then F is a pseudo-finite field. Proof. We need to show that

(i) F is perfect (i.e., if char(F) = p > 0, then every element is a p-th power). (ii) Gal(F^{alg}/F) ≃ ℤ̂ ≃ lim_{←} ℤ/nℤ ≃ ∏_{p prime} ℤ_{p}. (iii) F is pseudo-algebraically closed (PAC), i.e., every variety defined over F

has a point with coordinates in F. Item (i) is no problem: if the characteristic is p > 0, then every element a has a unique p-th root, denoted by a^{1/p}. Hence σ(a) = a implies σ(a^{1/p}) = a^{1/p}, and F is perfect. Item (iii) is no problem either: let U be a variety defined over F, and consider the diagonal subvariety V ⊆ U × U, i.e., V is defined by x̄ ∈ U, ȳ ∈ U, and x̄ = ȳ. Then U = U^{σ}, and U, V satisfy the hypotheses of axiom (3) of ACFA, so that there is ā ∈ K with (ā, σ(ā)) ∈ V, i.e., ā ∈ U and σ(ā) = ā.

Let us now look at item (ii). First of all, if L is a finite Galois extension of F, then σ(L) = L: if α generates L over F, then σ fixes the coefficients of the minimal monic polynomial of α over F, so that σ(α) ∈ L. Hence σ restricts to an element of Gal(L/F). By Galois theory, F = F ix(σ) is the subfield of L fixed by the group generated by σ|_{L}, and this shows that Gal(L/F) is cyclic, generated by σ|_{L}.

We will now show that for every n, F has at most one Galois extension of degree n. Given a finite Galois extension L of F, there is a 1-1 correspondence between algebraic extensions of F contained in L, and subgroups of Gal(L/F), under which the extensions which are Galois over F correspond to the normal subgroups of Gal(L/F). Every finite algebraic extension of F is contained in a finite Galois extension of F, and all subgroups of a cyclic group are normal. Hence every algebraic extension of F is Galois.

Assume now by way of contradiction that F has two Galois extensions, L and M, of the same degree n over F. Consider Gal(LM/F). Then Gal(LM/L) and Gal(LM/M) are subgroups of Gal(LM/F) of order d = [LM : F]/n. But Gal(LM/F) is cyclic, and therefore has only one subgroup of order d: this implies that L = M.

To show (ii), it will be enough to show that for every n, F has at least one Galois extension of degree n. But this is easy: consider the difference field extension L = K(X_{1}, …, X_{n}) of K (X_{1}, …, X_{n} indeterminates), with σ(X_{i}) = X_{i+1} for i = 1, …, n − 1, and σ(X_{n}) = X_{1}. Then

L ⊧ ∃x σ^{n}(x) = x ∧ ⋀_{1≤i<n} σ^{i}(x) ≠ x,

so that K satisfies the same sentence. Let a ∈ K be such that σ^{n}(a) = a, σ^{i}(a) ≠ a for 1 ≤ i < n. By Galois theory, a generates a Galois extension of degree n of F.

4.2. An example We saw earlier that if A is an algebraically closed difference field, then ACFA ∪ qftp(A) is complete. Hence, in a sense, ACFA is close to having quantifier-elimination. However, it does not. Here is an example:

Let a, b ∈ F ix(σ) be transcendental elements, and assume that a does not have a square root in F ix(σ) (so, we assume that char(F ix(σ)) ≠ 2). Then we have:

K ⊧ ∀y y^{2} = a → σ(y) ≠ y
K ⊧ ∃y y^{2} = b^{2} ∧ σ(y) = y.

However, a and b^{2} have the same quantifier-free type, as the quantifier-free type of a transcendental element of the fixed field is unique: it simply says that the element satisfies no non-trivial equation (over ℚ or over 𝔽_{p}).

4.3. Proposition. Let K ⊧ ACFA, and consider F = F ix(σ) = {x ∈ K | σ(x) = x}. Every definable subset of F^{n} is definable in the pure field F. In other words, the structure on F induced by K is the one of the pure field F. Proof. We assume K sufficiently saturated. Let S ⊂ F^{n} be definable (in K). Then σ(S) = S. By elimination of imaginaries, this implies that there is a tuple c̄ of elements of F, and a formula φ(x̄, ȳ) such that φ(x̄, c̄) defines S. Hence, we have shown that S is definable over F. To finish the proof, it suffices to show that there is a “small” subset C of F such that every field-automorphism of F which fixes C is an elementary map within the structure K (exercise).

Let C be a countable elementary substructure of the field F. Then C^{alg} and F are linearly disjoint over C because C ≺ F. If C_{n} is the unique algebraic extension of C of degree n, then F C_{n} is an algebraic extension of F of degree n also, and therefore is the unique algebraic extension of F of degree n. Hence F^{alg} = F C^{alg}.

Let ρ be a field-automorphism of F which is the identity on C. Because C^{alg} and F are linearly disjoint over C, we can extend ρ to a field-automorphism ρ̃ of F C^{alg} which is the identity on C^{alg}. Since F^{alg} = F C^{alg}, we get that ρ̃ is defined on F^{alg} = acl_{σ}(F). We have

σρ̃|_{F} = ρ̃σ|_{F}

because σ|_{F} = id_{F}, and

σρ̃|_{C^{alg}} = ρ̃σ|_{C^{alg}}

because ρ|_{C^{alg}} = id_{C^{alg}}. Hence, ρ̃ commutes with σ on C^{alg} and on F, so that ρ̃ is an automorphism of the difference field acl_{σ}(F). By 2.15, ρ̃ is an elementary map.

4.4. Exercise 9. Let K be a model of ACFA, F its fixed field.

(1) Let E be a subfield of F such that E^{alg}F = F^{alg}, and ā a tuple of elements of F. Show that SU(ā/E) = tr.deg(E(ā)/E).

(2) Let S be a definable subset of F^{n}. Show that SU(S) equals the algebraic dimension of the Zariski closure (in K) of S.

§5. Orthogonality and modularity. In this section we will show that we can reduce the study of types of finite SU-rank to the study of types of SU-rank 1. We will always be working in a model K of ACFA, which we will assume to be sufficiently saturated.

5.1. Definitions. Let A and B subsets of K, and p, q types over A and B respectively.

(1) If A = B, then we say that p and q are almost orthogonal (denoted by p ⊥^{a} q) if whenever ā realises p and b̄ realises q, then ā and b̄ are independent over A.

(2) We say that p and q are orthogonal (denoted by p ⊥ q) if whenever C contains A ∪ B, and ā is a realisation of p which is independent from C over A, b̄ is a realisation of q which is independent from C over B, then ā and b̄ are independent over C.

(3) Recall that S ⊆ K^{n} is ∞-definable over E, or type-definable over E, if there is a partial n-type Φ over E (i.e., a consistent set of L(E)-formulas in the variables (x_{1}, …, x_{n})) such that S is the set of n-tuples from K satisfying Φ.

(4) Let S be a set which is (∞-) definable over B. We say that p is orthogonal to S (denoted by p ⊥ S, or by p ⊥ φ if S is defined by φ) if p is orthogonal to all types over supersets of B which are realised in S, i.e.: for all C containing A ∪ B, and ā realising p and independent from C over A, and b̄ ∈ S, we have that ā and b̄ are independent over C. One also says that p is foreign to S or to φ.

5.2. Remark and example. Note that in (4) above, we do not require that b̄ be independent from C over B. Here is an example which shows that orthogonality to a set and to a type are different.

Let E = acl_{σ}(E) ⊆ K, and let p be the type of a transformally transcendental element a over E, let P be the set of realisations of p over E.

Then p is orthogonal to all types of finite SU-rank over E: indeed, assume that a is transformally transcendental over F = acl_{σ}(F) ⊃ E, and that SU(b/F) < ω, but a and b are not independent over F. This means that I_{σ}(a/acl_{σ}(F b)) is non-zero, and therefore that a is transformally algebraic over acl_{σ}(F b), i.e., SU(a/F b) < ω. But this contradicts Lascar’s inequality: on the one hand, we have SU(a, b/F) ≥ SU(a/F) = ω, and on the other, we have SU(a, b/F) ≤ SU(a/F b) ⊕ SU(b/F) < ω.

However, let c = σ(a)−a, let a′ be a realisation of tp(a/acl_{σ}(Ec)), independent from a over Ec, and let b = a − a′. Then σ(b) = σ(a) − σ(a′) = (a + c) − (a′ + c) = a − a′ = b, so that SU(b/E) = 1 = SU(b/Ea′). So we have that tp(b/E) ⊥̸ tp(a/Ea′), and therefore tp(b/E) ⊥̸ P.

5.3. Another comment about orthogonality. Orthogonality or non-orthogo-nality tell us about interactions between sets. For instance, let D_{1} and D_{2} be two infinite definable sets, defined over some E = acl_{σ}(E).

If all types realised in D_{1} are orthogonal to all types realised in D_{2}, then this means that whenever you take a tuple ā of elements in D_{1} and a tuple b̄ of elements of D_{2}, then they are independent over any set containing E. In particular, any definable map f : S → D_{2}^{m}, where S ⊆ D_{1}^{n}, for some integers m, n, will take only finitely many values.

Assume in addition that all types realised in D_{1} and in D_{2} are stationary (Recall that a type over a set A is stationary if it has a unique non-forking extension to any set containing A). It will then follow that any definable subset of D_{1} × D_{2} is a finite union of rectangles, i.e., is of the form ⋃_{i=1}^{n} S_{i} × T_{i}, where S_{i} is a definable subset of D_{1} and T_{i} is a definable subset of D_{2}.

5.4. Exercise 10. Let E = acl_{σ}(E) ⊂ K, ā, b̄ tuples from K, and assume that SU(ā/E) = SU(b̄/E) = 1. Show that tp(ā/E) ⊥̸ tp(b̄/E) implies that tr.deg(cl_{σ}(Eā)/E) = tr.deg(cl_{σ}(Eb̄)/E).

5.5. Exercise 11. Let p, q, r be types over algebraic closed sets A, B, and C respectively. Assume that p, q and r have SU-rank 1, and that A | ⌣_{B}C. Show that if p ⊥̸ q and q ⊥̸ r, then p ⊥̸ r. [Warning: your proof should use all the assumptions on A, B, C: their being algebraically closed, and the independence hypothesis. Non-orthogonality is not an equivalence relation on types of SUrank 1. One can however show that the relation E defined by E(p, q) if and only if there is r of SU-rank 1 such that p ⊥̸ r and q ⊥̸ r defines an equivalence relation on types of SU-rank 1.]

5.6. Exercise 12. Let E = acl_{σ}(E) ⊂ K, and a ∈ K with SU(a/E) = 1. Assume that tp(a/E) ⊥̸ (σ(x) = x). We know that tr.deg(cl_{σ}(Ea)/E) = 1, so that E(a)^{alg} = acl_{σ}(Ea). Show that there exists an integer N such that [E(a, σ^{k}(a)) : E(a)] ≤ N for every k ∈ ℤ. [Hint: take E′ = acl_{σ}(E′) independent from a over E such that there is some b ∈ acl_{σ}(E′a) \ E′, with σ(b) = b. Then for every k one has [E(a, σ^{k}(a)) : E(a)] = [E′(a, σ^{k}(a)) : E′(a)], so we may assume that E = E′. Let m = [E(a, b) : E(a)], n = [E(a, b) : E(b)]. Show that [E(a, b, σ^{k}(a)) : E(a)] ≤ mn.]

5.7. Remark. One can show that the converse is true, but the proof is much harder.

5.8. Exercise 13/Example. Let E = acl_{σ}(E) ⊂ K, let a ∈ E, and let b ∈ K \ E satisfy σ(x) − x = a.

(a) Show that tp(b/E) ⊥̸ (σ(x) = x).

(b) Show that tp(b/E) is not almost orthogonal to some type containing σ(x) = x, if and only if there is an integer m > 0 and α ∈ E satisfying σ^{m}(x) = x+a+⋯+σ^{m−1}(a). [Hint: assume that c ∈ (acl_{σ}(Eb)\E)∩F ix(σ); looking at the coefficients of the minimal polynomial of c over cl_{σ}(Eb), we may in fact assume that c ∈ cl_{σ}(Eb), so that c = g(b) for some g(X) ∈ E(X). Look at the sets S_{0} of poles and S_{1} of zeroes of g, and use the equation g(b) = g^{σ}(b + a).]

5.9. Proposition. Every finite SU-rank type is non-orthogonal to a type of SU-rank 1.

Proof. Let E = acl_{σ}(E) ⊂ K, and let ā be a tuple in K with SU(ā/E) = n < ω. We want to find F = acl_{σ}(F) independent from ā over E, and b ∈ acl_{σ}(F ā) \ F such that SU(b/F) = 1.

By definition of the SU-rank, there is some tuple d̄such that SU(ā/Ed̄) = n−1. Given such a tuple, we may always write it as (b̄, c̄), with c̄ independent from ā over E. Find (b̄, c̄) such that c̄ and ā are independent over E, SU(ā/Eb̄c̄) = n−1, and SU(b̄/Ec̄) is least possible.

Let ā′ realise tp(ā/Eb̄c̄) and independent from ā over Eb̄c̄. Then SU(ā/Eb̄c̄ā′) = SU(ā/Eb̄c̄) = n − 1. Claim. ā and ā′ are not independent over Ec̄.

Otherwise, assume that ā and ā′ are independent over Ec̄, and let c̄′ = (c̄, ā′). Since ā′ realises tp(ā/Eb̄c̄), we have SU(ā′/Eb̄c̄) < SU(ā′/Ec̄); hence, by symmetry we get SU(b̄/Ec̄ā′) < SU(b̄/Ec̄), so that the pair (b̄, c̄′) contradicts the minimality of SU(b̄/Ec̄).

Hence ā′ and ā are not independent over Ec̄, so that SU(ā/Ec̄ā′) = SU(ā′/Ec̄ā) = n − 1. It follows that ā′ is independent from b̄ over Ec̄ā. As it was independent from ā over Ec̄b̄ by definition, we get that ā′ is independent from (āb̄) over acl_{σ}(Ec̄ā) ∩ acl_{σ}(Ec̄b̄) (by 3.22). Let d̄∈ acl_{σ}(Ec̄ā) ∩ acl_{σ}(Ec̄b̄) be such that ā′ and āb̄ are independent over Ec̄d̄. By the claim, we know that d̄ ∉ acl_{σ}(Ec̄), and this implies that

SU(ā/Ec̄d̄) < n,

as d̄∈ acl_{σ}(Ec̄ā). We now obtain

n − 1 = SU(ā/Ec̄b̄) ≤ SU(ā/Ec̄d̄) < SU(ā/Ec̄) = n,

so that SU(ā/Ec̄d̄) = n − 1, which implies that SU(d̄/Ec̄) = 1.

5.10. Definitions. Let S be an (∞-) definable set, defined over some E = acl_{σ}(E). We say that S is modular if, for every m and n, and ā ∈ S^{m}, b̄ ∈ S^{n}, we have that ā and b̄ are independent over acl_{σ}(Eā) ∩ acl_{σ}(Eb̄).

A type (over some set E) is modular if the set of its realisations is modular.

Remarks. (1) In general, one requires that ā and b̄ are independent over acl^{eq}(Eā) ∩ acl^{eq}(Eb̄). We use the fact that T h(K) eliminates imaginaries.

(2) This notion is also sometimes referred to as “1-basedness” in the case of stable theories.

(3) There is no harm in extending the notion to sets which are invariant under E-automorphism of K, i.e., sets of realisations of a set of types over E.

5.11. Canonical bases. Let E = cl_{σ}(E), ā a tuple. Since I_{σ}(ā/E) is finitely generated as a σ-ideal, there is an integer m such that the (algebraic) locus V of (ā, …, σ^{m}(ā)) over E completely describes qftp(ā/E), i.e.,

If b̄ is such that (b̄, …, σ^{m}(b̄)) has locus V over E, then qftp(b̄/E) = qftp(ā/E). Let k_{0} be the field of definition of V, and let k be the difference field generated

by k_{0}. Then tp(ā/E) does not fork over k, and k is in a sense smallest with that property (not quite true in positive characteristic). We call k the canonical base of tp(ā/E), and denote it by Cb(ā/E).

5.12. Comments. (1) It is well-known that if c̄_{i}, i ∈ ω, is an independent sequence of generics of V over E, then k_{0} is contained in the field generated by c̄_{0}, …, c̄_{N} for some N. Hence, the same is true in our case provided E is algebraically closed: if ā_{i}, i ∈ ω, is a sequence of independent realisations of tp(ā/E), then k is contained in cl_{σ}(ā_{0}, …, ā_{N}) for some N.

(2) Our definition does not quite agree with the usual definition of canonical bases for types in simple theories.

(3) What we are really interested in, is that k^{alg} is the smallest algebraically closed difference field over which tp(ā/E) does not fork. We could also have defined Cb(ā/E) as k^{alg}. What really matters, is that by 1), there is an integer N such that if ā_{1}, …, ā_{N} are independent realisations of tp(ā/E) then Cb(ā/E) ⊂ acl_{σ}(ā_{1}, …, ā_{N}). See Remark 3.23 for a proof.

5.13. Corollary. Let S be modular. Then for every n, m, and ā ∈ S^{n}, b̄ ∈ K^{m}, we have that ā and b̄ are independent over C = acl_{σ}(Eā) ∩ acl_{σ}(Eb̄). Proof. Let ā_{i}, i ∈ ω, be a sequence of realisations of tp(ā/acl_{σ}(Eb̄)), independent over Eb̄, and with ā_{0} = ā. Then C = acl_{σ}(Eā_{i}) ∩ acl_{σ}(Eb̄) for i ∈ ω because tp(ā_{i}/Eb̄) = tp(ā/Eb̄). Since ā is independent from {ā_{i} | i > 0} over Eb̄, we have

C = acl_{σ}(Eā) ∩ acl_{σ}(Eā_{i} | i > 0).

By the above, acl_{σ}(Eā_{i} | i > 0) contains the canonical base of tp(ā/acl_{σ}(Eb̄)). By modularity, we get that this canonical base is contained in C, and therefore that ā and b̄ are independent over C.

5.14. Remark. This is a particularly nice way of phrasing modularity: a set S is modular if and only if, for every tuple ā of elements of S and set B, we have that acl_{σ}(Eā) ∩ acl_{σ}(EB) contains the canonical base of tp(ā/acl_{σ}(EB)). It coincides with the definition of 1-basedness.

5.15. What is modularity? Or rather, what is it not? Modularity forbids the existence of complicated sets. For instance, in the theory of algebraically closed fields, no infinite set is modular. This comes from the typical counterexample to modularity: let a, b, c be algebraically independent over ℚ, say, and let d = ac + b. Then

ℚ(a, b)^{alg} ∩ ℚ(c, d)^{alg} = ℚ^{alg},

but (a, b) and (c, d) are not independent.

In case the elements of S have SU-rank 1, modularity is equivalent to the non-existence of a SU-rank-2 family of “curves”, that is, a set C(e), e ∈ D, of definable subsets of S^{2} of SU-rank 1, where D is definable, has SU-rank 2, and is such that if d ≠ d′ ∈ D, then C(d) ∩ C(d′) is finite.

When we are in a stable situation and there is a group around, there is a remarkable theorem of Hrushovski-Pillay ([14]) which says: Theorem. Let G be a stable group (maybe with additional structure), and assume that G is modular. Then for every n, any definable subset of G^{n} is a Boolean combination of cosets of definable subgroups of G^{n}, and these subgroups are defined over acl(∅).

Thus in particular, a stable modular group has essentially only one group law. Moreover, one can show that it is necessarily abelian by finite.

The completions of ACFA are unstable, and we will not be able to apply directly the result of Hrushovski-Pillay. In fact, in positive characteristic this result will be false. However, all the quantifier-free formulas are “stable” and they control independence (I will not define what a stable formula is, let me just say that if T is a stable theory then all formulas are stable). It will follow that some restricted version of H-P’s theorem holds in positive characteristic, while the full result holds in characteristic 0.

5.16. Lemma. Let T be a complete theory satisfying the conclusion of 3.20(2), that if A | ⌣_{C}B and A | ⌣_{B}C then A | ⌣_{acl(B)∩acl(C)}BC (e.g., eliminating imaginaries, and stable or with stable forking), let A, B, C be algebraically closed sets containing E = acl(E), and assume that C is independent from (A, B) over E. Then acl(AC) ∩ acl(BC) = acl((A ∩ B)C). Proof. Let α ∈ acl(AC) ∩ acl(BC). By hypothesis, C | ⌣_{E}AB, so that C | ⌣_{A}B. Since α ∈ acl(AC), we get (C, α) | ⌣_{A}B. Similarly, (C, α) | ⌣_{B}A. By 3.20, we obtain (C, α) | ⌣_{A∩B}(A, B), which gives α ∈ acl((A ∩ B)C).

5.17. Proposition. Let E = acl_{σ}(E) ⊂ F = acl_{σ}(F) ⊂ K, let ā be a tuple in K which is independent from F over E, and let S be a set of realisations of a set of types of SU-rank 1 over E.

(1) If tp(ā/E) is modular, then so is tp(ā/F).

(2) S is modular if and only if all types realised in S are modular.

(3) If SU(ā/E) = 1 and tp(ā/F) is modular, then tp(ā/E) is modular.

Proof. (1) Let b̄ be a finite set of realisations of tp(ā/F), and let c̄ be a finite tuple of elements of K. By modularity, b̄ is independent from acl_{σ}(F c̄) over D = acl_{σ}(Eā) ∩ acl_{σ}(F c̄). This implies that b̄ is independent from acl_{σ}(F c̄) over acl_{σ}(F D) ⊆ acl_{σ}(F b̄) ∩ acl_{σ}(F c̄), and shows that tp(ā/F) is modular.

(2) One direction is clear: if S is modular and P ⊂ S, then P is modular. For the other direction, assume that this is not true, and let n be minimal such that there are ā_{1}, …, ā_{n} ∈ S, a tuple b̄ in K and C = acl_{σ}(E, ā_{1}, …, ā_{n}) ∩ acl_{σ}(Eb̄), with (ā_{1}, …, a_{n}) and b̄ not independent over C.

Note that, by minimality of n and because all tuples have SU-rank 1, we have that the elements ā_{1}, …, ā_{n} are independent over E. Moreover, the types tp(ā_{i}/E) are pairwise non-orthogonal: indeed, assume by way of contradiction that tp(ā_{n}/E) is orthogonal to tp(ā_{1}/E). By minimality of n and transitivity of forking, we have that (ā_{1}, …, ā_{n−1}) and b̄ are independent over C′ = acl_{σ}(E, ā_{1}, …, ā_{n−1}) ∩ acl_{σ}(Eb̄), and that ā_{n} and b̄ are not independent over (C′, ā_{1}, …, ā_{n−1}) = (E, ā_{1}, …, ā_{n−1}). Hence C = C′, and ā_{n} ∈ acl_{σ}(E, ā_{1}, …, ā_{n−1}, b̄) \ acl_{σ}(E, ā_{1}, …, ā_{n−1}). Since tp(ā_{n}/E) is orthogonal to tp(ā_{1}/E), we get that ā_{n} ∈ acl_{σ}(E, ā_{2}, …, ā_{n−1}, b̄), which contradicts the minimality of n.

By definition and because SU(ā_{i}/E) = 1, this means that for each i ≥ 2 there is F_{i} containing E and independent from ā_{i} over E, such that acl_{σ}(F_{i}ā_{i})\F_{i} contains a realisation ā′_{i} of tp(ā_{1}/E). Moving the F_{i}’s by an E-automorphism, we may choose them such that (F_{2}, …, F_{m}) is independent from (ā_{1}, …, ā_{n}, b̄) over E. Setting F = acl_{σ}(F_{2}, …, F_{m}) and using 5.16, we then have acl_{σ}(F, ā_{1}, …, ā_{n})∩ acl_{σ}(F b̄) = acl_{σ}(F C), but (ā_{1}, …, ā_{n}) is not independent from b̄ over acl_{σ}(F C).

As each ā_{i} is equi-algebraic over F with the realisation ā′_{i} of tp(ā_{1}/E), we get that:

— acl_{σ}(F, ā_{1}, ā′_{2}, …, ā′_{n}) = acl_{σ}(F, ā_{1}, …, ā_{n}),

— SU(ā_{1}, …, ā_{n}/F) = n = SU(ā_{1}, ā′_{2}, …, ā′_{n}/F) = SU(ā_{1}, ā′_{2}, …, ā′_{n}/E). Therefore (by transitivity of independence), we get that

(∗): F and (ā_{1}, ā′_{2}, …, ā′_{n}, b̄) are independent over E.

Hence, letting D = acl_{σ}(E, ā_{1}, ā′_{2}, …, ā′_{n}) ∩ acl_{σ}(Eb̄), we have (by 5.16)

acl_{σ}(F, a′_{2}, …, ā′_{n}) ∩ acl_{σ}(F b̄) = acl_{σ}(F D) = acl_{σ}(F C).

By modularity of tp(ā_{1}/E), we also have that (ā_{1}, ā′_{2}, …, ā′_{n}) and b̄ are independent over D, and by (∗), this gives that they are independent over acl_{σ}(F D). But, since each ā′_{i} is equi-algebraic with ā_{i} over F, we have that ā_{1}, ā′_{2}, …, ā′_{n} and b̄ are not independent over acl_{σ}(F C) = acl_{σ}(F D): this gives us the desired contradiction.

(3) Assume that tp(ā/F) is modular. Note that if φ is an E-automorphism of K, then φ(tp(ā/F)) is also modular. Let ā_{1}, …, ā_{n} be realisations of tp(ā/E), b̄ a tuple of elements of K, and C = acl_{σ}(Eā_{1}, …, ā_{n}) ∩ acl_{σ}(Eb̄). For each i, let F_{i} be such that tp(ā_{i}, F_{i}/E) = tp(ā, F/E); then each tp(ā_{i}/F_{i}) is modular. Using (1), and moving the F_{i} by some E-automorphism of K, there is F′ independent from (ā_{1}, …, ā_{n}, b̄) over E, such that tp(ā_{i}/F′) is modular for i = 1, …, n. By 5.16, we have acl_{σ}(F′ā_{1}, …, ā_{n}) ∩ acl_{σ}(F′b̄) = acl_{σ}(F′C). By (2), we get that (ā_{1}, …, ā_{n}) and b̄ are independent over acl_{σ}(F′C). Since F′ is independent from (ā_{1}, …, ā_{n}, b̄) over E, this implies that (ā_{1}, …, ā_{n}) is independent from b̄ over C and therefore that tp(ā/E) is modular.

5.18. Corollary. Let p and q be types over algebraically closed sets, which are of SU-rank 1, and are non-orthogonal. Then p is modular if and only if q is modular. Proof. Let p′ and q′ be non-forking extensions of p and q to some set A = acl_{σ}(A) containing the sets over which p and q are defined, and such that p′ is non-almost-orthogonal to q′. Use 5.17.

5.19. Non-example. Let E = acl_{σ}(E) ⊂ K, let ā be a tuple, and assume that SU(ā/E) ≥ ω. We claim that tp(ā/E) is not modular.

Indeed, we know that some element of the tuple ā, say a_{1}, is transformally transcendental over E. Take b̄ realising tp(ā/E) and independent from ā over E, and let c be transformally transcendental over acl_{σ}(Eāb̄), let d = a_{1}c + b_{1}. Then certainly (ā, b̄) and (c, d) are not independent over E. Since d = a_{1}b+b_{1}, we have that (c, d) is independent from (ā, b̄) over acl_{σ}(Ea_{1}b_{1}), and so it is enough to show that (a_{1}, b_{1}) and (c, d) are not independent over C = acl_{σ}(Ea_{1}b_{1}) ∩ acl_{σ}(Ecd).

We claim that the canonical base of tp(c, d/Ea_{1}b_{1}) contains a_{1}b_{1}. Indeed, let (c′, d′) be a realisation of tp(c, d/Ea_{1}b_{1}), independent from (a_{1}, b_{1}, c, d) over Ea_{1}b_{1}. Then a_{1}, b_{1}, c, c′ are independent over E. However (a_{1}, b_{1}) belongs to the field generated by (c, d, c_{1}, d_{1}). As (a_{1}, b_{1}) does not belong to acl_{σ}(Ecd), this implies that (a_{1}, b_{1}) and (c, d) are not independent over C = acl_{σ}(Ea_{1}b_{1}) ∩ acl_{σ}(Ecd).

5.20. Sets of SU-rank 1. Let us again concentrate on sets S of SU-rank 1, defined over E. Because all our tuples have SU-rank 1, we get a pre-geometry on S, where the closure operator is simply acl(E, −): given A ⊂ S and b ∈ S, either b ∈ acl(E, A), or b is independent from A over E. The exchange principle holds, and the SU-rank of a set is the dimension of this set, etc. Definition. A SU-rank-1 type p over E is called trivial if whenever ā_{1}, …, ā_{n}, ā realise p, then either ā ∉ acl_{σ}(E, ā_{1}, …, ā_{n}), or ā ∈ acl_{σ}(Eā_{i}) for some i. In other words, if P is the set of realisations of p, then acl_{σ}(EP) = ⋃_{ā∈P} acl_{σ}(Eā).

5.21. Exercise 14. Show that a trivial type is modular.

5.22. Some additional properties of modular types in models of ACFA.

(1) If the type p is non-orthogonal to a modular type of SU-rank 1, then p is non-almost-orthogonal to a (modular) type of SU-rank 1.

(2) If p is a non-trivial modular type of SU-rank 1, then p is non-orthogonal to the generic of a group of SU-rank 1 defined over the same set as p (see the definition of generic in the next chapter).

§6. Groups, generic types, stabilisers. In this section, contrary to my promise, I will use some results on simple theories. You may very well replace everywhere “simple”, or “supersimple”, by complete theory extending ACFA. The notion of generic is meant to be the analogue of a generic of an algebraic group (so, generic in the sense defined in 1.10). A good reference for generics and stabilisers in simple theories is Pillay’s paper [16]. But see also Wagner’s book [19].

6.1. Generics in groups. Let G be a group (maybe with extra structure), whose theory is simple [Or: let G be a group definable in some model of ACFA]. We call a type p over some set E = acl(E) a generic type of G, if p is realised in G, and whenever a ∈ G and b realising p are independent over E, then b · a is independent from a over E. An element of G is generic over E if it realises a generic type over E.

Let H be a definable subgroup of G, let a ∈ G, consider the coset a · H and let E = acl(E) be a set over which a · H is defined. We say that b is a generic of the coset a · H over E if b ∈ a · H and whenever h ∈ H is independent from b over E, then b · h is independent from h over E. Some facts.

(1) Generics exist. A non-forking extension of a generic type is generic.

(2) Assume that tp(b/E) is generic, and let a ∈ G be independent from b over E. Then a · b is independent from a over E, and a · b and b · a realise generic types over E. (The definition described a generic element of G in terms of its action on G by multiplication on the left. This shows that it can be defined also in terms of the action by multiplication on the right, and that a “left” generic is also a “right” generic.)

Below we will give some examples illustrating this notion. First we need some definitions from algebraic geometry.

6.2. Projective space, projective varieties. Let n be an integer, and consider the set of lines through the origin O in affine n + 1-space 𝔸^{n+1}(Ω). This set can be viewed as the set of equivalence classes of 𝔸^{n+1}(Ω) \ O for the equivalence relation

(x_{0}, …, x_{n}) ∼ (y_{0}, …, y_{n}) ⟺ ∃λ ≠ 0 ⋀_{i=0}^{n+1} x_{i} = λy_{i}.

We denote this set by ℙ^{n}(Ω), and call it the projective n-space. The equivalence class of (x_{0}, …, x_{n}) is usually denoted by (x_{0} : ⋯: x_{n}).

If F (x_{0}, …, x_{n}) is a homogeneous polynomial over Ω, i.e., for some d all monomials appearing in F have total degree equal to d, then the equation F (x_{0}, …, x_{n}) = 0 is compatible with the equivalence relation, and defines a subset of ℙ^{n}(Ω). The topology whose closed sets are finite intersections of sets of this form, is called the Zariski topology.

Note that we can view ℙ^{n}(Ω) as the union of n + 1 copies of affine n-space. Indeed, for i = 0, …, n, define

U_{i} = {(x_{0} : ⋯: x_{n}) | x_{i} ≠ 0}.

Then the map f_{i} : (x_{1}, …, x_{n}) ↦ (x_{1} : ⋯: x_{i−1} : 1 : x_{i} : ⋯: x_{n}) defines a bijection between 𝔸^{n}(Ω) and U_{i}. Moreover, ℙ^{n}(Ω) = ⋃_{i=1}^{n} U_{i}. Each U_{i} is open for the Zariski topology, and the maps f_{i} are homeomorphisms (for the Zariski topology on 𝔸^{n}(Ω) and the induced topology on U_{i}).

Closed subsets of projective n-space are called projective sets. If X is closed and irreducible (i.e., for each i, U_{i} ∩ X is a variety), then we call X a projective variety. One can show that the product of two projective varieties is a projective variety.

Morphisms between projective varieties are defined locally using the covering by affine spaces.

6.3. Algebraic groups.

We say that G is an algebraic group if G is an open subset of a projective variety, and we have a group operation G×G → G which is an morphism, i.e., is everywhere defined, and is locally defined by rational functions. We also require that the inverse map G → G is a morphism.

Here are some examples:

— The additive group, usually denoted 𝔾_{a} to distinguish it from the affine line 𝔸 (with no structure). It is an open subset of ℙ^{1}.

— The multiplicative group 𝔾_{m} (so 𝔾_{m}(K) = K \ {0}).

— Also, some projective group varieties (i.e., closed for the Zariski topology), which are called abelian varieties. Some examples are elliptic curves, and also certain groups of the form ℂ^{n}/Λ, where Λ is a 2n-dimensional lattice subgroup of ℂ^{n} generating the ℝ-vector space ℂ^{n}.

6.4. Some elementary facts about algebraic groups. We did not require in our definition that G be irreducible (by which I mean, that the Zariski closure of G in the projective space we are working in, is a variety).

If G is not irreducible, then one can verify that the irreducible components of G are disjoint, so that there is a unique irreducible component of G, denoted G^{0} and called the connected component of G, which contains the identity element e of G. One has [G : G^{0}] < ∞.

Let G be an algebraic group, and H a subgroup of G, H̄ its Zariski closure. Then H̄ is also an algebraic group.

6.5. Examples of generics. We let K be an algebraically closed field, sufficiently saturated, and G an algebraic group defined over some subfield E of K.

The first thing to remark, is that a generic of G(K) for the theory ACF of algebraically closed fields, is simply a generic in the sense of algebraic geometry. This is fairly immediate from the definition: let d = dim(G), assume that g is a generic of G, independent from h ∈ G over E. We know that g · h ∈ E(g, h), and (using the inverse map), that g ∈ E(g · h, h^{−1}) = E(g · h, h). Hence g and g · h are equi-algebraic over E(h). This implies

tr.deg(g · h/E(h)) = tr.deg(g/E(h)) = d ≤ tr.deg(g · h/E) ≤ d = dim(G),

so that equality holds and g · h is independent from h over E, as desired. Let us now assume that K is a model of ACFA, sufficiently saturated, and let H be a definable subgroup of G(K), defined also over E = acl_{σ}(E). Then the σ-closure H̃ of H is a σ-closed subgroup of G(K). Thus there is a natural notion of generic for H: an element g ∈ H is generic if and only if I_{σ}(g/E) = I_{σ}(H̃/E). Using the description of generics of algebraic groups, one can show (easily if SU(H) < ω, with a little more work in the general case) that this definition is the correct one. Furthermore, that [H̃ : H] < ∞, so that a generic of H is also a generic of H̃.

One interesting consequence of elimination of imaginaries, is the following: let H̃^{0} be the connected component of H̃ (i.e., the irreducible component in the sense of the σ-topology which contains the identity element). Then both H̃ and H̃^{0} are defined over E. The cosets of H̃^{0} in H̃ are then imaginary elements, and algebraic over E (since [H̃ : H̃^{0}] < ∞). By elimination of imaginaries, they are defined over E.

6.6. Lemma. Let G be a group, H a definable subgroup of G, defined over some E = acl(E), and g ∈ G a generic of G over E. Then g is a generic of g · H (over C = E ∪ c̄, where c̄ is the code of g · H). Proof. Let h be a generic of H over Eg. We claim that g′ = g · h is a generic of g · H over Eg. Indeed, let h_{1} ∈ H be independent from g′ over Eg. Then g′ · h_{1} ∈ g · H, and we need to show that h_{1} and g′ · h_{1} are independent over Eg. But, g′ · h_{1} is equi-algebraic with h · h_{1}over Eg, and it is therefore enough to show that h_{1} and h · h_{1} are independent over Eg, which follows because h is a generic of H over Eg. As C ⊂ acl(Eg), this implies that g′ is a generic of g · H over C.

By genericity of g, we know that g′ and h are independent over E, so that h is also a generic of H over Eg′. From g · H = g′ · H and the previous step, we deduce that g is a generic of gḢ over C.

6.7. Stabilisers. Let G be a group (maybe with extra structure), whose theory is simple, let p be a type over E = acl(E). Assume that the independence theorem is satisfied over E: i.e., given a and b independent over E, and any two non-forking extensions p_{1} to (E, a) and p_{2} to (E, b) of some type p over E, p_{1} ∪p_{2} extend to some type p′ over (E, a, b) which does not fork over E. [Or: let G be a group definable in a model of ACFA.]

We define S(p) to be the set of elements h ∈ G, such that there exists a realisation g of p such that h · g realises p, and h is independent from g and from h · g over E.

One then defines Stab(p) = S(p) · S(p). By the independence theorem over E, we have that if h_{1} and h_{2} are independent elements of S(p), then h_{1} · h_{2} ∈ S(p). This will imply that Stab(p) is a subgroup of G, with generics in S(p). One can show that Stab(p) is ∞-definable.

6.8. Example. Let K be a model of ACFA, G an algebraic group defined over E = acl_{σ}(E), and let p be a type over E, X ⊆ G(K) its set of realisations. By definition, if h ∈ S(p), then there are g, g_{1} ∈ X such that g | ⌣_{E}h, g_{1} | ⌣_{E}h, and h · g = g_{1}. In particular, h = g_{1} · g^{−1}, so that h ∈ X · X^{−1}. The requirement that h be independent from g and from g_{1} over E then implies that necessarily SU(h/E) ≤ SU(g/E) = SU(g_{1}/E).

Let Y be the σ-closure of X. Then Y is irreducible, and g, g_{1} belong to Y. This implies in particular that S(p) ⊂ Stab(Y) =_{def} {h ∈ G(K) | hY = Y }. Note that Stab(Y) is clearly a subgroup of G(K), and is quantifier-free definable, so that it is σ-closed. If I am not mistaken, one can then show that Stab(p) is the intersection of all definable subgroup of Stab(Y) which are defined over E and have finite index in Stab(Y).

6.9. Remarks/Exercise 15. Let us now assume that p has a unique nonforking extension to any set containing E, and let P be the set of realisations of p. [This hypothesis is verified for instance when p is definable over E and T is stable.]

(1) Then Stab(p) = S(p): this is because the quantifier “there exists” in the definition of S(p) can be replaced by the quantifier “for all”.

(2) If E′ ⊃ E, and p′ is the non-forking extension of p to E′, then Stab(p) = Stab(p′).

(3) Let h ∈ E. Then h ∈ S(p) ⟺ h · P = P.

No completion of ACFA is stable. However, in the language of “local stability theory”, all quantifier-free formulas are stable. We will see below some of the consequences this has.

6.10. Proposition. Let E = acl_{σ}(E) ⊂ K, K a sufficiently saturated model of ACFA.

(1) Let p be a type over E, and assume that whenever F contains E, then p has a unique non-forking extension to F. Then p is definable over E, i.e., given a formula φ(x̄, ȳ), there is a formula (over E) d_{φ}(ȳ) which defines in K the set of c̄ such that φ(x̄, c̄) belongs to the unique-non-forking extension of p to K.

(2) Let p be a quantifier-free type over E. Given a quantifier-free formula φ(x̄, ȳ), there is a quantifier-free formula (over E) d_{φ}(ȳ) which defines in K the set of c̄ such that φ(x̄, c̄) belongs to some/all non-forking extensions of p to K.

Proof. (1) is well-known, but we will give the proof. Assume by way of contradiction that p is not definable, i.e., that there is some formula φ(x̄, ȳ) such that the set X of tuples c̄ of K such that φ(x̄, c̄) belongs to the unique non-forking extension of p to K is not definable. Then, for every L(E)-formula ψ(ȳ), there are c̄_{1} ∈ X, c̄_{2} ∉ X which both satisfy ψ(ȳ). By compactness, there are c̄_{1}, c̄_{2} in K, realising the same type over E, and such that c̄_{1} ∈ X, c̄_{2} ∉ X. Let f be an automorphism of K which leaves E fixed and sends c̄_{1} to c̄_{2}. If q is the unique non-forking extension of p to K, then f(q) is also a non-forking extension of p to K, and so must equal q: but this contradicts the fact that φ(x̄, c̄_{1}) ∈ q, φ(x̄, c̄_{2}) ∉ q = f(q).

We now sketch a proof of (2). The crucial point is the following observation: since quantifier-free types correspond exactly to descriptions of the isomorphism type of the “difference field generated by”, it follows that if F = acl_{σ}(F) contains E, then there is a unique quantifier-free type q over F which extends p and is such that whenever ā realises q, then ā is independent from F over E.

This implies that, given p = qftp(ā/E) and c̄ independent from ā over E, we have:

qftp(ā/E) ∪ qftp(c̄/E) ∪ Σ(x̄, ȳ) ⊢ qftp(ā, c̄/E),

where Σ(x̄, ȳ) is the set of quantifier-free formulas over E expressing that the tuples x̄ and ȳ are independent over E. The proposition follows by compactness.

6.11. Remarks. (1) Let E = acl_{σ}(E), ā a tuple in K. Let C be the set of codes of all (quantifier-free) definitions of qftp(ā/E). Then certainly tp(ā/E) does not fork over C. Moreover, if D ⊂ E and tp(ā/E) does not fork over D then acl_{σ}(D) ⊃ C. Hence, we are getting that for quantifier-free types, our definition of a canonical base agrees with the classical one.

(2) Let G be a group definable in K by quantifier-free formulas (over some E = acl_{σ}(E)), and assume in addition that the group operation and the inverse map are piecewise definable by terms of the language {+, −, ·,^{−1}, σ, σ^{−1}, e(e ∈ E)}, so that if a, b ∈ G, then a · b ∈ cl_{σ}(E, a, b) and a^{−1} ∈ cl_{σ}(E, a). [By the multiplication being piecewise defined, we mean that there is a definable finite partition of G^{2}, and that on each piece of this partition multiplication is defined by a term.] If p is a quantifier-free type over E realised in G, then we may also define Stab(p) to be the set of elements b of G such that whenever a realises p and is independent from b over E, then b · a realises p and is independent from b over E. All remarks made in 6.9 go through to this particular case. Examples of such groups are algebraic groups defined over E.

6.12. Theorem. Let G be a group, quantifier-free definable in a model K of ACFA, and assume that G is modular, and the group operation and the inverse map are piecewise defined by terms of the language {+, −, ·,^{−1}, σ, σ^{−1}, e(e ∈ E)}. If X is a quantifier-free definable subset of G, then X is a Boolean combination of quantifier-free definable subgroups of G. Furthermore, if G is defined over E = acl_{σ}(E), then so are these subgroups. Proof. We will fist show that if p is a quantifier-free type defined over some A = acl_{σ}(A) containing E, then the set P of realisations of p is contained in a coset of some quantifier-free definable subgroup S of G, and p is the generic type of this coset. Without loss of generality, we will assume that A is the algebraic closure (over E) of the canonical base of p.

Define S = Stab(p), the set of elements b ∈ G, such that there is a ∈ P, independent from b over A and such that b · a ∈ P. Claim. S is quantifier-free definable (over A).

Let a ∈ P and consider I_{σ}(a/A). Then I_{σ}(a/A) completely determines p: let b ∈ G. Then

I_{σ}(b/A) = I_{σ}(a/A) ⟺ b ∈ P.

Let F (X) be a tuple of difference polynomials over A generating the σ-ideal I_{σ}(a/A) (as a σ-ideal).

By the piecewise quantifier-free definability of the group operation, there is a quantifier-free formula ψ(y, x) such that, if a ∈ P is independent from g ∈ G over A, then

K ⊧ ψ(g, a) ⟺ F (g · a) = 0.

Let φ be the formula F (x) = 0∧ψ(y, x). Then S is defined by the formula d_{φ}(y): Let g ∈ G. Then g satisfies d_{φ}(y) if and only if F (x) = 0 ∧ ψ(g, x) belongs to the non-forking extension of p to A ∪ g, if and only if whenever a ∈ P is independent from g over A, then b = g · a satisfies F (x) = 0. Note that because b and a are equi-algebraic over acl_{σ}(A, g), this implies that b belongs to P and is independent from g over A (because b cannot satisfy “more” equations than F (x) over acl_{σ}(A, g)). [In the classical stable (non-superstable) case, one cannot get S to be definable, only ∞-definable. There the defining formulas are of the form ∀z̄d_{φ}(z̄) ↔ d_{φ^{∗}}(z̄, y), where φ^{∗}(z̄, y, x) = φ(z̄, y · x).]

Fix a ∈ P, and g a generic of G over Aa. Then b = g · a is a generic of G over Aa, and is independent from a over A. Let G_{0} ≺ G contain A, g, let P_{0} be the set of realisations of the non-forking extension of p to G_{0}, and Q_{0} the set of realisations of the non-forking extension q_{0} of qftp(b/Ag) to G_{0}. Then Q_{0} = g · P_{0}. If τ ∈ Aut(G_{0}/A), then τ also acts on the set of formulas over G_{0}. By abuse of notation, if U is an (∞-) definable over G_{0} subset of G, we will denote by τ(U) the (∞-) definable subset of G defined by applying τ to the formulas defining U. Claim. Let τ ∈ Aut(G_{0}/A). Then τ(Q_{0}) = Q_{0} ⟺ τ(g · S) = g · S.

Since p is definable over A, we have that τ(P_{0}) = P_{0}, and τ(S) = S. Hence τ(Q_{0}) = Q_{0} ⟺ τ(g) · P_{0} = g · P_{0} ⟺ g^{−1} · τ(g) · P_{0} = P_{0} ⟺ g^{−1} · τ(g) ∈ S ⟺ τ(g · S) = g · S.

By elimination of imaginaries of ACFA, we get that the codes of Q_{0} and of g · S are equi-algebraic over A. Let c̄ be the code of g · S. Because g is a generic of G over A, and S is a subgroup of G defined over A, we get that g is a generic of g · S (see Lemma 6.6), and that c̄ is equi-algebraic with the canonical base of qftp(g/acl_{σ}(Ac̄)), so that SU(g/Ac̄) = SU(S). By the claim, we also get that the canonical base of q_{0} is equi-algebraic over A with c̄. Since q_{0} is the non-forking extension of qftp(b/acl_{σ}(Ag)) to G_{0}, it follows that c̄ is equi-algebraic over A with Cb(b/Ag). By modularity, we have:

acl_{σ}(Ac̄) = acl_{σ}(Ab) ∩ acl_{σ}(Ag)  (∗)

Since g is a generic of G over Aa, we have that a is independent from g and from b = g · a over A, and therefore also over Ac̄ by (∗). Hence, using the symmetry of independence, we have:

SU(P) = SU(b/Ag) = SU(b/Ac̄) = SU(b/Ac̄a) = SU(g/Ac̄a) = SU(g/Ac̄) = SU(S)

(The fourth equality is because b and g are equi-algebraic over Aa ⊂ Aac̄). Hence SU(P) = SU(S). If h ∈ S is independent from a over A, then b = h · a ∈ P, and is independent from a and from h over A. Hence the generics of S are of the form d · e^{−1}, for d, e ∈ P independent over A. This implies that Sd = Sa for all d ∈ P, and therefore that P ⊂ Sa. It remains to show that if b is a generic of Sa, then b ∈ P. Let b be a generic of Sa over A, and choose d ∈ P independent from b over A. Then Sd = Sa, so that there is h ∈ S such that b = h · d. From SU(S) = SU(b/A) = SU(b/Ad) = SU(h/Ad) ≤ SU(S), we deduce that h is independent from d over A, so that h · d ∈ P (by definition of S).

Hence the type p can be described as follows:

— x belongs to the coset Sa.

— If S′ is a definable subgroup of G, and the coset S′b is definable over A and strictly contained in Sa, then x does not belong to S′b.

Indeed, if b ∈ Sa is not a generic of Sa, then the realisations of qftp(b/A) are the generics of some S′b strictly contained in Sa and defined over A. It follows, by compactness, that every formula is equivalent to a Boolean combination of formulas of the form “x belongs to a coset of some definable subgroup of G”.

We will now show that S is in fact defined over E. By modularity, tp(g/Ac̄) is definable over acl_{σ}(Eg), and since g · S is a coset of a group, this implies that g · S is also defined over acl_{σ}(Eg). Hence so is S = g^{−1} · (g · S), and we obtain that S is defined over acl_{σ}(Eg) ∩ A = E.

§7. General results about models of ACFA. We state here some results without proofs. We are working in a model K of ACFA. Let us start with an easy one:

7.1. Proposition. Let ϕ be the identity if char(K) = 0, and the Frobenius automorphism x ↦ x^{p} if char(K) = p > 0. Let m ≥ 1 and n be integers. Then the difference field (K, σ^{m}ϕ^{n}) is also a model of ACFA. Proof. See below exercise 7.19.

7.2. The dichotomy theorem. Using Theorem 4.1, we obtain that, in positive characteristic the fields F ix(σ^{m}ϕ^{n}) are pseudo-finite fields. We will refer to these fields as the fixed fields of K.

Assume that (m, n) = 1 (or m = 1, n = 0), let F = F ix(σ^{m}ϕ^{−n}). One can then show that SU(F) = 1. From this, it follows that if S ⊂ F is definable and infinite, then F = a_{1}S + a_{2}S + ⋯ + a_{n}S for some elements a_{1}, …, a_{n} ∈ K. This implies that all non-algebraic 1-types containing the formula σ(x) = x [or σ^{m}(x) = ϕ^{n}(x) if the characteristic is positive] are non-orthogonal; since the field F is certainly non-modular, and has no induced structure from K, this implies the following:

A type p which is non-orthogonal to one of the formulas σ^{m}(x) = ϕ^{n}(x) cannot be modular.

In fact, the converse is also true:

Theorem. Let K be a model of ACFA, and let p be a type of SU-rank 1 over E = acl_{σ}(E) ⊂ K. Then p is modular if and only if p is orthogonal to all formulas σ^{m}(x) = ϕ^{n}(x).

If char(K) = 0, then p is modular if and only if p has a unique non-forking extension to any set containing E.

7.3. Discussion/Theorem. □□□ In characteristic 0, we therefore get that modular types of SU-rank 1 are definable, and hence stable. Using proposition 5.9, one can then show that if a (finite SU-rank) formula φ(x̄) is orthogonal to σ(x) = x (i.e., to all types containing the formula σ(x) = x), then all types containing φ(x̄) are stable and definable, the set S defined by φ is stably embedded, i.e., all subsets of S^{m} which are definable in K are definable with parameters from S, so that S with the structure induced from K is superstable of finite U-rank and one-based.

In characteristic p > 0, if a finite SU-rank formula is orthogonal to all formulas defining fixed fields, then the set it defines is also modular, but is usually not stable. □□□

7.4. For p a prime and q a power of p, consider the difference field F_{q} = (𝔽_{p}^{alg}, σ_{q}), where σ_{q} : x ↦ x^{q}. Let U be a non-principal ultrafilter on the set Q of all prime powers.

Theorem (Hrushovski [7], Macintyre [8]). The difference field ∏_{q∈Q} F_{q}/U is a model of ACFA.

7.5. One can then show that the theory ACFA coincides with the set of sentences true in all but finitely many of the difference fields F_{q}. This is the analogue of Ax’s theorem, which states that the theory of pseudo-finite fields coincides with the set of sentences true in almost all finite fields. Hrushovski’s proof gives more: it gives estimates on the size of finite definable sets. Before stating his result, let me remark that one can replace the scheme of axioms (3) of the theory ACFA by the apparently weaker scheme of axioms (3’) If U and V are varieties of dimension d defined over K, with V ⊆ U ×U^{σ},

such that the projections of V to U and to U^{σ} are generically onto, then there is a tuple ā in K such that (ā, σ(ā)) ∈ V.

Let me explain why this (together with axiom schemes (1) and (2)), will give us an axiomatisation of ACFA. The proof that every difference field K embeds in a model of ACFA actually shows that every difference field embeds in a model L of ACFA, all of whose elements are transformally algebraic over K. This means that given a tuple ā ∈ L, there is an integer m such that acl_{σ}(Kā) = K(ā, …, σ^{m}(ā))^{alg}. Hence, every particular instance of axiom (3) is implied by finitely many instances of axiom (3’).

7.6. Theorem (Hrushovski). Let F (X̄, Z̄) and G(X̄, Ȳ , Z̄) be tuples of polynomials over ℤ. There is a positive constant C with the following property:

For all prime p and power q of p, and tuple c̄ in 𝔽_{p}^{alg}, if the equations F (X̄, c̄) = 0 and G(X̄, Ȳ , c̄) = 0 define varieties U and V satisfying the requirements of axiom (3’), then

|Card({ā ∈ 𝔽_{p}^{alg} | (ā, ā^{q}) ∈ V }) − cq^{d}| ≤ Cq^{d−1/2},

where

d = dim(U) = dim(V), and c = [K(V) : K(U)]/[K(V) : K(U^{σ})]_{ins}.

This theorem is a strengthening of a theorem of Lang-Weil for the number of points of varieties defined over finite fields: take U defined over 𝔽_{q}, and V the intersection of the diagonal with U × U. The above formula then gives you an estimate on the number of points of U with their coordinates in 𝔽_{q}.

7.7. Some applications of these theorems. You may recall the proof of Ax that if V is a variety defined over ℂ, and f : V → V is a morphism which is injective on the set V (ℂ) of points of V with their coordinates in ℂ, then f defines a bijection of V (ℂ) (i.e., f is also surjective). This is done as follows: this statement will hold in ℂ if and only if it holds in all algebraically closed fields of characteristic p > 0, if and only if it holds in all 𝔽_{p}^{alg}, p a prime. So, let V be a variety and f : V → V a morphism, both defined over 𝔽_{p}^{alg}. Then for some q = p^{m}, they are defined over 𝔽_{q}. Thus, for every n ≥ 1, f is a map V (𝔽_{q^{n}}) → V (𝔽_{q^{n}}), is injective, and therefore also surjective since V (𝔽_{q^{n}}) is finite. Because 𝔽_{p}^{alg} = ⋃_{n} 𝔽_{q^{n}}, we get that f defines a permutation of V (𝔽_{p}^{alg}).

Corollary. Let B be a group of finite SU-rank defined in a model K of ACFA, and assume that f is a definable endomorphism of B. Then

[B : f(B)] = Card(Ker(f)).

Proof. Without loss of generality, we may assume that K is countable. Let φ(x̄, ū), ψ(x̄, ȳ, ū) be L-formulas and b̄ ∈ K be such that φ(x̄, b̄) defines B and ψ(x̄, ȳ, b̄) defines the graph of f. By 7.4, the difference field K embeds into an ultraproduct ∏_{q} F_{q}/U of the difference fields F_{q}. Take a representative (b̄_{q})_{q} of b̄. Then for almost all q (in the sense of the ultrafilter U), we have that φ(x̄, b̄_{q}) defines a group B_{q} in F_{q}, and ψ(x̄, ȳ, b̄_{q}) defines an endomorphism f_{q} of B_{q}. Note also that because B is of finite SU-rank, there is an integer m such that every element ā of B satisfies σ^{m}(ā) ∈ cl_{σ}(b̄)(ā, …, σ^{m−1}(ā))^{alg}. Hence the same is also true for almost all B_{q}. By 7.6, the B_{q} are therefore finite, and we obtain trivially that

[B_{q} : f_{q}(B_{q})] = Card(Ker(f_{q})).

Thus the same holds in K. Note that Ker(f) is infinite if and only if the size of the Ker(f_{q}) is unbounded.

7.8. Application to families of finite simple groups. With the exception of the sporadic finite simple groups and the alternating groups A_{n}, the finite simple groups are members of infinite families. Some of these families are of the form G(𝔽_{q}) for some linear algebraic group G (for instance SL_{n}(𝔽_{q})). Hence looking at G(F) for F a pseudo-finite field, one gets uniformity results on the family G(𝔽_{q}), q a prime power.

Some of these families however do not originate directly from a simple algebraic group, but their definition involves some automorphism of the field 𝔽_{q}. They are the so-called “twisted finite simple groups”. And they become definable in the structure (𝔽_{p}^{alg}, σ_{q}). Using 7.4, one then also gets uniformity results for these families.

Typical examples of uniformity results: we know that in a finite simple group H, every non-trivial conjugacy class X generates the whole group. The uniformity gives a bound on the number N such that H = (X · X^{−1})^{N}.

7.9. The Jacobi conjecture for difference fields.

Let n ≥ 1, let u_{1}(X_{1}, …, X_{n}), …, u_{n}(X_{1}, …, X_{n}) ∈ K[X_{1}, …, X_{n}]_{σ} (K a difference field, and we will assume it is a model of ACFA). The equations

u_{1}(x_{1}, …, x_{n}) = ⋯ = u_{n}(x_{1}, …, x_{n}) = 0

define a σ-closed subset of K^{n}, and some of the irreducible components of this set will be of finite SU-rank. If Y is an irreducible component of finite SU-rank of this set, let us define the order of Y as Sup{tr.deg(cl_{σ}(Eā)/E) | ā ∈ Y }, where E = acl_{σ}(E) is such that the elements u_{1}, …, u_{n} are defined over E. The Jacobi’s conjecture gives an explicit bound H on the order of the irreducible components of finite SU-rank. This bound is defined as follows: for each k and i, define h_{k}^{i} to be the order of u_{k} when viewed as a difference polynomial in x_{i} (so, it equals m if x_{i}^{σ^{m}} appears in u_{k} and x_{i}^{σ^{m+1}} does not). One then sets

H = max_{θ∈Sym(n)} ∑_{k=1}^{n} h_{k}^{θ(k)}.

Hrushovski uses the fact that if an irreducible σ-closet set Y has order d, then in a structure F_{q}, it will have approximately cq^{d} points for some fixed constant c. Thus the order of Y will be lim_{q} log_{q}(Card(Y (F_{q})). One uses this remark to reduce the Jacobi’s conjecture to a problem about number of points in algebraic sets.

7.10. Modular subgroups. One can show that if one has an exact sequence

0 → A → B → C → 0

of definable groups (in a model K of ACFA), then B is modular [and stable] if and only if A and C are modular [and stable]. Hence the study of modular definable subgroups of an algebraic group reduces to the study of definable subgroups of simple algebraic groups, i.e., of 𝔾_{a}(K), 𝔾_{m}(K), or of a simple abelian variety.

In characteristic 0, one can show that no proper definable subgroup of 𝔾_{a}(K) is definable. Indeed, it is fairly easy to show that such a group is commensurable with one defined by an equation of the form ∑_{i=0}^{n} a_{i}σ^{i}(x) = 0, with a_{i} ∈ K (recall that two definable groups G_{1} and G_{2} are commensurable if [G_{1} : G_{1} ∩ G_{2}] and [G_{2} : G_{1} ∩ G_{2}] are both finite). Furthermore the operator ∑_{i=0}^{n} a_{i}σ^{i} can be written as a composition of operators of the form σ − b, and the sets defined by the equation σ(x) − bx are non-orthogonal to the fixed field.

Let A be an abelian variety, defined over the fixed field F of the model K of ACFA. Then σ(A) = A, and therefore, if f(T) = ∑_{i=0}^{n} a_{i}T^{i} ∈ ℤ[T], then the equation

f(σ)(x) = ∑_{i=0}^{n}[a_{i}]σ^{i}(x) = 0

defines a subgroup of A(K). (Here, [a_{i}]x denotes x + ⋯ + x a_{i} times). We will denote this group by Ker(f(σ)).

A beautiful result of Hrushovski states that in characteristic 0, f(T) is relatively prime to all cyclotomic polynomials T^{m} − 1 if and only if Ker(f(σ)) is modular, and therefore stable. [He actually has a complete characterisation of the modular definable subgroups of an arbitrary abelian variety.]

A similar statement holds for definable subgroups of 𝔾_{m}(K): if f(T) = ∑_{i} a_{i}T^{i} ∈ ℤ[T], then the equation ∏ σ^{i}(x^{a_{i}}) = 1 defines a subgroup of 𝔾_{m}(K) (denoted Ker(f(σ))), and this subgroup is modular if and only if f(T) is relatively prime to all cyclotomic polynomials T^{m} − 1, m ≥ 1.

7.11. Application to the Manin-Mumford conjecture.

The Manin-Mumford conjecture is a conjecture in number theory, of which various versions have been proved by various people. Here I will only state the version of which Hrushovski gives a proof using difference fields. It is weaker than the full Manin-Mumford conjecture, because of the restriction on the field of definition, but is stronger in the sense that the constant M below can be effectively computed from the data. If G is a group, let us denote by T or(G) the set of torsion elements of G. I will certainly not give the full proof, but will try to indicate the main steps of the proof and some of its ingredients.

Theorem. Let A be a commutative algebraic group defined over some number field k (= finite extension of ℚ), and let X be a subvariety of A. Then

X ∩ T or(A) = ⋃_{i=1}^{M} a_{i} + T or(A_{i}),

where the A_{i} are group subvarieties of A. The number M is bounded above by c deg(X)^{e}, where c and e are constants depending on A, and deg(X) is the degree of X given some embedding of A (and of X) in projective space.

7.12. A first idea for a proof. Show that there is σ ∈ Gal(k^{alg}/k) and a functional equation f(σ)(x̄) = 0 valid on T or(A) and such that: whenever K is a model of ACFA extending (k^{alg}, σ), then the equation f(σ)(x̄) = 0 defines a modular subgroup of A(K).

Assume that we have done that. Then:

(1) We get the qualitative version of the result, because of modularity. Indeed, let B be the subgroup of A(K) defined by the equation f(σ)(x̄) = 0. Then by the results we got above, X ∩ B is a Boolean combination of cosets of quantifier-free definable subgroups of B. We can therefore write X ∩ B as a finite union of sets of the form (a + C) \ U, where C is a σ-closed subgroup of B, irreducible as a σ-closed set, and U is a union of cosets which are strictly contained in a + C. The σ-closure of (a + C) \ U equals a + C, and must also be contained in the σ-closed set X ∩ B. This shows that X ∩ B is in fact a finite union of cosets of definable subgroups of B. Write

X ∩ B = ⋃_{i=0}^{n} a_{i} + B_{i}.

Now, if a_{i} + B_{i} intersects T or(A), then we may take a_{i} ∈ T or(A). Note also that a_{i} + B_{i} ⊂ X implies that a_{i} + A_{i} ⊂ X, where A_{i} is the Zariski closure of the subgroup B_{i} of A, and therefore is an algebraic subgroup of A. Hence we have that a_{i} + T or(A_{i}) ⊂ X, and this gives the result.

(2) If we have explicit bounds on the degree of f and the absolute values of its coefficients, then we get an explicit bound on the number of irreducible components of the σ-closed set X ∩ B, and hence on the number M of cosets.

Unfortunately, this strategy needs to be slightly modified. First of all, the commutative algebraic group A may have a vector subgroup, i.e., an algebraic subgroup V which is isomorphic to 𝔾_{a}^{n} for some n. As we saw above, no definable subgroup of 𝔾_{a}^{n}(K) is modular. So the first step of the proof is to show that it is enough to prove the result for A/V, where V is the maximal vector subgroup of A. This is done using an easy algebraic result, and a more complicated modeltheoretic one showing that if B is a definable subgroup of A(K) which is such that B/B ∩ V is modular, then the intersection of B with any set is of a special form (see below 7.15). So, one reduces the proof of the theorem to the case where A has no algebraic subgroups isomorphic to 𝔾_{a} (such an algebraic group is called a semi-abelian variety).

Also, it turns out that finding explicit bounds on coefficients and degree of the equation is not effective. One bypasses this difficulty by looking first at T or_{p′}(A), the prime-to-p torsion subgroup of A, proving the desired result there, and then using another prime ℓ. This produces explicit but very ugly bounds on the number n of cosets.

More precisely: We get a bound for the number of cosets in X ∩ T or_{p′}(A), and this bound only depends on the degree of X. Hence, it also works for (a+X)∩T or_{p′}(A), for any a ∈ T or(A). Now one uses that T or_{p′}(A)+T or_{ℓ′}(A) = T or(A). But there is definitely some work involved. For details, please see [6].

7.13. Why do we need a bound on the coefficients of the equation? Let f(T) = ∑_{i=0}^{ℓ} m_{i}T^{i} ∈ ℤ[T]. We are interested in the number of ir-

reducible components of the σ-closed set X ∩ Ker(f(σ)), and we can find a bound on this number as follows. We first look at the Zariski closed sets Y = (X×X^{σ}×⋯×X^{σ^{ℓ}}) and C = {(x_{0}, …, x_{ℓ}) ∈ A^{ℓ+1} | ∑_{i=0}^{ℓ} m_{i}x_{i} = 0}. Then the number of irreducible components of Y ∩C is bounded by deg(X)^{ℓ+1}deg(C), where the degree is computed via a certain embedding of A in projective space. Unfortunately, the degree of C will depend on the values of the |m_{i}| (and increase if they do – of course), which means that already to know the number of irreducible components of Y ∩ C, we need to know things about f(T).

7.14. Existence of the equation, and some bounds. Here Hrushovski chooses a prime p of good reduction (for A). Grosso modo, this means that, reducing modulo p the equations defining A, one gets an algebraic group Ā defined over some finite field 𝔽_{q}, and which resembles A. Moreover, let L_{p} be the field generated over k by the elements of T or_{p′}(A). Then L_{p} is an unramified extension of the field k, i.e., reduction “mod p”: O_{k} → 𝔽_{q} extends to a homomorphism O_{L_{p}} → 𝔽_{p}^{alg}, and this homomorphism defines a 1-1 map on T or_{p′}(A) (with values in T or_{p′}(Ā)).

The automorphism x ↦ x^{q} of 𝔽_{p}^{alg} lifts to an automorphism of L_{p} fixing k, and one takes for σ any automorphism of ℚ^{alg} extending this automorphism.

It then suffices to find the functional equation on Ā (because reduction mod p is injective on the torsion). Observing that if

0 ⟶ A_{1} ⟶ A_{3} ⟶ A_{2} ⟶ 0

is a short exact sequence of (connected) commutative algebraic groups, then T or(A_{3}) contains T or(A_{1}) and projects into T or(A_{2}), reduces to finding such an equation (and bounds on its coefficients and degree) for each of the simple factors of Ā: for the abelian ones, its existence and bound on the degree and absolute value of the coefficients are given by a result of Weil. For products of copies of 𝔾_{m}, it is more or less σ(x) − x^{q}, and 𝔾_{a}(K) has no torsion elements, so we do not need to look at what happens in 𝔾_{a}(𝔽_{p}^{alg}).

We have therefore taken care of the p′-torsion, and furthermore have shown that there is an effective bound on the number n such that

T or_{p′}(A) ∩ X = ⋃_{i=1}^{n} a_{i} + T or_{p′}(A_{i})

for some algebraic subgroups A_{i} of A and elements a_{1}, …, a_{n}. Take now another prime ℓ of good reduction. Proceeding as above, one gets an automorphism τ of ℚ^{alg}, such that τ satisfies a functional equation g(τ) on T or_{ℓ′}(A), and therefore also on T or_{p}(A) (the p-component of T or(A)). Let M_{p} = k(T or_{p}(A)). Then a result of Serre tells us that L_{p} ∩ M_{p} is a finite extension of k, say of degree m. Hence there is an automorphism ρ of L_{p}M_{p}, which extends σ^{m} on L_{p} and τ^{m} on M_{p}. Note the following: write f(T) = a ∏(T −α_{i}), where a ∈ ℤ, the α_{i} are in ℂ. Then Ker(f(σ)) is modular if and only if none of the α_{i} is a root of unity. Also, let h(T) = ∏(T − α_{i}^{m}), and choose b ∈ ℤ such that bh(T) ∈ ℤ[T]; then bh(σ^{m}) vanishes on T or_{p′}(A), and is relatively prime to all cyclotomic polynomials, hence defines a modular subgroup of A(K). Reasoning similarly with τ, we get that there is a modular subgroup B of some model of ACFA extending (L_{p}M_{p}, ρ), which contains T or(A). This gives us the qualitative version of Manin-Mumford conjecture: we have shown that T or(A) ∩ X is a finite union of cosets of the torsion subgroups of some connected algebraic subgroups of A. This fact is used in the proof.

Unfortunately, we do not know a priori that the constant m of Serre’s result has an effective bound. This means that we do not know a bound on the complexity of the set B defining our modular subgroup, and hence cannot derive a bound on the number of irreducible components of the σ-closed set B ∩X. However, we do have explicit bounds (depending on p and ℓ) on the number of cosets appearing in the decompositions of T or_{p′}(A)∩X and T or_{ℓ′}(A)∩X, and this will be enough to apply the strategy alluded to at the end of (7.12).

7.15. The reduction to the semi-abelian variety case. We will state here the two results which are needed to reduce the problem from the commutative algebraic group A to the semi-abelian variety A/V. We will not state the modeltheoretic result in its full generality, only the particular case that is of interest to us.

Definition. Let A be a commutative algebraic group, V its maximal vector group. We call a subvariety X of A special if X = Y +C, where Y is a subvariety of V, and C is a coset of a (connected) algebraic subgroup E of A.

We call a definable subset D of A(K) special, if X = Y + C, where Y is a definable subset of V (K), C is a coset of a definable subgroup of A(K).

7.16. Proposition. Let A be a commutative algebraic group, and V its maximal vector subgroup, π : A → A/V. Let B be a definable subgroup of A(K) of finite SU-rank, and assume that π(B) is modular. Then every definable subset of B is a Boolean combination of special sets.

The proof uses the following crucial ingredients:

(1) The fact that any definable subset of F ix(σ)^{n} is definable with parameters from F ix(σ) (see 4.3).

(2) The stability and the modularity of π(B) with the structure induced by K.

(3) The fact that B ∩ V is strongly related to F ix(σ). [In fact, there is a definable bijection between B ∩ V and F ix(σ)^{k}]

(4) The fact that any definable subset of (V ∩B)×π(B) is a Boolean combination of “rectangles”, i.e., sets of the form U_{1} ×U_{2} where U_{1}, U_{2} are definable and included in B ∩ V and π(B) respectively.

(5) If Y is a definable subset of B, then there is a group H, contained in (Y · Y^{−1})^{n} for some n (and hence definable), such that Y H/H is finite. This is a property of groups of finite SU-rank.

7.17. Corollary. Let B be as above, and X a subvariety of A. Then X ∩ B is contained in a finite union of finitely many special varieties which are contained in X. Proof. Go to the Zariski closure of X ∩ B.

7.18. Lemma. Let A be a commutative algebraic group, T the group of torsion points of A (or the group of torsion points of order prime to p, for some prime p), and assume that X ∩ T is contained in the union of the special subvarieties D_{i} of X, i = 1, …, M. Then the Zariski closure of X ∩ T is the union of at most M cosets of connected algebraic subgroups of A. Proof. It is enough to show that if D is a special subvariety of X, then the Zariski closure of D ∩ T is a coset of a group subvariety of A.

Write D = C + Y, where C is a coset of the connected group variety E, and Y is a subvariety of V. We will first show that we may assume that Y is contained in an algebraic subgroup V_{1} of V which intersects E in (0). Indeed, because V is a vector space, there is a definable endomorphism π : V → V, with π^{2} = π and Ker(π) = V ∩E. Then E +Y = E +π(Y) because Ker(π) ⊆ E, and C = C +E because C is a coset of E. Therefore

D = C + Y = C + E + Y = C + E + π(Y) = C + π(Y),

and we may replace Y by π(Y) ⊆ π(V) = V_{1}, and V_{1} ∩ E = (0).

Fix d_{0} ∈ D∩T, and for d ∈ D∩T define f(d) = d−d_{0}. Then f(d) ∈ (E+V_{1})∩T. Write f(d) = e + y, with e ∈ E, y ∈ V_{1}. Since f(d) is a torsion element and V_{1} ∩ E = (0), we get that necessarily y = 0 (V has no torsion elements), so that f(d) ∈ E, and d ∈ E + d_{0}. Hence

D ∩ T = (E + d_{0}) ∩ T = (E ∩ T) + d_{0}.

The only thing remaining to be shown, is that the Zariski closure B of E ∩ T is a connected subgroup of A. Let B^{0} be the connected component of B. Because E ∩ T is divisible, E ∩ T has no subgroup of finite index, which implies that B^{0} ∩ T = B, so that B = B^{0}.

7.19. Exercise 16. Let (E, σ) be an algebraically closed difference field, let m ≥ 1 and let (L, τ) be a difference field extending (E, σ^{m})). For each i = 1, …, m − 1, choose L_{i} realising σ^{i}(tp_{ACF} (L/E)), linearly disjoint from the composite field of L_{0} = L, …, L_{i−1}) over E. Let f_{0} = id_{L}, and for i = 1, …, m− 1 let f_{i} : L → L_{i} be an isomorphism extending σ^{i}. For i = 0, …, m − 2, define σ_{i} : L_{i} → L_{i+1} by

σ_{i} = f_{i+1}f_{i}^{−1}

and define σ_{m−1} : L_{m−1} → L_{0} by

σ_{m−1} = τf_{m−1}^{−1}.

(1) Show that the σ_{i} agree with σ on E.

(2) Show that σ_{m−1} ⋯ σ_{0} = τ.

(3) Show that there is a difference field (M, σ) containing (E, σ) and such that M contains L, and σ^{m} agrees with τ on L.

(4) Deduce that if K is a model of ACFA, then (K, σ^{m}ϕ^{n}) is also a model of ACFA (m ≥ 1, n ∈ ℤ).

(5) Deduce that if K is a model of ACFA, and F = F ix(σ^{m}ϕ^{n}), then F is a pseudo-finite field.

(6) Let F be as in (5). Show that every definable (in K) subset of F^{k} is definable with parameters from F. If furthermore m = 1, then show that the structure induced on F is the pure field structure. [Repeat the proof of Proposition 4.3. This result does not extend to the case m > 1, as σ defines then an automorphism of F, which is not definable in the pure field language.]

References

Algebraic results

[1] R.M. Cohn, Difference algebra, Tracts in Mathematics 17, Interscience Pub. 1965.

[2] S. Lang, Introduction to algebraic geometry, Addison-Wesley Pub. Co., Menlo Park 1973.

Model theoretic results on difference fields

[3] A. Macintyre, Generic automorphisms of fields, APAL 88 Nr 2-3 (1997), 165 – 180.

[4] Z. Chatzidakis, E. Hrushovski, Model theory of difference fields, Trans. Amer. Math. Soc. 351 (1999), pp. 2997-3071.

[5] Z. Chatzidakis, E. Hrushovski, Y. Peterzil, Model theory of difference fields, II: Periodic ideals and the trichotomy in all characteristics, preprint 1999. To appear in Proc. of the LMS

[6] E. Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, APAL 112 Nr 1 (2001), 43 – 115.

[7] E. Hrushovski, The first-order theory of the Frobenius, preprint (1996).

[8] A. Macintyre, Nonstandard Frobenius, in preparation.

The papers [5] to [7] definitely fall into the category of “further advanced reading”. Some of the easy parts of [6] appear in survey papers, see below.

Related model-theoretic results on finite fields.

[9] J. Ax, The elementary theory of finite fields, Annals of Math. 88 (1968), 239 – 271.

[10] E. Hrushovski, Pseudo-finite fields and related structures, manuscript 1991.

[11] E. Hrushovski, A. Pillay, Groups definable in local fields and pseudo-finite fields, Israel J. of Math. 85 (1994), 203 –262.

[12] E. Hrushovski, A. Pillay, Definable subgroups of algebraic groups over finite fields, J. reine angew. Math. 462 (1995), 69 –91.

The proofs of many of the results of the first few chapters of these notes are essentially translations of proofs appearing in [10]. See also [11]. In [12], one uses results of [11] to obtain very pretty results on groups definable in finite fields.

Other model-theoretic results

[13] L. van den Dries, K. Schmidt, Bounds in the theory of polynomials rings over fields. A non-standard approach. Invent. Math. 76 (1984), 77 – 91.

[14] U. Hrushovski, A. Pillay, Weakly normal groups, in: Logic Colloquium 85, North Holland 1987, 233 – 244.

[15] B. Kim, A. Pillay, Simple theories, APAL 88 Nr 2-3 (1997), 149 – 164.

[16] A. Pillay, Definability and definable groups in simple theories, JSL 63 (1998), 788 – 796.

Stability is a vast subject, with an abundant literature. Here are some books for the interested reader, but there are many others.

[17] S. Buechler, Essential stability theory, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1996.

[18] A. Pillay, An introduction to stability theory, Oxford Logic Guide 8, Clarendon Press, Oxford, 1983.

[19] F. Wagner, Simple theories, Kluwer Academic Pub., Dordrecht 2000.

Survey papers

[20] E. Bouscaren, Théorie des modèles et conjecture de Manin-Mumford, [d’après Ehud Hrushovski], Séminaire N. Bourbaki exposé 870 (Mars 2000), to appear in Astérisque.

[21] Z. Chatzidakis, Groups definable in ACF A, in: Algebraic Model theory, B. Hart et al. ed., NATO ASI Series C 496, Kluwer Academic Publishers 1997, 25 – 52.

[22] Z. Chatzidakis, A survey on the model theory of difference fields, Proc. Workshop, Haskell, Pillay, Steinhorn Ed., MSRI Publications 39, Cambridge University Press, 65 – 96.

[23] A. Pillay, ACF A and the Manin-Mumford conjecture, in: Algebraic Model theory, B. Hart et al. ed., NATO ASI Series C 496, Kluwer Academic Publishers 1997, 195 – 205.

UFR DE MATHÉMATIQUES

UNIVERSITÉ PARIS 7 - CASE 7012

2, PLACE JUSSIEU

75251 PARIS CEDEX 05

FRANCE

E-mail: zoe@logique.jussieu.fr