Minimal types in separably closed fields

Zoé Chatzidakis Université Paris 7, CNRS Carol Wood Wesleyan University §0. Introduction.

In [1], examples of types of U-rank 1 (i.e., minimal types) in the theories of separably closed fields were constructed, en route to displaying certain dimension phenomena. We construct here additional examples with U-rank 1 and of various transcendence degrees over arbitrary separably closed fields. Our examples include ones which are minimal but of infinite transcendence degree, i.e., not thin. Our interest in building new examples was piqued after seeing the role played by minimal types over separably closed fields in Hrushovski’s analysis of abelian varieties. In [2] Delon shows, in the setting of separably closed fields of finite degree of imperfection, that there is a Zariski geometry on the set of realizations of any minimal type. Her proof generalizes to the setting of our examples as well. This article is the result of several working sessions between the authors at Wesleyan University and Paris 7, and was completed during the Model Theory of Fields program at MSRI in 1998. We are grateful for the hospitality and support of all three institutions. We thank Elisabeth Bouscaren and Françoise Delon for reading an earlier version of this paper, providing useful suggestions and corrections.

Background on separably closed fields. Let K be a field of characteristic p > 0. Then K^{p} is a subfield of K, isomorphic to K via the Frobenius map x ↦ x^{p}. Thus K is naturally a K^{p}-vector space, of dimension either infinite or p^{e} for some natural number e. The degree of imperfection is taken to be infinite in the first case, and equal to e otherwise.

Fix a prime number p, and let L = {+, −, ·, 0, 1} be the language of rings. For each element e of N ∪ {∞}, we take SCF_{e} to be the L-theory of separably closed fields of characteristic p and degree of imperfection e. Ershov [4] showed that each SCF_{e} is complete, and that all completions of the theory of separably closed fields are of this form. Furthermore, any complete theory of separably closed fields is stable (Wood [5]), and is superstable only when it is the theory of algebraically closed fields (i.e., when e = 0).

Let K be a field of characteristic p. We say that B ⊆ K is p-independent if the set of all p-monomials in B, i.e., monomials of the form b_{1}^{i(1)} ⋯ b_{n}^{i(n)} with b_{1}, …, b_{n} ∈ B and 0 ≤ i(1), …, i(n) ≤ p − 1, is linearly independent in the K^{p}-vector space K. A maximal p-independent subset B of K is called a p-basis, and in this case K = K^{p}[B].

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

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

Note that these functions depend on the field K, and that the above properties define them uniquely. These functions are also definable in the pure ring language. Consider the language L_{λ} = L ∪ {λ_{i,n} | n ∈ N, 0 ≤ i < p^{n}}, and let SCF_{e,λ} be the theory in the language L_{λ} obtained by adjoining to SCF_{e} axioms expressing the defining properties of the functions λ_{i,n}. Then SCF_{e,λ} is complete and eliminates quantifiers but does not eliminate imaginaries (Delon [3]).

Fix a separably closed field K. When the degree of imperfection of K is finite, one may fix a p-basis {b_{1}, …, b_{e}}, and only consider the one variable functions λ_{i,e}(b_{1}, …, b_{e}, x). Then T h(K) eliminates quantifiers and imaginaries in the language L∪{b_{1}, …, b_{e}, λ_{i,e}(b_{1}, …, b_{e}, −) : 0 ≤ i < p^{e}} (Delon [3]).

Let A ⊆ K be a set. Then the definable closure of A, dcl(A), is the smallest subfield of K containing A and closed under the functions λ_{i,n}. The algebraic closure of A, acl(A), is the separable closure of dcl(A). (Again, see Delon [3] for details.)

If K ≺ L and c̄ is a tuple of elements of L we denote the field dcl(K, c̄) by K⟨c̄⟩. By quantifier elimination, the type of c̄ over K is completely determined by the L-isomorphism type of K⟨c̄⟩ over K.

Let B ⊂ K be p-independent, and assume that c ∈ L is such that c ∈ L^{p^{n}}[B] for every n ∈ N. Consider the infinite tuple obtained from c by iterating the functions λ_{i,n}(b_{1}, …, b_{n}, −) for n ∈ N, b_{1}, …, b_{n} ∈ B, and 0 ≤ i < p^{n}. This infinite tuple, denoted c_{∞}, is called the B-tree of c. This tree c_{∞} has a natural notion of level, defined by induction: the level of c is 0, and if t is an element of level m, and n ∈ N, b_{1}, …, b_{n} ∈ B, then for each i, 0 ≤ i < p^{n}, the element λ_{i,n}(b_{1}, …, b_{n}, t) has level m+1. For m ∈ N we also define c_{≤m} to be the tuple consisting of the elements of c_{∞} of level ≤ m.

Claim. K⟨c⟩ = K(c_{∞}).

Proof. It suffices to show that K(c_{∞}) is closed under the functions λ_{i,n}. Extend B to a p-basis B′ of K. Then the elements of the B′-tree of c all belong to the B-tree of c, and we may therefore assume that B is a p-basis of K.

By construction of c_{∞}, for every n ∈ N we have: K(c_{≤n}) ⊆ (K(c_{≤n+1}))^{p}[B]. Hence K(c_{∞}) = (K(c_{∞}))^{p}[B], from which we deduce successively: B is a p-basis of K(c_{∞}); L^{p} and K(c_{∞}) are linearly disjoint over (K(c_{∞}))^{p}; L is a separable extension of K(c_{∞}); K(c_{∞}) is an L_{λ}-substructure of L.

Thus the type of c over K is completely described by the L-isomorphism type of c_{∞} over K. If we define tp(c/K)|_{m} to be the subset of tp(c/K) describing the quantifier-free L-type of the tuple c_{≤m} over K, then T h(K) ∪ ⋃_{m} tp(c/K)|_{m} ⊢ tp(c/K). We say that a partial type q(x_{1}, …, x_{n}) is complete up to level m if q implies a complete quantifier-free L-type over K in the variables (x_{1,≤m}, …, x_{n,≤m}). By definition, if q is a complete type over K then q|_{m} is complete up to level m.

§1. Constructions of types of arbitrary transcendence degree.

Conventions. Throughout this section we fix p and we consider types over K a countable model of SCF_{e} for nonzero e. We further fix an element u of K \ K^{p}, and assume L to be a sufficiently saturated elementary extension of K.

In the constructions of this section, all types under consideration will be p-dependent on u, i.e., will lie in L^{p^{n}}[u] for every n ∈ N. Thus an element will be given by its {u}-tree. For ease of notation, we write a realization a of such a type as

a = ∑_{i=0}^{p−1} a_{i}^{p}u^{i}.

In particular, the subscript 0 corresponds to the coefficient of 1 (= u^{0}) and the index 1 corresponds to the coefficient of u. We concatenate subscripts as we iterate this process, for example

a_{σ} = ∑_{i=0}^{p−1} a_{σi}^{p}u^{i}.

For n ∈ N, we let p^{n} denote the set of sequences of length n of elements of {0, …, p−1}, and set p^{≤n} = ⋃_{0≤m≤n} p^{m}, p^{<ω} = ⋃_{m∈N} p^{m}. Then K⟨a⟩ = K(a_{σ} | σ ∈ p^{<ω}). If q = tp(a/K), then tr.deg.(q) (the transcendence degree of q) denotes the (field) transcendence degree of K⟨a⟩ over K. We denote by a_{≤m} the tuple of elements defined by stage m, i.e., the tuple (a_{σ})_{σ∈p^{≤m}}, and by tr.deg.(q|_{m}) the transcendence degree of K(a_{≤m}) over K. We make repeated use of the following easy consequence of the nature of definable closure:

Fact. If there exist v_{0} and v_{1} ∈ L such that v_{0}^{p} + v_{1}^{p}u ∈ K, then v_{0}, v_{1} ∈ K. By working in the setting of a fixed u, we see easily that the constructions of [1] are valid for separably closed fields of any degree of imperfection, including the infinite case. Indeed, over any subfield k_{0} of L containing u, [1] provides an infinite family of pairwise orthogonal types of U-rank 1 for any (nonzero) value of e.

Lemma 1 (cf. Th. 3.2 of [1]) Assume that a, b, a_{1}, b_{1} ∈ L satisfy

\begin{aligned}a & = a_{1}^{p} + b^{p}u \\ b & = b_{1}^{p} + a^{p}u. \end{aligned}

Then

(1) K(a, b, a_{1}, b_{1}) is separably algebraic over K(a_{1}, b_{1}).

(2) (a, b) and (a_{1}, b_{1}) are equi-algebraic over K.

(3) a ∈ K ⟺ b ∈ K ⟺ K(a, b, a_{1}, b_{1}) = K. Proof. (1) The equation X = a_{1}^{p} + (b_{1}^{p} + X^{p}u)^{p}u is separable in X, and has a for solution. Hence a is separable over K(a_{1}, b_{1}), and similarly for b = b_{1}^{p} + a^{p}u.

(2) Clearly a_{1} and b_{1} are algebraic over K(a, b). By (1) we know that a and b are algebraic over K(a_{1}, b_{1}).

(3) Assume that a ∈ K. By the Fact above, a_{1} and b are also in K. Thus b_{1} (= (b − a^{p}u)^{1/p})∈ K. Similarly, b ∈ K implies that a, a_{1} and b_{1} are in K.

Notation. Fix an integer n, and consider the system of equations Σ(X_{i,j}), 1 ≤ i ≤ n, 1 ≤ j ≤ 2n − 1, defined by the equations (k) below, for k = 1, …, 2n − 2, where we adopt the convention that X_{ℓ,j} stands for the variable X_{ℓ−n,j} whenever ℓ > n:

(k)  \begin{aligned}X_{k,k} & = X_{k,k+1}^{p} + X_{k+1,k}^{p}u \\ X_{k+1,k} & = X_{k+1,k+1}^{p} + X_{k,k}^{p}u, \\ X_{i,k} & = X_{i,k+1}^{p}  if i ≠ k, k + 1, k − n, k − n + 1. \end{aligned}

Lemma 2. With notation as above,

(1) Σ(X_{i,j}) is consistent.

(2) If (a_{i,j}) is a solution of Σ(X_{i,j}) in L, then K(a_{i,j})_{i,j} is separably algebraic over K(a_{1,2n−1}, …, a_{n,2n−1}).

(3) If, in addition, a_{i,1} ∈ K for some i, then all the a_{i,j}’s lie in K.

Proof. (1) Let a_{1,2n−1}, …, a_{n,2n−1} be n elements of L. By reverse induction on k and by Lemma 1, each system (k) is consistent.

(2) By Lemma 1, for 0 ≤ k ≤ 2n − 2 the elements a_{i,k}, i = 1, …, n, are separably algebraic over K(a_{1,k+1}, …, a_{n,k+1}). Hence the tuple (a_{i,j}) is separably algebraic over K(a_{1,2n−1}, …, a_{n,2n−1}).

(3) Assume that a_{i,1} ∈ K. Then the systems of equations (1) through (i − 2) give: a_{i,i−1} = a_{i,1}^{1/p^{i−2}} ∈ K.

By Lemma 1 applied to the equations (i − 1), we deduce from a_{i,i−1} ∈ K that a_{i−1,i−1}, a_{i−1,i} and a_{i,i} are in K. Proceeding in the same manner with the equations (i), …, (2n − 2), we obtain successively that for k = i − 1, …, 2n − 2, the elements a_{k+1,k}, a_{k,k+1} and a_{k+1,k+1} are in K. For k ranging from i − 1 to 2n − 2, we have a_{k,k+1} ∈ K. (Recall our convention that a_{ℓ,j} stands for a_{ℓ−n,j} when ℓ > n). Therefore a_{k,2n−1} = a_{k,k+1}^{1/p^{2n−k−2}} ∈ K. All possible first subscripts occur in the range of values we have for k, and so a_{1,2n−1}, …, a_{n,2n−1} ∈ K. Since K is separably closed, assertion (2) suffices to show that the a_{i,j}’s lie in K.

Definition. For n as above and 1 ≤ j ≤ 2n − 1, let η_{j} ∈ p^{j−1} be the sequence of zeros of length j − 1 (so η_{1} = ∅), and let Φ_{n}(z_{1}, …, z_{n}) be the partial type given by: the tuple (z_{i,η_{j}})_{1≤i≤n,1≤j≤2n−1} solves the system Σ(X_{i,j}).

The resulting Φ_{n} is consistent, and becomes complete up to level 2n − 2 once one requires that z_{1}, …, z_{n} be algebraically independent over K.

Lemma 3. Let p|_{m} be a 1-type complete up to level m, and assume that σ(1), …, σ(k) ∈ p^{m} are such that if a realizes p then {a_{σ(1)}, …, a_{σ(k)}} is a separating transcendence basis of K(a_{≤m}) over K.

Given a k-type q|_{n} complete up to level n and realizable by a k-tuple of algebraically independent elements, consider the partial type r(x) = p|_{m}(x)∪q|_{n}(x_{σ(1)}, …, x_{σ(k)}). Then r is consistent, and complete up to level m + n.

Proof. Let b_{1}, …, b_{k} ∈ L realize q|_{n}. Choose a_{≤m} (in some algebraic closure of L) such that a_{σ(j)} = b_{j} for j = 1, …, k, and such that for any c realizing p|_{m}, the tuples a_{≤m} and c_{≤m} have the same quantifier-free L-type over K.

Then a_{≤m} is separably algebraic over K(b_{1}, …, b_{k}), and therefore is in L. Hence, a_{≤m+n} ∈ K(b_{1,≤n}, …, b_{k,≤n}, a_{≤m}), which gives the result.

Theorem 1. For any d ≥ 1, there exists a type over K of U-rank 1 and of transcendence degree d.

Proof. By results of [1], we may choose p_{1}, …, p_{d} to be pairwise orthogonal 1-types over K, each of U-rank 1 and transcendence degree 1, and such that all elements definable over realizations of p_{1}, …, p_{d} are p-dependent on u. For d = 1 there is nothing new to be proved. We assume d > 1.

Let Ȳ be a countably infinite tuple of variables, and let X_{∞} = {X_{σ} | σ ∈ p^{<ω}}, X_{≤n} = {X_{σ} | σ ∈ p^{≤n}}. Fix an enumeration f_{n}, n ∈ N, of the elements of K[X_{∞}, Ȳ] such that f_{n} ∈ K[X_{≤n}, Ȳ] for every n.

We will build the type q by induction on n. We assume that at stage n we have built a type q_{n} = q_{n}|_{N(n)} complete up to level N(n), and satisfying:

(a) tr.deg.(q_{n}) = d and q_{n} extends q_{n−1},

(b) for any tuple c̄ from L and a realizing q_{n}, if f_{n−1}(a_{≤n−1}, c̄) = 0, then either the polynomial f_{n−1}(X_{≤n−1}, c̄) vanishes on all realizations of q_{n}, or a ∈ acl(K, c̄).

For n = 0, the condition is vacuous. Choose N(0) so that p^{N(0)} ≥ d. Let σ(1), …, σ(d) be distinct elements of p^{N(0)}, and consider the partial type q_{0} which says that x_{σ(1)}, …, x_{σ(d)} are algebraically independent over K, and that x_{σ} = 0 for σ ∈ p^{N(0)} \ {σ(1), …, σ(d)}. Then q_{0} is complete up to level N(0).

Assume that we have built q_{n} satisfying the induction hypothesis. We will now build q_{n+1}, in two steps. Choose τ(1), …, τ(d) ∈ p^{N(n)} such that if a is a realization of q_{n} then {a_{τ(1)}, …, a_{τ(d)}} is a separating transcendence basis of K(a_{≤N(n)}) over K.

Step 1.

Consider f_{n}(X_{≤n}, Ȳ). By our choice of τ(1), …, τ(d), we know that a_{≤N(n)} is separably algebraic over K(a_{τ(1)}, …, a_{τ(d)}). Working modulo the theory of fields and using compactness, there are polynomials h_{j} ∈ K[Z_{1}, …, Z_{d}, Ȳ], j = 1, …, m, such that if a realizes q_{n} and c̄ is any tuple such that f_{n}(X_{≤n}, c̄) does not vanish on all realizations of q_{n}, then

f_{n}(a_{≤n}, c̄) = 0 → ⋁_{j=1}^{m}(h_{j}(a_{τ(1)}, …, a_{τ(d)}, c̄) = 0 ∧ h_{j}(Z_{1}, …, Z_{d}, c̄) ≢ 0).

Consider now the d-type p_{1}(z_{1}) ∪ ⋯ ∪ p_{d}(z_{d}). Since the p_{i}’s are pairwise orthogonal and of rank 1, for any tuple c̄ in L we have

p_{1}(z_{1}) ∪ ⋯ ∪ p_{d}(z_{d}) ∪ { ⋁_{j=1}^{m}(h_{j}(z_{1}, …, z_{d}, c̄) = 0 ∧ h_{j}(Z_{1}, …, Z_{d}, c̄) ≢ 0)}

→ ⋁_{i=1}^{d} “z_{i} ∈ acl(K, c̄)”.

Note that “z ∈ acl(K, c̄)” is equivalent modulo T h(K) to an infinite disjunction of formulas. By compactness, there is an integer M and a formula θ(z, ȳ), with parameters in K, such that for every tuple c̄′ in L, the formula θ(z, c̄′) defines a finite (possibly empty) set, and such that

p_{1}|_{M}(z_{1}) ∪ ⋯ ∪ p_{d}|_{M}(z_{d}) ∪ { ⋁_{j=1}^{m}(h_{j}(z_{1}, …, z_{d}, ȳ) = 0 ∧ h_{j}(Z_{1}, …, Z_{d}, ȳ) ≢ 0)}

→ ⋁_{i=1}^{d} θ(z_{i}, ȳ).

Define q′′_{n} = q_{n} ∪ p_{1}|_{M}(x_{τ(1)}) ∪ ⋯ ∪ p_{d}|_{M}(x_{τ(d)}). By Lemma 3, this partial type is consistent and complete up to level N(n) + M. For i = 1, …, d, choose ρ(i) ∈ p^{(N(n)+M)} such that a_{σ} is separably algebraic over K(a_{ρ(i)}) whenever a realizes q′′_{n} and σ ∈ p^{(N(n)+M)} extends τ(i). Note that this implies that a_{τ(i)} is separably algebraic over K(a_{ρ(i)}), and that {a_{ρ(1)}, …, a_{ρ(d)}} is a separating transcendence basis of K(a_{≤N(n)+M}) over K . Step 2.

Let N(n + 1) = N(n) + M + 2d − 2, and q_{n+1} = q′′_{n} ∪ Φ_{d}(x_{ρ(1)}, …, x_{ρ(d)}). By Lemma 3, q_{n+1} is consistent and complete up to level N(n + 1).

We need to show that q_{n+1} satisfies the induction hypothesis. Let a realize q_{n+1} and assume that f_{n}(a_{≤n}, c̄) = 0 for some tuple c̄ such that f_{n}(X_{≤n}, c̄) does not vanish on all realizations of q_{n+1}. Then for some j = 1, …, m, h_{j}(a_{τ(1)}, …, a_{τ(d)}, c̄) = 0 and h_{j}(Z_{1}, …, Z_{d}, c̄) ≢ 0. By our choice of M, there is i such that θ(a_{τ(i)}, c̄) holds. Thus a_{τ(i)} lies in acl(K, c̄), and therefore so does a_{ρ(i)}, since a_{τ(i)} and a_{ρ(i)} are equi-algebraic over K. From Φ_{d}(a_{ρ(1)}, …, a_{ρ(d)}) and Lemma 2 we then deduce that a_{ρ(1),≤2d−2}, …, a_{ρ(d),≤2d−2} ∈ acl(K, c̄). Thus a_{≤N(n+1)} ⊆ acl(K, c̄).

Assume now that a realizes q = ⋃_{n} q_{n}. Then tr.deg.(K⟨a⟩/K) = d. Assume that b̄ is a tuple of elements of L, and that tp(a/K⟨b̄⟩) forks over K. Then there is a polynomial f ∈ K[X_{∞}, Ȳ] and a tuple c̄ from K⟨b̄⟩ such that f(a_{∞}, c̄) = 0 and f(X_{∞}, c̄) does not vanish identically on the set of realizations of q. Then f = f_{n} for some n, and by the definition of q_{n+1}, we obtain that a ∈ acl(K, b̄). This shows that U(q) = 1.

Theorem 2. There exists a type of U-rank 1 and of infinite transcendence degree over K. Proof. Conventions and notation are as before. By results of [1], we may select an infinite sequence (p_{n})_{n∈N} of types of U-rank and transcendence degree 1, which are pairwise orthogonal.

Let Ȳ be a countably infinite tuple of variables, and fix an enumeration f_{n}, n ∈ N, of the elements of K[X_{∞}, Ȳ] such that f_{n} ∈ K[X_{≤n}, Ȳ] for every n.

We will build the type q by induction on n. We assume that at stage n we have built a type q_{n} = q|_{N(n)} complete up to level N(n), and satisfying:

(a) tr.deg.(q_{n}) = n and q_{n} extends q_{n−1},

(b) for any tuple c̄ from L and a realizing q_{n}, if f_{n−1}(a_{≤n−1}, c̄) = 0, then either the polynomial f_{n−1}(X_{≤n−1}, c̄) vanishes on all realizations of q_{n}, or a ∈ acl(K, c̄).

For n = 1, let N(1) = 0, and q_{1} the type of a transcendental element. Then q_{1} satisfies trivially the induction hypothesis.

Assume that we have built q_{n} satisfying the condition. We will now build q_{n+1}, in three steps. Step 1. Increasing the transcendence degree.

Let σ(1), …, σ(n) ∈ p^{N(n)} be such that whenever a realizes q_{n} then {a_{σ(1)}, …, a_{σ(n)}} is a separating transcendence basis of K(a_{≤N(n)}) over K. Consider the type

q′_{n} = q_{n} ∪ {x_{σ(1)} = x_{σ(1) 0}^{p}, …, x_{σ(n−1)} = x_{σ(n−1) 0}^{p}, x_{σ(n)} = x_{σ(n) 0}^{p} + x_{σ(n) 1}^{p}u}.

By Lemma 3, this type is consistent and complete up to level N(n) + 1. Furthermore, if a realizes it then {a_{σ(1) 0}, …, a_{σ(n−1) 0}, a_{σ(n) 0}, a_{σ(n) 1}} is a separating transcendence basis of K(a_{≤N(n)+1}) over K. Steps 2 and 3. Apply Steps 1 and 2 of the preceding construction to the partial type q′_{n}, with d = n + 1, to get a type q_{n+1} satisfying the induction hypothesis.

We conclude as in the preceding construction that U(q) = 1 for q = ⋃_{n} q_{n}, while Step 1 of the present construction guarantees that q has infinite transcendence degree.

§2. Additional remarks on the case of infinite degree of imperfection.

Let {v_{n}: n ∈ N}, be a p-independent subset of K, with v_{0} = u. The construction in Theorem 2 can easily be modified to give a type q of U-rank 1 over K, and such that if a realizes q, then all the v_{i}’s are definable over {a, u}. In order to incorporate the infinitely many p-independent elements in this type of U-rank 1, we no longer assume, as in §1, that definable elements in our types be p-dependent on u. It suffices to change step 1 in Theorem 2 as follows: define

q′_{n} = q_{n}∪{x_{σ(1)} = x_{σ(1) 0}^{p}+v_{n}^{p}u, x_{σ(n)} = x_{σ(n) 0}^{p}+x_{σ(n) 1}^{p}u, x_{σ(i)} = x_{σ(i) 0}^{p} for i = 2, …, n−1}.

Then apply steps 1 and 2 of the first construction to the type q′_{n} and the variables x_{σ(1) 0}, …, x_{σ(n−1) 0}, x_{σ(n) 0}, x_{σ(n) 1} (with d = n + 1).

This construction shows how infinitely many p-basis elements from K can be defined over {a, u}, where a realizes a minimal type over K. In contrast, the next Lemma shows one can never define an element p-independent over K via a minimal type. Lemma 4. Let a be an element of U-rank 1 over K. Then K⟨a⟩ ⊆ KL^{p^{n}} for every n, where K⟨a⟩ is the definable closure of a over K. Proof. If this were false, then there would be an element b ∈ K⟨a⟩, such that b is pindependent from K over L^{p}. Let c ∈ L^{p^{∞}} be transcendental over K⟨b⟩ = K(b), and consider tp(b/K(bc)). This type forks over K, but is non-algebraic over K, which shows that tp(b/K) does not have U-rank 1. Since b is definable over K(a), a cannot have U-rank 1 either.

Delon’s work showing that the set of realizations of a minimal type has a Zariski geometry generalizes to the infinite invariant case. We will indicate below the changes which are needed; the interested reader should then be able to follow the arguments given by Delon [2] in this broadened context.

We fix a model K of SCF_{∞} with a countable p-basis B, and a saturated elementary extension L of K. We also fix a minimal type q over K, and let Q be the set of its realizations in L. By Lemma 4, Q ⊂ KL^{p^{n}} (= L^{p^{n}}[B]) for every n ∈ N. By the discussion in §0 about quantifier elimination, the type over K of an n-tuple (a_{1}, …, a_{n}) of elements of Q is given by the L-isomorphism type over K of the associated n-tuple of B-trees (a_{1,∞}, …, a_{n,∞}).

We want to define a topology on the cartesian powers of Q. Observe that in general an equation of the type λ_{i,n}(a_{1}, …, a_{n}, x) = 0 defines a set containing all elements which are not in L^{p}[a_{1}, …, a_{n}], and thus is much too large. Here Lemma 4 allows us, in working inside Q^{n}, to circumvent this problem. Fix an enumeration B = {b_{1}, b_{2}, …}. Since for every m ≥ 0 and a ∈ Q, the tuple a_{≤m} contains only finitely many distinct elements, we may fix natural numbers n(1) ≤ n(2) ≤ … such that Q ⊂ L^{p^{m}}[b_{1}, …, b_{n(m)}] for every m > 0.

We define by induction on d the notion of a {Q, B}-allowed term (in the variable x) of level d as follows: x is {Q, B}-allowed of level 0; if t(x) is a {Q, B}-allowed term of level d and 0 ≤ i < p^{n(d+1)}, then λ_{i,n(d+1)}(b_{1}, …, b_{n(d+1)}, t(x)) is {Q, B}-allowed of level d + 1.

Claim. Let D ⊂ L^{n} be a K-definable set (in the language L). Then D ∩ Q^{n} is a Boolean combination of sets defined by equations of the form f(t_{1}(x_{1}), …, t_{n}(x_{n})) = 0, where f is a polynomial over K and each t_{i}(x_{i}) is a tuple of {Q, B}-allowed terms in the variable x_{i}.

Proof. If a_{1}, …, a_{n} ∈ Q, the type of (a_{1}, …, a_{n}) over K is determined by the Lisomorphism type over K of the n-tuple of B-trees (a_{1,∞}, …, a_{n,∞}). Every element of a_{i,∞} is either 0, or the evaluation of some {Q, B}-allowed term at a_{i}. The result follows, using compactness and the saturation of L, and noting that the L-isomorphism type over K of a tuple is given by the set of all polynomial equalities and inequalities with coefficients in K which are satisfied by the tuple.

Now to define the topology. Fix n > 0. A closed subset of Q^{n} is a set defined by a (possibly infinite) conjunction of formulas of the form f(t_{1}(x_{1}), …, t_{n}(x_{n})) = 0, where f is a polynomial over K and each t_{i}(x_{i}) is a tuple of {Q, B}-allowed terms in the variable x_{i}.

We then define K[X_{∞}] to be the polynomial ring over K in the variables X_{t} for t a {Q, B}-allowed term in x. For m ∈ N, we define K[X_{≤m}] to be the polynomial ring over K in the variables X_{t} for t a {Q, B}-allowed term in x of level ≤ m. Similarly for K[X_{1,∞}, …, X_{n,∞}]. Note that for every m ≥ 0, K[X_{1,≤m}, …, X_{n,≤m}] is finitely generated over K.

Theorem 3. Let K be a separably closed field. The set of realizations of a minimal type over K forms a Zariski geometry.

This theorem is due to Hrushovski when the type has finite transcendence degree over K (“ thin types”) and when e is finite. Delon’s proof extends the finite e case to the non-thin setting, i.e, allowing the type to have infinite field transcendence degree over K. With no additional ideas or definitions beyond what we have given, and with only very minor modifications, the proofs of Section 5 of Delon [2] go through to give Theorem 3 for the infinite invariant case. We leave the details to the interested reader.

References.

[1] Z. Chatzidakis, G. Cherlin, S. Shelah, G. Srour, C. Wood, Orthogonality of types in separably closed fields, in: Classification theory, Proc. of the 1985 Chicago Conference, Springer Verlag, Lecture Note 1292.

[2] F. Delon, Separably closed fields, in Model Theory and Algebraic Geometry, E. Bouscaren ed., Springer Verlag, Lecture Note 1696, Berlin 1998.

[3] F. Delon, Idéaux et types sur les corps séparablement clos, Supplément au Bull. de la S.M.F, Mémoire 33, Tome 116, 1988.

[4] Y. Ershov, Fields with a solvable theory, Doklady 174 (1967), pp. 19-20; English translation in Soviet Mathematics 8 (1967), pp. 575-576.

[5] C. Wood, Notes on the stability of separably closed fields, Journal of Symbolic Logic 44 (1979), pp. 412-416.