The trichotomy theorem in dierence elds

Transcription

1 The trichotomy theorem in dierence elds Thomas Scanlon UC Berkeley December 2011 Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

2 Trichotomy theorem Theorem Every minimal type relative to the theory ACFA of dierence closed elds is either modular or almost internal to a denable minimal xed eld. Moreover, the class of modular minimal types further decomposes into the trivial types and those which are almost internal to a generic of a modular group. The main goal of today's lecture will be the explanation of the terms appearing in the trichotomy theorem. With the following lectures we shall see parts of three proofs. The proof of the trichotomy theorem in all characteristics appears in a paper of Chatzidakis, Hrushovski and Peterzil. The rst proof of the trichotomy theorem in characteristic zero is due to Chatzidakis and Hrushovski. Pillay and Ziegler gave a more geometric proof in characteristic zero using jet spaces. Each of these proofs yields further information beyond the raw statement of the trichotomy theorem. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

3 Paradigmatic dierence equations Fix(σ) is the denable set in one variable given by the equation σ(x) = x. Consider G, the denable set, again in one variable, dened by σ(x) = x 2 & x 0. Finally, we dene T by σ(x) = x Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

4 The xed eld So that we may speak of the xed eld as an actual set, we shall work in a specic model U = ACFA of the theory of dierence closed eld and set F := Fix(σ)(U). Then as F is the subset of U xed by the eld automorphism σ, it is a subeld. In fact, Fix(σ) is minimal (denition to follow) and F is pseudonite. By Ax's axiomatization of the theory of nite elds, to check that F is pseudonite we should verify that F is perfect, F is quasinite, and F is pseudoalgebraically closed. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

5 Fix(σ) is perfect If char(f ) = 0, then there is nothing to check. Suppose now that char(f ) = p > 0 and let a F. As U is algebraically closed, there is some α U with α p = a. Applying σ, we have (σ(α)) p = σ(α p ) = σ(a) = a. Hence, σ(α) is a root of X p a = (X α) p and must be equal to α. That is, α F. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

6 Fix(σ) is quasininite Strictly speaking, to say that a eld is quasinite includes the assertion that it is perfect. We shall concentrate on proving that Gal(F alg /F ) = Ẑ. From the Galois correspondence, it is clear that Gal(F alg /F ) is topologically generated by σ (or to be more precise, the restriction of σ to F alg ). Let N Z + be a positive integer. Consider X := A N U = Spec(U[x 0,..., x N 1 ]). We write X σ as Spec(U[y 0,..., y N 1 ]). Let Γ X X σ be dened by the equations y i = x i+1 for i < N 2 and x 0 = y N 1. Let U Γ be the dense constructible subset dened by the inequalities x 0 y i for i < N 1. By the axioms for ACFA there is a point a = (a 0,..., a N 1 ) X (U) with (a, σ(a)) U(U). From these equations, we see that the orbit of a 0 under σ has size exactly N. Hence, [Fix(σ N )(U) : Fix(σ)(U)] N, while the opposite inequality holds on general grounds. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

7 Fix(σ) is PAC Let V be an absolutely irreducible variety over F. Then V U is an irreducible variety over U, (V U ) σ = V U, and the diagonal VU (V U ) (V U ) is an irreducible subvariety of the product which projects dominantly in both directions. Hence, there is a point a V (U) with (a, σ(a)) V (U). That is, σ(a) = a so that a V (F ). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

8 Other xed elds If char(u) = p > 0, then map τ : U U given by x x p is a denable eld automorphism which commutes with σ. For what follows, if char(u) = 0, then we dene τ(x) := x For any pair of integers m, n Z the map ρ := σ n τ m is, thus, a denable eld automorphism of U and the xed eld of ρ Fix(σ n τ m )(U) := {x U : σ n (x) = τ m (x)} is a denable subeld of U which is pseudonite provided that n 0. Moreover, every denable eld is denably isomorphic to Fix(σ n τ m ) for some choice of n and m. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

9 Induced structure on G dened by σ(x) = x 2 and x 0 If char(u) = 2, then G is the set of nonzero elements of the xed eld of ρ = στ 1, and, thus, has the structure of a pseudonite eld (with zero removed). If char(u) 2, then G is a subgroup of the multiplicative group having the property that every denable subset of G n (for any n Z + ) is a nite Boolean combination of cosets of subgroups. Let us prove this for quantier-free denable sets. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

10 Quantier-free induced structure on G, char(u) 2 If X G n is a quantier-free denable set, then it is a nite Boolean combination of sets dened by dierence equations of the form f (x 1,..., x n ; σ(x 1 ),..., σ(x n );... ; σ l (x 1 ),..., σ l (x n )) = 0 for some polynomial f in n(l + 1) variables. Using the fact that σ(x) = x 2 on G, we may replace such a dierence equation with a strictly algebraic equation f (x 1,..., x n ; x 2 1,..., x 2 n;... ; x 2l 1,..., x 2l n ) = 0 After having reduced to such algebraic equations, we may assume that X is the intersection of an irreducible algebraic variety in which X is Zariski dense. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

11 Quantier-free induced structure on G, char(u) 2, continued From our reduction, we conclude that for any m that X σm = [2 m ] Gm n(x ) where we have written [2 m ] n Gm for the selfmap of algebraic groups given by (x 1,..., x n ) (x 2m,..., x 2m 1 n ). Let d := dim(x ). Then 2 m(n d) deg(x ) = 2 m(n d) deg(x σm ) = deg([2 m ] 1 G m n(x σm ) = deg( ξ µ n ξx ) = (2mn /#{ξ µ n 2 m 2m : ξx = X }) deg(x ). Hence, if we set Stab(X ) := {γ G n m : γx = X }, we have # Stab(X )(U) µ n = 2 m 2md. Thus, dim(stab(x )) = d and X is a coset of its stabilizer. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

12 The structure on T dened by σ(x) = x If char(u) = 2, then T is a denable principal homogenous space for the eld Fix(στ 1 ) under addition. However, there is no denable function Fix(στ 1 ) T dened without parameters as, for example, Fix(στ 1 ) has denable elements (consider 0) but neither 0 nor 1 belongs to T. Regardless of the characteristic of U, since T is dened over Fix(σ), the restriction of σ to T gives a denable function σ : T T. If char(u) 2, then there is no more structure. Theorem If char(u) 2, then every denable (with parameters) subset of T m is denable (with parameters from T ) in the structure (T, σ). It would be an interesting exercise to prove this theorem directly, but we shall do so only as a consequence of the general trichotomy theorem combined with a theorem of Ritt on polynomial decompositions. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

13 Induced structure versus quantier-free induced structure As we have seen, ACFA does not admit outright quantier elimination, as, for example, the set {x Fix(σ) : ( y)y 2 = x} is not quantier-free denable outside of characteristic two. Consequently, to describe the full induced structure on a denable set it does not suce to consider only quantier-free formulas. In characteristic zero, every modular minimal type is stable and stably embedded. Thus, to describe the induced structure it suces to describe the induced structure over a small set of parameters over which the type is dened. In positive characteristic, there are modular minimal types which are not stably embedded. A simple example is given by the equation σ(x) = x p x. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

14 Some notational conventions U is a xed dierence closed eld, assumed to be saturated and much larger than any other dierence eld or parameter set we shall mention. All sets of parameters will be tacitly assumed to be small subsets of U. Often, a tuple will be written without an explicit reference to its components or to its length. We may write expressions like a A to mean that a is a nite tuple whose components are elements of the set A. We may write expressions such as p S x (A) to mean that p is a complete type in the variable x (which again may be properly a nite tuple of variables or a variable ranging over some interpretable set) over the parameter set A. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

15 σ-degree Denition Given an inversive dierence eld K and a tuple a (from U) we dene K(a) σ to be the inversive dierence eld generated by a over K, which as a eld, is K({σ j (a i ) : j Z, j n}) where a = (a 1,..., a n ). Denition We dene deg σ (a/k) := tr. deg(k(a) σ /K). If deg σ (a/k) is innite, then it is necessarily ℵ 0, but we shall simply write deg σ (a/k) =. Over a general set B, we dene deg σ (a/b) to be deg σ (a/k) where K is the inversive dierence eld generated by B. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

16 σ-degree for denable sets and types Remark Since the isomorphism type of K(a) σ over K, we may regard deg σ (a/k) as a function of qftp(a/k), the quantier-free type of a over K, or equivalently, of I (a/k) := {f K[x] σ : f (a) = 0} the ideal of dierence polynomials over K which vanish on a. Denition For a denable set X dened over some set of parameters K we dene deg σ (X ) := sup{deg σ (a/k) : a X (U)} With this denition we are implicitly asserting that if X is also dened over L, then deg σ (X ) = sup{deg σ (a/l) : a X (U)}. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

17 Independence of σ-degree on eld of denition We may reduce immediately to the case that K L and both L and K are inversive dierence elds. We should then show that if there is some a X (U) with deg σ (a/k) n, then there is some b X (U) with deg σ (b/l) n. To simplify the presentation, I shall assume that a is a singleton leaving the reduction to this case as an exercise. The partial type {x X } {f (x, σ(x),..., σ n 1 (x)) 0 : f L[x 0,..., x n 1 ] : f 0} is consistent as otherwise (by taking products) there would be some f L[x 0,..., x n 1 ] for which ACFA x X f (x, σ(x),..., σ n 1 (x)) = 0. Applying any automorphism ρ of U over K, we see that ACFA x X f ρ (x,..., σ n 1 (x)) = 0. Taking conjunctions and using Noetherianity, we would nd a an order < n dierence equation over the denable closure of K (which is contained in the algebraic closure of K in the usual algebraic sense) which is implied by x X. Taking a product of the nitely many conjugates of this equation over K, we contradict the hypothesis that deg σ (a/k) n. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

18 Independence in ACFA Denition For an extension K L of inversive dierence elds and a tuple a, we say that a is free from L over K, written a K L if K(a) σ is algebraically independent from L over K. Visibly, the relation a K L depends only on qftp(a/l). While we have dened independence algebraically, it is an instance of general non-forking independence. Since algebraic independence is insensitive to algebraic extensions, we could replace K and L with their algebraic closures without changing independence. If A, B and C are general sets then we dene A B C if for each nite tuple a from A we have a K L where K is the inversive dierence eld generated by B and L is the inversive dierence eld generated by B C. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

19 Lascar rank We dene Lascar rank, or SU-rank (or even just U-rank) for a a tuple from U and B U a small subset by transnite recursion. SU(a/B) 0 always. SU(a/B) α + 1 if there is some C so that a B C and SU(a/C) α. SU(a/B) λ for λ a limit ordinal just in case SU(a/B) α for all α < λ. Denition We say that p(x) = tp(a/b) is minimal if SU(a/B) = 1. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

20 Comparison of deg σ and SU Theorem Let a be a tuple and K a small inversive dierence eld. SU(a/K) deg σ (a/k) SU(a/K) < ω deg σ (a/k) < SU(a/K) = 0 deg σ (a/k) = 0 K(a) K alg Remark Strictly speaking, the inequality SU(a/K) deg σ (a/k) is not meaningful when either side is innite as SU(a/K) is an ordinal and deg σ (a/k) is a cardinal. We shall understand the inequality to mean only that deg σ (a/k) is innite if SU(a/K) is innite. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

21 Proof that SU(a/K) deg σ (a/k) Working by induction on n we show that for any such pair (a, K) if SU(a/K) n, then deg σ (a/k) n. The base case of n = 0 is trivial. If SU(a/K) n + 1, then we may nd an inversive dierence eld L with a K L and SU(a/L) n. By induction, deg σ (a/l) n. By denition of, if tr. deg(k(a) σ /K) <, then tr. deg(l(a) σ /L) < tr. deg(k(a) σ /K). Thus, deg σ (a/k) n + 1. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

22 Proof that SU(a/K) = 0 deg σ (a/k) = 0 a K alg From the denition of deg σ (a/k) := tr. deg(k(a) σ /K) it is clear that deg σ (a/k) = 0 a K alg. (Note: K is an inversive dierence eld! If a K alg, then σ n (a) K alg for all n.) We have already shown that deg σ (a/k) = 0 = SU(a/K) = 0. If a / K alg, then K(a) σ K K(a) σ but SU(a/K(a) σ ) 0. Hence, SU(a/K) 1. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

23 Lemma: Additivity of deg σ and SU-rank It is clear from additivity of transcendence degree that for tuples a and b and inversive dierence elds K one has deg σ (ab/k) = deg σ (a/k) + deg σ (b/k(a) σ ). The corresponding additivity result for SU-rank goes under the name of the Lascar inequalities (which reduce to outright equalities when all of the quantities in question are nite) and holds in all generality for SU-rank. Theorem Given tuples a and b and a small parameter set C, one has SU(a/Cb) + SU(b/C) SU(ab/C) SU(a/Cb) SU(b/C) where denotes Cantor's natural sum of ordinals. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

24 Monotonicity lemmata about SU-rank Lemma For tuples a and b and any small parameter set C one has SU(b/C) SU(ab/C). For a tuple a and parameter sets B C one has SU(a/C) SU(a/B). We argue by induction on α that if SU(b/C) α, then SU(ab/C) α with the base case of α = 0 being trivial and the limit case immediate. If SU(b/C) α + 1 witnessed by some D C with SU(b/D) α and b C D, then by induction we have SU(ab/D) α while clearly ab C D. Hence, SU(ab/C) α + 1 as required. We leave the second monotonicity assertion as an exercise. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

25 Proof of the Lascar inequalities Working by induction on α we show that if SU(b/C) α, then SU(a/Cb) + α SU(ab/C). The case of α = 0 is the content of the lemma from the last slide and limit case is immediate. If SU(b/C) α + 1 witnessed by D C with b C D and SU(b/D) α, then by induction we have SU(ab/D) SU(a/Db) + α and visibly ab C D so that SU(ab/C). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

26 Proof of Lascar inequalities, continued Now working by induction on α we show that if SU(ab/C) α, then SU(a/Cb) SU(b/C) α. Again, the case of α = 0 is trivial and the limit case is immediate. Suppose that D C witnesses SU(ab/C) α + 1 in the sense that ab C D and SU(ab/D) α. By induction, SU(a/Db) SU(b/D) α. From the dependence, we conclude that either a Cb Db or b C D. Thus, either the righthand side of our inequality is or SU(a/Db) < SU(a/Cb) or SU(b/D) < SU(b/C). In any case, we see that SU(a/Cb) SU(b/C) α + 1. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

27 Proof that SU(a/K) ω deg σ (a/k) = We already know that SU(a/K) ω = deg σ (a/k) =. Writing a = (a 1,..., a n ) as an n-tuple of elements of U, using additivity of deg σ, we see that deg σ (a i /K(a 1,..., a i 1 ) σ ) = for some i n. Hence, we may assume that n = 1. Using the fact that Fix(σ N ) is an N-dimensional vector space over Fix(σ), one sees that SU(Fix(σ N )) N. It follows that upon setting b N := σ N (a) a one has SU(a/b N ) N. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

28 Minimality and σ-degree Recall that tp(a/k) is minimal if SU(a/K) = 1. We have seen that deg σ (a/k) = 1 = SU(a/K) = 1. It may happen that tp(a/k) is minimal, but deg σ (a/k) > 1. Consider, for example, a satisfying σ 2 (a) = a 3 with a / K alg. However, SU(a/K) = 1 does imply that deg σ (a/k) <. We shall use this fact to give a geometric criterion for minimality. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

29 σ-varieties Denition Let K be a dierence eld. By a pre-σ-variety over K we mean a pair (X, Γ) consisting of an algebraic variety X over K and a subvariety Γ X X σ. We say that (X, Γ) is a σ-variety if X and Γ are irreducible and the projection maps Γ X and Γ X σ are dominant. We say that (X, Γ) is nitary if the projection maps restricted to Γ are nite. Sometimes the term σ-variety is reserved for the case where Γ is the graph of a regular map f : X X σ. It makes sense to relax the requirement that X be a variety, allowing for example σ-schemes. We shall consider σ-varieties generically. As such, the nitary hypothesis may be relaxed to require only that the projections are generically nite. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

30 (X, Γ) Denition Given a σ-variety (X, Γ) over a dierence eld K and an extension dierence eld L, we dene (X, Γ) (L) := {a X (L) : (a, σ(a)) Γ(L)} The geometric axioms for ACFA assert that if K is dierence closed and (X, Γ) is a σ-variety over K for which Γ is irreducible, then (X, Γ) (K) is Zariski-dense in X and indeed the set {(a, σ(a)) : a (X, Γ) (K)} is Zariski-dense in Γ. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

31 Reduction to σ-varieties Proposition Every quantier-free denable set D is in denable bijection (via a quantier-free denable function) with a nite Boolean combination of sets of the form (X, Γ) for some σ-variety (X, Γ). If deg σ (D) <, then (X, Γ) may be taken to be nitary. We may express D as the solutions to some quantier-free formula φ(x) = φ(x 1,..., x m ) which we may further express as φ(x, σ(x),..., σ n (x)) for some quantier-free formula in the language of rings. Since every quantier-free formula in the language of rings is a nite Boolean combination of equations, we may assume that φ is given by a polynomial equation, F (y 0, y 1,..., y n ) = 0. Moreover, since σ is invertible, we may assume that both y 0 and y n appear in F. The map x (x, σ(x),..., σ n 1 (x)) induces a bijection between D and (A nm, Γ) where Γ is dened by y i = y i+1 (for i < n) together with F (y 0,..., y n 1, y ) = 0. n 1 Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

32 Generic types of σ-varieties If (X, Γ) is an absolutely irreducible σ-variety over K, then by a generic type in (X, Γ) we mean any complete (quantier-free) type extending the partial type determined by x (X, Γ) (x, σ(x)) / Y for each proper closed subvariety (not necessarily absolutely irreducible) Y Γ dened over K. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

33 A geometric criterion for minimality If K is an inversive dierence eld and a is a tuple from U for which tp(a/k) is minimal, then deg σ (a/k) < so that qftp(a/k) is interdenable with a generic type of a nitary σ-variety (X, Γ). Question For which σ-varieties (X, Γ) is a generic type of (X, Γ) minimal? Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

34 A geometric rank We dene a rank GU on σ-varieties with the usual conditions that GU(X, Γ) 0 as long as X (we could dene GU(, ) := 1) and GU(X, Γ) λ a limit provided that GU(X, Γ) α for all α < λ. For the successor case, we require the notion of a normal family of sub-σ-varieties. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

35 Sub-σ-varieties A morphism of σ-varieties g : (Y, Ξ) (X, Γ) is given by a map of algebraic varieties g : Y X for which the restriction of (g g σ ) to Ξ maps to Γ. Note that g takes (Y, Ξ) to (X, Γ). A sub-σ-variety (Y, Ξ) (X, Γ) of a σ-variety is a σ-variety for which the inclusion map ι : Y X induces a morphism of σ-varieties. Observe that if Y X is any subvariety, then (Y, (Y Y σ ) Γ) is sub-pre-σ-variety of (X, Γ), but it is possible that the projection maps from (Y Y σ ) Γ are not dominant. In the special case that Γ is the graph of a regular function f : X X σ, then (Y, Ξ) is a sub-σ-variety just in case the restriction of f to Y maps Y to Y σ. In particular, the sub-σ-varieties of (X, Γ) are determined by certain subvarieties of X. Note that a point {a} X can be the underlying variety of a σ-variety of (X, Γ) only if a (X, Γ) (K). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

36 Families of sub-σ-varieties By a family of sub-σ-varieties of a σ-variety (X, Γ) we mean a sub-σ-variety (Y, Υ) (X, Γ) (B, Ξ) of the base change of (X, Γ) over some σ-variety (B, Ξ). For b (B, Ξ) the (projection to X ) of the bre (Y b, Υ b ) is a sub-σ-variety of (X, Γ). We say that the family is normal if for distinct b, b B one has (Y b, Υ b ) (Y b, Υ b ). We say that the family is generically covering if for a generic a (X, Γ) (U) there is some generic b (X, Ξ) (U) for which a (Y b, Υ b ) (U). For a normal family, we dene dimension to be the dimension of B. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

37 Denition of GU completed We dened GU(X, Γ) for (X, Γ) a σ-variety by recursion. GU(, ) = 1 GU(X, Γ) 0 provided X is nonempty. GU(X, Γ) α + 1 if there is some positive dimensional, normal generically covering family (Y, Υ) (X, Γ) (B, Ξ) of sub-σ-varieties of (X, Γ) for which GU(Y b, Υ b ) α for generic b (B, Ξ) (U). GU(X, Γ) λ for λ a limit ordinal just in case GU(X, Γ) α for all α < λ. Proposition If qftp(a/k) is a generic type of the σ-variety (X, Γ), then SU(a/K) = GU(X, Γ). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

38 Canonical bases Denition For a tuple a and inversive dierence eld K, we dene the canonical base of tp(a/k), Cb(a/K), to be the smallest inversive dierence eld over which I σ (a/k) is dened. In stable theories, one denes canonical bases only for stationary types. In a general simple theory, the canonical base makes sense only for Lascar strong types. If qftp(a/k) is a generic type of a σ-variety (X, Γ), then Cb(a/K) has the same algebraic closure as the eld of denition of (X, Γ). We sometimes wish to treat a given inversive dierence eld L as a common base. In that case, we dene Cb L (a/k) to be the compositum of L and Cb(a/K). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

39 A geometric interpretation of the canonical base Suppose now that qftp(a/k) is a generic type of the σ-variety (X, Γ). Let k be the prime eld and set X := loc(a/k) = V (I (a/k)) and Γ := loc((a, σ(a))/k). Then (X, Γ) is a σ-subvariety of ( X K, Γ K ). Realize (X, Γ) as the base change to K of a generic bre (Z b, Υ b ) of some family (Z, Υ) (B, Ξ) of sub-σ-varieties of ( X, Γ). Then Cb(a/K) is interalgebraic with the function eld k(b) for (B, Ξ) irreducible of minimal possible rank. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

40 Modularity Denition Let p(x) be a not necessarily complete type over some small inversive dierence eld K. We say that p is modular if for any nite tuple a of realizations of p and any small inversive dierence eld L extending K we have Cb K (a/l) K(a) alg σ. What is called modularity here is one of several related notions, such as weak normality, one-basedness, matroid modularity, linearity, etc. Modularity has a clean geometric interpretation for types of nite σ-degree (an, in fact, every modular type has nite σ-degree). If p is a generic type of the nitary dierence variety (X, Γ), then p is modular just in case whenever Y (X, Γ) n (B, Ξ) is a generically covering normal family of sub-σ-varieties of some power of (X, Γ), then dim(y ) n dim(x ). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

41 Examples of modular types: σ(x) = x 2 Let K be a small inversive dierence eld with char(k) 2 and a U K alg be some element satisfying σ(x) = x 2. Then tp(a/k) is modular. qftp(a/k) is a generic type of (G m, x x 2 ) We say that every irreducible σ-subvariety (Y, Γ) (G m, x x 2 ) n is a translate of an algebraic subgroup of G m n. Thus, if b = (b 1,..., b n ) is an n-tuple of realizations of qftp(a/k), L is an inversive dierence eld extending K and Y =: loc(b/l) G m n, then Y = bg for some algebraic torus G G m n. Then Y is dened over K(b) K(b) σ. Hence, I σ (b/l) is dened over K(b) alg so that Cb K (b/l) K(b) alg. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

42 A trivial example of a modular type If char(k) 2, then the induced structure on the set T dened by σ(x) = x is denable in the structure (T, σ). In this structure, one sees easily that all families of denable come from varying a coordinate so that from a single point in the set one can compute a eld of denition. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

43 Internality and almost-internality Denition A denable set X is internal to the denable set Y if there is a denable function f (possibly dened over parameters not appearing in the denitions of X and Y ) from some Cartesian power of X onto Y. We say that Y is almost-internal to X if there is a denable set Z X n Y (again, we allow new parameters for the denition of Z ) so that the projection of Z to Y is all of Y and for each a X n the bre Z a is nite. In many cases of interest, the internalizing function f : X n Y requires new parameters for its denition. (Consider, for example, X dened by σ(x) = x + 1, Y = Fix(σ) and K the prime eld.) Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

44 Alternate forms of almost-internality (Almost-)internality might be dened at the level of (quantier-free) types. We would have that tp(a/k) is internal to tp(b/k) if there is some L extending K and b 1,..., b n realizing tp(b/k) for which a K L, b i K L, and a is denable from L(b 1,..., b n ). For almost internality we would ask only that a L(b 1,..., b n ) alg σ. One could rene the denition of internality for quantier-free denable sets/quantier-free types to ask that the internalizing function also be quantier-free denable. In general, even for quantier-free denable X and Y this is a proper renement. Almost-internality is closely related to the notion of non-orthogonality: We say that tp(a/k) and tp(b/k) are orthogonal, written tp(a/k) tp(b/k), if for any L extending K for which a K L and b K L we have a L b. If tp(b/k) is minimal, then it is non-orthogonal to some type p(x) just in case it is almost internal to p. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

45 Recalling the trichotomy theorem We have now dened all of the terms of the rst part of the trichotomy theorem. Theorem Every non-modular minimal type relative to ACFA is almost internal to a minimal xed eld Fix(ρ). For the second cut in the trichotomy theorem we need a precise denition of the notion of triviality and a description of modular groups. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

46 Triviality Denition We say that the type p(x) S x (A) is trivial if for any nite sequence a 1,..., a n of realizations of p if a i A a j for all i < j, then a i A {a j : i j} for all i < n. That is, a type is trivial if all dependencies on realizations of p are essentially binary. Proposition A trivial minimal type is modular. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

47 Modular dichotomy Theorem Every non-trivial modular minimal type is non-orthogonal to a modular group. If a denable group G is modular, then every (relatively) quantier-free denable subset of any Cartesian power is a nite Boolean combination of cosets of denable subgroups. This theorem may be proven as a special case of the group conguration theorem asserting under appropriate stability theoretic hypotheses that the presence of a denable group may be detected solely from dependence-theoretic data. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

48 Key ideas in the proof of the modular dichotomy theorem Working over parameters, the non-triviality of a minimal type p may be detected by the existence of a 1, a 2 and a 3 realizing p which are pairwise independent but for which a 3 is algebraic over {a 1, a 2 }. We may regard a 2 as a code for a many-to-many correspondence f a2 : p p. Composing such correspondences as a 2 varies through the realizations of p we obtain an ostensibly two-parameter family of p-curves, but by modularity this family must actually be one dimensional. It follows that the resulting family of maps is closed under composition and thus forms a group non-orthogonal to p. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

49 Some complications in the proof of the modular dichotomy theorem Since the correspondence f a2 is not actually a function, one must rst reduce to the case that it is a function by replacing the elements a 1, a 2 and a 3. For example, one begins by replacing a 3 with its nite set of conjugates over a 1 and a 2. The proof outlined above uses stability in some crucial ways. The theory ACFA is not stable, but working with quantier-free formulas, one has enough stability to push the proof through. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

50 Canonical base property Theorem If K is any inversive dierence eld of characteristic zero, a and c are tuples for which deg σ (a/k) < and K(c) alg σ = Cb K (a/k(c) σ )) alg, then qftp(c/k(a) σ ) is internal to Fix(σ). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

51 Dichotomy theorem as a consequence of the canonical base property Theorem If K is any inversive dierence eld of characteristic zero and p is a minimal type over K, then either p is modular or p is almost internal to Fix(σ). If p were non-modular, then there would exist some tuple a of realizations of p and tuple b so that K(b) alg σ = Cb K (a/k(b) alg ) but b / K(a) alg σ. By the canonical base property theorem, tp(b/k) is internal to Fix(σ). Since b is not algebraic over K, tp(b/k) is non-orthogonal to p. Hence, p is non-orthogonal to (and, thus, almost internal to) Fix(σ). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

52 Geometric reformulation of canonical base property As the hypotheses and conclusions are insensitive to replacements of a and c by interdenable tuples, we may assume that qftp(a/k) is a generic type of a nitary σ-variety (X, Γ) and that qftp(a/k(c) σ ) is a generic type of a sub-σ-variety (Y c, Υ c ) where (Y, Υ) (X, Γ) (C, Ξ) is a normal, irreducible family of sub-σ-varieties for which qftp(c/k) is a generic type of (C, Ξ). The theorem asserts that in reversing the roles of X and C, so that now (Y, Υ) gives a family of sub-σ-varieties of (C, Ξ), the bre above a may be identied with a quotient of the Fix(σ)-points of some algebraic variety. In fact, we shall realize this bre as a certain denable subset of a Grassmannian. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

53 Jet spaces Denition For X an algebraic variety over a eld K, a X (K) a rational point on X, and n Z + a positive integer, we dene to be the n th jet space of X at a. Jet n X a (K) := Hom(m X,a /m n+1 X,a, K) Jet n X a is represented by a linear space Jet n X X over X and the association from X to this bundle is functorial. There are other natural interpretations of Jet n X a (K), as, for example, the germ of the sheaf of order at most n linear dierential operators on O X at a. Jet 1 X X is the Zariski tangent bundle. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

54 Jets determine subvarieties Proposition Let X be an algebraic variety over a eld K, a X (K) a K -rational points, and Y and Z two irreducible subvarieties of X both passing through a. Then X = Y if and only if for every n, Jet n Y a (K) = Jet n Z a (K) as subspaces of Jet n X a (K). Passing to a dense open ane neighborhood of a, we may assume that X is ane. (For me, variety means separated, reduced scheme of nite type over a eld.) For any given n, the image of Jet n Y a (K) in Jet n X a (K) is {ψ : m X,a /m n+1 K : ( g I (Y ))ψ(g) = 0} (and likewise for Z ). X,a If Y and Z dier, then, possibly at the cost of exchanging Y and Z, we can nd f I (Y ) I (Z). Since O X,a is noetherian, there is some n for which f / m n+1. Hence, X,a we can nd some ψ : m X,a /m n+1 K which vanishes on the image of X,a I (Z) but not on f. Then ψ Jet n Z a (K) Jet n Y a (K) contradicting our hypothesis. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

55 Jets of étale maps Proposition If f : X Y is a étale map of algebraic varieties over a eld K, then for any point a X (K) and any n Z +, the map Jet n (f ) : Jet n X a (K) Jet n Y f (a) (K) is an isomorphism of K -vector spaces. Proof. f induces an isomorphism between ÔY,f (a) = lim O Y,f (a) /fm n+1 Y,f (a) and Ô X,a. Truncating and taking duals we conclude the proof. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

56 σ-jet spaces Suppose now that (X, Γ) is a nitary σ-variety over some dierence eld K. Provided that the projection maps π 1 : Γ X and π 2 : Γ X σ are separable (which is automatic in characteristic zero), then for a suciently general point a X (L) (where L might be a proper extension of K ) the projections are étale in a Zariski neighborhood of (a, σ(a)). Hence, for any n Z + we obtain an invertible linear map γ n,a := Jet n (π 2 ) (a,σ(a)) Jet n (π 1 ) 1 (a,σ(a)) : Jet n X a Jet n X σ σ(a) Note that (Jet n X a, γ n,a ) is a nitary σ-variety. We dene Jet n (X, Γ) a := (Jet n X a, γ n,a ). Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

57 Properties of the σ-jet space Proposition Let n Z + be a positive integers and (X, Γ) be a nitary σ-variety over a dierence closed eld U for which both projections π 1 : Γ X and π 2 : Γ X σ are separable. Then for a suciently general point a (X, Γ) (U) the σ-jet space (Jet n X a, γ n,a ) (U) is a nite dimensional Fix(σ)-vector space which is Zariski dense in Jet n X a. Since γ n,a is invertible, Zariski density is an immediate consequence of the geometric axioms for ACFA. That Jet n (X, Γ) a is a Fix(σ)-vector space follows from the fact that γ n,a and σ are Fix(σ)-linear. The dimension of this space is equal to dim J n X a from our general result on the σ-degree of sharp-sets. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58

58 Proof of canonical base property Let (X, Γ) be a nitary σ-variety over U = ACFA 0, a (X, Γ) (U) a sharp point and (Y, Ξ) (X, Γ) (B, Ξ) an irreducible, normal family of sub-σ-varieties for which Y b is irreducible and a (Y b, Υ b ) (U) for b (B, Ξ) (U). By compactness, for n large enough, we have that if Jet n Y b = Jet n Y c, then b = c. As Jet n (Y b, Υ b ) a is Zariski dense in Jet n Y b, we see that b = c if and only if Jet n (Y b, Υ b ) a (U) = Jet n (Y c, Υ c ) a (U) for b, c (B, Ξ) (U). Hence, we have a denable embedding of (B, Ξ) into a Grassmannian of subspaces of Jet n (X, Γ) a establishing that (B, Ξ) is Fix(σ) internal. Thomas Scanlon (UC Berkeley) The trichotomy theorem in dierence elds December / 58