May 26, 2000 - Ðκ. There is no harm in working in ¯Meq in which there are additional sorts SE whenever E(x, y) is a formula of L defining an equivalence ...
Supersimple theories Steven Buechler∗ Dept. Mathematics University of Notre Dame A. Pillay† Dept. Mathematics University of Illinois at Urbana-Champaign F. Wagner‡ Maths. Institute Oxford University May 26, 2000
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Introduction
Suppose M is a structure in the sense of model theory. It is natural to consider definable quotients, X/E where X is a definable set in M and E a definable equivalence relation. An element a/E of X/E is called an imaginary element of M . Shelah introduced the formalization M eq , a many-sorted first order structure where such quotients and their elements can be treated on an equal footing with ordinary definable sets and elements. The passage to M eq was required in stability theory to understand “canonical bases” of types (the stability-theoretic generalization of the smallest field of definition of a variety). ∗
Supported by NSF grant. Supported by NSF grant. Thanks to MSRI for its hospitality ‡ Supported by DFG grant Wa 899/2-1. Thanks to MSRI for its hospitality †
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One can go further and consider sets X/E where E is now a “typedefinable” equivalence relation, namely defined by a possibly infinite conjunction of first order formulas. We call an element a/E of X/E a “hyperimaginary”, and X/E a hyperdefinable set. To really see X/E one should assume the structure M to be sufficiently saturated. Hyperimaginaries naturally arise in nonstandard analysis, by taking E(x, y) to be the relation “|a − b| is infinitesimal”. On the other hand if M is stable, then it was pointed out in [14] that a type-definable equivalence relation is essentially a conjunction of definable equivalence relation, whereby a hyperimaginary is no more than a possibly infinite sequence (or projective limit) of imaginaries (we call this phenomenon elimination of hyperimaginaries). In the past few years, the class of simple theories, a far-reaching generalization of the class of stable theories, has been intensively studied. As vector spaces and algebraically closed fields are natural examples of stable theories (or rather structures), so vector spaces over finite fields equipped with bilinear forms, pseudofinite fields, and algebraically closed fields with a generic automorphism, are examples of simple unstable theories. Simple theories were introduced by Shelah in his fundamental paper [16], and defined using his notion of “forking”. In [7], major progress was made by showing that in simple theories, forking behaves very much like in stable theories, in particular provides a good notion of independence. (See also [8], [9]). Further advances were made in [12], where among other things the Independence Theorem for Lascar strong types was proved. This is a theorem concerning the compatibility of various free extensions, and was motivated by earlier work of Hrushovski ([5]) in a certain finite-dimensional context. The interesting phenomenon was that, in place of the definable equivalence relations with finitely many classes (corresponding to strong types) which occur in stability theory, we seemed to be forced to consider type-definable equivalence relations with a small number of classes (corresponding to Lascar strong types). In a similar vein, in [4], in trying to develop the theory of canonical bases (something like fields of definition of varieties) in the simple theory context, one was forced to consider hyperimaginaries, rather than just imaginaries. (Related work is in [2].) However the mystery was that we did not know (and still do not know) any example of a simple theory in which hyperimaginaries are not eliminable. Supersimple theories (or structures) are simple structures in which the independence relation given by forking is finitely based. All the examples mentioned earlier are supersimple. The 2
purpose of the current paper is to prove that in a supersimple theory (or structure) hyperimaginaries are eliminable. Equivalently, any equivalence relation defined by a possibly infinite conjuction of formulas is essentially an intersection of definable equivalence relations. A special case of our result, when E has a small number of classes, is that (in a supersimple structure), any hyperdefinable compact group G which acts definably as a group of automorphisms of some hyperdefinable set X, has to be profinite. We now mention related results as well as the genesis of the current paper (which is a bit complicated). Kim [10] proved that in any countable theory with only countably many types, any type-definable equivalence relation (over a finite set) is a conjunction of definable equivalence relations.Hrushovski ([6]) showed that if one was willing to work in suitable non first order contexts (so-called Robinson structures) then there are supersimple theories which do not eliminate (even bounded) hyperimaginaries. Buechler [1] proved that in simple theories satisfying an additional technical property (lowness) any type-definable equivalence relation with few classes is a conjunction of definable finite equivalence relations (at least on the set of realizations of a complete type). This was also proved by Shami [15]. Buechler generalized his methods in [2], suggesting a proof that for supersimple low theories, canonical bases (slightly different from those in the present paper) exist as sets of imaginaries. Hart and Pillay [3] shortened the proof as well as translating it into the context of [4]. (However both [2] and [3] had gaps.) Using these ideas, as well as an an analysability strategy developed a year earlier at the Fields Institute, Wagner suggested a proof for elimination of hyperimaginaries in the general supersimple context. This was subsequently simplified by Pillay and Wagner and is essentially the proof presented here. The remainder of this paper assumes some acquaintance with model theory and its language. In section 2 we give definitions and background results on simple theories and hyperimaginaries, as well as additional machinery required. The proof of the main result (elimination of hyperimaginaries for supersimple theories) is given in section 4. The proof is quite technical and depends on the availability of a lot of machinery (such as analysability, orthogonality) in the hyperimaginary context. So for the sake of motivation and for the benefit of the reader, in section 3 we outline a rather elementary case of the argument, which in particular yields the main result in the supersimple low case. The reader familiar with the essentials of stability and simplicity theory could read section 3 first if he or she so desired. 3
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Preliminaries
We start by recalling definitions and results regarding simple theories and hyperimaginaries. The basic material on simple theories is from [16], [7], [12] and also [1] for low theories. The material on hyperimaginaries and canonical bases is from [4] and [13]. But there are also a few new observations. ¯ aκ T denotes a complete theory in a language L, and M ¯ - saturated model of T for some suitable large κ ¯ . “Small” means of cardinality strictly less than ¯ eq in which there are additional sorts SE κ ¯ . There is no harm in working in M ¯ n. whenever E(x, y) is a formula of L defining an equivalence relation on M Until we say otherwise x, y etc. will denote possibly infinite but small tuples of variables, each variable ranging over a sort SE . Similarly, a, b denote small ¯ (or M ¯ eq ). A partial type in tuples of elements and A, B small subsets of M variables x is a small consistent collection of formulas in some variables x and various parameters. We write Σ(x, a) for such a partial type if we want to specify the parameters as coming from a. Definition 2.1 Σ(x, b) divides over A if there is an A-indiscernible sequence S (bi : i < ω) in tp(b/A) such that {Σ(x, bi ) : i < ω} is inconsistent (is not ¯ ). realized in M Definition 2.2 T is simple if whenever a is a finite tuple and B any set then there is a subset A of B of cardinality at most |T | such that tp(a/B) does not divide over A. Remark 2.3 (i) For simple theories, dividing is equivalent to a certain apriori weaker notion, forking. For simple theories we use these words interchangeably. (ii) For T simple, forking is a good independence relation, satisfying symmetry, transitivity, and existence of nonforking extensions, even for infinite tuples. For example symmetry means tp(a/Ab) does not fork over A iff tp(b/Aa) does not fork over A. Existence of nonforking extensions means, for any a, A and A ⊆ B there is a0 such that tp(a0 /A) = tp(a/A) and tp(a0 /B) does not fork over A. We also use the expression: a is independent from b over A to mean tp(a/Ab) does not fork over A. (iii) Note that tp(a/B) does not fork over A iff for each finite subtuple a0 of a, tp(a0 /B) does not fork over A. 4
(iv) Suppose {ai : i ∈ I} is a set of tuples. We call this set A-independent if for each i ∈ I, ai is independent from {aj : j ∈ I, j 6= i} over A. Definition 2.4 (i) Let ψ(x) be a formula with maybe additional parameters, φ(x, y) and L-formula, and k < ω. D(ψ(x), φ(x, y), k) is defined to be ≥ 0 if ψ(x) has a solution, and ≥ n + 1 if there are bi for i < ω such that {φ(x, bi ) : i < ω} is k-inconsistent (every k-sized subset is inconsistent) and for each i, D(ψ(x) ∧ φ(x, bi ), φ, k) ≥ n. (ii) For Σ(x) a partial type closed undr conjunctions, D(Σ, φ(x, y), k) is defined to be min{D(ψ(x), φ(x, y), k) : ψ(x) ∈ Σ(x)} if this min. exists (and ∞ otherwise). Fact 2.5 (i) If T is simple, then for any partial type Σ(x) (in finitely many variables x), φ(x, y) ∈ L and k < ω, D(Σ, φ, k) exists (namely has value < ω. Moreover, if p(x) ⊂ q(x) are complete types over A ⊂ B respectively, then q does not fork over A (is a nonforking extension of p) iff for each φ(x, y) and k, D(p, φ, k) = D(q, φ, k). (ii) For fixed ψ(x, z), φ(x, y), k, n, there is a set Γ(z) of formulas such that D(ψ(x, c), φ, k) ≥ n iff |= Γ(c). (iii) It follows that for T simple, and p(x) a complete type (even in infinitely many variables) over A, there is a partial type Σ(xi )i∈I over A such that for any set {ci : i ∈ I} of tuples, this is an A-independent set of realizations of p iff |= Σ(ci )i . (iv) (T simple.) Let Σ(x) be a partial type over B ⊃ A which does not fork over A. Then Σ(x) extends to a complete type p(x) over B which does not fork over A. Definition 2.6 (T simple.) Let p(x) ∈ S(A). By a Morley sequence in p we mean an infinite A-independent A-indiscernible sequence (ci : i < ω) of realisations of p(x). Fact 2.7 (T simple.) Let Σ(x, b) be a partial type, and A a set. Then Σ(x, b) does not fork over A iff for any Morley sequence (bi : i < ω) in tp(b/A), S {Σ(x, bi ) :< ω} is inconsistent, iff for some Morley sequence (bi : i < ω) in S tp(b/A), {Σ(x, bi ) : i < ω} is inconsistent. Definition 2.8 The formula φ(x, y) is said to be low, if there is some k < ω such that whenever (bi : i < ω) is indiscernible and {φ(x, bi ) : i < ω} is 5
inconsistent, then it is k-inconsistent. T is low iff all formulas φ(x, y) are low. (Note the similarity to the non fcp.) Definition 2.9 (T simple, or more generally “G-compact”.) Let a, b be possibly infinite tuples, and A a set. Then a and b are said to have the same Lascar strong type over A, Lstp(a/A) = Lstp(b/A) if whenever E(x, y) is an equivalence relation with a small number of classes which is defined by a partial type over A, we have E(a, b). The Independence Theorem for Lascar strong types is: Fact 2.10 (T simple.) Suppose A ⊆ B and A ⊆ C with B independent from C over A. Suppose a1 , a2 are tuples such that a1 is independent from B over A, a2 is independent from C over A, and Lstp(a1 /A) = Lstp(a2 /A). Then there is a such that tp(a/B) = tp(a1 /B), tp(a/C) = tp(a2 /C) and Lstp(a/A) = Lstp(a1 /A). Here is a convenient reformulation: Fact 2.11 Let A be a set. Let a be independent from b over A, and suppose that Σ(x, a) ∪ Γ(x, b) does not fork over A. Let a0 be such that Lstp(a/A) = Lstp(a0 /A) and a0 is independent from b over A. Then Σ(x, a0 ) ∪ Γ(x, b) does not fork over A. Proof. Let c realise Σ(x, a) ∪ Γ(x, b) such that tp(c/A, a, b) does not fork over A. (Thus {a, b, c} is A-independent.) So a is independent from c over A, a0 is independent from b over A, and Lstp(a/A) = Lstp(a0 /A). By Fact 2.10, we can find a00 such that tp(a00 /Ab) = tp(a0 /Ab) and tp(a00 /Ac) = tp(a/Ac), and {a00 , b, c} is A-independent. Let f be an A-automorphism fixing b and taking a00 to a0 . f (c) solves the problem. Definition 2.12 (i) T is supersimple if any complete type p(x) ∈ S(B) where x is a finite tuple of variables, does not fork over some finite subset A of B. (ii) (T simple.) The (ordinal or ∞-valued) SU rank on complete types is defined by: SU (p) ≥ 0 for all p, SU (p) ≥ α + 1 if p has a forking extension q with SU (q) ≥ α, and for limit ordinal δ, SU (p) ≥ δ iff SU (p) ≥ α for all α < δ. (We sometimes write SU (a/A) for SU (tp(a/A)).) (iii) (T simple.) The ordinal or ∞-valued D rank is defined on formulas 6
by D(φ(x)) ≥ 0 if φ(x) is consistent, D(φ(x)) ≥ α + 1 if for some set A over which φ(x) is defined, there is a formula ψ(x) which implies φ(x) and forks over A. As in (i) for limit δ. For Σ(x) a partial type closed under conjunctions D(Σ) = min{D(φ(x) : φ ∈ Σ}. Fact 2.13 T is supersimple iff [T is simple and for all complete types p(x) in finitely many variables, SU (p) < ∞] iff [T is simple and for each formula φ(x), D(φ(x)) < ∞]. Remark 2.14 Suppose T is supersimple. Let a be a possibly infinite tuple, and A a set. Then SU (tp(a/A)) < ∞ iff there is a finite subtuple a0 of a such that a ⊆ acl(A, a). Now we pass to hyperimaginaries. By a type-definable equivalence relation over A we mean an equivalence relation E(x, y) on possibly infinite tuples, which is defined by a partial type Φ(x, y) with parameters from A. By a hyperimaginary we mean an object of the form a/E (i.e. the class of a mod E) where E is type-definable over ∅. It is not in general possible to formally adjoin sorts SE for hyperimaginaries, and obtain a saturated (many-sorted) first order structure. Nevertheless, much of the machinery of model theory applies to hyperimaginaries. Note that everything we have dealt with so far is a hyperimaginary (real elements, imaginary elements, and ¯ ) acts on everybody. even infinite tuples of imaginaries). Note that Aut(M ¯ ) ⊆ Aute (M ¯ ), If d, e are hyperimaginaries, then we say e ∈ dcl(d) if Autd (M ¯ and e ∈ bdd(d) if e has a bounded number of images under Autd (M ). Note that if e is an imaginary element, then e ∈ bdd(d) iff e ∈ acl(d) (namely has only finitely many images under d automorphisms). We will say that hyperimaginaries e, d are equivalent (or interdefinable) if dcl(e) = dcl(d). Similarly for interbounded. Definition 2.15 We say that T has elimination of hyperimaginaries if any ¯ eq . hyperimaginary e is equivalent to a tuple of elements of M The main point of this paper is to prove that supersimple theories eliminate hyperimaginaries. What is crucial is that we can talk about tp(e/d) if e, d are hyperimaginaries: Suppose that e = a/E and d = b/F . Let φ(x, y) be an L-formula. Let 7
φE,F (x, y) be the partial type “∃x0 , y 0 (E(x, x0 ) ∧ F (y, y 0 ) ∧ φ(x0 , y 0 )”. Actually the above is an infinitary formula but by compactness (saturation) is equivalent to the set of its “approximations” which is a partial type. We define tp(e/d) to be the union of all those partial types φE,F (x, b) such that |= φE,F (a, b). Although as a set of formulas this has parameter b, a different choice of b0 would yield an equivalent partial type. We have: ¯ which fixes Fact 2.16 tp(e/d) = tp(e0 /d) iff there is an automorphism of M 0 d and takes e to e . The following will be useful in this paper: ¯ eq , and e is a hyperimaginary Lemma 2.17 Suppose that a is a tuple from M such that e ∈ dcl(a) and a ∈ bdd(e). Then e is equivalent to a tuple from ¯ eq . M Proof. Suppose a = (ai : i ∈ I). For each finite J ⊆ I, let aJ = (ai : i ∈ J). So each such aJ ∈ acl(e). Let bJ be the finite set of e conjugates of aJ , viewed ¯ eq . So b = (bJ : J finite subset of I) is a sequence as a single element of M of imaginaries. Clearly b ∈ dcl(e). We want to see that e ∈ dcl(b). Let f be an automorphism fixing b, so fixing each bJ . It follows by definition of bJ that tp(aJ /e) = tp(f (aJ )/e) (for each finite J ⊆ I), and thus tp(a/e) = tp(f (a)/e). As e ∈ dcl(a), f has to fix e. So b is equivalent to e. Another useful result is: Lemma 2.18 Let e, d be hyperimaginaries and suppose e ∈ bdd(d). Then the set of d conjugates of e is equivalent to a hyperimaginary e0 . (That is, an automorphism f fixes setwise the set of d-conjugates of e iff f (e0 ) = e0 .) ¯ eq . For each (symmetric) Proof. Let e = a/E, where a is a tuple from M nφ 0 formula φ(x, y) ∈ E(x, y), let aφ , .., aφ be a maximal set of realizations of tp(a/d) such that |= ¬φ(aiφ , ajφ ) for i 6= j. Let r(z, xiφ : φ ∈ E, i ≤ nφ ) be the partial type saying tp(z, xiφ ) = tp(d, a) for all φ, i, and ¬φ(xiφ , xjφ ) for i 6= j. Let F (z1 , z2 ) be the type-definable relation ∃¯ x(r(z1 , x¯) ∧ r(z2 , x¯)) on ¯ , F (d, f (d)) tp(d) × tp(d). It is easy to see that for any automorphism f of M holds iff f fixes setwise the set of d-conjugates of e. So F is a type-definable equivalence relation on tp(d), and d/F is equivalent to the set of d-conjugates of e. 8
It is important to note that if d, e, f are hyperimaginaries, then the notion “tp(e/d) divides over f ” is well-defined. Fact 2.19 Suppose T is simple. (i) Remark 2.3 (ii), and Facts 2.7, 2.10 and ¯ eq . (Also Fact 2.5 2.11 hold for hyperimaginaries in place of tuples from M holds when suitably modified.) (ii) For e, d, f hyperimaginaries, Lstp(d/f ) = Lstp(e/f ) iff tp(d/bdd(f ) = tp(e/bdd(f )). (iii) Suppose T is supersimple, Suppose e is a hyperimaginary element which ¯ eq . Then SU (tp(e/A)) < ∞. is in bdd(aA) for a some finite tuple from M From now, a, b, , e, ... will range over hyperimaginaries, unless we say otherwise. Note that a set A of parameters, will, under a given enumeration also be a hyperimaginary, but we stick with notation A, B, .. for sets of parameters. Definition 2.20 (T simple.) By an amalgamation base, we mean a complete type p(x) over a set A (possibly of hyperimaginaries) such that the Independence Theorem holds for p over A, namely whenever B, C ⊆ A are independent over A and p1 (x), p2 (x) are complete nonforking extensions of p over B, C respectively, then p1 (x) ∪ p2 (x) extends to a nonforking extension q(x) ∈ S(B ∪ C) of p. Note that any Lascar strong type (complete type over a bdd-closed set) is an amalgamation base. Definition 2.21 (T simple.) Let p(x) ∈ S(A) be an amalgamation base. Then the canonical base of p, Cb(p) is defined to be the smallest (up to equivalence) hyperimaginary e ∈ dcl(A) such that p does not fork over e and p|e is an amalgamation base. Fact 2.22 (T simple) Canonical bases exist. Moreover Cb(p) has the following alternative description: Let Pp be the family of amalgamation bases q(x) for which there are amalgamation bases p = p0 , p1 , .., pk = q (some k) such that pi and pi+1 have a common nonforking extension. Then an auto¯ fixes Cb(p) pointwise iff it fixes Pp setwise. In particular morphism f of M Cb(p) = Cb(q) whenever q ∈ Pp .
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Remark 2.23 T is stable just if T is simple and for any complete type p(x) ¯ ), p has a unique nonforking over a model M (elementary substructure of M extension over any B ⊃ M . If T is stable, then any type over a model is definable (for each φ(x, y) ∈ L there is ψ(y) over M such that φ(x, b) ∈ p(x) iff |= ψ(b) for all b ∈ M .) In this (stable) case, Cb(p) is precisely the sequence ¯ eq . of of these defining formulas, considered as elements of M Fact 2.24 (T simple.) Let p(x) ∈ S(A) be an amalgamation base. Let (ai : i < ω) be a Morley sequence in p. Then Cb(p) ∈ dcl(ai : i < ω). Moreover if T is supersimple and SU (p) < ∞ then Cb(p) ∈ dcl(ai : i < k) for some k < ω. ¯ eq . Let a be Proof. We will prove it assuming that x is a finite tuple from M such that (ai : i < ω) _ (a) is indiscernible. So q(x) = tp(a/A ∪ {ai : i < ω}) is a nonforking extension of p(x). But clearly q(x) is finitely satisfiable in (ai : i < ω) so does not fork over (ai : i < ω). Let r(x) = tp(a/(ai : i < ω)). We claim that r(x) implies Lstp(a/(ai : i < ω)): Let E(x, y) be a typedefinable equivalence relation over (ai : i < ω) with a small number of classes. Let φ(x, y) be a formula in E(x, y). φ(x, y) has parameters from a0 , .., an say. As there do not exist infinitely many tuples each distinct pair from which satisfies ¬φ(x, y) it follows from, indiscernibility, that φ(ai , a) holds for all i > n. It easily follows (as E is an equivalence relation) that E(a, b) for any other realization b of r(x). Thus r(x) implies Lstp(a/(ai : i < ω)) as claimed, so by the Independence Theorem, r(x) is an amalgamation base. Thus Cb(p) = Cb(r) ∈ dcl(ai : i < ω) as required. We now pass to the theory of analysability for hyperimaginaries. Definition 2.25 (T simple.) Let A be a set and p(x) a complete type over ¯ ). We A. Let Σ be a family of partial types which is invariant under AutA (M say that p(x) is Σ-internal if there is some B ⊇ A and a sequence (bi : i < α) of realizations of types in Σ which are over B, and there is some realization a of p(x) such that a ∈ dcl(B, bi : i < α). Recall that ( for T simple), complete types p(x), q(y) are orthogonal if whenever p0 (x), q 0 (y) are nonforking extensions of p, q respectively over a common domain C, then for any realizations a of p0 and b of q 0 , a is independent from b over C. 10
Lemma 2.26 (T simple.) Suppose tp(a/A) is nonorthogonal to some type q. Then there is a0 ∈ dcl(a, A) \ bdd(A) such that tp(a0 /A) is internal in the family of A-conjugates of q. Proof. The situation allows us to assume that q is over a set B0 where a is independent from B0 over A and moreover there is a realization b0 of q such that tp(a/B0 , b0 ) forks over B0 . Let (Bi , ni : i < ω) be a Morley sequence in tp(B0 , b0 /A, a). Let e = Cb((B0 , b0 /bdd(A, a))). By Fact 2.24, e ∈ dcl(Bi , bi : i < ω). Also e ∈ bdd(A, a). On the other hand e ∈ / bdd(A) (for otherwise (B0 , b0 ) would be independent from a over A). Note that a and thus also e is independent from (Bi : i < ω) over A. Thus (as bi realises a conjugate of q over Bi ), e is internal in the family of A-conjugates of q. Remember that e ∈ bdd(A, a). By Lemma 2.18, let e0 be a hyperimaginary equivalent to the set of (A, a) conjugates of e. Then tp(e0 /A) is still internal in the family of A-conjugates of q and e0 ∈ / bdd(A). Now we introduce a certain “analysability rank” for supersimple theories, which will be crucial for the proof of our main result in section 4. For the remainder of this section we assume T to be supersimple. Lemma 2.27 Let SU (a/A) < ∞. Then there are (hyperimaginaries) a0 , a1 , ..., an ∈ dcl(A, a) (some n < ω) and ordinals αi for i < n such that a0 = (some enumeration of ) A, a ∈ bdd(an ) and for each i < n, ai ∈ dcl(ai+1 ), tp(ai+1 /ai ) is internal in a family of formulas of D-rank αi , and tp(a/ai ) is orthogonal to all types of D-rank < αi . Proof. Put a0 to be some enumeration of A. If SU (a/A) = 0 then a ∈ bdd(a0 ) and we are finished. Suppose otherwise. It is clear that there is some ¯ eq such that a forks with b over A. For any such b, D(tp(b/A)) < ∞, b∈M so let α0 be the least ordinal such that tp(a/A) is nonorthogonal to some type of D-rank α0 . By Lemma 2.26, there is a1 ∈ dcl(A, a) \ bdd(A) such that tp(a1 /a0 ) is internal in the family of A-conjugates of q, so in particular internal in the family of formulas of D-rank α0 . As a forks with a1 over A, SU (a/A, a1 ) < SU (a/A), so continuing we finish after finitely many steps. Definition 2.28 Given a and A such that SU (a/A) < ∞, we define R(a/A) ( = R(tp(a/A))) to be the least ordinal α such that for some n < ω there are a0 , .., an and ordinals α0 , .., αn−1 as in Lemma 2.27, with αi ≤ α for all i < n. 11
Lemma 2.29 Let a = (e1 , .., er ) and suppose R(ei /A) < α for each i. Then R(a/A) < α. Proof. It is enough to show that if tp(a/A) is internal in a family Σ of partial types (where Σ is A-invariant), then tp(a/B) is also internal in Σ, for any B ⊇ A. So for some C ⊃ A such that a is independent from C over A, a ∈ dcl(b) where b is a tuple of realizations of types in Σ which are over C. Let C 0 realise a nonforking extension of tp(C/A, a) over (B, a). Then a is independent from C 0 over B and a ∈ dcl(b0 , A) where b0 is the image of b under some (A, a)-automorphism taking C to C 0 . As Σ is A-invariant. tp(a/B) is internal to Σ. Corollary 2.30 Suppose a = (ei : i ∈ I). Suppose SU (a/A) < ∞ and R(ei /A) < α for each i. Then R(a/A) < α. Proof. As SU (a/A) < ∞, a ∈ bdd(ei1 , .., ein ) for some i1 , .., in ∈ I. So the result easily follows from the previous lemma. Lemma 2.31 Suppose SU (a/A) < ∞. Let a0 be some enumeration of {b ∈ dcl(A, a) : R(b/A) < α}. Then R(a0 /A) < α and tp(a/a0 ) is orthogonal to all formulas of D-rank < α. Proof. R(a0 /A) < α is by Corollary 2.30. Suppose for a contradiction that tp(a/a0 ) is nonorthogonal to a formula of Drank < α. By 2.26 there is c ∈ dcl(a, a0 )\bdd(a0 ) such that tp(c/a0 ) is internal in some family of formulas of D-rank β for some β < α. Let a0 = A, a1 , .., an witness R(a0 /A) < α. In particular a0 ∈ bdd(an ). So easily tp(c/an ) is internal in the family of formulas of D-rank β. It follows that R(c, an /A) < α. As (c, an ) ∈ dcl(a, A) it follows that c ∈ dcl(a0 ) a contradiction. Finally, we make a rather useful observation on type-definable equivalence relations. Lemma 2.32 (T supersimple.) Let a be a possibly infinite tuple of elements ¯ eq . Let q(x) = tp(a). Assume SU (q) < ∞. Let E be a type-definable from M (over ∅) equivalence relation on q. Then there is a formula ψ(x, y) ∈ E(x, y) such that whenever b, c realize q and |= ψ(b, c) then b/E and c/E are interbounded. 12
Proof. As SU (q) < ∞, a ∈ acl(a0 ) for a suitable finite subtuple a0 of a. x0 denotes the corresponding finite subtuple of x. Among formulas θ(x0 , a) ∈ E(x, a) choose one of least D-rank. Call it θ(x0 , a) and suppose its D-rank is α. Note that α is smallest among the D-ranks of formulas φ(x00 , a) ∈ E(x, a) for x00 a finite subtuple of x containing x0 . Now, by compactness, find a symmetric formula θ1 (x, y) ∈ E(x, y) implying θ and such that |= θ1 (x, y) ∧ θ1 (y, z) → θ(x, z). Now let ψ(x, y) ∈ E(x, y) be symmetric such that |= ψ(x, y) ∧ ψ(y, z) ∧ ψ(z, w) → θ1 (x, w). Now suppose that b realizes q, and |= ψ(a, b). Claim. b/E ∈ bdd(a/E). Proof of claim. Suppose not. Then we can find an infinite indiscernible sequence (bi : i < ω) of realizations of tp(b/(a/E)) such that ¬E(bi , bj ) for i 6= j. In fact by indiscernibility there is a formula χ(x, y) ∈ E(x, y) such that ¬χ(bi , bj ) for each i 6= j. Without loss of generality, ψ(a, b0 ). For each i there is ai such that E(a, ai ) and ψ(ai , bi ). So by the choice of ψ each bi satisfies θ1 (x, a). Let φ(x, y) ∈ E(x, y) be symmetric, such that |= φ(x, y) → θ1 (x, y), and |= φ(x, y) ∧ φ(y, z) → χ(x, z). It follows that (i) for each bi , φ(x, bi ) → θ(x, a), and (ii)φ(x, bi ) ∧ φ(x, bj ) is inconsistent for i 6= j. We may rechoose the bi such that they still each realize q, (i), (ii) remain true and (bi : i < ω) is indiscernible over a. But then φ(x, b0 ) forks over a, and so by (i) D(φ(x, b0 ) < D(θ(x, a)) contradicting the choice of θ. The Claim is proved. By symmetry, a/E ∈ bdd(b/E) too, and the Lemma is proved.
3
The basic argument
We assume T to be supersimple in this section. Let a be a finite tuple of real elements, and E a type-definable equivalence relation over ∅ on q = tp(a). Let us assume that tp(a/(a/E)) |= Lstp(a/(a/E)). By Lemma 2.23, let ψ0 (x, y) be a formula in E such that for a, b realising 13
q, if ψ(a, b) then a/E and b/E are interbounded. Let ψ1 (x, y) ∈ E be such that if b, c, c0 realise q, E(c0 , c) and ∃x(ψ1 (x, c) ∧ ψ(x, b) then ψ0 (c0 , b) (by compactness). Let Ψ(x, y) be the set of formulas in E which imply ψ1 (x, y). With the above notation, we make: Definition 3.1 Let ψ(x, y) ∈ Ψ(x, y), and let a, b realise q. We define Sψ (a, b) to hold if there exists c realizing q such that (i) E(c, a), and tp(c/b, (a/E)) does not fork over (a/E), (ii) ψ0 (c, b) holds, and ψ(x, c) ∧ ψ(x, b) does not fork over a/E. Remark 3.2 (i) If E(a, a0 ) then Sψ (a, b) iff Sψ (a0 , b). (ii) E(a, b) iff Sψ (a, b) holds for all ψ ∈ Ψ. Proof. (i) is immediate from the definition of Sψ . (ii) Left to right is trivial. Assume now the right hand side. So for each ψ ∈ Ψ, there is c such that |= E(c, a) and ψ(x, c) ∧ ψ(x, b) is consistent. As Ψ axiomatizes E, there is, by compactness, c such that |= E(c, a) and E(x, c) ∧ E(x, b) is consistent. Thus E(a, b). Remark 3.3 Suppose for each ψ ∈ Ψ, Sψ is definable (on q × q). Then E is equivalent on q to a conjunction of definable equivalence relations. Proof. Fix ψ, and suppose ρψ (x, y) is a formula defining Sψ on q × q. By Remark 3.2(i), and compactness there is a formula π(x) ∈ q such that E(x, y) |= ∀z(π(z) → (ρψ (x, z) ↔ ρψ (y, z)). Let Eψ (x, y) be the formula on the right hand side, which note defines an equivalence relation. Suppose that Eψ (a, b) holds for all ψ ∈ Ψ (a, b realising q). As ρψ (a, a) holds for all ψ it follows that ρψ (a, b) holds for all ψ, so by Remark 3.2, E(a, b) holds. Thus E is equivalent on q × q to the conjunction of the Eψ . Thus the main issue is the definability of the relations Sψ . Lemma 3.4 Sψ is type-definable (on q × q). Proof. For any L-formula φ(x, w) and k < ω, let nφ,k = D(E(x, a), φ, k). Then a, c, b satisfies (i) and (ii) of Definition 3.1 iff a, b, c realise q, E(a, c), D(tp(c/b, (a/E)), φ, k) ≥ nφ,k for all φ, k, ψ0 (c, b) and (*) there is a Morley sequence ((ci , bi ) : i < ω) in tp(c, b/(a/E)) such that 14
{ψ(x, ci ) ∧ ψ(x, bi ) : i < ω} is consistent. Indiscernibility over a/E of the sequence (ci , bi )i is expressible. As for the independence: note that, as ψ0 (c, b), a/E and b/E are interbounded. So independence of (ci , bi )i over a/E is equivalent to (bi )i is b/E-independent, and for each i, ci is independent from (cj : j < i) ∪ (bj : j < ω) over a/E. These are again expressible using the D(−, φ, k). (See Fact 2.5(ii).) We now see what can be said regarding the type-definability of ¬Sψ (x, y). Lemma 3.5 Let a, b realise q, and ψ ∈ Ψ. Then ¬Sψ (a, b) holds iff there is c realising q such that (i) E(c, a) and tp(c/b, (a/E)) does not fork over a/E, and (ii) either ¬ψ0 (c, b) or ψ(x, c) ∧ ψ(x, b) forks over a/E. Proof. Left to right is immediate, for suppose that ¬Sψ (a, b). Choose c satisfying (i). Then from Definition 3.1 we see that (ii) in the lemma holds. Conversely, suppose Sψ (a, b), witnessed by c0 as in Definition 3.1. Now choose any c satisfying (i) in the lemma. We want to show that (ii) fails. By Fact 2.11 (asLstp(c/(a/E)) = Lstp(c0 /(a/E))), ψ(x, c)∧ψ(x, b) does not fork over a/E, so in particular is consistent. As |= ψ → ψ1 we see from the choice of ψ1 (x, y) that ψ0 (c, b). Lemma 3.6 Let ψ(x, y) ∈ Ψ(x, y). Suppose that there is some k such that whenever c, b realise q, c/E, b/E are interbounded, and c is independent from b over c/E, and ψ(x, c)∧ψ(x.b) forks over c/E, then for any c/E-independent sequence ((ci , bi ) : i < k) of realizations of tp(c, b/(a/E)), {ψ(x, ci )∧ψ(x, bi ) : i < k} is inconsistent. Then Sψ (x, y) is definable (on q × q). Proof. Given the assumptions of the lemma, we see from Lemma 3.5 that ¬Sψ (x, y) is type-definable: (i) is clear and (ii) says if ψ0 (c, b) then ψ(x, c) ∧ ψ(x, c) forks over a/E. The latter is, by the assumptions, equivalent to the existence of a Morley sequence ((ci , bi )i 1. Proof of sublaim. Clearly cˆi is independent from bi over cˆ1 . Also tp(ˆ ci /ˆ c1 ) being a nonforking extension of tp(ˆ ci /(ˆ a/F ) is (by Fact 4.4 (v)) orthogonal to all types of D-rank < α. Also tp(bi /bdd(ˆ c1 )) does not fork over b/E (see proof of Lemma 4.6) and so by assumption is orthogonal to all types of D-rank < α. Thus the subclaim follows. By the subclaim and the fact that ((ˆ ci , bi ) : i = 2, .., n + 2) is cˆ1 -independent, we easily get a contradiction to each (ˆ ci , bi ) forking with (d1 , .., dn ) over cˆ1
19
(i = 2, .., n + 2). This contradiction proves the claim. It is now easy, as in the proof of Lemma 4.6 to show the type-definability of ¬Sψ on qˆ × q. We now complete the proof of Lemma 4.3. By Lemmas 4.6, 4.7 and compactness, there is, for each ψ ∈ Ψ a formula ρψ (ˆ x, y) defining Sψ on qˆ × q. Note that F (ˆ x, yˆ) ∪ q(z) |= ρψ (ˆ x, z) ↔ ρψ (ˆ y , z). Thus there is a formula πψ (z) ∈ q such that F (ˆ x, yˆ) |= ∀z(πψ (z) → (ρψ (ˆ x, z) ↔ ρψ (ˆ y , z)). Let Eψ (ˆ x, yˆ) be the definable equivalence relation given by the formula on the right hand side. So a ˆ/Eψ ∈ dcl(ˆ a/F ) for every ψ ∈ Ψ. Note that if ˆ Eψ (ˆ a, b) holds for all ψ, then Sψ (ˆ a, b) holds for all ψ whereby a/E = b/E. Let a ˜ = ((ˆ a/Eψ ) : ψ ∈ Ψ). So a/E ∈ dcl(˜ a). As we saw a couple of lines above, a ˜ ∈ dcl(ˆ a/F ), whereby R(˜ a/(a/E)) ≤ R((ˆ a/F )/(a/E)), and the latter is by Fact 4.4(iv) < α. So by induction hypothesis (as a ˜ is a sequence of imaginaries and SU (˜ a) < ∞), a/E is equivalent to a sequence of imaginaries. This completes the proof of Lemma 4.3. We give some consequences of Theorem 4.1. Corollary 4.8 (T supersimple.) Let p(x) be an amalgamation base, where x is a single imaginary variable (for example a finite tuple of real or imaginary variables). Then Cb(p) is equivalent to a sequence of imaginary elements (in ¯ eq ). other words Cb(p) exists in M Proof. By 2.24, Cb(p) ∈ dcl(a1 , .., an for some n where (ai )i is a Morley sequence in p. So the result follows from Theorem 4.1 Corollary 4.9 (T supersimple.) Any hyperimaginary element e is equivalent to a sequence of imaginaries. (Namely any type-definable over ∅, equivalence relation E(x, y) on a possibly infinitary complete type q(x) is equivalent on q(x) to a conjunction of definable equivalence relations.) Proof. We may assume that tp(a/(a/E)) |= Lstp(a/(a/E)) (by 3.8). Let a = (ai : i ∈ I) and for each finite subset J of I let aJ = (ai : i ∈ J). Easily tp(aJ /(a/E)) |= Lstp(aJ /(a/E)) for each J. By Corollary 4.8, Cb(tp(aJ /(a/E)) is a set CJ of imaginary elements. It is easy to see that S J CJ = Cb(tp(a/(a/E)). 20
Corollary 4.10 (T supersimple.) For any amalgamation base p, Cb(p) ex¯ eq . ists in M Corollary 4.11 (T supersimple.) Any complete type p(x) (in possibly in¯ eq finitely many real or imaginary variables) over an algebraically closed in M is an amalgamation base. (Restatements: (1) The Independence Theorem holds over any algebraically ¯ eq . closed set in M ¯ eq and any tuple a of real or imaginary parameters, (2) for any set A ⊂ M stp(a/A) implies Lstp(a/A).) Proof. We prove (2). Let e be a hyperimaginary in bdd(A). By Corollary 4.9, e is equivalent to a sequence (ai )i of imaginaries, each of which must be in acl(A). Thus acleq (A) is equivalent to bdd(A). So stp(a/A) implies tp(a/acleq (A)) implies tp(a/bdd(A)) impplies Lstp(a/A)). In recent work ([11]) B. Kim has used Corollary 4.10 to give a nice description of canonical bases in supersimple theories: Suppose, for example, that p(x) is a canonical type (namely Cb(p) = domain of p). Let P be the family of ¯ . Let Φ be the collection of L-formulas nonforking extensions of p over M φ(x, y) such that for all q, r ∈ P, q|φ = r|φ. Then one can show that for φ(x, y) ∈ Φ, this common φ(x, y) type is definable. What Kim shows is that Cb(p) is exactly the set of these φ(x, y)-definitions, for φ ∈ Φ. The nice phenomenon here is that, in this supersimple context one recovers a trace of definability of types, going via first elimination of hyperimaginaries and then existence of canonical bases as imaginaries. On the other hand in classical stability theory, the order was reversed: definability of types was the starting point, from which first existence of canonical bases, and then elimination of hyperimaginaries were deduced.
References [1] S. Buechler, Lascar strong types in some simple theories, to appear in Journal of Symbolic Logic. [2] S. Buechler, Canonical bases in some supersimple theories, preprint 1998. 21
[3] B. Hart and A. Pillay, A note on canonical bases, preprint 1998. [4] B. Hart, B. Kim and A. Pillay, Coordinatization and canonical bases, to appear in Journal of Symbolic Logic. [5] E. Hrushovski, Pseudofinite fields and related structures, preprint 1991. [6] E. Hrushovski, Simplicity and the Lascar group, preprint 1997. [7] B. Kim, Simple first order theories, Ph. D. thesis, University of Notre Dame, 1996. [8] B. Kim, Forking in simple unstable theories, Journal of London Math. Soc., 1999. [9] B. Kim, Recent results on simple first order theories, Proceedings Blaubeuren, London Math. Soc. Lecture Notes, Cambridge Univ. Press, 1997. [10] B. Kim, A note on Lascar strong types in simple theories, Journal of Symbolic Logic, 63 (1998), 926-936. [11] B. Kim, Simplicity and stability in there, preprint 1999. [12] B. Kim and A. Pillay, Simple theories, Annals of Pure and Applied Logic, 88 (1997). [13] D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, preprint 1998. [14] A. Pillay and B. Poizat, Pas d’imaginaires dans l’infini, Journal of Symbolic Logic, 52 (1987), 400-403. [15] Z. Shami, A natural finite equivalence relation definable in low theories, to appear. [16] S. Shelah, Simple unstable theories, Annals of Math. Logic, 19 (1980), 177-203.
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