aa r X i v : . [ m a t h . L O ] S e p AMPLE THOUGHTS
DANIEL PALAC´IN AND FRANK O. WAGNER
Abstract.
Non- n -ampleness as defined by Pillay [20] and Evans[6] is preserved under analysability. Generalizing this to a moregeneral notion of Σ-ampleness, this gives an immediate proof forall simple theories of a weakened version of the Canonical BaseProperty (CBP) proven by Chatzidakis [5] for types of finite SU-rank. This is then applied to the special case of groups. Introduction
Recall that a partial type π over a set A in a simple theory is one-based if for any tuple ¯ a of realizations of π and any B ⊇ A the canonicalbase Cb(¯ a/B ) is contained in the bounded closure bdd(¯ aA ). In otherwords, forking dependence is either trivial or behaves as in modules:Any two sets are independent over the intersection of their boundedclosures. One-basedness implies that the forking geometry is partic-ularly well-behaved; for instance one-based groups are bounded-by-abelian-by-bounded. The principal result in [26] is that one-basednessis preserved under analyses (i.e. iterative approximations by some othertypes): a type analysable in one-based types is itself one-based. Thisgeneralized earlier results of Hrushovski [12] and Chatzidakis [5]. One-basedness is the first level in a hierarchy of possible geometric behaviourof forking independence first defined by Pillay [20] and slightly mod-ified by Evans [6], n -ampleness, modelled on the behavior of flags in n -space. Not 1-ample means one-based; not 2-ample is equivalent to anotion previously introduced by Hrushovski [13], CM-triviality. Fields Date : 4 September 2012.2000
Mathematics Subject Classification.
Key words and phrases. stable; simple; one-based; CM-trivial; n -ample; internal;analysable; closure; level; flat; ultraflat; canonical base property.The first author was partially supported by research project MTM 2008-01545of the Spanish government and research project 2009SGR 00187 of the Catalangovernment. He also would like to thank Amador Mart´ın Pizarro for interestingdiscussions around the Canonical Base Property, and suggesting a simplification inthe proof of Theorem 3.6.The second author was partially supported by ANR-09-BLAN-0047 Modig. are n -ample for all n < ω , as is the non-abelian free group [17]. In [20]Pillay defines n -ampleness locally for a single type and shows that asuperstable theory of finite Lascar rank is non n -ample if and only ifall its types of rank 1 are; his proof implies that in such a theory, atype analysable in non n -ample types is again non n -ample.We shall give a definition of n -ampleness for invariant families ofpartial types, and generalize Pillay’s result to arbitrary simple theo-ries. Note that for n = 1 this gives an alternative proof of the mainresult in [26]. Since for types of infinite rank the algebraic (bounded)closure used in the definition is not necessarily appropriate (for a reg-ular type p one might, for instance, replace it by p -closure), we alsogeneralize the notion to Σ-closure for some ∅ -invariant collection of par-tial types (thought of as small), giving rise to the notion of n -Σ-ample.This may for instance be applied to consider ampleness modulo typesof finite SU-rank, or modulo supersimple types. Readers not interestedin this additional generality are invited to simply replace Σ-closure bybounded closure. However, this will only marginally shorten the proofs.As an immediate Corollary of the more general version, we shall derivea weakened version of the Canonical Base Property CBP [23] shownby Chatzidakis [5], where analysability replaces internality in the def-inition. We also give a version appropriate for supersimple theories.Finally, we deduce that in a simple theory with enough regular types,a hyperdefinable group modulo its approximate centre is analysable inthe family of non one-based regular types; the group modulo a nor-mal nilpotent subgroup is almost internal in that family. This can bethought of as a general version of the properties of one-based groupsmentioned above.Our notation is standard and follows [25]. Throughout the paper,the ambient theory will be simple, and we shall be working in M heq ,where M is a sufficiently saturated model of the ambient theory. Thustuples are tuples of hyperimaginaries, and dcl = dcl heq .2. Internality and analysability
For the rest of the paper Σ will be an ∅ -invariant family of partialtypes. Recall first the definitions of internality, analysability, foreign-ness and orthogonality. Definition 2.1.
Let π be a partial type over A . Then π is MPLE THOUGHTS 3 • ( almost ) Σ -internal if for every realization a of π there is B | ⌣ A a and a tuple ¯ b of realizations of types in Σ based on B , such that a ∈ dcl( B ¯ b ) (or a ∈ bdd( B ¯ b ), respectively). • Σ -analysable if for any realization a of π there are ( a i : i < α ) ∈ dcl( Aa ) such that tp( a i /A, a j : j < i ) is Σ-internal for all i < α ,and a ∈ bdd( A, a i : i < α ).A type tp( a/A ) is foreign to Σ if a | ⌣ AB ¯ b for all B | ⌣ A a and ¯ b realizingtypes in Σ over B .Finally, p ∈ S ( A ) is orthogonal to q ∈ S ( B ) if for all C ⊇ AB , a | = p ,and b | = q with a | ⌣ A C and b | ⌣ B C we have a | ⌣ C b .So p is foreign to Σ if p is orthogonal to all completions of partialtypes in Σ, over all possible parameter sets.The following lemmas and their corollaries are folklore, but we addsome precision about non-orthogonality. Lemma 2.2.
Suppose a | ⌣ b and a ⌣ b c . Let ( b i : i < ω ) be an in-discernible sequence in tp( b ) and put p b = tp( c/b ) . Then p b i is non-orthogonal to p b j for all i, j < ω .Proof: We prolong the sequence to have length α . As a | ⌣ b and ( b i : i <α ) is indiscernible, by [25, Theorem 2.5.4] we may assume ab ≡ ab i forall i < α and a | ⌣ ( b i : i < α ). Let B = ( b i : i < ω ), so ( b i : ω ≤ i < α )is independent over B and a | ⌣ B . Choose c i with b i c i ≡ a bc and c i | ⌣ ab i ( b j : j < α )for all ω ≤ i < α . Then ab i c i | ⌣ B ( b j : j = i ) for all ω ≤ i < α . Byindiscernability, if p b i were orthogonal to p b j for some i = j , then theywould be orthogonal for all i = j . As c i | ⌣ b i ( b j : j = i ), the sequence( b i c i : ω ≤ i < α ) would be independent over B . However, a ⌣ B b i c i forall ω ≤ i < α , contradicting the boundedness of weight of tp( a/B ). (cid:3) Lemma 2.3.
Suppose a | ⌣ b and a ′ = Cb( bc/a ) . Let P be the familyof bdd( ∅ ) -conjugates of tp( c/b ) non-orthogonal to tp( c/b ) . Then a ′ ∈ bdd( a ) is P -internal and bdd( ab ) ∩ bdd( bc ) ⊆ bdd( a ′ b ) .Proof: If a | ⌣ bc then a ′ ∈ bdd( ∅ ) and bdd( ab ) ∩ bdd( bc ) = bdd( b ),so there is nothing to show. Assume a ⌣ b c . Clearly a ′ ∈ bdd( a ); as bc | ⌣ a ′ a we get c | ⌣ a ′ b a and hence bdd( ab ) ∩ bdd( bc ) ⊆ bdd( a ′ b ). Let( b i c i : i < ω ) be a Morley sequence in Lstp( bc/a ) with b c = bc . Then a ′ ∈ dcl( b i c i : i < ω ); since b | ⌣ a we get ( b i : i < ω ) | ⌣ a , whence DANIEL PALAC´IN AND FRANK O. WAGNER ( b i : i < ω ) | ⌣ a ′ . So a ′ is internal in { tp( c i /b i ) : i < ω } . Finally,tp( c i /b i ) is non-orthogonal to tp( c/b ) for all i < ω by Lemma 2.2. (cid:3) Corollary 2.4. If a | ⌣ b and tp( c/b ) is (almost) Σ -internal, then Cb( bc/a ) is (almost) Σ -internal. The same statement holds with analysable in-stead of internal .Proof: Let d | ⌣ b c and ¯ e realize partial types in Σ over bd such that c ∈ dcl( bd ¯ e ) (or c ∈ bdd( bd ¯ e ), respectively). We may take d ¯ e | ⌣ bc a .Then d | ⌣ b ac , whence a | ⌣ bd . So Cb( bd ¯ e/a ) is Σ-internal by Lemma2.3. But a | ⌣ bc d ¯ e and c ∈ dcl( bd ¯ e ) implies Cb( bc/a ) ∈ dcl(Cb( bd ¯ e/a ));similarly c ∈ bdd( bd ¯ e ) implies Cb( bc/a ) ∈ bdd(Cb( bd ¯ e/a )).The proof for Σ-analysability is analogous. (cid:3) Definition 2.5.
Two partial types π and π are perpendicular , de-noted π ⊥ π , if for any set A containing their domains and any tuple¯ a i | = π i for i = 1 , a | ⌣ A ¯ a .For instance, orthogonal types of rank 1 are perpendicular. Corollary 2.6.
Suppose a | ⌣ b , and a ∈ bdd( ab ) is (almost) Π -internalover b for some b -invariant family Π of partial types. Let Π ′ be the fam-ily of bdd( ∅ ) -conjugates π ′ of partial types π ∈ Π with π ′ π . Thenthere is (almost) Π ′ -internal a ∈ bdd( a ) with a ∈ bdd( a b ) . Thesame statement holds with analysable instead of internal .Proof: If tp( a /b ) is Π-internal, there is c | ⌣ b a and ¯ e realizing par-tial types in Π over bc such that a ∈ dcl( bc ¯ e ); we choose them with c ¯ e | ⌣ ba a . So c | ⌣ b a , whence a | ⌣ bc . Furthermore, we may assumethat e ⌣ bc a for all e ∈ ¯ e , since otherwise ec | ⌣ b a and we may justinclude e in c , reducing the length of ¯ e . Now a ∈ bdd( abc ) ∩ bdd( bc ¯ e ),so by Lemmas 2.2 and 2.3 there is Π ′ -internal a ∈ bdd( a ) with a ∈ bdd( bca ). Since a | ⌣ b c implies a | ⌣ a b c , we get a ∈ bdd( a b ).For the almost internal case, we replace definable by bounded closure;for the analysability statement we iterate, adding a to the parameters. (cid:3) To finish this section, a decomposition lemma for almost internality.
Lemma 2.7.
Let
Σ = S i<α Σ i , where (Σ i : i < α ) is a collection ofpairwise perpendicular ∅ -invariant families of partial types. If tp( a/A ) is almost Σ -internal, then there are ( a i : i < α ) interbounded over A with a such that tp( a i /A ) is Σ i -internal for i < α . MPLE THOUGHTS 5
Clearly, if a is a finite imaginary tuple, we only need finitely many a i . Proof:
By assumption there is B | ⌣ A a and some tuples ( b i : i < α ) suchthat b i realizes partial types in Σ i over B , with a ∈ bdd( B, b i : i < α ).Let a i = Cb( Bb i /Aa ). Then a i ∈ bdd( Aa ) and tp( a i /A ) is Σ i -internalby Corollary 2.4.Put ¯ a = ( a i : i < α ). Then a | ⌣ Aa i Bb i implies a | ⌣ B ¯ a b i ; since b i | ⌣ Ba ( b j : j = i ) by perpendicularity we obtain b i | ⌣ B ¯ a ( a, b j : j = i )for all i < α . Hence ( a, b i : i < α ) is independent over B ¯ a , and inparticular a | ⌣ B ¯ a ( b i : i < α ) . Since a ∈ bdd( B, b i : i < α ) we get a ∈ bdd( B ¯ a ); as a | ⌣ A B implies a | ⌣ A ¯ a B we obtain a ∈ bdd( A ¯ a ). (cid:3)
3. Σ -closure, Σ -closure and a theory of levels In his proof of Vaught’s conjecture for superstable theories of finiterank [3], Buechler defines the first level ℓ ( a ) of an element a of fi-nite Lascar rank as the set of all b ∈ acl eq ( a ) internal in the family ofall types of Lascar rank one; higher levels are defined inductively by ℓ n +1 ( a ) = ℓ ( a/ℓ n ( a )). The notion has been studied by Prerna BihaniJuhlin in her thesis [1] in connection with a reformulation of the canon-ical base property. We shall generalise the notion to arbitrary simpletheories. Definition 3.1.
For an ordinal α the α -th Σ -level of a over A is definedinductively by ℓ Σ0 ( a/A ) = bdd( A ), and for α > ℓ Σ α ( a/A ) = { b ∈ bdd( aA ) : tp( b/ [ β<α ℓ β ( a/A )) is almost Σ-internal } . Finally, we shall write ℓ Σ ∞ ( a/A ) for the set of all hyperimaginaries b ∈ bdd( aA ) such that tp( b/A ) is Σ-analysable. Remark 3.2.
Clearly, tp( a/A ) is Σ-analysable if and only if ℓ Σ ∞ ( a/A ) =bdd( aA ) if and only if ℓ Σ α ( a/A ) = bdd( aA ) for some ordinal α , and theminimal such α is the minimal length of a Σ-analysis of a over A . Lemma 3.3. If a | ⌣ b , then ℓ Σ α ( ab ) = bdd( ℓ Σ α ( a ) , ℓ Σ α ( b )) for any α .Proof: Let c = ℓ Σ α ( ab ). Clearly, ℓ Σ α ( a ) ℓ Σ α ( b ) ⊆ c . Conversely, put a =Cb( bc/a ). Then tp( a ) is internal in the family of bdd( ∅ )-conjugates DANIEL PALAC´IN AND FRANK O. WAGNER of tp( c/b ) by Corollary 2.4; since even tp( c ) is Σ-analysable in α steps,so is tp( a ). Thus a ⊆ ℓ Σ α ( a ). Now bc | ⌣ a a implies c | ⌣ ℓ Σ α ( a ) b a, whence c ⊆ bdd( ℓ Σ α ( a ) , b ). By symmetry, c ⊆ bdd( ℓ Σ α ( b ) , a ), that is, ℓ Σ α ( ab ) ⊆ bdd( ℓ Σ α ( a ) , b ) ∩ bdd( ℓ Σ α ( b ) , a ) . On the other hand, a | ⌣ b yields a | ⌣ ℓ Σ α ( a ) ℓ Σ α ( b ) b. Thus,bdd( ℓ Σ α ( a ) , b ) ∩ bdd( ℓ Σ α ( b ) , a ) = bdd( ℓ Σ α ( a ) , ℓ Σ α ( b )) , whence the result. (cid:3) Corollary 3.4. If ( a i : i ∈ I ) is an ∅ -independent sequence, then ℓ Σ α ( a i : i ∈ I ) = bdd( ℓ Σ α ( a i ) : i ∈ I ) .Proof: Let c = ℓ Σ α ( a i : i ∈ I ) and set b J = Cb( c/a i : i ∈ J ) for eachfinite J ⊆ I . Note for each finite J ⊆ I that tp( b J ) is Σ-analysable in α steps. Thus b J ⊆ ℓ Σ α ( a i : i ∈ J ). On the other hand, ℓ Σ α ( a i : i ∈ J ) = bdd( ℓ Σ α ( a i ) : i ∈ J )by Lemma 3.3 and induction, since J ⊆ I is finite. Therefore c | ⌣ ( ℓ Σ α ( a i ): i ∈ I ) ( a i : i ∈ I )by the finite character of forking, whence c ⊆ bdd( ℓ Σ α ( a i ) : i ∈ I ). (cid:3) We shall see that the first level governs domination-equivalence.
Definition 3.5.
An element a Σ -dominates an element b over A , de-noted a ☎ Σ A b , if for all c such that tp( c/A ) is Σ-analysable, a | ⌣ A c implies b | ⌣ A c . Two elements a and b are Σ -domination-equivalent over A , denoted a (cid:3) Σ A b , if a ☎ Σ A b and b ☎ Σ A a . If Σ is the set of all types,it is omitted.The following generalizes a theorem by Buechler [3, Proposition 3.1]for finite Lascar rank. Theorem 3.6.
Let Σ ′ be an ∅ -invariant family of partial types. (1) a and ℓ Σ1 ( a/A ) are Σ -domination-equivalent over A . (2) If tp( a/A ) is Σ -analysable, then a and ℓ Σ1 ( a/A ) are domination-equivalent over A . (3) If tp( a/A ) is Σ ∪ Σ ′ -analysable and foreign to Σ ′ , then a and ℓ Σ1 ( a/A ) are domination-equivalent over A . MPLE THOUGHTS 7
Proof: (1) Since ℓ Σ1 ( a/A ) ∈ bdd( Aa ), clearly a dominates (and Σ-dominates) ℓ Σ1 ( a/A ) over A .For the converse, suppose tp( b/A ) is Σ-analysable and b ⌣ A a . Con-sider a sequence ( b i : i < α ) in bdd( Ab ) such that tp( b i /A, b j : j < i ) isΣ-internal for all i < α and b ∈ bdd( A, b i : i < α ). Since a ⌣ A b thereis a minimal i < α such that a ⌣ A, ( b j : j
If tp( a/A ) is Σ -analysable and Σ is a subfamily of Σ such that tp( a/A ) remains Σ -analysable, then ℓ Σ ( a/A ) ⊆ ℓ Σ ( a/A ) ⊆ bdd( aA )and ℓ Σ ( a/A ) and ℓ Σ ( a/A ) are both domination-equivalent to a over A . In fact it would be sufficient to have Σ such that tp( ℓ Σ ( a/A ) /A )is Σ -analysable. Question 3.8.
When is there a minimal (boundedly closed) a ∈ bdd( aA ) domination-equivalent with a over A ?If T has finite SU-rank, one can take a ∈ bdd( aA ) \ bdd( A ) with SU ( a /A ) minimal possible. Definition 3.9. • A type tp( a/A ) is Σ -flat if ℓ Σ1 ( a/A ) = ℓ Σ2 ( a/A ).It is A -flat if it is Σ-flat for all A -invariant Σ. It is flat if for all B ⊇ A every nonforking extension to B is B -flat. A theory T is flat if all its types are. • A type p ∈ S ( A ) is A -ultraflat if it is almost internal in any A -invariant family of partial types it is non-foreign to. It is ultraflat if for any B ⊇ A every nonforking extension to B is B -ultraflat. DANIEL PALAC´IN AND FRANK O. WAGNER
Flatness and ultraflatness are clearly preserved under non-forking ex-tensions and non-forking restrictions, and under adding and forgettingparameters.
Remark 3.10.
If tp( a/A ) is Σ-flat, then ℓ Σ α ( a/A ) = ℓ Σ1 ( a/A ) for all α >
0. Clearly, ultraflat implies flat.
Example. • Generic types of fields or definably simple groupsinterpretable in a simple theory are ultraflat. • Types of Lascar rank 1 are ultraflat. • If there is no boundedly closed set between bdd( A ) and bdd( aA ),then tp( a/A ) is A -ultraflat. • In a small simple theory there are many A -ultraflat types over fi-nite sets A , as the lattice of boundedly closed subsets of bdd( aA )is scattered for finitary aA .Next we shall prove that any type internal in a family of Lascar rankone types is also flat. Lemma 3.11. It tp( a/A ) is flat (ultraflat), then so is tp( a /A ) for any a ∈ bdd( Aa ) .Proof: Consider a set B extending A with B | ⌣ A a ; we may assumethat B | ⌣ Aa a ,whence B | ⌣ A a .Firstly, the flat case is clear since ℓ Σ α ( a /B ) = ℓ Σ α ( a/B ) ∩ bdd( Ba )for any α > B -invariant family Σ. Assume now tp( a/A )is ultraflat, a ∈ bdd( Aa ) and tp( a /B ) is not foreign to some B -invariant family Σ. Then tp( a/B ) is not foreign to Σ, hence almostΣ-internal, as is tp( a /B ). (cid:3) Corollary 3.12. If tp( a/A ) is almost internal in a family of types ofLascar rank one, then it is flat.Proof: Assume there is some B | ⌣ A a and some tuple ¯ b of realizationsof types of Lascar rank one over B such that a ⊆ bdd( B ¯ b ). We mayassume ¯ b is an independent sequence over B since all its elements haveSU-rank one. Hence ¯ b is an independent sequence over any C ⊇ B with C | ⌣ B ¯ b , so tp(¯ b/B ) is flat by Corollary 3.4. Thus, tp( a/B ) is flatby Lemma 3.11, and so is tp( a/A ). (cid:3) Question 3.13.
Is every (finitary) type in a small simple theory non-orthogonal to a flat type?
Question 3.14.
Is every type in a supersimple theory non-orthogonalto a flat type?
MPLE THOUGHTS 9
Problem 3.15.
Construct a flat type which is not ultraflat.We shall now recall the definitions and properties of Σ-closure from[24, Section 4.0] in the stable and [25, Section 3.5] in the simple case(where it is called P -closure: our Σ corresponds to the collection of all P -analysable types which are co-foreign to P ). Buechler and Hoover[2, Definition 1.2] redefine such a closure operator in the context ofsuperstable theories and reprove some of the properties [2, Lemma2.5]. Definition 3.16.
The Σ -closure
Σcl( A ) of a set A is the collection ofall hyperimaginaries a such that tp( a/A ) is Σ-analysable. Remark 3.17.
We think of partial types in Σ as small. We alwayshave bdd( A ) ⊆ Σcl( A ); equality holds if Σ is the family of all boundedtypes. Other useful examples for Σ are the family of all types of SU -rank < ω α for some ordinal α , the family of all supersimple types ina properly simple theory, or the family of p -simple types of p -weight 0for some regular type p , giving rise to Hrushovski’s p -closure [10]. Fact 3.18.
The following are equivalent: (1) tp( a/A ) is foreign to Σ . (2) a | ⌣ A Σcl( A ) . (3) a | ⌣ A dcl( aA ) ∩ Σcl( A ) . (4) dcl( aA ) ∩ Σcl( A ) ⊆ bdd( A ) .Proof: The equivalence of (1), (2) and (3) is [25, Lemma 3.5.3]; theequivalence (3) ⇔ (4) is obvious. (cid:3) Unless it equals bounded closure, Σ-closure has the size of the mon-ster model and thus violates the usual conventions. The equivalence(2) ⇔ (3) can be used to cut it down to some small part. Fact 3.19.
Suppose A | ⌣ B C . Then Σcl( A ) | ⌣ Σcl( B ) Σcl( C ) . More pre-cisely, for any A ⊆ Σcl( A ) we have A | ⌣ B Σcl( C ) , where B =dcl( A B ) ∩ Σcl( B ) . In particular, Σcl( AB ) ∩ Σcl( BC ) = Σcl( B ) .Proof: This is [25, Lemma 3.5.5]; the second clause follows from Fact3.18. (cid:3)
Lemma 3.20.
Suppose C ⊆ A ∩ B ∩ D and AB | ⌣ C D . (1) If Σcl( A ) ∩ Σcl( B ) = Σcl( C ) , then Σcl( AD ) ∩ Σcl( BD ) =Σcl( D ) . (2) If bdd( A ) ∩ Σcl( B ) = bdd( C ) , then bdd( AD ) ∩ Σcl( BD ) =bdd( D ) .Proof: (1) This is [25, Lemma 3.5.6], which in turn adapts [18, Fact2.4].(2) This is similar to (1). By Fact 15Σcl( BD ) | ⌣ Σcl( B ) ∩ dcl( AB ) AB ;since AD | ⌣ A AB we obtainCb(bdd( AD ) ∩ Σcl( BD ) /AB ) ⊆ bdd( A ) ∩ Σcl( B ) = bdd( C ) . Hence bdd( AD ) ∩ Σcl( BD ) | ⌣ C AB and by transitivity bdd( AD ) ∩ Σcl( BD ) | ⌣ D ABD, whence the result. (cid:3)
The following lemma tells us that we can actually find a set C withΣcl( A ) ∩ Σcl( B ) = Σcl( C ) as in Lemma 3.20(1), even though the Σ-closures have the size of the monster model. Lemma 3.21.
Let C = bdd( AB ) ∩ Σcl( A ) ∩ Σcl( B ) . Then Σcl( A ) ∩ Σcl( B ) = Σcl( C ) .Proof: Consider e ∈ Σcl( A ) ∩ Σcl( B ) and put f = Cb( e/AB ). Then e | ⌣ f AB ; since tp( e/A ) is Σ-analysable, so is tp( e/f ), and e ∈ Σcl( f ).If I is a Morley sequence in tp( e/AB ), then f ∈ dcl( I ). However, since e is Σ-analysable over A and over B , so is I , whence f . Hence f ∈ bdd( AB ) ∩ Σcl( A ) ∩ Σcl( B ) = C. The result follows. (cid:3)
However, for considerations such as the canonical base property, oneshould like to work with the first level of the Σ-closure rather than withthe full closure operator.
Definition 3.22.
The Σ -closure of A is given byΣ cl( A ) = ℓ Σ1 (Σcl( A ) /A ) = { b : tp( b/A ) is almost Σ-internal } . Unfortunately, unless tp(Σcl( A ) /A ) is Σ-flat, Σ -closure is not a clo-sure operator, as Σ cl(Σ cl( A )) ⊃ Σ cl( A ). MPLE THOUGHTS 11
Lemma 3.23.
Suppose A | ⌣ B C with B ⊆ A ∩ C . Then Σ cl( A ) | ⌣ Σ cl( B ) C. More precisely, Σ cl( A ) | ⌣ Σ cl( B ) ∩ bdd( C ) C .Proof: Consider a ∈ Σ cl( A ) and put c = Cb( Aa/C ). Then tp( c/B ) isalmost Σ-internal by Corollary 2.4, and c ∈ bdd( C ) ∩ Σ cl( B ). (cid:3) Question 3.24. If A | ⌣ B C , is Σ cl( A ) | ⌣ Σ cl( B ) Σ cl( C ) ?4. Σ -ampleness and weak Σ -ampleness Let Φ and Σ be ∅ -invariant families of partial types. Definition 4.1.
Φ is n - Σ -ample if there are tuples a , . . . , a n , with a n a tuple of realizations of partial types in Φ over some parameters A ,such that(1) a n ⌣ Σcl( A ) a ;(2) a i +1 | ⌣ Σcl( Aa i ) a . . . a i − for 1 ≤ i < n ;(3) Σcl( Aa . . . a i − a i ) ∩ Σcl( Aa . . . a i − a i +1 ) = Σcl( Aa . . . a i − )for 0 ≤ i < n . Remark 4.2.
Pillay [20] requires a n | ⌣ Aa i a . . . a i − for 1 ≤ i < n initem (2). We follow the variant proposed by Evans and N¨ubling [6]which seems more natural and which implies a n . . . a i +1 | ⌣ Σcl( Aa i ) a . . . a i − . Lemma 4.3. If Σ ′ is a Σ -analysable family of partial types, then n - Σ -ample implies n - Σ ′ -ample, and in particular n -ample.Proof: As in [20, Remark 3.7] we replace a i by a ′ i = Cb( a ′ n . . . a ′ i +1 / Σcl( Aa i ))for i < n , where a ′ n = a n . Then a ′ n . . . a ′ i +1 | ⌣ a ′ i Σcl( Aa i ) and a ′ n . . . a ′ i +1 | ⌣ Σcl( Aa i ) Σcl( Aa . . . a i )by Fact 3.18, whence a ′ n . . . a ′ i +1 | ⌣ a ′ i Σcl( Aa . . . a i ) . Put A ′ = Σcl( A ) ∩ bdd( Aa ′ ). Then A ⊆ A ′ ⊆ Σcl( A ), whence Σcl( A ) =Σcl( A ′ ), and a ′ | ⌣ A ′ Σcl( A ). Now a n ⌣ Σcl( A ′ ) a implies a ′ n ⌣ Σcl( A ) a ′ ,whence a ′ n ⌣ Σ ′ cl( A ) a ′ . Clearly a ′ i +1 | ⌣ a ′ i Σcl( Aa . . . a i ) implies a ′ i +1 | ⌣ Σ ′ cl( A ′ a ′ i ) a ′ . . . a ′ i − for 1 ≤ i < n . Finally, A ′ a ′ . . . a ′ i a ′ i +1 | ⌣ Σ ′ cl( A ′ a ′ ...a ′ i − ) Σcl( Aa . . . a i − )yieldsΣ ′ cl( A ′ a ′ . . . a ′ i − a ′ i +1 ) , Σ ′ cl( A ′ a ′ . . . a ′ i ) | ⌣ Σ ′ cl( A ′ a ′ ...a ′ i − ) Σcl( Aa . . . a i − ) , so Σ ′ cl( A ′ a ′ . . . a ′ i − a ′ i +1 ) ∩ Σ ′ cl( A ′ a ′ . . . a ′ i ) ⊆ Σcl( Aa . . . a i − )impliesΣ ′ cl( A ′ a ′ . . . a ′ i − a ′ i +1 ) ∩ Σ ′ cl( A ′ a ′ . . . a ′ i ) ⊆ Σ ′ cl( A ′ a ′ . . . a ′ i − ) . (cid:3) This also shows that in Definition 4.1 one may require a , . . . , a n − to lie in Φ heq , and a i +1 | ⌣ a i Σcl( Aa . . . a i ). Remark 4.4. [20, Lemma 3.2 and Corollary 3.3] If a , . . . , a n witness n -Σ-ampleness over A , then a n ⌣ Σcl( Aa ...a i − ) a i for all i < n . Hence a i , . . . , a n witness ( n − i )-Σ-ampleness over Aa . . . a i − . Thus n -Σ-ample implies i -Σ-ample for all i ≤ n . Remark 4.5.
It is clear from the definition that even though Φ mightbe a complete type p , if p is not n -Σ-ample, neither is any extension of p , not only the non-forking ones.For n = 1 and n = 2 there are alternative definitions of non- n -Σ-ampleness: Definition 4.6. (1) Φ is Σ -based if Cb( a/ Σcl( B )) ⊆ Σcl( aA ) forany tuple a of realizations of partial types in Φ over some pa-rameters A and any B ⊇ A .(2) Φ is Σ -CM-trivial if Cb( a/ Σcl( AB )) ⊆ Σcl( A, Cb( a/ Σcl( AC ))for any tuple a of realizations of partial types in Φ over someparameters A and any B ⊆ C such that Σcl( ABa ) ∩ Σcl( AC ) =Σcl( AB ). Lemma 4.7. (1) Φ is Σ -based if and only if Φ is not - Σ -ample. MPLE THOUGHTS 13 (2) Φ is Σ -CM-trivial if and only if Φ is not - Σ -ample.Proof: (1) Suppose Φ is Σ-based and consider a , a , A with Σcl( Aa ) ∩ Σcl( Aa ) = Σcl( A ). Put a = a and B = Aa . By Σ-basednessCb( a/ Σcl( B )) ⊆ Σcl( Aa ) ∩ Σcl( B ) = Σcl( A ) . Hence a | ⌣ Σcl( A ) Σcl( B ), whence a | ⌣ Σcl( A ) a , so Φ is not 1-Σ-ample.Conversely, if Φ is not Σ-based, let a, A, B be a counterexample. Put a = Cb( a / Σcl( B )) and a = a . Then a / ∈ Σcl( Aa ). Now take A ′ = bdd( Aa a ) ∩ Σcl( Aa ) ∩ Σcl( Aa ) . Then Σcl( A ′ a ) ∩ Σcl( A ′ a ) = Σcl( A ′ ) by Lemma 3.21.Suppose a | ⌣ Σcl( A ′ ) a . Since Σcl( A ′ ) ⊆ Σcl( Aa ) ⊆ Σcl( B ) we have a | ⌣ a Σcl( A ′ ). As a = Cb( a / Σcl( B )), this implies a ⊆ Σcl( A ′ ) ⊆ Σcl( Aa ) , a contradiction. Hence a , a , A ′ witness 1-Σ-ampleness of Φ.(2) Suppose Φ is Σ-CM-trivial and consider a , a , a , A withΣcl( Aa ) ∩ Σcl( Aa ) = Σcl( A ) , Σcl( Aa a ) ∩ Σcl( Aa a ) = Σcl( Aa ) , and a | ⌣ Σcl( Aa ) a . Put a = a , B = a and C = a a . Then a | ⌣ Σcl( Aa ) Σcl( Aa a ) , so Cb( a/ Σcl( AC )) ⊆ Σcl( Aa ). MoreoverΣcl( ABa ) ∩ Σcl( AC ) = Σcl( AB ) , whence by Σ-CM-trivialityCb( a/ Σcl( AB )) ⊆ Σcl( A, Cb( a/AC )) ∩ Σcl( AB ) ⊆ Σcl( Aa ) ∩ Σcl( Aa ) = Σcl( A ) . Hence a | ⌣ Σcl( A ) a , so Φ is not 2-Σ-ample.Conversely, if Φ is not Σ-CM-trivial, let a, A, B, C be a counterex-ample. Put a = Cb( a/ Σcl( AB )) , a = Cb( a/ Σcl( AC )) , a = a,A ′ = bdd( Aa a ) ∩ Σcl( Aa ) ∩ Σcl( Aa ) ⊆ Σcl( AB ) . Then a | ⌣ Σcl( A ′ a ) a and a / ∈ Σcl( Aa ); by Lemma 3.21Σcl( A ′ a ) ∩ Σcl( A ′ a ) = Σcl( A ′ ) . Moreover, a | ⌣ a Σcl( AB ) impliesΣcl( A ′ a a ) | ⌣ Σcl( A ′ a ) Σcl( AB ) . Thus Σcl( A ′ a a ) ∩ Σcl( A ′ a a ) ⊆ Σcl(
ABa ) ∩ Σcl( AC )= Σcl( AB ) ∩ Σcl( A ′ a a ) = Σcl( A ′ a ) . Suppose a | ⌣ Σcl( A ′ ) a . Since a | ⌣ a Σcl( A ′ ) we obtain a = Cb( a/ Σcl( AB )) = Cb( a/a Σcl( A ′ )) ⊆ Σcl( A ′ ) ⊆ Σcl( Aa ) , a contradiction. Hence a , a , a , A ′ witness 2-Σ-ampleness of Φ. (cid:3) In our definition of Σ-ampleness, we only consider the type of a n over a Σ-closed set, namely Σcl( A ). This seems natural since the ideaof Σ-closure is to work modulo Σ. However, sometimes one needs astronger notion which takes care of all types. Let us first look at n = 1and n = 2. Definition 4.8. • Φ is strongly Σ -based if Cb( a/B ) ⊆ Σcl( aA )for any tuple a of realizations of partial types in Φ over some A and any B ⊇ A . • Φ is strongly Σ -CM-trivial if Cb( a/AB ) ⊆ Σcl( A, Cb( a/AC ) forany tuple a of realizations of partial types in Φ over some A andany B ⊆ C with Σcl( ABa ) ∩ bdd( AC ) = bdd( AB ). Remark 4.9.
Cb( a/ Σcl( B )) ⊆ bdd(Cb( a/B ) , a ) ∩ Σcl(Cb( a/B )).
Proof:
By Fact 3.19 the independence a | ⌣ Cb( a/B ) B implies a | ⌣ dcl( a, Cb( a/b )) ∩ Σcl(Cb( a/B )) Σcl( B ) . The result follows. (cid:3)
Conjecture.
Cb( a/B ) ⊆ Σcl(Cb( a/ Σcl( B ))).If this conjecture were true, strong and ordinary Σ-basedness andΣ-CM-triviality would obviously coincide. Since we have not been ableto show this, we weaken our definition of ampleness. Definition 4.10.
Φ is weakly n - Σ -ample if there are tuples a , . . . , a n ,where a n is a tuple of realizations of partial types in Φ over A , with MPLE THOUGHTS 15 (1) a n ⌣ A a .(2) a i +1 | ⌣ Aa i a . . . a i − for 1 ≤ i < n .(3) bdd( Aa . . . a i − a i ) ∩ Σcl( Aa . . . a i − a i +1 ) = bdd( Aa . . . a i − )for i < n .Note that (3) implies that tp( a i /Aa . . . a i − ) is foreign to Σ by Fact3.18 for all i < n , and so is tp( a i /Aa i − ) by (2). If Σ is the familyof bounded partial types, then weak and ordinary n -Σ-ampleness justequal n -ampleness. Lemma 4.11. An n - Σ -ample family of types is weakly n - Σ -ample. If Σ ′ is Σ -analysable, then a weakly n - Σ -ample family is weakly n - Σ ′ -ample, and in particular n -ample.Proof: If a , . . . , a n witness n -Σ-ampleness over A , we put a ′ n = a n , a ′ i = Cb( a ′ n . . . a ′ i +1 / Σcl( Aa i )) ⊆ Σcl( Aa i ) for n > i and A ′ = bdd( Aa ′ ) ∩ Σcl( Aa ′ ) ⊆ Σcl( Aa ) ∩ Σcl( Aa ) = Σcl( A ) . As in Lemma 4.3 we have for i < na ′ n . . . a ′ i +1 | ⌣ a ′ i Σcl( Aa . . . a i ) . For 0 < i < n we obtain a ′ i +1 | ⌣ A ′ a ′ i a ′ . . . a ′ i − ; moreoverbdd( A ′ a ′ . . . a ′ i − a ′ i ) ∩ Σcl( A ′ a ′ . . . a ′ i − a ′ i +1 ) ⊆ Σcl( A ′ a ′ . . . a ′ i − a ′ i ) ∩ Σcl( A ′ a ′ . . . a ′ i − a ′ i +1 ) ⊆ Σcl( Aa . . . a i − a i ) ∩ Σcl( Aa . . . a i − a i +1 )= Σcl( Aa . . . a i − ) . But then a ′ i | ⌣ A ′ a ′ ...a ′ i − Σcl( Aa . . . a i − ) yieldsbdd( A ′ a ′ . . . a ′ i − a ′ i ) ∩ Σcl( A ′ a ′ . . . a ′ i − a ′ i +1 ) = bdd( A ′ a ′ . . . a ′ i − ) , while bdd( A ′ a ′ ) ∩ Σcl( A ′ a ′ ) = bdd( A ′ ) follows from the definition of A ′ . Finally a n ⌣ Σcl( A ) a implies a ′ n ⌣ Σcl( A ) a ′ , whence a ′ n ⌣ A ′ a ′ astp( a ′ /A ′ ) is foreign to Σ and Σcl( A ) = Σcl( A ′ ).The second assertion is clear, since Σ ′ cl( A ) ⊆ Σcl( A ) for any set A . (cid:3) This also shows that in Definition 4.10 one may require a , . . . , a n − to lie in Φ heq . Lemma 4.12. (1) Φ is strongly Σ -based iff Φ is not weakly - Σ -ample. (2) Φ is strongly Σ -CM-trivial iff Φ is not weakly - Σ -ample.Proof: This is similar to the proof of Lemma 4.7, so we shall be concise.(1) Suppose Φ is strongly Σ-based and consider a , a , A withbdd( Aa ) ∩ Σcl( Aa ) = bdd( A ) . Put a = a and B = Aa . By strong Σ-basednessCb( a/B ) ⊆ Σcl( Aa ) ∩ bdd( B ) = bdd( A ) , whence a | ⌣ A a , so Φ is not weakly 1-Σ-ample.Conversely, if Φ is not strongly Σ-based, let a, A, B be a counterex-ample. Put a = Cb( a /B ) and a = a . Then a / ∈ Σcl( Aa ). Nowtake A ′ = bdd( Aa ) ∩ Σcl( Aa ). Clearly A ′ = bdd( A ′ a ) ∩ Σcl( A ′ a ).Suppose a | ⌣ A ′ a . Since a = Cb( a /B ) implies a | ⌣ a A ′ , we obtain a ⊆ bdd( A ′ ) ⊆ Σcl( Aa ) , a contradiction. Hence a , a , A ′ witness weak 1-Σ-ampleness of Φ.(2) Suppose Φ is strongly Σ-CM-trivial and consider a , a , a , A withbdd( Aa ) ∩ Σcl( Aa ) = bdd( A ) , bdd( Aa a ) ∩ Σcl( Aa a ) = bdd( Aa ) , and a | ⌣ Aa a . Put a = a , B = a and C = a a . Then Cb( a/AC ) ⊆ bdd( Aa ).Moreover Σcl( ABa ) ∩ bdd( AC ) = bdd( AB ) , whence by strong Σ-CM-trivialityCb( a/AB ) ⊆ Σcl( A, Cb( a/AC )) ∩ bdd( AB ) ⊆ Σcl( Aa ) ∩ bdd( Aa ) = bdd( A ) . Hence a | ⌣ A a , so Φ is not 2-Σ-ample.Conversely, if Φ is not strongly Σ-CM-trivial, let a, A, B, C be acounterexample. Put a = AB, a = Cb( a/AC ) , a = a,A ′ = bdd( Aa ) ∩ Σcl( Aa ) . Then a | ⌣ A ′ a a and Cb( a /AB ) / ∈ Σcl( Aa ) = Σcl( A ′ a ); moreoverbdd( A ′ a ) ∩ Σcl( A ′ a ) = bdd( A ′ ) . MPLE THOUGHTS 17
Clearly Σcl( A ′ a a ) ∩ bdd( A ′ a a ) ⊆ Σcl(
ABa ) ∩ bdd( AC )= bdd( AB ) = bdd( A ′ a ) . Suppose a | ⌣ A ′ a . Then Cb( a /AB ) ∈ bdd( A ′ ) ⊆ Σcl( Aa ), a con-tradiction. Hence a , a , a , A ′ witness weak 2-Σ-ampleness of Φ. (cid:3) Lemma 4.13. If Φ is not (weakly) n - Σ -ample, neither is the familyof ∅ -conjugates of tp( a/A ) for any a ∈ Σcl(¯ aA ) , where ¯ a is a tuple ofrealizations of partial types in Φ over A .Proof: Suppose the family of ∅ -conjugates of tp( a/A ) is n -Σ-ample, aswitnessed by a , . . . , a n over some parameters B . There is a tuple ¯ a ofrealizations of partial types in Φ over some ∅ -conjugates of A inside B such that a n ∈ Σcl(¯ aB ); we may choose it such that¯ a | ⌣ a n B a . . . a n − . Then ¯ a | ⌣ a n − a n B a . . . a n − , and hence¯ a | ⌣ Σcl( a n − a n B ) a . . . a n − . As a n | ⌣ Σcl( a n − B ) a . . . a n − impliesΣcl( a n − a n B ) | ⌣ Σcl( a n − B ) a . . . a n − by Fact 3.19, we get ¯ a | ⌣ Σcl( a n − B ) a . . . a n − . We also have ¯ a | ⌣ a ...a n − a n B a n − , whence(1) Σcl( a . . . a n − ¯ aB ) | ⌣ Σcl( a ...a n − a n B ) Σcl( a . . . a n − a n − B );since Σ-closure is boundedly closed,Σcl( a . . . a n − ¯ aB ) ∩ Σcl( a . . . a n − a n − B ) ⊆ Σcl( a . . . a n − a n B ) ∩ Σcl( a . . . a n − a n − B )= Σcl( a . . . a n − B ) . Finally, ¯ a | ⌣ Σcl( B ) a would imply Σcl(¯ aB ) | ⌣ Σcl( B ) a by Fact 3.19, andhence a n | ⌣ Σcl( B ) a , a contradiction. Thus ¯ a ⌣ Σcl( B ) a , and a , . . . , a n − , ¯ a witness n -Σ-ampleness of Φ over B , a contradiction. Now suppose a , . . . , a n witness weak n -Σ-ampleness over B , andchoose ¯ a as before. Then easily ¯ aa n | ⌣ Ba n − a . . . a n − , yielding (2)from the definition. Moreover, equation (1) impliesΣcl( a . . . a n − ¯ aB ) ∩ bdd( a . . . a n − a n − B ) ⊆ Σcl( a . . . a n − a n B ) ∩ bdd( a . . . a n − a n − B )= bdd( a . . . a n − B ) . Finally suppose ¯ a | ⌣ B a . Since tp( a /B ) is foreign to Σ, so is tp( a /B ¯ a ).Then a | ⌣ B ¯ a Σcl( B ¯ a ) by Fact 3.18, whence a | ⌣ B a n , a contradiction.Thus ¯ a ⌣ B a , and a , . . . , a n − , ¯ a witness weak n -Σ-ampleness of Φover B , again a contradiction. (cid:3) Lemma 4.14.
Suppose B | ⌣ A a . . . a n . If a , . . . , a n witness (weak) n - Σ -ampleness over A , they witness (weak) n - Σ -ampleness over B .Proof: Clearly B | ⌣ a ...a i − A a . . . a i +1 A , so Lemma 3.20 yieldsΣcl( Ba . . . a i − a i ) ∩ Σcl( Ba . . . a i − a i +1 ) = Σcl( Ba . . . a i − )in the ordinary case, andbdd( Ba . . . a i − a i ) ∩ Σcl( Ba . . . a i − a i +1 ) = bdd( Ba . . . a i − )in the weak case, for all i < n .Next, a i +1 | ⌣ Aa ...a i B , whence a i +1 | ⌣ Σcl( Aa ...a i ) Σcl( Ba i ) by Lemma3.19. Now a i +1 | ⌣ Σcl( Aa i ) a . . . a i − implies a i +1 | ⌣ Σcl( Aa i ) Σcl( Aa . . . a i ),whence a i +1 | ⌣ Σcl( Ba i ) a . . . a i − for 1 ≤ i < n by transitivity. In the weak case, a i +1 | ⌣ Aa i a . . . a i − im-plies a i +1 | ⌣ Aa i Ba . . . a i − by transitivity, whence a i +1 | ⌣ Ba i a . . . a i − .Finally, a n | ⌣ Σcl( A ) Σcl( B ) by Fact 3.19, so a n | ⌣ Σcl( B ) a would imply a n | ⌣ Σcl( A ) a , a contradiction. Hence a n ⌣ Σcl( B ) a . In the weak case, a n | ⌣ A B and a n ⌣ A a yield directly a n ⌣ B a . (cid:3) Lemma 4.15.
Let Ψ be an ∅ -invariant family of types. If Φ and Ψ arenot (weakly) n - Σ -ample, neither is Φ ∪ Ψ .Proof: Suppose Φ ∪ Ψ is weakly n -Σ-ample, as witnessed by a , . . . , a n = bc over some parameters A , where b and c are tuples of realizations ofpartial types in Φ and Ψ, respectively. As Ψ is not n -Σ-ample, we MPLE THOUGHTS 19 must have c | ⌣ A a . Put a ′ = Cb( bc/a A ). Then tp( a ′ /A ) is internalin tp( b/A ) by Corollary 2.4. Put a ′ n = Cb( a ′ /a n A ) . Then tp( a ′ n /A ) is tp( a ′ /A )-internal and hence tp( b/A )-internal. Notethat a n ⌣ A a implies a n ⌣ A a ′ , whence a ′ n ⌣ A a ′ and a ′ n ⌣ A a . Moreover a ′ n ∈ bdd( Aa n ), so a , . . . , a n − , a ′ n witness weak n -Σ-ample-ness over A .As tp( a ′ n /A ) is tp( b/A )-internal, there is B | ⌣ A a ′ n and a tuple ¯ b ofrealizations of tp( b/A ) with a ′ n ∈ dcl( B ¯ b ). We may assume B | ⌣ Aa ′ n a . . . a n − , whence B | ⌣ A a . . . a n − a ′ n . Hence a , . . . , a n − , a ′ n witness weak n -Σ-ampleness over B by Lemma 4.14. As a ′ n ∈ dcl( B ¯ b ), this contradictsnon weak n -Σ-ampleness of Φ by Lemma 4.13.The proof in the ordinary case is analogous, replacing A by Σcl( A ). (cid:3) Corollary 4.16.
For i < α let Φ i be an ∅ -invariant family of partialtypes. If Φ i is not (weakly) n - Σ -ample for all i < α , neither is S i<α Φ i .Proof: This just follows from the local character of forking and Lemma4.15. (cid:3)
Lemma 4.17.
If the family of ∅ -conjugates of tp( a/A ) is not (weakly) n - Σ -ample and a | ⌣ A , then tp( a ) is not (weakly) n - Σ -ample.Proof: Suppose tp( a ) is (weakly) n -Σ-ample, as witnessed by a , . . . , a n over some parameters B , where a n = ( b i : i < k ) is a tuple of realiza-tions of tp( a ). For each i < k choose B i | ⌣ b i ( B, a . . . a n , B j : j < i )with B i b i ≡ Aa . Then B i | ⌣ b i , whence ( B i : i < k ) | ⌣ Ba . . . a n . Then a , . . . , a n witness (weak) n -Σ-ampleness over ( B, B i : i < k ) by Lemma4.14, a contradiction, since tp( b i /B i ) is an ∅ -conjugate of tp( a/A ) forall i < k . (cid:3) Remark 4.18.
In fact, in the above Lemma it suffices to merely assumethat the single type tp( a/A ) is not (weakly) n -Σ-ample in the theory T ( A ), using Corollary 4.16. It follows that ampleness is preserved underadding and forgetting parameters. Corollary 4.19.
Let Ψ be an ∅ -invariant family of types. If Ψ is Φ -internal and Φ is not (weakly) n - Σ -ample, neither is Ψ .Proof: Immediate from Lemmas 4.13 and 4.17. (cid:3)
Theorem 4.20.
Let Ψ be an ∅ -invariant family of types. If Ψ is Φ -analysable and Φ is not (weakly) n - Σ -ample, neither is Ψ .Proof: Suppose Ψ is n -Σ-ample, as witnessed by a , . . . , a n over someparameters A , where a n is a tuple of realizations of Ψ. Put a ′ n = ℓ Φ1 ( a n / Σcl( A ) ∩ bdd( Aa n )). Then a n and a ′ n are domination-equivalentover Σcl( A ) ∩ bdd( Aa n ) by Theorem 3.6; moreover a n and hence a ′ n are independent of Σcl( A ) over Σcl( A ) ∩ bdd( Aa n ) by Fact 3.18, so a n and a ′ n are domination-equivalent over Σcl( A ). Thus a , . . . , a ′ n witnessnon-Σ-ampleness over A , contradicting Corollary 4.19.For the weak case we put a ′ n = ℓ Φ1 ( a n /A ). So a n and a ′ n are domination-equivalent over A , whence a ′ n ⌣ A a . Thus a , . . . , a ′ n witness weaknon-Σ-ampleness over A , contradicting again Corollary 4.19. (cid:3) Analysability of canonical bases
As an immediate Corollary to Theorem 4.20, we obtain the following:
Theorem 5.1.
Suppose every type in T is non-orthogonal to a regulartype, and let Σ be the family of all n -ample regular types. Then T isnot weakly n - Σ -ample.Proof: A non n -ample type is not weakly Σ-ample by Lemma 4.11.So all regular types are not weakly n -Σ-ample. But every type isanalysable in regular types by the non-orthogonality hypothesis. (cid:3) Corollary 5.2.
Suppose every type in T is non-orthogonal to a regulartype. Then tp(Cb( a/b ) /a ) is analysable in the family of all non one-based regular types, for all a , b .Proof: This is just Theorem 5.1 for n = 1. (cid:3) Note that a forking extension of a non one-based regular type ofinfinite rank may be one-based.
Remark 5.3.
In fact, the proof shows more. For every type p let Σ( p )be the collection of types in Σ not foreign to p . Then tp(Cb( a/b ) /a ) isanalysable in Σ(tp(Cb( a/b ))). In particular, if tp( a ) or tp( b ) has rankless than ω α , so does tp(Cb( a/b )). Hence tp(Cb( a/b ) /a ) is analysablein the family of all non one-based regular types of rank less than ω α . MPLE THOUGHTS 21
Corollary 5.2 is due to Zo´e Chatzidakis for types of finite SU-rankin simple theories [5, Proposition 1.10]. In fact, she even obtainstp(Cb( a/b ) / bdd( a ) ∩ bdd( b )) to be analysable in the family of nonone-based types of rank 1, and to decompose in components, each ofwhich is analysable in a non-orthogonality class of regular types. Ininfinite rank, one has to work modulo types of smaller rank. So let Σ α be the collection of all partial types of SU-rank < ω α , and P α be thefamily of non Σ α -based types of SU-rank ω α . Note that the Σ α -basedtypes of SU-rank ω α are precisely the locally modular types of SU-rank ω α . Theorem 5.4.
Let b = Cb( a/ Σ α cl( b )) be such that SU( b ) < ω α +1 for some ordinal α and let A = Σ α cl( b ) ∩ Σ α cl( a ) . Then tp( b/A ) is (Σ α ∪ P α ) -analysable.Proof: Firstly, if a ∈ Σ α cl( b ) then a = b ∈ A . Similarly, if b ∈ Σ α cl( a )then b ∈ A ; in both cases tp( b/A ) is trivially (Σ α ∪ P α )-analysable.Hence we may assume a Σ α cl( b ) and b Σ α cl( a ).Suppose towards a contradiction that the result is false and con-sider a counterexample a, b with SU( b ) minimal modulo ω α and thenSU( b/ Σ α cl( a )) being maximal modulo ω α . Note that this implies ω α ≤ SU( b/a ) ≤ SU( b/A ) ≤ SU( b ) < ω α +1 . Clearly (after adding parameters) we may assume A = Σ α cl( ∅ ). Thenfor any c the type tp( c ) is (Σ α ∪ P α )-analysable iff tp( c/A ) is. Claim.
We may assume a = Cb( b/ Σ α cl( a )) .Proof of Claim: Put ˜ a = Cb( b/ Σ α cl( a )) and ˜ b = Cb(˜ a/ Σ α cl( b )). Then˜ a ∈ Σ α cl( a ) and a | ⌣ ˜ a b . Hence Σ α cl( b ) = Σ α cl(˜ b ) by [25, Lemma3.5.8], and tp(˜ b ) is not (Σ α ∪ P α )-analysable either. Thus the pair ˜ a, ˜ b also forms a counterexample. Moreover, SU( b ) equals SU(˜ b ) modulo ω α and SU( b/ Σ α cl( a )) = SU( b/ Σ α cl(˜ a )) equals SU(˜ b/ Σ α cl(˜ a )) modulo ω α . (cid:3) Since a is definable over a finite part of a Morley sequence in Lstp( b/a )by supersimplicity of tp( b ), we see that SU( a ) < ω α +1 . On the otherhand, a / ∈ Σ α cl( b ) implies SU( a/b ) ≥ ω α .Let ˆ a ⊆ bdd( a ) and ˆ b ⊆ bdd( b ) be maximal (Σ α ∪ P α )-analysable.Then a / ∈ Σ α cl(ˆ a ) and b Σ α cl(ˆ b ), and tp( a/ ˆ a ) and tp( b/ ˆ b ) are foreignto Σ α ∪ P α . Since Cb(ˆ a/b ) and Cb(ˆ b/a ) are (Σ α ∪ P α )-analysable, we obtain a | ⌣ ˆ a ˆ b and b | ⌣ ˆ b ˆ a. Claim. tp( b/ ˆ b ) and tp( a/ ˆ a ) are both Σ α -based.Proof of Claim: Let Φ be the family of Σ α -based types of SU-rank ω α . Then tp( a/ ˆ a ) is (Σ α ∪ P α ∪ Φ)-analysable, but foreign to Σ α ∪ P α .Put a = ℓ Φ1 ( a/ ˆ a ) and b = ℓ Φ1 ( b/ ˆ b ). Then a (cid:3) ˆ a a and b (cid:3) ˆ b b byLemma 3.6(3); as a | ⌣ ˆ a ˆ b and b | ⌣ ˆ b ˆ a we even have a (cid:3) ˆ a ˆ b a and b (cid:3) ˆ a ˆ b b .Since a ⌣ ˆ a ˆ b b we obtain a ⌣ ˆ a ˆ b b . Moreover, tp( a / ˆ a ) and tp( b / ˆ b ) areΣ α -based by Theorem 4.20 (or [26, Theorem 11]).On the other hand, as a ⌣ ˆ b b , we see that b ′ = Cb( a / Σ α cl( b ))is not contained in ˆ b and hence is not (Σ α ∪ P α )-analysable. So a , b ′ is another counterexample; by minimality of SU-rank b and b ′ havethe same SU-rank modulo ω α , whence b ∈ Σ α cl( b ). Hence tp( b/ ˆ b ) isΣ α -based, as is tp( a/ ˆ a ) since a = Cb( b/a ) and a | ⌣ ˆ a ˆ b . (cid:3) Claim. Σ α cl( a, ˆ b ) = Σ α cl( b, ˆ a ) = Σ α cl( a, b ) .Proof of Claim: As tp( a/ ˆ a ) is Σ α -based, we have a | ⌣ Σ α cl( a ) ∩ Σ α cl(ˆ ab ) ˆ ab, whence Σ α cl( a ) | ⌣ Σ α cl( a ) ∩ Σ α cl(ˆ ab ) b by Fact 3.19. Thus a = Cb( b/ Σ α cl( a )) ∈ Σ α cl(ˆ ab ). Similarly b ∈ Σ α cl(ˆ ba ). (cid:3) Let now ( b ) ⌢ ( b j : j < ω ) be a Morley sequence in tp( b/a ) and letˆ b j represent the part of b j corresponding to ˆ b . Then (ˆ b j : j < ω ) is aMorley sequence in tp(ˆ b/ ˆ a ) since a | ⌣ ˆ a ˆ b . As SU(ˆ b ) < ∞ there is someminimal m < ω such that ˆ a = Cb(ˆ b/ ˆ a ) ∈ Σ α cl(ˆ b, ˆ b j : j < m ). Then m >
0, as otherwise Σ α cl( b ) = Σ α cl(ˆ a, b ) ∋ a , which is impossible.Moreover, a ∈ Σ α cl(ˆ a, b j ) for all j < m by invariance and hence, a ∈ Σ α cl(ˆ b, b j : j < m ).Put b ′ = Cb( b j : j < m/ Σ α cl( b )). Then ( b j : j < m ) | ⌣ b ′ ˆ b Σ α cl( b ), soby Fact 3.19 Σ α cl(ˆ b, b j : j < m ) | ⌣ Σ α cl( b ′ , ˆ b ) Σ α cl( b ) . MPLE THOUGHTS 23
Then a | ⌣ Σ α cl( b ′ , ˆ b ) Σ α cl( b ), so b = Cb( a/ Σ α cl( b )) ∈ Σ α cl( b ′ , ˆ b ). As b Σ α cl(ˆ b ) we obtain b ′ Σ α cl(ˆ b ). Claim. tp( b ′ / Σ α cl( b ′ ) ∩ Σ α cl( b j : j < m )) is not (Σ α ∪ P α ) -analysable.Proof of Claim: Note first that ( b j : j < m ) | ⌣ a b impliesΣ α cl( b j : j < m ) | ⌣ Σ α cl( a ) Σ α cl( b )by Fact 3.19, whenceΣ α cl( b ′ ) ∩ Σ α cl( b j : j < m ) ⊆ Σ α cl( b ) ∩ Σ α cl( a ) = Σ α cl( ∅ ) . As b ∈ Σ α cl( b ′ , ˆ b ) and tp( b/ ˆ b ) is not (Σ α ∪ P α )-analysable, neither istp( b ′ / ˆ b ), nor a fortiori tp( b ′ / Σ α cl( ∅ )). (cid:3) As b ′ = Cb( b j : j < m/ Σ α cl( b ′ )), the pair ( b j : j < m ) , b ′ formsanother counterexample. By minimality SU( b ) equals SU( b ′ ) modulo ω α , which implies Σ α cl( b ) = Σ α cl( b ′ ).As tp( b j / ˆ b j ) is foreign to Σ α ∪ P α and ˆ b is (Σ α ∪ P α )-analysable, weobtain ˆ b | ⌣ (ˆ b j : j
Corollary 5.5.
Let b = bdd(Cb( a/b )) be such that SU( b ) < ω . Then tp( b/ bdd( b ) ∩ bdd( a )) is analysable in the family of all non one-basedtypes of SU -rank . Applications and the Canonical Base Property
In this section and the next, Σ nob will be the family of non one-based regular types (seen as partial types). For the applications onewould like (and often has) more than mere strongly Σ nob -basedness ofcanonical bases:
Definition 6.1.
A supersimple theory T has the Canonical Base Prop-erty CBP if tp(Cb( a/b ) /a ) is almost Σ nob -internal for all a , b . Remark 6.2.
In other words, in view of Corollary 5.2 a theory has theCBP if for all a, b the type tp(Cb( a/b ) /a ) is Σ nob -flat.It had been conjectured that all supersimple theories of finite rankhave the CBP, but Hrushovski has constructed a counter-example [14]. Remark 6.3.
Chatzidakis has shown for types of finite SU -rank thatthe CBP implies that even tp(Cb( a/b ) / bdd( a ) ∩ bdd( b )) is almost Σ nob -internal [5, Theorem 1.15]. Example.
The CBP holds for types of finite rank in • Differentially closed fields in characteristic 0 [23]. • Generic difference fields [23, 5]. • Compact complex spaces [4, 7, 22].Moreover, in those cases we have a good knowledge of the non one-based types.Kowalski and Pillay [16, Section 4] have given some consequences ofstrongly Σ-basedness in the context of groups. In fact, they work in atheory with the CBP, but they remark that their results hold, with Σ -analysable instead of almost Σ -internal , in all stable strongly Σ-basedtheories. Fact 6.4.
Let G be an ∅ -hyperdefinable strongly Σ -based group in astable theory. (1) If H ≤ G is connected with canonical parameter c , then tp( c ) is Σ -analysable. (2) G/Z ( G ) is Σ -analysable. MPLE THOUGHTS 25
An inspection of their proof shows that mere simplicity of the am-bient theory is sufficient, replacing centers by approximate centers andconnectivity by local connectivity. Recall that the approximate center of a group G is ˜ Z ( G ) = { g ∈ G : [ G : C G ( g )] < ∞} . A subgroup H ≤ G is locally connected if for all group-theoretic ormodel-theoretic conjugates H σ of H , if H and H σ are commensurate,then H = H σ . Locally connected subgroups and their cosets havecanonical parameters; every subgroup is commensurable with a uniqueminimal locally connected subgroup, its locally connected component .For more details about the approximate notions, the reader is invitedto consult [25, Definition 4.4.9 and Proposition 4.4.10]. Proposition 6.5.
Let G be an ∅ -hyperdefinable strongly Σ -based groupin a simple theory. (1) If H ≤ G is locally connected with canonical parameter c , then tp( c ) is Σ -analysable. (2) G/ ˜ Z ( G ) is Σ -analysable.Proof: (1) Take h ∈ H generic over c and g ∈ G generic over c, h . Let d be the canonical parameter of gH . Then tp( gh/g, c ) is the generic typeof gH , so d is interbounded with Cb( gh/g, c ). By strongly Σ-basedness,tp( d/gh ) is Σ-analysable. But c ∈ dcl( d ), so tp( c/gh ) is Σ-analysable,as is tp( c ) since c | ⌣ gh .(2) For generic g ∈ G put H g = { ( x, x g ) ∈ G × G : x ∈ G } , and let H lcg be the locally connected component of H g . Then g ˜ Z ( G )is interbounded with the canonical parameter of H lcg , so tp( g ˜ Z ( G )) isΣ-analysable, as is G/ ˜ Z ( G ). (cid:3) Theorem 6.6.
Let G be an ∅ -hyperdefinable strongly Σ -based groupin a simple theory. If G is supersimple or type-definable, there is anormal nilpotent ∅ -hyperdefinable subgroup N such that G/N is almost Σ -internal. In particular, a supersimple or type-definable group G ina simple theory has a normal nilpotent hyperdefinable subgroup N suchthat G/N is almost Σ nob -internal.Proof: G/ ˜ Z ( G ) is Σ-analysable by Proposition 6.5. Hence there is acontinuous sequence G = G ⊲ G ⊲ G ⊲ · · · ⊲ G α ⊲ ˜ Z ( G ) of normal ∅ -hyperdefinable subgroups such that successive quotients Q i = G i /G i +1 are Σ-internal for all i < α , and G α / ˜ Z ( G ) is bounded.Now G acts on every quotient Q i . Let N i = { g ∈ G : [ Q i : C Q i ( g )] < ∞} be the approximate stabilizer of Q i in G , again an ∅ -hyperdefinablesubgroup. If ( q j : j < κ ) is a long independent generic sequence in Q i and g , g ′ are two elements of G which have the same action on all q j for j < κ , there is some j < κ with q j | ⌣ g, g ′ . Since g − g ′ stabilizes q j it lies in N i , and gN i is determined by the sequence ( q j , q gj : j < κ ).Thus G/N i is Q i -internal, whence Σ-internal.Put N = T i<α N i . Since Q i<α G/N i projects definably onto G/N ,the latter quotient is also Σ-internal. In order to finish it now sufficesto show that N is virtually nilpotent. In particular, we may assumethat N is ∅ -connected.Consider the approximate ascending central series ˜ Z i ( N ). Note that N centralizes G α / ˜ Z ( G ) by ∅ -connectivity. Moreover, N approxima-tively stabilizes all quotients ( G i ∩ N ) / ( G i +1 ∩ N ). Hence, if G i +1 ∩ N ≤ ˜ Z j ( N ), then G i ∩ N ≤ ˜ Z j +1 ( N ). If G is supersimple, we may as-sume that all the Q i are unbounded, so α is finite and N = ˜ Z α +2 ( N ).In the type-definable case, note that ˜ Z i ( N ) is relatively ∅ -definableby [25, Lemma 4.2.6]. So by compactness the least ordinal α i with G α i ∩ N ≤ ˜ Z i ( N ) must be a successor ordinal, and α i +1 ≤ α i − < α i .Hence the sequence must stop and there is k < ω with N = ˜ Z k ( N ).But then N is nilpotent by [25, Proposition 4.4.10.3]. (cid:3) Remark 6.7.
In a similar way one can show that if G acts definablyand faithfully on a Σ-analysable group H and H is supersimple or type-definable, then there is a hyperdefinable normal nilpotent subgroup N ⊳ G such that
G/N is almost Σ-internal.7.
Final Remarks
We have seen that for (weak) Σ-ampleness only the first level of anelement is important. However, the difference between strong Σ nob -basedness and the CBP is precisely the possible existence of a second(or higher) Σ nob -level of Cb( a/b ) over a , i.e. its non Σ nob -flatness.One might be tempted to try to prove the CBP replacing Σ nob -closureby Σ nob -closure. In fact it is possible to define a corresponding notionof Σ -ampleness, and to prove an analogue of Theorem 4.20. However, MPLE THOUGHTS 27 since Σ -closure is not a closure operator, the equivalence between Σ nob -basedness (i.e. the CBP) and non 1-Σ nob -ampleness breaks down. Sofar we have not found a way around this.A possible approach to circumvent the failure of the CBP in generalcould be to use Theorem 6.6 in the applications, rather than establishthe CBP for particular theories and use Fact 6.4 (or Proposition 6.5),but we have not looked into this.Finally, it might be interesting to look for a variant of amplenesswhich does take all levels into account, as one might hope to obtainstronger structural consequences. References [1] Prerna Bihani Juhlin.
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Universitat de Barcelona; Departament de L`ogica, Hist`oria i Filosofiade la Ci`encia, Montalegre 6, 08001 Barcelona, Spain
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Universit´e de Lyon; CNRS; Universit´e Lyon 1; In-stitut Camille Jordan UMR5208, 43 bd du 11 novembre 1918, 69622Villeurbanne Cedex, FranceUniversit´e de Lyon; CNRS; Universit´e Lyon 1; Institut Camille Jor-dan UMR5208, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex,France
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