forking, normalization and canonical bases

Annals of Pure and Applied Logic 32 (1986) 61-81 North-Holland

61

FORKING, N O R M A L I Z A T I O N A N D CANONICAL BASES

Anand PILLAY University of Notre Dame, Notre Dame, IN 46556, USA

Communicated by K. Kunen Received 28 March 1984; revised 15 February 1985

O. Introduction

The aim and content of this paper are threefold: first, to show that all of the theory of forking can be developed in an arbitrary (maybe unstable) theory, with respect to any 'Booleanly closed' set of stable formulas; secondly, to give a simple proof of a general normalization theorem; and thirdly, to prove some new facts about and characterizations of forking in stable theories, relating the general theory (hopefully) more closely to the natural (especially the algebraic) examples. We also try to show the close relations between the notions in the title.

Concerning the first point, the question of what are the minimum assumptions needed to develop a 'smooth' theory of forking and how much of the theory can be developed in an unstable theory, was raised and examined by Harnik and Harrington [3], and also by Hodges (4). We discuss this background, as well as our approach, in Section 1.

Normalization theorems were first proved by Lachlan [5], and have been an important tool in much work (e.g. Cherlin, Harrington and Lachlan [2]). Normalization gives in a sense a 'quantifier elimination theorem' for stable theories. For example, call the definable set X, MR-normalized (MR = Morley rank) if MR(X) = ac < ~ and for any conjugate X' of X if MR(X V X ' ) < oc, then X = X' . Then, in fact, if T is totally transcendental, then any definable set is a Boolean combination of MR-normalized sets of Morley degree 1. Harnik and Harrington [3] proved an abstract normalization theorem. We reprove their theorem by reducing the problem of finding normalizations to that of finding 'canonical bases' for suitable collections of types, and proving some results on the existence of canonical bases.

In those concrete examples of stable theories which are fully worked out, forking turns out to mean the following: p ~ S(B) does not fork over A iff certain 'important' formulae in p are almost over A (i.e. depend on some finite A-definable equivalence relation). For example, if T is a theory of R-modules (R some ring) and p ( 2 ) e S ( B ) , then p does not fork over A iff every positive primitive ~(.~) e p is almost over A. (See Pillay and Prest [10]. A similar result for 'normal' theories was observed by Rothmaler [11].) If T is the theory of

016843072/86/$3.30 (~ 1986, Elsevier Science Publishers B.V. (North-Holland)

62 A. PiUay

algebraically closed fields of some given characteristic, then any type p($)e S(K) corresponds to a variety V~ defined over K. Again p does not fork over k if V, is almost over k. In Section 5, given a stable theory T and p ( $ ) e S(B) we define what it means for a formula (or definable set) in p to be p-normalized, and we show that p does not fork over A iff every p-normalized formula in p is almost over A. (In particular, there are 'enough' p-normalized formulas.)

In a sense this paper builds on that of Harnik and Harrington [3], although we do not use their notation of 'fragment' (we use C eq instead). We should mention that our aim is to give new results rather than just new proofs.

I began thinking about some of the problems dealt with in this paper in the Autumn of 1982 while visiting McGill University, supported by a grant from NSERC. I thank Mihaly Makkai and Gabriel Srour for some helpful discussions on these matters at that time. Thanks are also due to John Baldwin for organizing several model theory meetings at Chicago Circle in the Autumn of 1983, which stimulated some of the work presented here. Finally, I wish to thank John Vaughn for pointing out an error in an earlier proof of Proposition 4.5.

1. Preliminaries

T will be a complete theory, not necessarily stable. L is the language of T. As usual we work in C w a large saturated model of T. M, N denote elementary substructures of C of cardinality less than Icl, and are called models. A, B etc. denote subsets of C of cardinality less than [C[.

0 denotes the empty set. By a definable set we mean a subset X of C" for some n < to such that for some

L-formulas ~ ( x l , - . . , x,, iT) and ~ from C.

x = c":

X, Y etc. will denote definable sets. We will often identify a definable set X with the formula ¢(2, a) which defines

it. By a conjugate of X we mean an image of X under an automorphism of C.

Note that if X is defined by ¢(~, ~), then X ' is a conjugate of X iff X ' is defined by q)(2, ~') for some ~' with tp(~') = tp(~).

Recall than an L-formula q)(~, jT) is said to be stable if there do not exist ~i, 6 , i < to such that ~ q)(ai,/~j) iff i ~< j for i, j < w.

Note that calling a formula ¢ stable presupposes a division of the free variables of ~p into two sets ~ and )7.

If 4(2, Y) is stable, then by compactness there is a greatest m < w such that there exist ~ii,/~, i < m with ~¢(ai,/~j) iff i<~]. Following Hodges [4] we call m the ladder index of cp(g, )7).

Forking, normalization and canonical bases 63

It is convenient to introduce the following definition:

Definition 1.1. The definable set X is stable if there do not exist conjugates X~, i < to of X and t~i for i < to such that aie .Xj iff i <~ j for all i, j < to.

Note that by compactness, X is stable if and only if X is defined by a formula ~($,/~) such that ~($, )7) is stable.

By a type p we mean a consistent set of definable sets (of a given finite arity) which is closed under conjunctions and disjunctions. If the arity is n, we will often write p = p ( $ ) where $ = ( x l , . . . , Xn). (So note that a type is not necessarily complete.)

Given an L-formula dp($, )7) we may wish to distinguish between 'special' variables, and parameter variables. Usually .~ will be the special variables and )7 the parameter variables, but occasionally it will be the other way round.

Following Harnik and Harrington [3] we make the following definition:

Definition 1.2. The set of L-formulas A(:~) is said to be Booleanly closed if: (i) Every member of A($) is of the form ~(:~, )7) for some )7 where we

designate $ as the special variables and )7 as the parameter variables. (ii) A(:~) is closed under Boolean combinations and substitution of parameter

variables by other parameter variables.

Note that: (a) If F($) is a collection of stable formulas of the form ~(:~, )7) some )7, then

every member of the Boolean closure of F(:~) is stable, and thus: (b) The collection of all stable formulas of the form ~($, )7) for $ fixed, is

Booleanly closed. A or A($) will from now on, always denote a Booleanly closed set of formulas,

and we say that A($) is stable if every dp($, )7) e A is stable. Given A(:~), the definable set X is said to be A-definable if X is of the form

~($, ti) for some ti and some ~(,~, )7) • A. The definable set X is A-definable if X is of the form ~($, ti) for some t~(:~, )7) e L and some ti in A. X is said to be A-A-definable if X is A-definable and A-definable. This is a slight but useful departure from usual terminology. For example, let T be the theory of an equivalence relation E with two infinite classes. Let A(x) be the Boolean closure of E(x, y). Then the universe C is A-0-definable in our sense, for the universe is both 0-definable (by x = x ) and A-definable (by E(x, a ) v E(x, b) for a, b in different E-classes). However, clearly there is no ~(x) in A (without parameters) which defines the universe.

Given A and A($), S,~(A) denotes the collection of maximal consistent sets of A-A-definable sets.

S(A) denotes the collection of complete types (of some finite arity) over A. We will occasionally work in C ~ , especially in Sections 4 and 5. The reader is

referred to Shelah [12], Makkai [8] and Harnik-Harr ington [3] for the definitions.

64 A. Pillay

The important thing from our point of view is that a definable set of C is represented by an element of C ~. Namely, for any X there is an element IX] of C ~q such that an automorphism f of C eq leaves X invariant ifff( rx] ) = i x ] . ix ] is not unique, but is unique up to interdefinability. IX] can be chosen as follows: Let X be defined by t#(g,~). Then let IX] be ~/E, where E,(Yl, Y2) is (W)(~(~, y l ) ~ ~(~, Y2)).

As usual dcl(A) denotes the set of elements which are definable over .4. Thus for any X there is (in C eq) a unique set A = dcl(A) such that an automorphism fixes X setwise iff it fixes A pointwise.

acl(A) has the usual meaning--namely acl(A) is the set of elements which are in finite A-definable sets.

As usual X is said to be over A iff X is A-definable (equivalently X is left invariant by any A-automorphism of the universe). X is said to be almost over .4 if X has only finitely many conjugates under A-automorphisms. Two important facts are:

(i) X is almost over A iff X is over acl(A) (in ceq), and (ii) X is almost over A iff X is over M for any M D A. The notion of forking was defined by Shelah [12] (in terms of 'dividing').

Various other notions or definitions of forking have been given, which in a stable theory are all equivalent. For example, let T be stable (i.e. every tp(£, )7) is stable), p(x) • S(B) and A c B, then the following are equivalent:

(i) p has an extension q ( f ) • S(M) for some M ~ B such that q is definable almost over A (i.e. such that for any tp(£, )7) there is ~p~(y) almost over A such that VC" in M, ~(,~, e) • q iff ~ ~p~(~),

(ii) p is almost finitely satisfiable in A (i.e. for any M ~ A and ~(f,/~) e p there is C" in M such that ~ ~(~,/~)).

(iii) For any ¢($, )7) • L if ~(,~, )7) is represented in every complete extension of p to a model, then also ~(f , )7) is represented in every complete extension of p IA to a model.

Then p does not fork over A (or p is a nonforking extension of p r A) ff (i), (ii) or (iii) holds. There is an extensive literature on forking for stable theories, beginning with Shelah [12], Lascar [6] and Lascar-Poizat [7]. The equivalence of (i), (ii) and (iii) above can be found in Baldwin [1], Hamik-Harrington [3], Pillay [9] and Rothmaler [11] to name but a few.

Again, for T stable, and p(~), q($) complete types, let p r- q mean that q is a nonforking extension of p. Harnik-Harrington [3] noted the followinR properties as characteristic of the relation r-:

(A0) c is preserved by elementary maps and p r- q =>p c q. (A1) If p c q ~ r, then p c r ff and only if p r- q and q r- r. (A2) If p • S(A), and A c B, then there is q • S(B) with p r- q. (A3) If p • S(A), then there is A o c A , IAI ~< Irl such that p IAor-p. (A4) Given p, there is t such that there are at most ~ mutually contradictory q

with p c q.


In Section 5 our aim is to present some new facts about forking for stable theories, and thus in Section 5 we freely use facts about forking contained in the above references. In particular, we make use of the ranks R(- , q~, No).

In Sections 2 and 3, however, our aim is to develop 'local' forking in a not necessarily stable theory. So here we assume nothing which is not mentioned explicily. In particular, the reader should be careful not to a priori assume that such equivalences as between (i), (ii) and (iii) above hold in this context. It should be emphasised that by forking for an unstable theory T we do not here mean a notion of independence suited to unstable theories, but a theory of forking with respect to the stable formulas of T, which satisfies analogous properties to (A0)-(A4) above. Some work was done on this problem by Harnik-Harrington [3] and Hodges [4] but the matter was left unresolved. What we do is as follows: for an arbitrary theory T, and for any Booleanly closed stable A($) i fp e Sa(B) and A c B, we define what it means forp not to fork over A (or p to be a nonforking extension ofp IA). I f p ~ S(B) we say that p does not A-fork over A (or p is a A-nonforking extension of p IA) if p IA does not fork over A (where p I A is the collection of A-definable sets in p, and p IA is the collection of A-definable sets in p).

Now interpreting r- as the relation between complete A-types p, q: q is a nonforking extension of p ; we show that r- satisfies (A0)-(A4). Similarly, if we interpret c as the relation between complete types p, q: q is a A-nonforking extension of p; we show that r- satisfies (A0), (A1), (A2), (A3) and (A4a) where (A4zx) says: for any p e S(A) there is 2 such that there are at most 2 mutually A-contradictory q with p r-q.

The proof of the 'uniqueness' of a relation satisfying r- is routine and will be left to the reader. But what does uniqueness mean in this context. It will mean that for any T and Booleanly closed A(:~) there is at most one relation r- between complete A-types p($), q(:~) satisfying (A0)-(A4), and that there is at most one relation r- between complete types p(,~), q(:~) satisfying (A0)-(A3) and (A4A).

2. Forking

Definition 2.1. (i) Let X be a definable set of n-tuples, which is stable. We say that X does not fork over A if for every model M ~ A, X t7 M n :~ 0.

(ii) Let p (:~) be a type, all members o f which are stable. We say that p does not fork over A if every X e p does not fork over A.

(iii) Let p(.~)e S(B), and let A($) be stable. We say that p does not A-fork over A if p IA does not fork over A.

Remarks. Definition 2.1(i) and (ii) agree with the usual notion of 'not forking over' in the case where T is stable (by PiUay [9, Proposition 6.4] for example).

Definition 2.1(ii) says p($) does not fork over A if p is 'almost finitely satisfiable in A', in the terminology of Rothmaler [11].

66 A. PiUay

If A($) is stable, p(Y) e Sa(B), A c B and q = p IA, then we say that p is a nonforking extension of q, i fp does not fork over A. Similarly, ifp(Y) ~ S(B) and q = p IA (where again A c B) we say that p is a Anonforking extension o f q i fp does not A-fork over A (where A(Y) is again stable).

For T unstable, the reader should be careful not to immediately identify the above definitions with any of the 'alternative' definitions of nonforking (for example involving definability almost over).

Proposition 2.2. Let ¢#(~, y) be stable, with ladder index m. Suppose that ~($, fi)f ')M" = 0 (i.e. dp($, fi) forks over M). Let [to, f i l , . . . , film be such that tp(fi/M) = tp(fii/M) Vi <<- m and tp(fij/M U fi0, .- •, fij-1) is finitely satisfiable in M Vj <<-m. Then

~ "--l(3X ) (i~=o $ (X, fii) ) (i.e., {@(~, fii):i <-m} is inconsistent)

Proof. Assume, by way of contradiction that

(i) (,__/30 ,) We will define, by downward induction on i (starting with i = m + 1) tuples f i ' ,

- - ? - - f - - F - - ? - - ? Cm, a m - ~ , Cm--~, • • • , a~, c i in M such that

( i i) ~32 (k6i $(~) fik) A i<~k<~mA ¢(X)fik))) (iii) ~-7(p (e~,, fij)

(iv) ~ ~(~,, fi~)

For i = m + 1,

V i i i , i<~k<-m,

iff k<~j for i<~k, j<<-m.

these conditions reduce to (i). Now suppose that fi ' , ~ ' , . . . , fi~, ~' in M have been defined satisfying (ii), (iii) and (iv). We show how to define fi '-l, c~-1 in M.

Firstly, as tp ( f i /_QMOfi0 , . . . , fii-2) is finitely satisfiable in M (and as a~,, C'~, e M for i ~< k ~< m), we can find fi'-i in M such that (ii), (iii) and (iv) hold with fii_l replaced by fi'-l.

In particular we have,

(ii)'

and

(~)' ~-~(~'i, fi'-1) for all k with i <~ k ~< m.

By (ii)', as fi~ e M for i - I <~ k ~ n, we can pick ~'~_i in M such that

(iv)' ~ A ¢(~'-1, fi~,). i-- l ak<~m


As ~b(~, t:) N M n = 0 and tp(t:) = tp(t~j) Vj we have

(iii)" ~m~(e '_l , t:j) Vj < i - 1.

Now the holding of (ii), (iii), (iv) for a ' , U , . . . , a[_l, C[--1 follows from (ii)', (iii) and (iii)", and (iv), (iii)' and (iv)'.

Now when t i ' , U , . . . , t i ~ , ~'~ have been defined, we get (by (iv)) a contradiction to m being the ladder index of ~(~, )7). This proves the proposition.

Note that Proposition 2.2 needs stability of the formula tp($, )7). For example, let T = Th(Q, <) , M = (~), <). Let tp(x, y) be x > y and a be some element >M. Then (x > a) t ] M = 0, but no conjunction of M-conjugates of "x > a" is inconsistent.

Proposition 2.3. Let A($) be stable, and p($) e SA(A). Let Z = 2 2M~'(~l'laq~. Then p has at most ~ distinct nonforking extensions q ~ Sa(C). Similarly, if p e S(A), then there are at most ), distinct A-types among the A-nonforking extensions q ~ S(C) ofp.

Proof. This is basically given us free by virtue of our definitions. Let M D A, Igl=max(IAI, ITI). Let q eSA(C) be a nonforking extension of p. Then q is finitely satisfiable in M (by Definition 2.1(ii)). As is well known, this implies that q is determined once we know { X t3 M : X ~ q}. The proposition clearly follows.

Corrollary 2.4. Let A($) be stable and p ~ Sa(C). Then p does not fork over A, if and only if the number o f A-conjugates of p / s < [C[. Similarly p($) ~ S(C) does not A-fork over A iff the number o f A-conjugates o f p I A/s < Icl.

Proof. The left to right directions follow from Proposition 2.3 (as dearly p e Sa(C) does not fork over A iff every A-conjugate of p does not fork over A, and likewise for p e S(A) and A-nonforking). For the other direction, suppose for example that p e S(C) A-forks over A. Let dp($, (~) e p, dp($, y) e A($), be such that ~(~, t:) forks over A. So let M ~ A be such that dp($, ~) f] M n = 0. It is easy to construct t~, tr < Icl such that tp(ti~) = tp(ti) Vtr and tp ( t i JM O { ~ :fl < a~}) is finitely satisfiable in M for all tr < ICI . For each tr, let f~ be an M-automorphism of C such that f (a ) =t:~ and let p,~ =f,~(p). So Va~, $($, a~) ep~ In. Let m be the ladder index of ~p(.~, )7). By Proposition 2.2 each {~p(.~, a~o) , . . . , dp($, ti,,,)}, tr0 < trl- • - < tr,,,, is inconsistent. Thus among the p~ there are I C]-many distinct A -types.

In order to prove the existence of nonforking extensions we will need Harnik and Harrington's 'needing lemma' (the proof of which is attributed to Ziegler). In fact, we will use this lemma in an 'inverted' form.

68 A. Pillay

Recall that if p07) e S(A) and ~p07, ~) is an L-formula, the p needs ~p if for any M = A and extension p ' (y) • S(M) of p, there is ti • M such that ap07, ti) e p ' .

The following is immediate:

Observation 2.5. Let dp(£, 37) be stable. Then ¢p($, 4) does not fork over A i f and only if tp(ti/A) needs dp($, 37) (where now 37 is the type variables and £ the parameter variables).

Lemma 4.8 of Hamik and Harrington [3] says:

Lemma 2.6. Let p(~) • S(A), and let ~Pl(Y, u), ~PE(Y, u) be stable L-formulas. I f p needs ~Pl v ~P2, then p needs ~Pl or p needs ~2.

We thus can conclude:

Lemma 2.7. Let X, Y be stable definable sets (of the same arity). I f X v Y does not fork over A, then either X does not fork over A or Y does not fork over A.

Proof. We may assume, by adding dummy variables, that X is defined by ~(~, ti) and Y by ~p(£, ti), where tp($, 37) and ~p(£, 37) are stable. Then X v Y does not fork over A ==> tp(d/A) needs ~(~, 37) v ap($, 37) ($ now parameters variables) tp(ti/A) needs tp(£, 37) or tp(ti/A needs ~p($, 37)~ X does not fork over A or Y does not fork over A.

Proposition 2.8. Let A(£) be stable. Let p ( £ ) • S(A), A c B. Then p has an extension q ( £ ) e S ( B ) such that q does not A-fork over A (i.e., p has a A-non forking extension in S(B)).

Proof. Let F = {X: X is A-B-definable and X forks over A }. It is enough to show that p U {-nX:X • F} is consistent (for then any extension of this set to S(B) is the required q). If p U { -n X:Xe F} is inconsistent, then by compactness there are Y • p and X1, • • •, Xk • F such that

Y ~ _ X I V . . . v X k .

As Y is consistent and over A, Y is satisfied in any M ~ A. Thus X1 v • • • v Xk does not fork over A. By Lemma 2.7, some X~ does not fork over A, giving a contradiction.

Similarly we have:

Proposition 2.9. Let A($) be stable. Let p • Sa(B) and p does not fork over A (where A c B). Then for any C D B, p has an extension q($) e S(C) which does not A-fork over A.


Proof. Again we must show that p U { ~ X : X is A-C-definable and X forks over A) is consistent. If not

Y c X l v" " v X,,

where Y e p, and each X~ forks over A. As Y does not fork over A, we again get a contradiction as in the proof of 2.8.

Let Proposition 2.9' be just as Proposition 2.9 but replacing the hypothesis 'p ~ Sa(B) and p does not fork over A' by 'p ~ S(B) and p does not A-fork over A'. Is Proposition 2.9' true? The reader will see that the above proof breaks down in this case, as the Y one obtains need not be a-definable. In fact Proposition 2.9' will drop out of the material in Section 3 (verification of axiom (A1)), and is one of the nontrivial things to be proved in our set-up.

Let p(~) ~ S(M) and tp(~, )7) ~ L (or p(~) ~ Sa(M) and tp(,f, )7) ~ A). By a cp-definition of p we mean a formula ~p0 7, ~), ~ e M such that for all fi in M, tp(,f, t i )~p iff ~p(~i, t~). (So to say that p has a tp-definition means that {ti~ M: q~(,f, ti) ¢p} is an M-definable set.)

I fp($) e SA(M), then p is said to be definable over (almost over) A if for every tp(:L )7) e A, p has a tp-definition which is over (almost over) A.

It is now a standard fact that, if p($) e S(M) and $(,f, )7) is stable, thenp has a tp-definition. The information we will require is contained in the following fact from Harnik and Harrington [3] (their Corollary 2.8). Rather than waste paper by giving another proof, we just state the result.

Lemma 2,10. Let M = A be (IAl+lTl)+-saturated. Let p($)=tp(b /M) and let cp($, y) be stable. Then p has a alp-definition ~p(y) say, which is (equivalent to) a positive Boolean combination of A-conjugates of ¢p(b, y).

Lemma 2.11. Let M = A be (]Al+lTl)+-saturated. Let a(~) be stable and p ~ Sa(M). I f p does not fork over A, then p is definable almost over A.

Proof. Let ~p(.~, y) ¢ A, and let (by 2.10), Y be the ~p-definition for p (i.e. Y is M-defnable and ~i ¢ Y fq M" iff ~p($, 4) ¢p). It is enough (as M is saturated) to show that Y is over N for any A c N ,--M. So let N be such. So p is finitely satisfable in N. It follows that if r~, th2 ¢ M, and ¢p($, thl) Cp, ~p(x, m2) CP, then tp(thl/N) :/:tp(rh2/N). So for th~ M whether or not rh E Y depends only on tp(r~/N). It follows easily that Y is over N.

3. Further characterizations and verification of the axioms

T is still an arbitrarily complete theory (maybe unstable). The following appears to us now to be the combinatorial essence both of the

theory in general and of normalization in particular.

70 A. P//lay

P r o p o s i t i o n 3.1. Let X be stable, A any set o f parameters. Then there is a set Y, a positive Boolean combination o f A-conjugates of X, which is over A. Moreover, i f X does not fork over A , then Y can be chosen to be consistent.

Proof. Let X be tp(i, ti) where tp(:L )7) is stable. Let p07) = tp(~/A). Let A07 ) be the Boolean closure of tp($, )7) (we view )7 as the type variable now). Let M ~ A be (IAI + ITI)+-saturated, and by Proposition 2.8, let q(y) e S(M) be an extension of p, which does not A-fork over A. Let ti' realize q. (So tp(ti '/A) = tp(ti/A).) By Lemma 2.10 there is a formula ~/,($), a positive Boolean combination of A-conjugates of ~p($, ti), which is a tp(.~, y)-definition of q(y). Moreover, by Lemma 2.11, ~/,($) is almost over A. Let Y be the union of the (finitely many) A-conjugates of ~/,(.~). Then clearly Y is over A, and Y is a positive Boolean combination of A-conjugates of X(= ~p($, a)).

If X does not fork over A, then ~(:L ti') does not fork over A. Thus ~p(.~) is consistent, and so is Y.

We thus obtain the following characterization of a stable set forking.

Coronary 3.2. Let X be stable. Then X does not fork over A i f and only i f some positive Boolean combination o f A-conjugates o f X is consistent and over A.

ProoL The left to right direction is given by Proposition 3.1. Conversely suppose Y is a positive Boolean combination of A-conjugates of X, which is consistent and over A. Thus Y = Y~ v • • • v Yk where each Y~ is a conjunction of A-conjugates of X. Now clearly Y does not fork over A (Y is over A and consistent). So by Lemma 2.6, some Y~ does not fork over A. But Y~ c X' for some A-conjugate X '

of X. Thus X' does not fork over A. So clearly X does not fork over A.

P r o p o s i t i o n 3.3. Let A($) be stable, A c B and p($) ~ Sa(B). Then p does not fork over A if and only i f for every X e p, there is Y e p such that Y is a positive Boolean combination o f A-conjugates of X and is over A.

P r o o f . First assume that p does not fork over A. By Proposition 2.9, let q($) e Sa(C) be an extension o f p which does not fork over A. By Lemma 2.11, q is definable over acl(A). (Here we work in ceq.) Now let X e p. By Corollary 3.2, there is Y', a positive Boolean combination of acl(A)-conjugates of X, which is over acl(A). Now X ~ q, and as q is definable over acl(A), any acl(A)-conjugate of X is also in q. So clearly Y' ~ q. Now let Y be the union of the (finitely many) A-conjugates of Y'. Thus Y is over A and Y e q.

As p = q IB and A c B, it follows that Y e p. This proves the left to right direction.

The right to left direction is an immediate consequence of Corollary 3.2 (as if Y ~ p, then Y is consistent).


Thus we also have:

Proposition 3.4. Let A(£) be stable. Let A c B and p($) • S(B). Then p does not A-fork over A if and only i f for every A-definable X • p there is Y e p such that Y is over A and Y is a positive Boolean combination o f A-conjugates o f X.

Proof. This is immediate by Proposition 3.3, for p does not A-fork over A just means that p IA does not fork over A. Note that Y as given by the proposition is A-A-definable.

We now verify the 'transitivity' axioms. (Remember that if p e S(C) or p • S•(C) and B c C, then p IB is the set of B-definable sets in p.)

Proposition 3.5. Let A(g) be stable, and A c B c C. Then (i) I f p ~ SA(C), then p does not fork over A iff p does not fork over B and p IB

does not fork over A.

(ii) I f p($) e S(C), then p does not A-fork over A iff p does not A-fork over B and p IB does not A-fork over A.

Proof. It is clearly enough to prove (ii). First suppose that p e S(C) does not A-fork over A. Thus any A-definable

X • p does not fork over A. In particular, any A-definable X • p does not fork over B (as B ~ A ) and any A-B-definable X e p (i.e. any A-definable X e p IB) does not fork over A.

Conversely suppose that p does not A-f0rk over B and p IB does not A-fork over A. We want to show that p does not A-fork over A. We will use Proposition 3.4. So let X e p IA. Thus (by 3.4), there is Y e p such that Y is over B and Y is a positive Boolean combination of B-conjugates of X. So Y e p IB and Y is A-definable. Again (as p IB does not A-fork over A), there is Z e p IB, Z over A and Z a positive Boolean combination of A-conjugates of Y. Clearly Z is a positive Boolean combination of A-conjugates of X. As X was arbitrary in p IA it follows that p does not A-fork over A.

Note that now Proposition 2.9' follows from 2.8 and 3.5(ii). For given p(g) • S(B) such that p does not A-fork over A, let q(£) e S(C) b e a n extension of p such that q does not A-fork over B. Thus (by 3.5(ii)) q does not A-fork over A.

We now wish to verify axiom (A3). In order do this, we first show the equivalence of our definition of forking with some familiar notions.

Proposition 3.6. Let A c M where M is (IAI + ITI) ÷-saturated. Let A($) be stable and p(g) ~ Szx(M). Then p does not fork over A if and only if p is definable almost over A.

2 A. Pillay

'roof. (=>) is precisely Lemma 2.11. ( ~ ) So suppose that p is definable almost over A. It is enough to show that for

ny model N with A c N c M, p is finitely satisfiable in N. So let N be such a model. Let/~ realize p and suppose that ~p(£, 37) • A, fi • M

nd ¢ q~(/~, a). We must find/~' • N such that ~ ~p(/~', ~). Let us suppose that there is no such/~' and obtain a contradiction. The proof is now basically the same as that of Proposition 2.2 in Pillay [9]. Let

~07) be the ~p-definition of p. So as ap(y) is almost over A, ~p is over N. We define inductively fii, /~i, in N for i < co such that

(i) ~ ~(bi , aj) if j ~ i, and

(ii) ~ ¢(/~, fij) for all j.

Suppose ai, bi E N are defined for i ~< n satisfying (i) and (ii). Note that (by ssumption),

~-ncp(6i, ~) '¢i ~< n.

Now as ~ cp(/~, t~) we have ~ W(fi). Thus

N ~ (3y)(~P(Y) ̂ A -¢(/;. y)). i<~n

Let tin + 1 • N be such that

^ A i ~ n

In particular, we have W ~p(/~, an+l)" Now using (ii) we find/~n+l • N such that

i ~ n + l

Thus dearly (i) and (ii) hold for i ~< n +-1. But then (i) contradicts the stability f ¢(£, y), proving the proposition.

roposition 3.7. Let A(2) be stable.

(i) Let p • Sa(B), A c B. Then p does not fork over A i f and only i f f o r some f D B, p has an extension q • S~(M) which is definable almost over A. (ii) Let p(2) • S(B) , A c B. Then p does not A-fork over A if and only if for

~me M ~ B, p has an extension q(2) • S ( M ) such that q IA is definable almost ver A.

roof. (i) (=>) Suppose p e Sz~(B), A c B and p does not fork over A. Let f = A be ([A[ + [T[)+-saturated, and let q • SA(M) be an extension of p which 3es not fork over A (by Proposition 2.9). By 3.6, q is definable almost over A.

( ~ ) Let q • Sa(M) be an extension of p, which is definable almost over A. Let " = M be ([A[ + [T[)+-saturated. Clearly by applying the f-definitions (for

Forking, normalization and canonical bases 7 3

tp • A) to M', we obtain q' • Sa(M'), such that q' ~ q and q' is definable almost over A. By 3.6, q' does not fork over A. So by 3.5(i), q' IB = p does not fork over A.

The proof of (ii) is similar (using 2.9' in place of 2.9).

Corollary 3.8. Let A($) be stable, p • SA(A). Then there is A o c A , IA01 ITI such that p does not fork over Ao. Likewise for complete types and A-nonforking.

Proof. Let M DA ad let q • Sa(M) be a nonforking extension ofp. By 3.7(i), q is definable over acl(A). But clearly (as there are at most I Tl-many formulas in A), q is definable over some model Mo c M with [Mo] ~< [TI. Thus q is definable over Mo N acl(A) (i.e. for each tp • A the ~-definition of q is over M0 fq acl(A)). Let Ao c A be such that IAol I TI and M0 N acl(A) = M0 N acl(A0). Clearly q is definable over acl(A0). By 3.7(i) again, p does not fork over Ao.

We now check the verification of axioms (A0)-(A4). So first let us fix stable A(,f). Let p r- q be the relation between types p • SA(A) and q • Sa(B) for some A, B such that p c q and q does not fork over A. (A0) clearly holds. (A1) is Proposition 3.5(i). (A2) follows from Proposition 2.8 (or 2.9). (A3) is given by Corollary 3.8, and (A4) follows from Proposition 2.3.

Similarly if p E q is interpreted by: for some A c B, q • S(B), p • S(A) and q is a A-nonforking extension of p, we see that (A0), (A1), (A2), (A3)and (A4a) are true.

It is routine to check for example that, if r- is a relation between complete types p c q satisfying (A0)-(A3) and (A4a), then A($) is stable (count types) and r- is the A-nonforking relation.

Finally in this section we show how the finite equivalence relation theorem holds in the local context.

Proposition 3.9. Let A(~) be stable. Let A c M, Pi • Sa(M) and pi does not fork over A for i = 1, 2 and suppose that p l IaclA =P21 aclA (in ceq). Then Pl =P2.

Proof. Let ql, q2 • SA(C) be nonforking extensions of Pl, P2 respectively. So qi does not fork over A for i = 1, 2 (by 3.5). By 3.7 both ql and q2 are definable over aclA. In particular (clearly), ql and q2 are both invariant under aclA- automorphism of C. It is enough to show that ql - - q2 (assuming the hypotheses). Let X • ql. Let Y be a positive Boolean combination of aclA-conjugates of X such that Y • ql and Y is over acl A (by 3.3). Let Y = Yx v • • • v Y,, where each Y~ is a conjunction of aclA-conjugates of X. Now Y • p ~ IaclA. So Y e q2. Thus (as q2 • Sa(C)) some Y/• q2. Thus X ' • q2 where X ' is some acl A-conjugate of X. But then also X • q2 (as q2 is left invariant by acl A-automorphisms). Similarly X • q2==>X • ql. So ql = q2, whereby PI =P2.

74 A. Pillay

Corollary 3.10. Let A($) be stable. Let Pi E Sz~(M), and Pi does not fork over A for i = 1, 2. I f p l =k p2, then there is X almost over A such that X ~ pl, X q~ p2.

Proof. If for all X almost over A we have X Epl iff X Ep2, then (in C eq) Pl I aclA =P21 aclA. By 3.9, Pl =P2-

By similar means we can show that i f pl , p2eS~(M) where M D A and p I I A = p E I A , M is (IAI + ITI)+-saturated and P~,p2 do not fork over A, then there is an A-automorphism f of M such that f (P0 = P2.

4. Canonical bases and normalization

Here we prove a result concerning the existence of 'canonical bases' for families of types. We use this plus previous results to deduce Harnik and Harrington's general normalization theorem. We also mention an alternative general notion of normalization. S. Buechler [13] independently arrived at a similar formulation and we thank him for discussions and communications.

Again T is an arbitrary complete theory.

Definition 4.1. Let P be a collection of (maybe incomplete) types (in ~) over C. We say that A is a canonical base o f P, A = Cb(P) if

(i) A is definably closed, and (ii) For any automorphism f of the universe, f permutes (i.e. induces a

permutation of) P if and only if f (a) = a Va ~ A.

A very general question (which Proposition 4.2 answers a special case of) is: What are necessary and sufficient conditions on P for P to have a canonical base (in ceq)? Note that a canonical base of P, if it exists, is unique.

Our definition above departs somewhat from Shelah's definition in [12]. For T stable and p (£ )e S(B) stationary, Shelah says that A is a canonical base of p just if [an automorphism f fixes A pointwise iff f ( p ' ) = p ' ] where p ' is the global nonforking extension of p. (So what Shelah calls Cb(p) we would call Cb({p'}).) Shelah [12] shows that an A exists in C ~ , A being dcl{ [d~p] • ~p e L}, where dtp is a tp-defmition for p.

We will generalize Shelah's result, by giving, for suitable P, a necessary and sufficient condition for P to have a canonical base.

First, given (arbitrary) P, let Gp denote the group of those automorphisms of C which permute P.

Proposition 4.2. Let A(.~) be stable and P c S,~(C). Suppose also that (i) IP[ < IC[, and

(ii) For any p e SA(C), i f p I# has an extension in P for all ¢p(~, y)~ A then p e P .


Then the following are equivalent. (iii) ~ has a canonical base (in c~q). (iv) For each p ~ P and dp($, )7) e A, the orbit of p I~p under Gp is finite.

Proof. First assume that P" has a canonical base A. Suppose by way of contradiction that for some ~ e A, and p e P the orbit of p I tp under Gp is infinite. We will conclude that P is unbounded.

Let fl(jT, ~) be a ~b-definition for p (as ~ is stable). For any x let ~, i < x, be new constants and let Z' say that tp(6i/A) = tp(6/A) Vi < x and also that fl0 7, ~i) and fl0 7, ~j) are inequivalent for all distinct i, j < r.

As the orbit of p l~ under Gp is infinite, 2" is consistent. (For tp(6i/A)= tp(~/A) means that there is an A-automorphism f//(which is thus in Gp) such that f/(e) = (~). And fl0 7, ~'i)<~ fi(Y, Cj) means that f~(p)Itp :/:fj(p)Itp.

Realizing Z gives us x many pairwise distinct conjugates of p Itp under Gp. For x big enough this contradicts the boundedness of P (i.e. (i)). Thus we prove (iv).

Conversely assume that (iv) holds. Let us first partition P into its Gp orbits P~. So ~'~i--" Ui~l~:Di and p, q are in the same ff~i just i f f (p) =q for some f e Gp.

Let us fix dp • A and i ~ I. By (iv) there are just finitely many ~p-types in P~. Let the tp-definitions of these types be f i i ( Y ) , . . . , fig(Y). Note that { r f i l ] , . . . , [fig]} is a finite (and thus definable) set in C eq, say X. Let a~,,i be IX] (in C eq again). Thus, an automorphism f fixes a,,~ iff f permutes the definable sets f i l ( Y ) , . . . , fig(Y) iff f permutes the tp-types in Pi.

We claim that A = dcl{a,,~:i e L ~b e A} is the canonical base of P. It is first clear from the above remarks that, if f e Gp, then f(a~,,i) = a~,.i Vdp, i

and so f fixes A pointwise. Conversely, suppose that f(a,,~)= a,.z Vdp, i. Let p ~ P. It is enough to show

that f ( p ) e P . Let p e P i and tpeA. Let, as above, f i l , . . . , f i k be the tp- definitions of the members of Pi. Say the tp-definition of p is fil. Thus, as f permutes { f i l , . . . , ilk}, the ~-definition of f(p) is fir say, some r ~ < k. Then clearly f (p) I (I) is a Gp conjugate of p I tp, and thus f(p) ItP extends to a member of P. By condition (ii), f (p) ~ P. This completes the proof.

Example 4.3. Let A(:~) be stable, p e S a ( B ) and let P p = { q ~ S A ( C ) : q is a nonforking extension of p}. Then P~, satisfies (i) of 4.2 (by Pioposition 2.3). Clearly Pp also satisfies (ii) (for if q e SA(C) and q I ¢ has an extension in Pp for all ~ e A, then clearly q Dp and q does not fork over B, whereby q e P~,). Finally an easy exercise shows that Pp satisfies (iv). Thus Pp has a canonical base.

Note that conditions (i) and (ii) alone do not imply that P has a canonical base. For example let T be the theory of an equivalence relation E with infinitely many classes. Let ai, i < to, be pairwise inequivalent elements. Let p / e SI(C) contain E(x, ai) and be nonalgebraic. Let p(x) ~ SI(C) say that x is in no E class of an element in C. Let P = {pi: i < to } U {t9}. Then P satisfies (i) and (ii), but not (iv).

76 A. Pillay

Let us now recall Harnik and Harrington's normalization set up: is a set of stable definable sets (of a given arity), closed under Boolean

combinations and conjugates. is an equivalence relation on • satisfying

(i) ~ is preserved by automorphisms. (ii) ¢ ~ p ~ Z and ¢'-~X~¢~P (for ¢, ~p, %E ~).

(iii) The ~ classes are closed under positive Boolean combination. For each ¢ e ~, let P , = the set of (/)-types (i.e. maximal consistent sets of

members of ~ ) p such that ¢ • p and for all 0 E p, ¢ J- ¢ ^ -10. (iv) IP I < Icl for all ¢ • ~. Let ¢(x, ~) • ~. All we will use from Harnik and Harrington are the following

facts.

Lemma 4.4. (i) For ¢, ~p • ~, cp --. ~p iff Pc = P~,. (ii) Given ¢(x , ~), there is a type t(f~, f~) (over O) such that for any

t(fz, ft') iff ~q,(~,a) = Pe~(x,a').

' =_~,

Proposition 4.5. (ceq). Under the above assumptions and notation, P~(~,a) has a canonical base.

Proof. Let us denote P~(x,a) by Pa. Let us introduce some notation. Let G denote the set of automorphisms of C which permute Pa. For any X • ~, let ~x denote the closure of {X} under Boolean combinations and conjugates. (So ~x ~ ¢ . )

For p • P~ let p r ~x = p N ~x. For x E • let ~ = (p r ~ x :p • pa}. For each X E • fix an L-formula ~Px(X, )~x) such that X is ~px-definable.

Claim (1). f f p • Pn, X • ~, then the orbit o f p I ~ x under G is finite.

Proof. Let X be defined by lPx(£,/~) and put r 0 7) = tp(/~). Let q(£) • S~,x(C ) be an extension of p r ~ x and let fl(y, ~) be a ~Px-definition of q. Consider the set of formulas _r =

(a) t(a, ti/), i < r,

(b) ai~ ' /- a~, i < x,

(c) r(y#), -~(fl(Yij, c'/) <-~ fl(Y#, •)) for all i < j < x

where x is some cardinal and ~, ~ are new constants. If the orbit of p I ~ x under G is infinite, then (using Lemma 4.4(ii)), we see that Z is consistent. But then a realization of 2' gives rise to x distinct members of P~ which for x big enough contradicts property (iv) of -~.

The following is easy.

(2). Let p be a ~-type such that p r ~ x has an extension in Pn for every X • ~. Then p • P~.


By claim (2) we obtain

(3) Let f be an automorphism of C. Then f ~ G iff for each X • • f permutes

Now for any X e • and p • ~ let

Qp = {q • S~,x(C) : q = p and R~,x(q) = R~,x(p) }

(where R~,x(- ) = R( - , lpx, ~0)). Then clearly, for each such p,

(4) Qp is finite.

Now let X • • and partition ~ into G-orbits Pi, i e Ix. For i • I x , let Q i X = U { Q p : p e P i } . By claims (1) and (4), QX is finite for each i. Thus, by considering the aPx-definitions of the q in Q/X as in the proof of Proposition 4.2, we obtain for each i • Ix, some element a x (in C eq) such that an automorphism f permutes QX iff f(ai x) = ai x.

Now let A = dcl{a/X:X• 4, i • I x } . It should be clear from (3) that f • G ifff fixes A pointwise. Thus A is the canonical base of Pa, completing the proof of Proposition 4.5.

A definable set (formula) X • • is said to be normalized (with respect to - ) if for any conjugate X' of X, X ~ X' iff X = X'.

Coronary 4.6 [3, 9.3]. I f X • 4, then there is a normalized Y • q~ with X--- Y. I f X is ~p(£, ~), then Y is ¢p*(£, ~) for some dp* and also Y can be chosen to be a positive Boolean combination of conjugates X ' o f X such that X -- X ' .

Proof. Let, by Proposition 4.5, A be the canonical base of Px. By Proposition 3.1, let Y be a positive Boolean combination of A-conjugates of X which is over A. Note (by 4.40) ) if X ' is an A-conjugate of X, then X - X ' , and thus by assumption (iii) on 4 , we have X - Y . Moreover again by 4.4(i) we have Px = Py, and so if Y' is a conjugate of Y by an automorphism f, we have: Y'--- Y iff Py, = Py itf f (Px) = Px itt f fixes A pointwise iff Y' = Y (as Y is over A).

So Y is also normalized. Let X be ¢p(x, ~). So Px clearly permuted by any ~-automorphism. As

A = Cb(Px), A is fixed by any ~-automorphism. Thus (as Y is A-definable) Y is also ~-definable and so of the form cp*(£, ~), some ~p*.

Perhaps a more natural general formulation of normalization is as follows: Let X be a definable set, and I the family of all conjugates of X. By a complete/-type, we mean a maximal consistent set of assertions and negations of members of I. Let P be a family of complete/-types which is permuted by any automorphism of C. For each Y • I let Py = {/9 • P: Y • p }.

For I11, Yz • I let Y~-~ Y2 mean PY1 = Pa,,. (This is clearly an equivalence relation.) We can clearly extend this notation to members of ~(I) the Boolean

78 A. PiUay

closure of L We call the pair (1, P) stable iff some (any) X e I is stable, and bounded if for some (any) X e I, IPxI < Icl.

In a similar (and easier) manner to Proposition 4.5 one can prove:

Proposition 4.7. Let (L P) be stable and bounded. Then for any X e L Px has a canonical base.

One can again obtain, as in Corollary 4.6, a normalization theorem for (L P). Namely, if (I, P) is stable and bounded, then for any X e / , there is Y a positive Boolean combination of conjugates of X which are all equivalent to X such that X - Y and such that for any conjugate Y' of Y

Y " - - Y iff Y ' = Y .

Normalization with respect to ranks falls easily out of his set-up. For example let T be superstable, R = R( - , L, oo) and let X be some definable set. Suppose R(X) = or. Let I be the family of conjugates of X and let P be the family of complete I-types p with R(p) = or. Then it is easy to check that for X, Y e I we have

P x = P y iff R ( X V Y ) < O r ( i f f X~ -Y )

(where X V Y denotes symmetric difference). Another kind of example was pointed out by John Vaughn. Let X be stable.

Let P be the family of those/ - types p which do not fork over 0. Then (L P) is stable and bounded, and moreover Px = Py(X ~- Y) just iff X V Y forks over 0.

5. p.normalization and nonforking

In this section we will assume that T is stable. For p e S(B), and A c B we wish to show that p not forking over A is equivalent to certain 'distinguished' sets X e P being almost over A. Here we make (apparently essential) use of the ranks R( - , q~, ~o), in particular of the fact that (for T stable) p(x) e S(B) and A c B, p does not fork over A if and only if for every ¢( , f , )7)eL, R(p, 9, Ro)= R(p IA, t~, Ro) (this is due to Shelah [12] but see also [9]).

Definition 5.1. Let p ( $ ) e S ( B ) , some B. By Pp we mean {q($)eS(C) :q is a nonforking extension of p }.

Definition 5.2. Let p e S(B), X e p. We say that X is p-normalized, if X is invariant under any automorphism of C which permutes Pp (i.e. X is invariant under Gpp in the notation of Section 4).

The p-normalized members of p wll play the role of the distinguished sets in p.


Proposition 5.3 (ceq), Let p • S(B). Then Pp has a canonical base.

Proof. This is precisely Example 4.3 (with L($) for A(:~)), which used Proposi- tion 4.2.

Note that Cb(Pp)= dcl(B) (for p • S(B)), as a B-automorphism of C clearly permutes pp.

Looking back at Definition 5.2 we see that X • p is p-normalized just if X is over Cb(Pp).

Proposition 5.4 (ceq). Let p • S(B) and A c B. Then p does not fork over A i f and only if Cb(Pp) ~ acl(A).

Proof. First suppose that p does not fork over A. Let q e Pp. So q does not fork over A. Thus q is definable over acl(A), whereby any acl(A)-automorphism fixes q. Thus Pp is permuted (in fact pointwise fixed) by any acl(A)automorphism. It clearly follows that Cb(Pp) c acl(A).

Conversely, suppose that p forks over A. Let q • Pp. So also q forks over A. Then q has [CI-many conjugates under acl(A)-automorphisms. As IPpl <lcl it follows that P~, cannot be permuted by all acl(A)-automorphisms and thus Cb(Pp ) ¢ acl(A).

Proposition 5.5. Let p • S(B) and A c B. Then p does not fork over A if and only if for every p-normalized X ~ p, X is almost over A.

Proof. First suppose that p does not fork over A. Let X e p be p-normalized. So X is over Cb(Pp). By Proposition 5.4, X is over acl(A), i.e., X is almost over A.

Conversely, suppose that every p-normalized X • p is almost over A. We are going to show that for every 4($, )7) • L, R(p, 4, I%) = R(p IA, 4, Ro).

By [9, Proposition 6.32], for example, it will follow that p does not fork over A. So let us fix 4(~,)7), and let R denote R(- , 4, Ro), and let Mult denote Mult(-, 4, R0). (See [12] or [9] for the properties of R and Mult.)

Let us suppose that R(p) = m. It follows that for each q ~ Pp, R(q) = m. Now let X be a definable set (over C) of least Rank and Mult such that

X e q Vq ePp.

Clearly R(AT) = m and suppose Mult(X) = k. Now let --- be defined (on all definable sets) as follows: Y1 -" Y2 iff R(Y1) V Y2) <

R ( ~ ) (i.e. R(Y~) = R(Y2) = R(Y1 ^ Y2) and Mult(Y~) = MUlt(Y2) = Mult(Y~ ~ Y2)). By Corollary 4.6, there is a --normalized Y such that X--- Y. Fix such a Y. So

in particular R ( Y ) = m, Mult(Y) = k.

Ciahn 1. Y ~ q V q ~ P p .

80 A. Pillay

Proof. Now R ( X ^ Y) = m and Mult(X ^ Y) = k. Let X ^ Y = Z1 v Z2 v . . . v Zk where R(Zi) = m, Mult(Zi) = 1 for i = 1 , . . . , k.

As X--- Y it follows that X = Z~ v -. • v Z~= where Z~ ~_ Z~ and R(Z ' ) = m, Mult(Z~) = 1 for i = 1 , . . . , k.

By choice of X, for each q • Pp there is (a unique) i ~< k such that Z~ • q. Thus (as R(Z~ - Zi) < m ) , for each q • Pp there is i such that Zi • q.

Thus X ^ Y • q Vq • Pp, and thus Y • q Vq • Pp, proving the claim.

Claim 2. Y is over Cb(Pj,).

Proof. Let f be a Cb(Pp)-automorphism. So f permutes Pp. Thus by Claim 1, f(Y) • q Vq • PI,, and so clearly Y ^ f ( Y ) • q Vq • P~,. Note that R(Y) = m, Mult (Y)=k. Thus by least choice of (m, k) we must have R ( Y ^ f ( Y ) ) = m , Mult(Y ^ f ( Y ) ) = k, namely Y. - - f (Y ) . As Y is --normalized, it follows that Y = f (Y ) . Thus Y is over CB(Pp).

Claim 3. Y is p-normalized (and in p).

Proof. First (as Cb(Pp) c dcl(B)), Y is over B. As Y • q '¢q • P~, and each such q extends p, we also have that Y •p . By Claim 2, Y is p-normalized.

Now by assumption, Y is almost over A. Let Y' be the union of the (finitely many) A-conjugates of Y. Thus Y' is over A, Y' • p and R ( Y ' ) = m. So Y' • p IA and thus R (19 IA) ~< m. So clearly R (p IA) = m = R (p). This is what we wanted to prove.

We should mention that the results of this section also hold (with suitable modification) for any arbitrary theory T and for complete A(£)-types where A is stable. All that is needed is to show: i fp • SA(B) and A c B, then p does not fork over A if and only iff for every 4(.~, )7) • A, R(p, 4, ~0) = R(p ~A, 4, b~o).

R e f e r e n c e s

[1] J.T. Baldwin, book on Stability Theory, to appear. [2] G. Cherlin, L. Harrington and A.H. Lachlan, l%-categorical, F,o-stable theories, Ann. Pure

Appl. Logic 28 (1985) 103-135. [3] V. Harnik and L. Harrington, Fundamentals of forking, Ann. Pure Appl. Logic 26 (1984)

245-286. [4] W.A. Hodges, Free composites and forking, Preprint. [5] A.H. Lachlan, Two conjectures on the stability of w-categorical theories, Fund. Math. 81 (1974)

133-145. [6] D. Lascar, Rank and definability in superstable theories, Israel J. Math. 23 (1976) 53-87. [7] D. Lascar and B. Poizat, An introduction to forking, J. Symbolic Logic 44 (1979) 330-350. [8] M. Makkai, A survey of basic stability theory, to appear. [9] A. Pillay, An Introduction to Stability Theory (Oxford Univ. Press, Oxford, 1983).

[10] A. Pillay and M. Prest, Forking and pushouts in modules, Proc. London Math. Soc. (3) 46 (1983) 365-384.


[11] P. Rothmaler, Another treatment of the foundation of forking theory, Proc. First Easter Conference on Model Theory, at Diedrichshagen, DDR, April 1983 (Humboldt-University, 1983).

[12] S. Shelah, Classification Theory (North-Holland, Amsterdam, 1978). [13] S. Buechler, A note on normalization, Preprint.

forking, normalization and canonical bases

Documents