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N ) ar bit ra rily. By the Compactness Theorem, 1" does admit a model A' = (A, ~ , (C;;') nEN) .
°
Let A = (A , ~) , th en A is a mod el of T, a nd hence is a well ordered set; however it contains t he non- em pt y subset
X={C;;': nEN} admit t ing no minimum , becau se, for every natural n, C;;~ l < c;;'. So we get a contradiction. Conseq uent ly K is not eleme ntary. In oth er words th ere a re linearl y ord ered sets which are not well ordered but sat isfy the same first order sentence as well ord ered sets. 5. Let now L = {O , 1, +, - , .} be our la nguage for fields. We cons ider in L t he class K of fields. K is elementary. In fact t he definition itself of field can be writ ten as a series of first order sentences (in most cases , of universal first order sentences) in L . For instance
says that a ny non zero element has an inver se. Also t he class of algebraically closed fields is elementary, although the corres ponding check is a lit tle subt ler. In fact what we have to say now is that , for every natural n , any (monic) polynomial of degree n + 1 has at least one root. So the point is how to quantify over polynomials of deg ree n + 1. However recall that such a polyn omi al is ju st an ordered sequence of length n + 2 of element s in th e field : the first is t he coefficient of degree 0, t he last is t he coefficient of degree n + 1 (and equals 1 if we deal with mon ic polynomials) ; so what we have to write is j ust , for every n ,
(where vi has the obvious meaning, for ever y i ~ n
+ 1).
26
CHAPTER 1. STRUCTURES T he logical consequences of t he sent ences liste d so far form the theory of algebraically closed fields, usually denoted ACF . Now let p be a pri me, or p = o. Also the class of (algebraically closed) fields of characteristic p is elementary; for, it suffices to add to the previous sentences the one saying that the sum of p t imes 1 is 0 when p is a prime, or, when p = 0, the negations of all t hese sentences. In conclusion, for every p prime or equal to 0, we can introduce the theory ACFp of algebraically closed fi elds of charact eristic p. Among the algebraically closed fields in characteris tic 0 recall the complex field C , as well as the (countable) field C o of complex algebraic numbers; th eir th eories contain ACF o, and one may wonder if they eq ual ACF o. Recall also that ever y field K has a (minimal) algebraically closed extension K; in particular, for p prime, Z / pZ is an example of algebraically closed field in characteristic p . Since we are t reating fields , let us consider agai n finite fields , a nd, mor e exactly, the infinite models of their t heory we met in exam ple 2: th e so ca lled pseudofinite fields. As observed before, one can ask which is the structure of these fields. J. Ax equipped them with a very elegant axiomatizatio n, explaining the essential nature of finite fields in t he first order setting: in fact , pseudofinite fields are just the fields K suc h that:
* K is perfect, * K has exactly one algebraic extension of every deg ree, * every absolutely irreducible variet y over K has a point in K. All these conditions ca n be written in a fi rst ord er way, although this is not imm ediat e to check. 6. A first order language for the class K of ordered fields is L = {O, 1, +, - , ., ::;}. K is elementary in L because it equals M od(T) where T is the set of the following sentences in L: (i) the field axioms (see Exa mple 5) ; (ii) those characterizing the linear orders (see Exam ple 3);
(iii) the sentence saying that sum s a nd prod ucts of non negative elements are non negative
1.5. ELEMENTA RY CLASSES AND THEORIES
27
Also t he class of real closed orde red fields (t hose satisfying t he Int ermediat e Value Property for polynomials of degr ee ~ 1) is eleme ntary, it suffices to add t he new sentences : (iv) for every natural n ,
+ v I . W + ... + Vn . ui" + w n +1 A u < w -+ u < v A v < w A Vo + vI . V + ... + Vn . v n + v n +1 = 0). A
-+
0 < Vo
T he logical consequences of (i), (ii), (iii) , (iv) form t he theory of real closed ordered fields, usu ally denoted R C F . Examples of real closed ord ered fields a re t he ord ered field of t he real numbers R , as well as t he (coun t abl e) ord ered field R o of real algebra ic numbers. T heir t heories includ e RC F , and one may wond er if act ua lly t hey equa l RC F . 7. Let R be a (countable) rin g wit h identi ty. Conside r t he lan gu age L R = {O, +, - , r (r E Rn of (left ) R -m odules. T he class of left R modules is elementary becau se it equ als t he class of mod els of t he following sentences in L R: (i) t hose ax iomatizing t he a belian groups in t he la nguage wit h 0, a nd - ;
+
(ii) for every r , s E R , if r +s and r ·s deno te t he sum and t he product (respecti vely) of rand s in R,
\lvo( (r
+ s) vo =
rvo + sVo) ,
\lvo((r · s )vo = r( svo)) , \lvO\lvl (r( Vo
+ VI) =
rvo + rvt} ,
(iii) finally, if 1 denotes th e identity element in R , \I Vo( 1Vo = vo).
T he logical consequences of t he previous sentences form t he t heory n T of left R-modules. Of course , t here is no reason to pr efer the left to t he right , at least in t his case ; indeed, one ca n check t hat even t he class of right R -modul es is element ary, and consequent ly one can introduce t he theory T n of right R-modules.
28
CHA P TER 1. STRUCTURES
Let us come back to our classification probl em for element ary, or also noneleme nt ary classes. The following fundam ental t heorem can suggest t hat , even in the elementary case, t his problem is not simple, as the class of models of a theory T ca n includ e ma ny pairwise non-isomorphic structures. Theorem 1.5.3 (Lowenheim-Skolem] Let T be a theory in a (countable) language L . Suppo se that T has some infinite model. T hen, for every infinite cardinal A, T admits some model of power A.
T he proof just uses Co mpactness in the extend ed fram ework of languages of a rbitrary cardinali ties. In fact one enlarges L by A many new constant sy mbols c; (i El , III = A) and one gets in t his way a n extended language L' . In L' one considers the following set of sentences
T'
= TU {--'(Ci = Cj) :
i, j E I, i # j }.
Any finite portion T~ of T' has a mod el; in fact it turns out that , for some finite subset 10 of I ,
so, in order t o obtain a mod el of T~ , it is sufficient to refer to a n infinite model A di T , as ensured by t he hypothesis, and t o interpret the finitely many cons t ants c; (i E 10 ) in pairwise different eleme nts of A. At this point , Com pact ness a pplies and gives a mod el of T ' (hence of T) of power :S: A. Bu t t his mod el has t o include the A many distint interp ret at ions of the c/s, a nd so its power is exactl y A. T herefore, if a theory T of L has at least an infinite model then T has a mod el in each infinite power (and two mod els with different cardinalit ies ca nnot be isomorp hic). Of cou rse, one may wond er how strong is the assu mp tion th at T has some infinit e mod el. Not so much , if one recalls that a th eory T ad mit t ing finite models of a rbitrarily large size must admit also some infinite mod els. Another reasonabl e question may concern how many mod els T adm it s in any fixed infinite cardinal A. One can check t hat t heir number cannot exceed 2\ bu t this upper bound can be reached, for every A, by some suit able T's. T he opposite case, when T has just one mod el in power A (up to isomorphism) , will be of some interest in the next chapters; we fix it in t he following definition. Definition 1.5.4 Let T be a theory with som e infinite model, A be an infinite cardinal. T is said to be A-categorical if and only it any two models of T of power A are isomorphic.
1.5. ELEMENTARY CLASSES AND THEORIES
29
We wish to devote some more lines t o t he Lowenh cim-Skol em th eorem . Among other things , it confirms th at elementary equivalence is a weaker relation than isomorphism. In fact, t ake a n infinite struct ure A , an d use the Lowenh eim-Skolem to build a model A' satisfying the same first order sentences as A but having a differen t cardinality, It is easily checked that A , A' are elementary equivalent; bu t , of course, t hey ca nnot be isomorphic. Now recall what we pointed out in 1.2 : the Induction P rinciple (in its usu al form) cannot be written in the fi rst order style in t he la nguage for (N , 0, s) because first order logic forbids quantification on set vari abl es. However, as far as we know , one might find an equivalent statement that ca n be expressed in t he first order set t ing; in this sense, Induction migh t becom e a first orde r statement. Well, the Lowenheim-Skolem theorem excludes this extreme possibility. For, t he Induction P rincip le cha racterizes (N , 0, s) up to isomorphism , while the Lowenh eim-Skol em theorem ens ures us that any tentat ive first order equivalent translation (even involving infinitely many se ntences) has some uncountable models. So this translation cannot exist . The Lowenh eim-Skol em t heorem emphasizes other similar expressiveness restrictions in first ord er logic. For instan ce, it is well known t hat t he ord ered field of teals is, up to isomorphism, t he only complete ord ered field (here completeness means t hat ever y non-empty upperly, lowerly bounded set of reals has a least upper bound , a greatest lower bound resp ectively). So completeness ca nnot be expressed in a first order way, because any tentative first order tran slation sh ould be true in some real closed field wit h a noncont inuum power. On t he ot her side, we will see that the Lowenh eim-Skolcm t heore m is a very useful and powerful technical tool in first ord er mod el theory (just as the Com pactness Theorem ). And actually the expressiveness restrictions rema rked before are only the other side of the picture of these technical advantages. This is j ust th e content of the Lindst rom theorem quo ted before in 1.2. Indeed , what Lind stro rn shows is t hat, if you have a logic (nam ely a reasona ble syst em offormulas a nd truth) and you demand t hat your logic satisfies the Compact ness Theorem and the weak er form of the Lowenh eim-Skolem Theorem , ca lled Downward Lowenh eim-Skolem Theorem , introduced in 1.4 and requiring -for countable lan gu ages- th at any set of sentences adm itting a mod el does hav e a count able model , then your logic is t he first ord er logic. In t his sense the first order framework is (Leibni zianl y) the best possible one .
30
1.6
CHAPTER 1. STRUCTURES
Complete theories
Let us deal again now with one of the main themes in Model Theory, Le. class ifying struct ures in a given class K . Due to our first order setting, we limit our analysis to elementary classes K = Mod(T), where T is a first order t heory. T his choice is not so partial and na rrow as it may ap pear. In fact, it certainly includes th e cases when T is exp licit ly given and eq uips K with an effect ive list of first order ax ioms , as in the positive examples of the last section; but it is also conce rned wit h oth er , and worse sit uations. For instance, think of the t heory T of finite sets, or groups, or fields, or, in general, of a class of finite arbitrarily la rge structures, so t hat T has also infinite models. Alternatively, think of the th eory T of a single infinite structure A: due to the Lowenheim-Skol em Th eorem, T has some models non-isomor phic to A. In t hese cases, T is introduced by sp ecifying some crucial models, but this does not determine in an exp licit way a priori which first order sentences belong to T , and which are excluded; inde ed we could just be interested in finding an effective ax iomatization as in the previous examples, and we could aim both at descri bing T and also -as a relat ed matter- at classifying its mod els. These are th e settings we wish to consider. Actually we should also admit that we have not clearly explained up to now which kind of classification we pu rsue; however we have ag reed t hat this classification should identify isomor phic models but disting uish non isomorphic structures. Also, we have seen that isomorphic mod els sat isfy t he same order orde r sentences. So a pr eliminary classification is j ust up to elementary equivalence, an d aims at distin guishing non elementarily equivalent structures. Once this is don e, we could restrict our analysis to structures satisfying t he same first ord er conditions; Le. fix a structure A and classify up to isomorphism t he models of its theory T = Th({ A}) (by t he way, let us a bbre viate for sim plicity 1'h( { A} ) by Th( A)). Which is an intrinsic syntactical characterization of such a theory 1'? Basica lly it is " complete" according to the following definit ion. Definition 1.6.1 A (consistent) theory T of L is said to be complete if, for every sentence
~o. By Vaught 's Theor em , KT ' is complete . • On t he cont rary, KT' may not be categorical in ~o . In fact , when JC is infinite, JC , JC2, .. . , JC{ No) are count abl e JC-vectorsp aces wit h distin ct dimensions, and so ca nnot be isomorphic, hen ce t hey are not even par tially isomorphic. So eleme ntary equivalence ca nn ot impl y pa rti al iso morphis m (a nd isomorphism ). T he read er may check directly what happens when JC is finite . We conclude t his section by introducing a nother noti on relat ed to completeness. It will be used in Chapter 3 t o show th at RC F is complete. Recall t hat a complete th eor y T equa ls T h(A) for every model A , and hence a t heory T is complet e if a nd onl y if a ny t wo mod els of T a re eleme nt arily eq uivalent . Definition 1.6 .9 A theory T is model complet e if every em bedding of models of T is elem entary.
It is eas y to exhibit t heories which ar e not mod el com plete . For instance, t he pr eviou s examples 1.3.1 4 ensure t hat t he t heory of (N , 1, and - (- m) if m < -1 ; simila rly for n). T his implies that the complet e extensions of ACF are fully determined by t he characterist ic of t heir models , a nd hence coincide wit h t he theories AC Fp where p is 0, or a pri me.
46
CHAPTER 2. QUANTIFIER ELIMINATION 4. Finally, let us deal with model completeness. Assume that T has quantifier elimination in L. We claim that, in this case, every embedding between models of T is elementary, in other words T is model complete. In fact, let A and B be models of T, f be an embedding of A into B. Given a formula 0) V
has s roots" 1\
" t he r -t h root Pr(v) satisfies !\ j5,m qj(Pr(v) , v) > 0") ,
where t he latter disjunct ca n be expressed by a quantifier free first ord er formula. So look at (b) . For every j ::; m a nd for every Sj ::; tj , t here a re qu antifier free formulas defining , for every real closed field 1< , the set of th e sequences bsuch that q(x , b) has Sj roots in x , and listing t hese roots
One can cornpu te the sign of qj (x, b) in t he intervals
(-00 , Pj, 1 (b)) , (pj,i(b), pj,i+db))
(1::; i < Sj),
(Pj, Sj (b) , +(0) in a uniform way (ind ependent of 1< and b) by looking at t he (sign) valu e of
qj(pj ,l(b) -1 , b) .(Pj,i(b) + (pj,i+l(b) b)
qJ
2
'
qj(p j,sj(b) + 1, b) resp ectively. List all the possible ord erin gs of t he roots (in x) of t he qj (x, b) 's when j ranges over the natural numbers ::; m, and divide in every case 1< int o finit ely many inter vals such that the qj (x , b) 's have a constant sign (with resp ect t o x) in eac h of t hem; check these signs (in the way suggested before) a nd for m a suitable disjunct ion picking the intervals where all t hese signs a re positive. This procedu re is ind epend ent of 1< a nd b and provides the required quantifier free formul a . • Corollary 2.5.2 RCF is model complete. Corollary 2.5.3 RC F is complete; in particular, RC F is the theory of the ordered field of reals (as well as of ever y real closed field).
66
CHAPTER 2. Q UANTIFIER ELIM INAT I ON
Proof. There is a min imal ordered real closed field , embedded in any mod el of RCF. This is the ordered field R a of real algebraic numbers. The model completeness of RCF ensures that every real closed field is an elementary extension of R a. In particular all the real closed fields are element a rily eq uivalent to R a and , con sequ ently, t o eac h other. '"
This is t he first completenes s proof we give a bout RCF; in fact Vau gh t 's criterion does not apply because RC F is not cate gorical in any infini t e power . We have see n th at real closed fields eliminate quantifiers in t heir lan gu age L = {+ , -, ' ,0, 1, ~}. Notably, they are fully charact erized by t his property : for, Macintyre, McKenna and Van den Dries showed t hat an ordered field, whose theory has the quantifier eliminat ion in L , must be real closed . We also noti ce that RCF doe s not pr eserve quantifier elimination in t he restricted lan guage L' = {+ , -, " 0, I} wit hout ord er . Actually on e can rem emb er t hat , even in checking solva bility of the popular equation ax 2 + bx + c = 0 wit h degree 2 and 1 unknown ove r t he reals (or over any real closed field), on e needs a disequ ation b2 - 4ac ~ 0 to ens ure roots, and henc e to eliminate 3 in the formul a 3w (V2 W2 +VIw +va = 0) . More form ally, recall that , with resp ect to the theory of the real field , th e formulas
',o( v):
v~ O,
',o' (v ) : 3w(v
= w 2)
a re equivalent. As RC F is com plete a nd hen ce eq ua ls the t heory of t he real field , t he same hold s in every real closed field . Consequently t he £I-formula (wit h t he quantifier 3) defines th e set of non-n egative eleme nt s in ever y real closed field. However ',o(v) cannot be equivalent in RCF to any quantifier free £I-formula ',o" (v ). In fact ',O (R ) is the half-line [0, + 00) of R , and so is both infinite a nd coinfini te , whil e ',O" (K ) is either finite or cofinite for ever y field K : see the proof of Corolla ry 2.4. 8. Now we discuss decid abili ty: as already said , t his was t he main conseq uence of elimina tion of qu an tifiers, according to t he ge ne ral feeling in t he fourties . Corolla r y 2.5 .4 RCF is decidabl e. Proof.
Ow ing to q uantifier elim ination, every L-sentence
(J
is equivalent in
RCF to a Boo lean combination of se nte nces m = n or m < n where m and
2.5. TARSKI AGAIN: REAL CLOSED FIELDS
67
n are integers. This quantifier free statement can be easily checked in our
framework.
..
We shall comment this result later in 2.9. Now we examine another remarkable consequence of quantifier elimination, namely definability. Recall that, in an ordered field K, every semialgebraic set (in other words, every finite Boolean combination of sets of solutions of disequations
q(x) 2:: 0 with q(x) E K[X]) is definable. Corollary 2.5.5 In a real closed ordered field K, the definable sets are exactly the semialgebraic ones.
Proof. Let n be a positive integer, X S;; K" be a set definable in K . So there are a formula 0) over K. Now take a transcendence basis tl, .. . , t s of 1-l over K, and split the embedding K ~ K(tl , .. . , t s ) by
where K, = K(t 1 , ... , ti) for every i = 1, ... , s. There is some i = 1, ... , s such that :3wcv(il, w) is true in K, and false in Ki-l. We can replace K ~ H. by Ki-l ~ x; and to assume
1-l
= K(t)
for a single transcendental element t over K. So :3wcv(il, w) is true in K(t) and false in K. Now take any algebraically closed extension K' of K having transcendence degree 2 1. Hence K' enlarges K (u) for some transcendental element u over K. Steinitz's analysis of algebraically closed fields tells us that K(t) and K (u) are isomorphic via a function enlarging the identity of K and mapping t into u. Then K (u) and, consequently, K' satisfy :3wcv( ii, w): our sentence is true in every algebraically closed extension K' of K with transcendence degree 2 1. Equivalently, in the language extending L(K) by a new constant c, :3wcv(il, tu) is a consequence of Th(KK) plus the infinitely many sentences ensuring that c is transcendental over K, i. e. does not solve any nonzero polynomial with 1 unknown and coefficients from K. Use compactness and deduce that finitely many sentences suffice (in addition to Th(KK)) to imply :3tucv( ii, w); in particular, c can be interpreted by a suitable element of K (out of the roots of a finite system of polynomials in K[x]). In conclusion, K satisfies :3wcv(il, w): a contradiction. Hence ACF is model complete. ..
3.3. MODEL COMPLETENESS AND ALGEBRA
93
Which are t he main algebraic ingredients of thi s Robinson proof? Basically, Steinitz 's analysis of algebraically closed fields . More specifically, two key points should be underlined: 1. every field has an algebraic closure , and this is unique up to isomorphism enlarging t he identity in the ground field ;
2. if K is an algebraically closed field and t is transcendental over J( , th en t he isomorphism class of K(t) over K is uniquely de t ermined (in t he sense that two extensions of this kind are isomorphic via a map extending the ident it y of J(). One can rea lize t hat real closed fields satisfy simila r propert ies: 1. every ord ered field has a real clos ure (a min imal real closed extension), and this is algeb ra ic over t he ground field , and un ique up to isomorphism enla rging th e identity in the gro und field ;
2. if K is a real closed ord ered field and t is t ra nscende ntal over J( , th en the isomorphism class of K(t) over K is fully charact erized by t he cut t det ermines over J(. So, when dealing with model completeness for R C F, one can reproduce the proof of the algebraically closed case (with some complicat ions du e to th e order) and deduce Theor em 3 .3.2 (A . Robin son) R CF is model complete . Robinson 's Test ca n also be used t o prove th e mod el completeness of sever al theories we met in t he previous chapter : discre te linea r orders , den se linear orders and so on . The reader may check this, as an exer cise. But here we prefer to discuss some very noteworthy applications of t he model complet eness of ACF and R C F to Algebra. T hey provide new elegant proofs of known algeb raic facts . First let us deal with algebraica lly closed fields and Hilbert 's Nu llstellensatz . Theorem 3.3.3 (Hilbert Nullstellensatz) Let K be an algebrai cally closed field, J be an ideal of the rin g K[X] (whe re x abridges, as usual, th e sequen ce of unknowns (Xl , ... , x n ) ) . Th en , fo r eve ry polynomial f( x) E J([X] , f(ii) = 0 f or every d E K " su ch that g(ii) = 0 fo r all g( x) E I if and on ly if
94
CHA P T ER 3. MODEL COMPLET ENESS
for some positive integer m f m(x ) El . Proof. The direction from right t o left is clea r. Converse ly, su ppose t owards a cont ra dict ion t hat t here exists some polynomial f( x ) in K[x] such that
f (ii) = 0 for every ii E K " s uch t hat g(ii) = 0 for all g(x) E I but
f m(x) rJ. I for every positi ve int eger m . Let J be an ideal of K[X] s uch th at J ;2 I , no power 1m (x) of 1(x) (with m a positive int eger ) lies in J, and J is maximal with resp ect t o th ese conditions (Zorn 's Lem ma ensures th at such a J exists). We claim that J is prime . In fact, take two polynomials go (x) , gl (x) in K[X] - J; t hen t he ideals
Jo generated by J and go(x) , J 1 generat ed by J and gl (x)
st rict ly includ e J ; accordingly t here are two positi ve int egers mo , m 1 such t hat f mo (x) E Jo, I": (x) E J 1 . So t here exist two polynomi als qo (x ), q1(x) E K[X] such t hat
F" (x) - s.(x) qi(x) Co nsequent ly
E
Ji' Vi
= 0 ,1.
r-r: (x)
is in t he ideal gener ated by J an d go(x ) . gl (x ), a nd so go(x) . gl (x ) rJ. J . Hence J is prime, and R = K[X] / J is an int egral domain exte nding K by th e fun ction mapping an y a E K t o a + J. Th en K em beds into the algebraic closure F of the field of quotients of R . As AC F is model complet e, this embedding is elementary. Now take any (finit e) set of generators 10(x), ... , 18(X) of I (I is finit ely gene rated becau se K[X] is Noetherian), an d noti ce that the L(K)-sentence
::Iv (/\ fi(V) = 0 1\ -, (J (v) = 0)) i~8
is t rue in F (owing t o t he seque nce (X l + J, ... , X n + J )). Consequent ly t his sentence is t rue also in K . So t here exists some ii in K " such t hat ii satisfies 10(x ), ... , f8(X) and consequent ly all the polyno mials in I , bu t ii do es not a nnihilate f (x ) -a contradictio n-. ..
3.3. MODEL COMPLE TENESS AND ALGEBRA
95
Now we deal with RCF, and with Hilbert 's Seventeenth Problem . This was solved by Artin in 1927. Indeed Artin himself and Schr eier developed t he algebraic notion of real closed field just to answer Hilbert 's qu estion. Later A. Robinson proposed a very nice a nd simple proof, founded on th e model com plete ness of RCF. Here we want t o report A. Robinson's a pproac h. First let us introduce Hilbert's problem in detail. Ind eed the seventeenth question in the celebrated Hilbert 1900 list ju st concerns ordered fields (more properly, the ordered field of reals ) . Recall t hat, in a ny ordered field K , a rational function f (x) E K (x) is said t o be sem idefinite positive if a nd only if, for every sequence ii in K (such that f(ii) is defined) , f(ii) ~ O. Of course, t he s ums of sq uares in K(x) a re semidefinite positi ve. I-Tilbert 's Seventee nt h Problem conjectures t hat the converse is also true when K is t he ordered field of real numbers. As already said, Art in solved positively t his question ; indeed he extended th e result to a rbit ra ry real closed ordered fields K . Now we pro vide A. Robinson 's proof of t his t heorem. Theorem 3 .3.4 (Ar tin) Let K be a real closed ordered fiel d, f (x ) a se midef inite positive rational fun ct ion in K (x). T hen f (x) can be expressed as a sum of squa res in K(x). Proof.
We need t he following algebra ic fact .
Fact 3 .3.5 Let K be a fiel d, an d assume that, fo r eve ry natural t an d for every choice of aa, .. . , at E K , if L i ~o be an inaccessible cardinal. saturated model of power A.
Then T has a
The existence of an uncountable inaccessible cardinal is a quite delicate matter. But assume momentarily that such a saturated model n exists (for a given inaccessible A > ~o). Recall that n is unique up to isomorphism. Furthermore
• n is A-universal:
every model of T of power < A can be embedded as an elementary substructure in n;
• n is
A-homogeneous: if X is a subset of n of power < A and a, a' are two tuples in n having the same type over X, then there is an automorphism of n fixing X pointwise and mapping a into a'.
It is a general agreement (and habit) in Model Theory to assume that such a model n exists. This makes things easier and, on the other hand, is sufficiently plausible; in particular, one can check that everything is shown inside n, so assuming that n exists, can be proved (at the cost of some major complications) even avoiding any reference to n. Which are the benefits of working in As we said, the cardinality of is very large, and so one can reasonably suppose that all the models we expect to handle have a smaller power. But this implies that they actually are elementary substructures of n of a smaller size (up to isomorphism). Under this perspective, the subsets of models of T can be directly viewed as subsets of n with power < 1nl: we will call these subsets small subsets of n, just to tell them from the other subsets of n admitting its same inaccessible cardinality. As already said, referring to small subsets of n instead of su bsets of arbitrary models makes our life, and also our notation simpler. For instance, take a
n?
n
5.4. SAT URA T ED MODELS
149
small X and a posit ive int eger n . W hen definin g Bn(X , A ) an d Sn(X , A ), we had t o explicitly refer to a model A of T (now an elementary substructu re of Q) where X is embedded . We also em phasized that Bn(X , A ) an d Sn(X, A ) do not de pen d directly on A , but on ly on the t heory of Ax : t he cho ice of A is qui t e arbitrary wit hin these bounds. But t hen it is convenient to refer to A = Q , as a uni versal mo del where X is embedded . T his is what we will do fro m now on. Ind eed we will write, when t here is no danger of misunde rstan di ng, Sn(X) to mean Sn(X, Q) for every small Xj so S(X) will denote t he union of t he spaces Sn(X) when n ra nges ove r posit ive int egers. At last , let us propose some algebraic examples concerning more or less sat urated st ruct ures.
Examp les 5 .4.10 1. Which algebraically closed fields K are sat urated in a given card inality A 2:: ~o ? To an swer , we may recall t hat, for any p = 0 or prime, t he t heory ACFp is categorical in every uncoun t abl e powe r; so, according to what we said before, eve ry uncou ntable algebraically closed field (in any characteristic) is sat urated. To confir m t his from t he algebraic point of view and to discuss t he existence of saturated mod els in t he co untable case, we ca n refer to Steinitz 's analysis of algebraically closed fields K and recall that such a K is the algeb raic closure of Ko(S) , where Ko is th e prime subfield and S is a t ranscendence basis of K , so a maximal algebraica lly independent subset. The isomorph ism ty pe of K is fully determ ined by it s characteristic and it s transcendence degr ee, i. e. t he power of S. Moreover, when S is infini t e, t his transcendence deg ree eq ua ls 1/(1. Now let 1/(12:: A and take a subset H of K of power < A. As we saw in t he last section , H ca n be replaced by t he subfield generated by H j t his does not change its size, except when H is finit e; however, even in t his case, eac h point in t his subfield is /i-definable. Every alge braic l -fype over H is clearly realized in K becau se K is algebraically closed . So t he point is: ca n we realize the remaining l-nype, the on e of t he eleme nts which are transcende nt al over H , for a ny H? If K has an infini t e t ransce nde nce degree, t hen t his degree is ju st 1/(12:: A > IHI, so it is st rictly larger t han t he t ransce nde nce degree of t he field ge nerated by If. T his mea ns t hat we can realize our type insid e K for every H . Otherwise, let H be generated by a finite t ra nscendence basis S of K , Clearly IHI < A, but now t here is no way to realize our ty pe inside K . In conclusion , for
1/(12:: A, K
is A-sat urat ed if and on ly if it s transcen-
150
CHAPTER 5. MORLEY RANK
dence degree is infinit e. In particula r every algebra ically closed field satisfying t his condit ion is satur ated (in it s own power ). So every uncountable algebra ically closed field is saturated (t his a pplies also t o t he complex field), while t he only countable saturated algebra ically closed field (in a fixed characterist ic) is t he algebraic closure of /Co(S ) where /Co is t he pri me subfield and S is a countable (infinite) algebraically ind epend ent set. 2. The real ord ered field is not ~o-saturated . This is perhaps remarkabl e, if one remembers t hat t he reals are a complete ord ered field (in t he sense t hat every Cau chy sequence has a limit) , and even the only complete ordered field up to isomorphism: in oth er words, given a Cauchy sequence (rn)n>o in R , the l -type defined by 1
Iv - rnl < - , n
when n ra nges over positi ve integers, has a (unique) realization in t he real field. However , look at t he ty pe of a positiv e infinitesim al non zero element . This is defined by t he cut 1
0 RM(B), which contradicts what we have observed before. Consequently there are only finitely many left cosets of S in G of the form with a E X (recall = bS {:} b- 1a E S {:} bp = ap for every a and b), as claimed. So S is a subgroup containing each Xi. Now we show S ~ BB-I. Let a E S, x FP. SO even ax realizes p, and both x and ax satisfy the formula "v E B " . Consequently a = axx- 1 satisfies
as
as
CHAPTER 6. w -STABILITY
192
"v E BB-I " in n and hence in g. In par t icular S is gener at ed by t he X i's (act ually, by a finit e subfa mily of them ). At t his point it rem ain s t o show t hat S is connected in g. So take a subgroup H of S definable in 9 and having a finit e ind ex in S . Accord ing ly, for eve ry i E I , X i overl aps onl y finit ely many left cosets of H , whence X i ~ H. Bu t t his for ces 1I = S. ..
6.3
w-st a b le fields
T he aim of this section is t wofold . F irst we want to a pply wh at we hav e seen a bout w-stable th eori es a nd , more particularly, w-stable groups to prove a beautifu l t heorem du e to Macintyre a nd characteri zing the fields having an w-stable theory. We already know t hat they include the algebraically closed fields ; bu t now we will show t hat no further exam ples arise, so t he w-st abl e co m plete theories of (pure) fields ar e just the t heories ACFp where p is 0 o r a prime. Secondly, we will provide a new proof of t he fact t hat algebr aically closed fields eliminate the imaginaries; t his alternative approach mainly refers t o t heir w-stability and , owing t o t his feature, a pplies t o other w-stable settings, including differen ti ally closed fields of cha racteristic 0 (as we will see lat er in t his chapter) . Now let us state Macintyre's T heo rem . Theorem 6.3.1 (Maci ntyre) Let I( be an infinite in tegra l do main with ident ity 1, and let I( ha ve an w -s table theory. T hen I( is an algebraicall y clo sed fi eld. Proof. First let us see t hat I( is a field. Take a E J( , a i= O. Let C denote prop er inclu sion. If «« C J( , t hen a n+1 J( C an J( for every positive integer n because I( is a dom ain and so, for b E J( - o.K, anb E anJ( - a n+ l J(. So I( is a n w-stable group (with resp ect to the sum op er ation) with a st rict ly decreasing infinite seq uence of definabl e subgroups J( J aJ( J a 2 J( J ... , whi ch cont radicts Theorem 6.2.2. Accord ingly o.K = K , whence there is some c E J( s uch t hat ac = 1. So I( is a field. Now suppose towards
a contrad iction th at I( is not alge braically closed . Hen ce I( has a G alois extension F of finite degr ee > 1. Conseq uently t here is some intermedi at e field E extending I( an d included in F such t hat t he Galois group of F over E is (cyclic) of pri me or de r q. O n t he other side E is an extension of finit e degree of 1( , a nd hence is definable in 1(: in fact , let d de note t he dimension of L as a vectors pace ove r 1( , t hen E ca n be viewed as I( d eq uipped with
193
6.3. w-STABLE FIELDS
s uitably defined field op erations. Hence E has an w-stable t heory, a nd an extension F with a (cyclic) Galois group of pri me order q. Wi th no loss of generality replace K by £ and hence ass ume that K it self has a Galoi s extension F of prime degree q. F ield t heory tells us t hat, in t his setting, F = K (a ) where t he minim um polynomial a over K is eit her xq -
a
wit h
a
E K , q =1=
ca r
K
or xq
-
x - a with a E K , q = car K .
To get a contradiction, we show that every poly nomial of this form must be redu cible . For this purpose, first recall th at K is an w-stabl e group with respect to addition +. Moreover J( coincid es with its connected component (with resp ect to +). In fact , take a E J( and look at t he multiplication by a. T his gives a n a ut omor phism of t he addit ive group of K , and indeed a n a ut omorphism definabl e in K. Consequent ly t he multiplication by a fi xes t he connected component So a J(° = J(o for every a E J( , in oth er words aJ(o is a n ideal of K . Hence J( o = J( as J( is infinite and so ca nnot eq ual {a } . Now conside r J(* = J( - { a}; J(* is an w-stable gro up wit h res pect to multipli cation . As J( is infinite, J(* has t he same Mo rley ra nk a nd degr ee as K , in particular J(* has degree 1, a nd consequent ly J(* eq ua ls its connect ed com ponent . Notice t hat t he fu nction of J(* int o J(* mapping any element k E J(* into k q is a definabl e group homomorphism having a finite kern el {a E J( : aq = I} . Owin g to Corollary 6.2.4, t he image of th is fun ction has a finite index in 1.. Accordingly DLO- is not classifiable. Mo re generally Theorem 7.1.6 applies, as we will see in the next section, to any complete theory of linearly ordered infinite structures. None of these theories is classifiable. In particular, no o-minimal theory is classifiable, although any such theory satisfies the conditions (Dl)-(D4) of the dependence relation a E acl(A) a -< A for a E n, A a small subset of n. This is a little surprising and will be discussed in more detail later in this chapter, and then in Chapter 9. Anyhow, if we (momentarily) agree with Shelah's definition of classifiable theory, then we have to take note that too many models (in other words 2'\ non-isomorphic models in every uncountable >.) exclude classifiability. On the other side, we would like also to determine the key criteria ensuring the classifiability of an arbitrary complete theory: this will be the matter of the forthcoming sections. To conclude the present one, let us spend some more words about the close relationship between classifying theories T (in the Shelah sense) and counting the models of T. This is already explicit in Theorem 7.1.6. More generally, for every countable complete T, one can define a function I(T, . ..) associating to every infinite cardinal >.
I(T, >.) = number of the isomorphim types of models of T of power >..
x f---7
I(T, >') is called the spectrum function ofT. Recall that 1 ::; I(T, >.) ::;
2'\ for every X, owing to the Lowenheim-Skolem Theorem and cardinal computations. The content of Theorem 7.1.6 is just that I(T, >.) = 2'\ for every uncountable >. excludes the classifiability of T. Indeed there was a conjecture of Morley, preceding Shelah's work and, in some sense, originating it, saying: Conjeet ure 7.1. 7 (Morley) Let T be a countable complete first order theory. Then the spectrum function>. f---7 I(T, >.) is increasing among uncountable cardinals: for ~o < < f-t, I(T, >.) < I(T, f-t).
x
Shelah positively answered this conjecture, as a non-minor consequence of his classification analysis. The problem of determining all the possible spectrum functions>' f---7 I(T, >.) when T ranges over countable complete theories (and>. is uncountable) was solved only in 2000 by Hart , Hrushovki
7.2. SIMPLE THEORIES
227
a nd Laskowski, who explicitly listed all these functions . Notably their proof involves some arguments from Descriptive Set Theory. Finally, what can we say whe n A = ~o? As we observed in th e last chapter when talking about ~o-categoricity, this case is a little oblique with respect to the general analysis and requi res peculiar approaches a nd techniques. A classical conject ure in this framework was raised by Vaught . Conjecture 7.1. 8 (Vaught) For a countable complete first order theory T , either I(T, ~o) ::; ~o or I(T, ~o) = 2No (apart from the Continuum Hypoth esis, of course) .
As in the case of Morl ey's Conjecture, this que stion is not only a mer e ca rdinal investigation; what is more relevant is to underst and the structure of the countable models of T. Shelah (together with Harrington and Makkai) positively answered Vaught's Conjecture for w-stabl e theories T . Other partial posit ive answers are known. Indeed in t he latest months the news of a counterexample (a theory with exactly ~l count a ble mod els) due to R. Knig ht [75] has been spreading, but this negative solut ion st ill seems (october 2002) under examination.
7. 2
Simple theories
All throughout this section T is a complete first order th eory in a coun table language L, T has no finite models and n denotes t he universe of T. We aim at determining which key properties ma ke T classifiable. A classification is very easy in the strongly minimal case . In fact, whe n T is strongly mini mal , every mod el of T is lab elled by a cardinal number - its dimens ion - classifying it up to isomorphism ; what ru les this dimension and its assignme nt is a notion of dep endence, based on the mod el theoretic algebraic closure ad . Unfortu nately the ad dependence does not work any more when we enlarge our setting and we leave the strongly minimal fram ework. So we need a more general notion of (in)depend ence , still including the classical cases of linear independence in vectorspaces, algebraic independence in algebraically closed fields a nd , definitively, ad independence in strongly minimal theories, but applying to a wider context. In other words we ai m at defining for any T ii is ind ependent fro m B over A
where ii is a tuple in n a nd A ~ B are small subsets of n. As not ions require a bstract symbols to be presented let us denote by I (I for independence) the set of all the triples (ii, B , A) in n such t hat
228
CHAPTER 7. CL A SSIF YING
a is independ ent from
B over A
in a sense t o be made more preci se. It is reason able t o ex pect t hat 1 satisfies the following proper ties:
(11) (invariance) for ever y (a, B, A) E 1 and a ut omorphism f of (1 (a), f(B) , f(A)) is still in 1; (12) (local character) for eve ry t hat
(a, E , A) E 1;
a a nd
B , th ere is a count able A
~
n,
B such
(13) (fini te character) for every a, A and B , (a, B , A ) E 1 if a nd onl y if, for all finit e tuples bin B , (a,AUb,A) E I;
a, A and B , t here is a t uple a' having t he same length and t he sa me ty pe over A as a suc h t hat (a', B , A) E 1;
(14) (ex tension) for every
(15) (symm etry) for every a,
(b, A U a, A)
b a nd
(a, A
A,
U b, A ) E 1 if a nd onl y if
E 1;
(16) (tran sitivity) for ever y a a nd A (a, B , A) E 1 and (a, C , B ) E 1.
~
B
~
C,
(a, C, A ) E 1
if and onl y if
Definition 7.2.1 A set 1 of triples (a, B , A) with a in n and A ~ B small subset ofn is called an independence system ofT if and only if I satisfies
(ll) - (16). A n eas y a pplicat ion of (13) and (15) shows t hat, if B , B' :::> A a re s mall subsets of n, t hen
(b, B' , A)
E1
VbE B
if and only if
(b , B , A) E 1 V;} E B'. ' We will say t hat B and B' a re ind epend ent over A when t his happens. Notice also t hat, if a' is a subseq uence of a and (a, B , A) E 1, t hen (a' , B , A) E 1 as well. By (13), it suffices t o check (a' , A Ub, A) E 1 for a ll bin B. We know (a, A U i, A) E 1. By sy mmet ry (15), (b, A U a, A ) E 1 , whence (b, AUa', A) E I by (13), and (a' , AUb, A) E 1 by (15) once again. Let us propose some exa mples of ind ep end ence systems, both t o illustrate t he meaning of (11) - (16) and to confirm t hat t hey pro vide a reason abl e axiomatic ground t o introdu ce a n a bstract notion of indep end ence.
229
7.2. SIMPLE THEORIES
Examples 7.2.2 1. First let us check that the old independence notion in strongly minimal theories T corresponds naturally to an indendence system in the new sense. In fact, let T be any theory. For a E nand A 1, t hen (iii) implies RM('l9(iJ) A'l9 j(iJ, xj )) = Q' for every j < d, j =1= i, a nd hence GM('l9 (nn) n 'l9 i (n n, xi )) < d. But t his cont radicts 'l9(iJ) A 'l9i(iJ, xi) E p. So d = 1, as claimed . ... Now we t urn ou r attent ion t o simplicity. Our aim is t o pro ve t hat w-stable th eorie s are simple. More specifically, we will show that , for an w-st ab le theor y T , th e t riples (ii, E , A) such that ii is a t uple in n , B 2 A a re small s ubsets of nand R M(tp(ii/A )) = R M (tp(ii/B )) form a good ind ependence system of T . This confirms t hat T is simple, and pro ves also t hat, for ii, A an d B as before, ii is ind ep endent of B over A ii -!-A B if a nd only if tp (ii]A) has the same Morley rank as its extension tp(ii/ B). Incident ally recall t hat, for an y ii, A and B , tp(ii/B) always includes tp(ii/A ), a nd consequent ly
RM(tp(ii/B))
< RM (tp(ii/A)) .
Theorem 7 .5.7 Let T be an cc-stable, and let I be the set of the triples (ii, B , A) where ii is a tuple in n, A ~ B are small subsets of n and R M (tp (ii/B )) = R M (tp(ii/A )). Then I is a good independence system of
T. Proof. We have to check t ha t I sat isfies th e condit ions (11)-(17) listed in Section 7.2. (11) is trivia lly t rue, as automorphisms preserve both Mo rley rank and Morley degree. (12) Fix ii, B , and pick a formula IAI. In fact, let r', r'' denote t he restricti on map s from N (NI, A) onto 8(A) and from N(M, A) in 8 (B) resp ecti vely ; it is an easy exe rcise t o check that the lat t er fun cti on maps N(M, A) onto N( B, A) and t hat r' is just t he com posit ion of r" a nd r . Conseq uent ly, if r' is open, then r is, as r" is cont inuous . Henc e we ca n replace B by M , as claimed; accordingly, let r denote from now on t he restriction map from N (M, A ) onto 8(A). Let U be an op en set of N(M , A) . We can assume t hat U is t he set of the types over M which do not fork over A and include a given formula , then Pt is orthogonal to all the non-algebraic
types over M s; (vi) )\1{ is prime over UsEc
u. .
Of course we are int erest ed in those t heories T such that every model gets such a presentation .
Definition 7.6.4 An to-sta ble theory T is called presentable when, fo r all models )\I{ o, M 1 , M 2 and )\I{ oj T such that • M o is an elementa ry substruct ure of both M 1 an d M
2,
• M 1 tMo M 2 (in other words ai t Mo M 2 f or every ai E M 1 , or, equzvalently, a2 tMo M 1 f or every a2 in M 2),
• M is prime over M 1 U M 2 , for eve ry non-algebraic type pES (M) there ex ists some type over M over M 2 that is not orthogonal to p .
1
or
Presentability is a new dichotomy within t he classification problem. In fact , on the one side, one shows :
7.6. CLASSIFIABLE THEORIES
265
Theorem 7.6.5 (Shelah) If T is an eo-stable theory and T is not presentable, then I(T, A) = 2-\ for every uncountable cardinal A. In particular, T is not classificable. On the other hand Theorem 7.6.6 (Shelah) If T is ea-stable and presentable, then every uncountable model M of T has a presentation. A model M may admit several presentations. Actually there is a "quasi uniqueness" theorem stating under which conditions two "different" presentations yield isomorphic models; but it is impossible to discuss it here shortly, so we omit its treatment, and we conclude our report about presentability and presentations by proposing some examples and, in particular, an w-stable theory which is not presentable. Examples 7.6.7 1. Let T be the theory of two equivalence relations El, E 2 such that any El-class and any E 2-class share infinitely many common elements. One checks that T is complete. Here are the nonalgebraic I-types over a model M of T: • for a E M, the formulas Et{v, a), E 2(v , a) and ,(v = b) for all b « M determine a type PM(a); • for a E M and i = 1,2, the formulas Ei(V , a) and ,E3-i(V, b) for all s « M give a new type PM(a, i); • finally there is a type qM determined by the formulas ,El (v, b) and ,E2 (v , b) for all s « M. In particular, if A1 is countable, then Sl (M) is. Hence T is w-stable. Moreover it is straightforward to check that all the types listed above are strongly regular and, for all a E M, PM(a, 1) ..1 PM(a, 2). Now fix a model Mo of T and a E 1110 • For i = 1, 2, let Xi realize PMo (a, i) and Mi denote MO(Xi): so Mi is built by taking Mo and adding a new E3_i-class (the class of Xi) having countably many common elements with any Ei-class in M o- One can see that both M 1 and M 2 are elementary extensions of Mo ; furthermore PM(a , 1) ..1 PM(a, 2) implies Xl -!-Mo X2, whence M 1 -!-Mo M 2 (see the proof of Theorem 7.5.19), Now form the model M of T prime over M 1 U M 2 ; M is obtained just by adding countably many elements to M 1 U M 2 in the intersection between the E 2-class of Xl and the El-class of X2. Let X be an element in this intersection , and consider p = PM(X) : P is
CHAPTER 7. CLASSIFYING
266
a non-algebraic type over M , and one can check t hat P is or thogon al (equivalent ly, is not "'RK-equivalent) to any ty pe over M 1 or M 2 • So T is not present abl e. Let us also check t hat, for every uncount abl e cardin al >. , J(T, >') j ust eq uals 2'\. In fac t take t he disjoint union I of tw o sets h and J2 of power >.; let R be an irreflexive sy mmet ric bin ary relation in I such t hat, if 81, 82 E I and ( 81 , 8 2 ) E R , th en 8 1 E h and 8 2 E h or conve rsely (so (1, R ) is a bipartite graph). Now build a model MR of T where the E i-classes correspond to th e elem ents of I; for ever y i = 1, 2 and , for every El -class Xl and E 2 -class X 2 ,
a nd
IX 1 n X 2 ! = ~o
otherwi se.
It is clear th at MR has power>. for ever y R. Furt hermore, for R =1= R' , A1 R i:- A1R' . So I (T , >') ~ 2.\ and conseq uent ly I (T , >. ) = 2.\: in fact , t here exist 2.\ man y relations R as befor e. 2. On t he contrary t he t heory T in Example 7.6.1 is present able. In fact take t hree models M o, A11> M 2 of T such t hat JV1 0 is a n elemen t ary substructu re of both M 1 a nd JV1 2 and M; -!-Mo M 2 . We observed t hat M 1 U M 2 is t he dom ain of a mod el M of T extending M 1 and M 2 ; of course, M is prime over M 1 U M 2 • Let p be a non-algebraic ty pe over j1;[ (and keep th e same notat ion as in 7.6.1 ). If P = PM( n , a) for some a E M and so me positiv e integer n , t hen P "' RK PMi (n , a) where i = 1, 2 sat isfies a E Ms, If P = qM , t he n P is '"RK-eq uivalent to both qM l and qM 2' So T is pr esentable, as claim ed . Nevertheless T is not classifiable, as it has 2.\ many pairwise non isomorphic models in every uncountable power >.. Let us see why. Ind eed , for every X, we ca n build 2.\ non isomorphic mod els sat isfying the furt her assum pt ion for every a E M a nd for some natural n f n(a) = O. Notice t hat such a mod el M ca n be viewed as a t ree of >. <w with resp ect t o t he relati on ~ defined as follows: \la , b E M , a ~ b if a nd only if r(b) = a for some natural n ; in parti cular a = b: if and only if f( b) = a , and < >= O. The isomorphism class of M clearl y det erm ines t he iso mo rphism ty pe of (M, ~) as a t ree . Moreover every
7.6. CLASSIFIABLE THEORIES
267
point in (M,~) has infinit ely many successors, a nd IMI = A. So it suffices to show t hat t here exist 2-\ pairwise non isomorphic trees of A<w sat isfying t he las t addit iona l proper tie s. Associate a t ree C (v) of A<w with any ordinal v < A as follows. (a ) F irst let v = 0: in C (O) th e roo t has A success ors, while an y fur t her vertex has ~o successors. (b) Now let v = J-l + 1: in C (v) has ~ o successors, and each of them is t he root of a t ree isomorphic t o C (J-l) . (c) Finally let v be a limit ordinal: in C (v), < > has a successor s/' for every J-l < v , a nd s/' is the root of a tree isomorphic to C (J-l) . At this point , let us bui ld for ever y S has a success or s., for every v < A and , for every t/ < A, S V is in it s turn th e root of a t ree isomorphic t o C( v) if v E 5 and t o w<w otherwise. It is clear t hat jC(5) I = A for every 5 ~ A and th at different subsets 5 I: 5 ' yield non isomorphic trees C (5 ) 'f!. C( 5' ). At t his poin t one may wonde r what is so wrong in t he las t example to exclude a ny classification of models. More generally one may as k why a present abl e t heory may be non-classifiabl e. Recall th at , if T is presentabl e, t hen any uncount abl e model A1 of T has a present ation , whose "skelet on" is a t ree C of IMI< w. Of course t his is a good feature t owa rds a genera l class ificat ion of mod els . Bu t th e poin t is t hat some involved t ree might be non well found ed . Let us recall what th is mean s. Let C be a ny t ree (say in A<W ). On e ass ociates with a ny poin t s in C a rank r (s) (an ordinal, or 00) in t he following way. First we define r (s) ~ a for any ordinal a . We pro ceed by induction on a :
1. r (s)
~
a if a = 0;
2. if a is limit , then r(s) ~ a means r (s) ~ {3 for every ordinal {3 < a; 3. if a = (3 + 1, th en r(s) ~ a means t hat , for some t E S with s = t r ; r(t) ~ (3 . If t here is an ordinal a for which r(s) 'l a, t hen t he least ordin al with t his property is necessarily a s uccessor ao + 1, and we pu t r( s) = ao. Otherwise (when r(s) ~ a for every ordinal a) we put r (s) = 00. We say t hat C is well found ed if r( 0 a model B v of T eleme ntarily extending Ba (and if) and even t he Bp's for p < u, properly including all of them a nd stilI satisfying a (Bv , if) = a( Bo, if) . Notice th at this is sufficient for our purposes becau se, pr oceedi ng in t his way, we eve nt ua lly build a mod el B of T of power f-l eleme ntarily extending Ba bu t satisfying a( B, if) = a( Bo, if), hence admitting a countable a( B, if); on t he other side , a sim ple use of Co m pactness Theorem yields another model C of T of power f-l such t hat , for every t u ple if in C ad mitting t he sa me type as if over t he em pty set , la (C , 01= f-l . Co nseq uently B and C cannot be isomorphic, a nd t his co ntradicts t he f-l-cat egoricity of T . So let us build the Bv's. vVe alr eady introduced B l . For a limit v, put B; = Up O. For simplicity we limi t ourselves t o p = 1, v = 2; indeed what we are going to say in t his case applies to any p > 0 as well, a nd so is ge nerally valid . First use ind ep endence t heory, mo re precisely (13) an d (14), and find an isomorph ic copy B~ of B l inside n, correspond ing t o B l by a n isom orphism fixin g Ba poin t wise, and satisfying B~ ../.-Bo Bi . To build B~ , consider a language L * enlargin g L by a constant b* for eve ry b E B l , and in L * t he t heo ry T * saying th at , for eve ry bin B l ,
b* satisfies t he non -forking extension of tp(bj Ba) over Bi . Any finit e por tion To of T* has a mode l; in fact , let bglue all t he t upIes from B l a rising in t he sent ences of To, use (14) and obtain a t upIe b* realizing t he non-forking extension of tp(bj Ba) over Bi ; recall that every subseq uence of b* has t he sa me prop erty. By com pactness , T* has a mod el. T he elements b' em bo dy ing t he constants b* in t his mod el form a st r uct ure B~ isomo rphic t o B l over Ba (as, for every L (Bo)-formul a 1 their class is larger t han in Example 1. Notice that an infinite group is trivially definable in A, but no field ca n be interpreted inside A . 3. At last , view A just as a n algebraically closed field. If p is its characteristic, then the complet e theory of A is ACFp and is strongly minimal. Moreover • for every subset X of A, acl(X) is t he algebraic closure of A in the field theoretic sense, • for every positive integer n, t he definabl e su bsets of A n a re the construct ible ones , in other words t he finite Boolean combinations of algebraic varieties of A n . Now take any strongly minimal st ruct ure A.
Definition 7.9 .4 A is called trivial if, for every X
~
A,
acl (X) = UxEXacl(X) .
Every (pu re) infinit e set is trivial. But, of course, vectorspaces and algebraically closed fields are not. Moreover no trivial st rongly minimal structure ca n interpret an infinite group.
Definition 7.9.5 A is called locally modular if, for eve ry choice of X , Y ~ A suc h that X n Y i= acl (0),
(*)
dim (X U Y)
+ dim (X n Y) =
dimX
+ dim Y.
CHA PTER 7. CLASSIFYING
284
Every t rivial structure A is locally modular (in fact , for every X and Y , acl (X U Y ) = acl (X ) U acl (Y ), so a basis of X U Y ca n be form ed by taking a basis a X n Y , extending it t o a basis of X a nd a basis of Y , and gluin g t hese bases together). Bu t also vect orspaces a re modular: in fact , in t his case , (*) is ju st t he Grassm an formula (and do es not need t he ass umpt ion X n Y :f: acl (0)) . On t he cont ra ry, no algebraically closed field A (of t ra nscende nce degree 2': 4) is locally modular. In fact, choose ao, aI , a2, a3 E A alge braically ind epend ent over the prim e subfield Aa , and form th e extensions
Then dim X = dim Y = 3 but dim(X n Y) = dim Ao(aa) = 1, whence X n Y :f: acl(0) , and dim(X U Y) = dim(Aa(ao , aI , a2, a3)) = 4. So (*) fails. Wha t is t he significa nce of local modularity? Basically a locally modular A eit her is triv ial or ca n define an infinite group. Furth ermore one obser ves th at a ny group 9 definabl e in a locally modular A is a belia n-by-finite (in other wor ds , it has a normal abelia n subgroup of finit e index); and every subset of any ca rtesia n power definabl e in A is a finite Boolean combination of cosets of definable subgroups of Accordingly, no infinite field is definable in A. In t his setting Zilber raised in 1984 t he following problem , generally called Zilb er Tricho tomy Conject ure.
en
en.
Conjecture 7.9.6 (Zilber ) Let A be a st rongly m in imal non locally modular struct ure. T hen A interprets an infi ni te fi eld K . Furthermore, for eve ry posit ive integer n , the subsets of K " definable in A are ju st thos e definable in K (and hence coin cide with the construciible ones). Recall that , owing to Macintyr e's Theor em , any infinite field interpretable in a n w-stabl e st ruct ure must be algebraically closed. Hence the importance of this conj ecture is clear: according to it any st rongly minim al st ruct ure A eit her is t rivial, and so looks like an infinite set (as in Exa mple 7.9.3.1) , or is locally modular and not t rivial, and t hen resembl es a module (as in Example 7.9.3.2), or looks like a n algebraically closed field , becau se it interpret s s uch a field (Example 7.9. 3.3). Hence t he conj ecture would pro vide a qui te sat isfactory classification of strongly minim al struct ures (and t heories) up to biinterpret abili ty. Bu t in 1993 Hru shovski showed t ha t Zilber 's Conjecture is false.
7.9. BIINTERPRETABILITY AND ZILBER CONJECTURE
285
Theorem 7 .9.7 (Hrushovski) Th ere do exist strongly minimal structures A which are not trivial but cannot int erpret any infinite group.
Clearly such a structure A is not locally modular and does not interpret any infinite field. However Zilber Conjecture (more precisely, a suitable restatem ent) do es hold in certain topological structures deeply related to st rongly minimal mod els: the so called Zariski geometries . To introduce them, let us come back to Example 7.9.3.3 , so dealing with algebraically closed fields A. We know that , for every positive integer n, the algebraic varieties of A n a re preserved und er finite union and arbit rary intersection , and form the closed sets in the Zariski topology on A n. These topologies a re Noeth eri an: none of them admits a ny infinite strictly decre asing sequence of closed sets. Moreover they sat isfy th e following properties (m and n denote below positive integer s) .
Ut, ... , f m) be a function from A n in A m . Assume that eac h component f i (1 ~ i ~ m), as a function from An in A , either projects A n onto A or is constant . Then f is continuous.
(Zl ) Let f =
( Z2 ) Every set {iI EA : a; = aj} with 1
~
i, j ~ n is closed.
(Z3) The projection of a closed set of A n+! onto A n is a constructible set in A n. (Z4) A , as a closed set, is irr educible.
(Z5) Let X be a closed irreducible subset of A n. For every ii E An-I , let X (iI) denote th e set of the elements b E A such that (iI, b) EX . T hen there is a natural N such t hat, for every ii E An -I, either IX (iI) I ~ N or XCiI) = A. In particular, when n = 1, every closed proper subset of A must be finite. (Z6 ) Let X be a closed irreducible subset of A n, d denote the (topological) dimension of X. Then, for every i, j a mong 1, . . . , n, X n {iI E A n : ai = aj} has dimension 2 d - 1. (Zl) - (Z6) restate in a topological style some properties which are well know n, or easy to check. For inst a nce (Z5) follows directly from some simple algebraic facts and the st rong minimality of A by a compactness argument: the reader may check this in detail as an exercise. Now it is easy to realize that even infinite sets a nd vectorspaces over a countable field satisfy (Zl) - (Z6) provided one takes as closed subsets the finite Boolea n com binations of t he following sets:
286
CHAPTER 7. CLASSIFYING
• the sets of the tuples ad mit t ing a fixed coordinate in a given plac e, or equal coordinates in two different places, when dealing with pur e infinite sets; • the cosets of pp-definabl e subgroups when dealing with vectorspaces. It is easy to control t hat in both cases t hese sets ar e actually t he closed sets in a su it a ble topology.
D efinition 7.9.8 A Za riski st ructure (or geometry ) is a collection (A, {Tn : n positive integer}) where A is a non-empty se t, for eve ry n T n is a No eth erian topology on A n and (Zl) - (Z6) hold. Hence the examples 7.9 .3 produce Zariski structures. Conversely, let (A, {Tn n positive integer} ) be any Zari ski structure. Assum e t hat A is t he domain of some st ruct ure A (in a language L) such that, for every positive integer n , the subsets of A n defin abl e in A are j ust the finit e Boolean combina t ions of closed sets in T'; (and so coincide with th e construct ible sets in T n ) . Then it is easy to check that A is st rongly minimal; mor eover t he possible triviality, or local modulari ty of A do es not depend on L , or on t he L-structure of A , bu t only relies upon th e charac terizat ion of th e definabl e sets of A and so, afte r all, upon (Zl ) - (Z6) . More notably, in t he restricted fram ework of Zari ski struct ures, t he Zilber Trichotomy Conjecture is t rue, as shown by Hru sho ski and Zilber himself.
T heor em 7.9.9 (Hr ushovski-Zilber) Let (A, {T n : n posit ive intege r}) be a Zarisk i struct ure, A be a strongly m inimal st ru cture with domain A such that, for eve ry posit ive in teger n , th e su bsets of A n definable in A are ju st th e cons iructible sets in T n . If A is not locally modular, th en A interprets an algebraically clos ed fi eld J( , and J( is unique up to definable isomorphism. Moreover, for eve ry positive integer n , th e subsets of K " definable in A coinc ide with the on es definable in J(.
7.10
Two algebraic examples
Let us s ummarize briefly some of the main notions introdu ced in t his chapter by examining t wo relevant classes of algebraic struct ures, and their first order th eori es: differentially closed fields of characteristic 0, and existent ially closed fields with an automorphism (again in charact eristic 0). Both these examples play an important role in the mod el t heoretic solution of some not abl e qu estions of Algebraic Geom et ry: we will describe t hese probl ems and t heir solut ion in t he next cha pter.
:
7.10. T WO A LGEBRA IC EXA M PLES
287
1. D C Fa . F irst let us deal wit h differenti ally closed fields of characterist ic O. Let us recall once again t hat t heir t heory D C Fo is complete and quantifier elimina ble in its natur al lan gu age L , containing the sy m0, 1, D and nothing mor e. So definabl e sets are easy t o bols classify: as we saw in Chapter 3, t hey includ e t he zero sets of (finite) systems of differenti al polyn omials - in oth er words, t he closed sets in t he Kolchin t opology - as well as t heir finite Boolean combinat ions t he construct ible sets in t his topology -, bu t nothin g else. As a ty pica l Kolchin closed set in a different ially closed field (K, D ) let us mention t he field of const an t s C (K ) = {a E K : D a = O}. T his is an algebraically closed field - ju st as K -, and is strongly minim al even in L; in fact , D is identically 0 on C (K ) a nd so adds no fur ther definabl e objects to th e field st ruct ure on C (K ).
+, " - ,
DC Fa eliminates the imagin aries, t oo. Mo reove r DC Fa is w-st able with Morley rank w, so ind ependence makes sense in D C Fa , and ind eed it is ruled by Morley rank: for ii , A and B as in Section 7.2,
s +A B
{:::=;>
R M (tp (il/ B )) = R M (tp (il/A )).
Of course t his raises t he question of characterizing algebraic ally Mo rley ra nk wit hin differenti ally closed fields of cha racterist ic O. Bu t t here are also oth er ways of describing forking an d inde pendence in DC Fa, having a pret ty algeb ra ic fl avour. For inst an ce, one can pr elimin arily obser ve t hat , for every small A , acl (A) - in t he model t heo ret ic sense - is ju st t he (field t heo retic) algebra ic closure of t he different ial subfield generated by A Q(Dia : a E A , i EN); at t his point , B ju st means one ca n realize t hat, for ii, A and B as usu al , ii t hat acl(AUil) and acl(B) are (algebraically) ind epend ent over acl( A). Among other things, this cha racterization sugges ts a n alternative rank notion, specifically concerning th e differential fr am ework: this is called differential degree or D-degree and denoted D-dg: for H a differential field a nd ii a tuple of elements in n, D-dg(il/H) is th e t ranscende nce degre of the differenti al field generated by H U ii over H . So, for ii, A an d B as before, and A and B differential subfields for simplicity,
+A
il
+AB
{:::=;>
D-dg(il/B ) = D-dg (il/A ).
However we have t o be ca reful here: t he last equivalence does not mean t hat R M a nd D -dg coincide . Their relationshi p, an algebra ic cha racterization of R M in D C Fa and t he con nection amo ng differenti al
288
CHAPTER 7. CLASSIFYING
degr ee, Morley rank and other possib le ranks in D C Fo a re described in the refer ences ment ioned at the end of th e chapte r. Now let us deal with biinterpret ability, in particula r let us consider st ro ngly minimal sets in differ entially closed fields IC of charact eristic O. They include the cons tant s ubfield C (IC ) (which also has different ial degree 1, as it is easy to check). C (IC ) is not locally modular, in fact the a rgume nt prop osed in t he last sect ion for algebra ically closed fields applies to C (IC ) as well. But what is most remarkabl e in this sett ing is a theorem of Hru shovski and Sokolovic sayi ng that Zilber Trichot omy Conjecture holds withi n strongly minimal sets in DCFo. In fact all t hese sets are Zariski structures, and so obey the Hru shov skiZilber Th eor em . We can say even more: any strongly minimal set S which is not locally modular, and henc e interprets an infinite field, doe s interpret t he field of constants C(IC) up to a definable isomorphism . We will provide more details about t hese mat te rs in Section 8.7 .
2. AC F A. Now we deal with existe nt ially closed fields with a n autom orphism . For simplicit y, we still work in cha racterist ic o. Let AD F A o denote the corresponding theory in the natural language L = {+ , ., - , 0, 1, O" } where 0" is the sy mbol representing the a utomorphism . Recall that, t his ti me , fixing t he characteristic is not sufficient to ensure com pleteness: in order to characterize a mod el of AC FAo up to element a ry equivalence, one has also to describe the act ion of t he a ut om orphism on th e prime subfield Q. Moreover ACF A o does not eliminate t he qu an tifiers in L, alt hough it is obviously model complete (as a mod el companion) . Accordingly definabl e sets exhibit some more complications than in the differential case. In fact , t hey include the zero sets of (finite) syst ems of difference polynomials , as well as their finite Boolean combinat ions; the former are the closed set s, a nd t he lat ter t he constructible ones in a suitable topology. But now , as qu an tifier eliminat ion fails, we have to conside r also the projections of const ruct ible sets - a nd nothing else, owing to model completeness - to ca pt ure the whole class of definabl e sets. An exam ple of a closed set in a mod el (IC , 0") of AC F A o is it s fixed su bfield Fix (O" ) = {a E I< : er(a) = a}. This is not algebraically closed (in particular it is not st rongly minimal); but one can see t hat it is a pseudofinite field , so an infinite model of the theory of finite fields. ACF A o elimi nat es the imaginaries. T his time no existentially closed field (IC, 0") with an au t omor phism is w-st able , or even st able. How-
289
7.11. REFERENCES
ever (K, a) is sim ple (as well as its fixed su bfield , a nd a ny pseudofinite field) . So ind ep end ence makes sense in ACFAa, and comes directl y from forkin g, bu t ca nnot be ruled by Morley ran k. Any how an explicit algebraic characterization ca n be don e as follows. We work for simplicity in a big saturated mod el (K , a) of AC F Aa. F irst one obser ves t hat, for every small A, acl(A) coincides wit h t he algebra ic closure in t he field t heoret ic sense - of t he differ ence subfield generated by A Q (a i (a ) : a E A, i E Z )
(here we use t he cha racterist ic 0 assu mp t ion; prim e cha racterist ics cau se some major trouble). At thi s point one shows th at, for ii, A a nd B as usu al , ii t A B ju st means that acl(A U il) and acl(B) are (algebraically) ind epend ent over acl (A). This yields an appropriate notion of rank , of a pret t y algebraic flavour , called difference degree or a -d egree and deno ted o-dq: for H a difference fi eld a nd ii a t uple of elements in n, a -dg (il/ H ) is t he t ra nsce nde nce degree of t he difference fi eld generated by H U ii over H . When finite, t he difference degree can reason abl y repl ace Mo rley rank and provides a good notion of dimension in t his unstable setting; on t he oth er side , when a-dg (il/ H ) is infinite, t hen clearly t he a i (a) 's (when i ra nges over integers) are algeb raic ally inde pe ndent over H. So, for d, A an d B as before, a nd A and B difference s ubfields for simp licity, ii
tA B
{:::::::?
a - dg (il/B ) = a - dg (il/A )
at least when t he la t ter degrees are finite. Wh en X is a ny definabl e set in f{ n , the difference degree of X over H a - dg (X/H) is t he maximal difference degree of a t uple ii in X over H . In particul ar the fixed subfield of K get s difference degr ee 1. In this sense, Fi x (a) is a "minimal" definabl e infinite set of K. Notably, an adapte d version of Zilber Trichotomy Conj ecture holds for these "minimal" sets in ACFAa, a nd even ensur es in part icular t hat, very roughly speaking, Fix (a) is t he only non "locally modul ar" exam ple among t hese structures.
7.11
References
T he classification issue fro m t he point of view of Descripti ve Set T heory is treated in [68]; t he par ticul ar and int riguin g case of torsionfree a belian groups of finit e rank is dealt wit h in [163]. Finite dim ension al vect orspaces
290
CHAPTER 7. CLASSIFYING
wit h two distinguished endomorphisms over a fixed field, and the wildness of t heir classification problem are described in [136] or, more specifically, in [137]. The mai n references on Shelah's classificat ion theory are just t he Shelah book [149] and its revised and up dated version [151]. Most of t he topics of this chapter are treated there in detail. Another good an d per haps mor e accessible source on classification and stability t heory is [8]. See also Makkai's paper [101] . Vaught's Conjecture is proposed in [174], and its solution in the w-stable fram ework ca n be foun d in [152]. Lascar's pap er [84] provides an enjoyable disc ussion of t his matter. The (uncountable) spectrum problem for complet e countable first order theor ies is fully solved in [53]. Simple theor ies were introduced in [149], but it was Kim who showed, together with Pill ay, t heir relevance within the classification program: see [71] and [73]. Kim again observed the key role of symmetry, transitivity a nd local character [72]. Wagner's book [175] provides a general a nd exhaustive report on this theme. As already said , stability, superst ability and the further dichotomies ari sing in t he classification program are treated in Shelah books [149, 151], in [8] or also in [101]. [85] provides a nice and terse introd uction to stability, using and emphasizing t he notion of heir and coheir. [83] pursues t his approach and deals in particular wit h Rudin-Keisler order and strong regu larity. The effectiveness aspects of Shelah 's classification progr am is discussed in [114], while [2] examines its connect ions with Stationary Logic. Turning our attention to t he algebraic examples, let us mention [136] or also [56] for modules, a nd [19] for pseudofinite fields . [110] treats differentially closed and separably closed fields, and includes a wide bibliographical list on them . [62] a nd [155] provide ot her key references on DCFo; see also [134]. Existentially closed fields wit h an automorphism are j ust t he su bject of [20]. An explicit exa mple of such a field can be found in [58] and [88]; see also [95]. Shelah's uniqueness theorem is in [148], Morley's theorem in [117]. Anot her proof of Morley 's Theorem is given in [9]; see also Sack's book [146], or [57]. T he Ehrenfeucht - Mostowski models quoted in Sect ion 7.8 are int roduced in [38]. Finally, let us deal with biinterpretability. Malcev 's correspondence is in [102] while Mekler's th eorem on nilpotent groups of class 2 is in [113]. Zilber 's program is developed in [182], where Zilber's Conjecture is also proposed . The negative solution of this conjecture is in [59], and the HrushovskiZilber theorem on Zar iski structures in [64].
Chapter 8
Model Theory and Algebraic Geometry 8 .1
Int roduction
We have often emphasized in t he pas t chapters t he deep relati onship bet ween Model Theor y and Algebra ic Geometry : we have see n, a nd we are going t o see also in t his cha pter t hat seve ra l relevan t notions a rising in Algebra ic Geom etry (like variet ies, morphisms, manifolds, algebraic gro ups over a field K ) are definabl e object s a nd are conseq uent ly concerne d wit h t he mod el t heor etic machinery develop ed in t he previous pages. For inst an ce, when K is algebraicall y closed , t hey a re w-stable st ruct ures.T his connection ca n yield, a nd is act ually yielding, significa nt frui ts in both Mod el T heory and Algebr aic Geom etry. On th e one hand , seve ral t echniques a nd ideas originated a nd employed within t he specific sett ing of Algebraic Geome t ry ca n inspire a more abstract model th eor etic treatment , applying to a rbit ra ry classes of st ruct ures. In this sense Algebraic Geometry over algebra ically closed fields can suggest new directions in th e study of w-stability: we will describe this connection in many sect ions of t his chapter. However a parallel analysis ca n be developed inside other relevant areas, like differentia lly closed fi elds (and Differential Algebraic Geom etry) , or existe nt ially closed fields with a n a ut omor phism , and so on . On t he oth er hand , it is right t o observe that t he benefits of t his relationship regard not only Model Theor y, bu t also, and releva nt ly, Algebraic Geomet ry. In particular, we will propose some prominent problems in Algebra ic Geometry, whos e solution do es profit by Model Theor y and its t echniques. This will be the aim of t he final section of this cha pte r.
291
292 CHAPTER 8. MODEL THEORY AND ALGEBRAIC GEOMETRY
8.2
Algebraic varieties, ideals, types
Let J( be a field , n be a positive integer. We already introduced in th e past chapters the (algebraic) varieties of K" : They a re t he zero sets of finite systems of polynomials of J([X] (where x abbreviates, as usual, (Xl , .. . , x n)), a nd so are definabl e in J(. Moreover they form the closed sets in the Zari ski t opology of K"; accordingly even the Zariski open or const ructible sets ar e defina ble in J( . But the varieties of K" are also closely related to the ideal s of J([X]. In fact one can define a function I from vari eties to ideals mapping a ny variety V of K" into the ideal I(V) of t he polynomials f(x) in J([X] such that f(a) = 0 for every a in V . Checking that I(V) is ind eed a n ideal is straight for war d; I(V) is even a radical ideal , in other words it coincides with its radical rad I (V ): if f(x) E [([X] and , for some positive int eger k, fk( x) E I(V) , t hen f (x) already occurs in I (V). In particular I is not onto. Bu t there is also a nother function V in the other direction , from ideals to varieties, mapping any ideal 1 of J([X] (in pa rt icular a ny ra dical ideal) into t he set V(I) of t hose elements a E K" a nnihilating all the polynomials of I
V (1) = {a E J(n : f (a)
= 0 vf (x)
El}.
Due to t he Hilbert Basis T heorem, I is finit ely generated , and so V(I) is a variety. Ind eed , any vari ety V can be ob t ained in this way by definition; in oth er word s V is onto. Not ice also t ha t V(I) = V(rad I) for every ideal I . The definition of I an d V trivially impli es t ha t, for every ideal I of J([X] , I(V(1)) "2 I. As I(V(I)) is a radical ideal , I(V(I)) "2 rad T, Hilbert 's Nullstellensatz (see Ch apt er 3) ensur es that , when J( is an algebraically closed field, eq uality hold s: fo r every ideal I of J([X] ,
I(V(1)) = rad I , It is easy t o deduce that , if J( is algebraically closed , t hen I a nd V deter mine t wo bijections, the one inverse of t he other, betw een varieties of K" and radical ideals of J([X] . We will st ill den ote these restricted bijections by I , V respectively. Notice t hat both reverse inclusion : for instan ce if V , W a re two varieties of K" , then
I(V) "2 I(W)
~
V
~
W.
We assume from now on that J( is a n algebraically closed field. By t he way recall that, under t his condit ion , what is definable in J( is w-stable of finite Morley rank , becau se J( is st rongly minimal.
8.2. ALGEBRAIC VARIETIES, IDEALS, TYPES
293
Let us restrict our at tention from radical ideal s of K[X] t o prime ideals (t hose ideals I in K[X] such that , if a product of two polynomials of K[XJ lies in I , th en at least one factor polynomial is in I as well) . Parallely we con sider, among vari eties of K" , the irreducible ones , so t he non-empty vari eties V t hat ca nnot decompose as a union of two proper subvariet ies . It is easy to check that the previous bijections I and V betwe en vari etie s of K" and rad ical ideals of K[X] link irreducible varieties of K" and prime ideals of K[XJ: for every vari ety V of K"; V is irreducible if and only if I(V) is prime. Notice also that I(0) = K[XJ . A closer relationship links irreducible varieties and prime ideals. For inst a nce, it is known that every non-empty variety V of K " ca n be expressed as a finite irredundant union of irreducible varieties, and that this decomposit ion is unique up to permuting the involved irreducible varieties (which ar e con sequ ently called the irreducible components of V) ; th e irredundancy of the decomposit ion just means that no irreducible component of V is included in t he union of the other components. Sp ecularly, every proper radi cal ideal I of K[XJ can be expressed as a finit e int ersection of prime ideals minimal with resp ect t o inclusion ; even t his representation is uniqu e up to permuting the involved minim al prime ideals. On e ca n also realiz e th at , und er t his poi nt of view , for every non- empty vari ety V in K " , t he irredu cible components of V correspond to t he minimal prime ideals occurring in th e decomposit ion of I (V ). So far we have summa rized -some very famili ar t opics of basic Algebraic Geom etry. Now let Model Theor y intervene. As we saw in Section 5.3, prime ideals of K[XJ - a nd hence, through t hem , irreducible variet ies of K" - directly and na turally corres pond to n-types over I< . In fact , for a n algebraically closed field K , th er e a re t wo bijections i and p , one inverse of the other, between n-types over I< and prime ideals of K[XJ. Basically, for every n-type p over I. r on O .
Just to underline th e power of this result, let us recall that , owing to Corollary 8.3.4, every non- empty op en subset 0 of K" has th e sa me Morley rank n as K" ; while th e Morley rank of K.. n-o is smaller (for , K" is an irreducible vari ety of rank n). Secondly, it is worth recalling th e general fact that , if f is a fun ction from a vari ety V of K " into K.. and , for every a E V , there is an op en neighbourhood 0 of a such t hat f equa ls some ration al funct ion in O n V , t hen f ca n be globa lly expressed as a polynomial fun ction . Now let us show Theor em 8.4. l. Proof. Let Q be the uni verse of the theory of K.. , t} , .. . , t« E Q be algebraically independent over K.. . As f is J(-definabl e , an y a utomorphism of n fixing K.. and t l , • • • , t n pointwise acts identically also on f(i) (f denote s here th e tuple (t}, ... , tn)). Whence j(i) is in dclt K n i) . So, if K.. has cha racterist ic 0, then f(i) = r(i)
for some suitable rational function r with coefficients in prime characteristic p , then
J(,
while, if K.. has
where r is as before and hp is a positive integer. Put s = r when car K.. = 0, s = r otherwise. The eleme nts in K " where f coincides with s form a set X definable in K.. (just as f and s) , and th e form ula "s (v) = f (v)" defining it is in th e type of f over J(. As t} , ... , t n are algebraically independent over K.. , t his t ype has Morley rank n . Hence RM(X ) 2: n . As X ~ K" ; t he Morl ey rank of X must equa l n , a nd RM (J( n - X ) < n. It follows th at t he
r -:» .
299
8.5. MA NIFOLDS Zari ski closure of K " - X has Morley ra nk an open set 0 in X of Morley rank n where
8.5
< n , and so its complement is f = s. •
Manifolds
Throughout this section IC still denotes an algebra ically closed field and n a positive integer. We deal here with (ab stract) manifolds in IC n a nd we show that th ey are defin abl e objects. F irst let us recall their definition . Definition 8 .5.1 A manifold of IC n is a structure V = (V, (Vi, f i) i5: m ) where m is a natural and • V is a subset of K " (called atlas); • V o, ... , V m are subsets of V , and V is th e union Ui<m Vi ; • fo r every i ~ m , f i is a bijection of Vi onto a Zariski closed set Vi (a coordinate chart of th e atlas V) ; • for i ,j ~ m and i
i- i , f i(Vi n Vj)
=
o.,
is an open sub set of Vi;
• for i, j ~ m and i i- i . f ij = fi . f T 1 (a biject ion between Vji and V ij ) can be locally exp ressed as a tu ple of rat ional fu n ct ions.
Manifolds include several familiar exa mples. Examples 8.5.2 1. Every algebraic vari ety (so every Zari ski closed set ) V of IC n is a manifold , provid ed we set m = 0 a nd choose Ui, = Vo = V as the only coordinat e cha rt of t he atlas; the res ulting man ifold is called affine . 2. Let 0 be an op en principal set of IC n; 0 is defined by a single inequ at ion -' ''g(v) = 0" ; notice that the formula "g(v) . V n +l = 1" defines a closed V in IC n+l , a nd it is easy to control t hat the projection of K n +l onto K " by t he first n coordina te s determines a bijection f of V onto O . Accordingly (V , (V , f -l )) is a manifold with t he only chart O . Such a manifold is called sem ia ffine . 3. Also the projective space p n(IC) can be equipped with a manifold structure. In fact, view p n(IC) as t he quo tient set of K n+l - {a} with resp ect to the equivalence relation rv linking two non zero tu ples x = (xo , Xl, . . . , x n) a nd fl = (Yo, YI , . .. , Yn) in K n+l if and only
300 CHAPTER 8. MODEL T HEORY AND ALGEBRAIC GEOMETRY
if t here is some k E J( such that Xi = kYi for every i ~ n. For x E J( n+l - {O} , let (xo : X l : •• . : x n ) be t he class of x with resp ect t o t his relation . Moreover , for i ~ n , let
• A i de note th e set of th e elements (x o : such that Xi i= 0,
Xl : • • • :
x n ) in p n(!C )
• f i be the function of A i int o K " mapping any (xo : X l
.In (3Zll. Xi '
Xi _l
· · · , ~,
5.±.!. xi
~)
, •• . , xi
: ... :
xn )
•
It is st ra ight forward to check that (pn(!C), (A i, Ji)i5:. n ) is a manifold .
Notice t hat affine and semiaffine var iet ies are definable -even as structuresin /C . Moreover p n(!C ) is interpretable in !C both as a set (since t he relation rv is 0-definable) and as a manifo ld . But a lgebraically closed fields uniformly eliminate th e imaginaries, so we can view p n(!C) even as a definable st ructur e in !C. Mor e gen er ally one can show T heo rem 8.5.3 Let V = (V, (Vi , Ji) i5:. m ) be a manifold of a structure defina ble in !C.
«» .
T he n V is
Proof. As algebraically closed fields have t he uniform eliminat ion of imaginari es , it is sufficient t o show t hat V is a struct ure interpretable in !C . In fact, every map U, of t he atlas V (i ~ m) is definable in !C . V ca n be rega rded as the quotient set of t he disjoint union of t he cha rts U, (wit h i ~ m) with respect to the equivalence relat ion ident ifying Uij and Uji via f ij for every i < j ~ m; moreover , for every i ~ m , Vi can be definably recovered as t he image of U, by th e projection into the quotient set V , and f i is given by the invers e funct ion of this projection (restrict ed to Ui) . So our claim is proved if we show t hat, for every i, j ~ m with i i= i , f ij is definable. But f ij can be locally expressed as a rat ional function, and its domain Uij is an open subset of U, and accordingly can be written as a finit e union of principal open sets. So th e th eor em is a direct con sequence of the next resu lt.
Lemma 8.5.4 Let 0 be a prin cipal open of !C n, and let q(x ) E [([X] be a polynomial satisfying 0 = {ii E K " : q(ii) i= O}. Let f be a funct ion of 0 in to K" which can be locally expressed as a ratio nal fun ct ion. T hen there are a polynomial r( x) E [([X] and a positive integer m such that f(ii) = r(ii)/qm(ii) for eve ry ii EO . In particular f is defi nabl e.
8.6. ALGEBRAIC GROUPS
301
Proof. We know that 0 is canonically homeomorphic to the closed subset V of K n+l defin ed by q(v) . Vn+l = 1. Under this per spective f can be replaced by the functi on 1* of V into 1< mapping any tuple (ii, an+d E V in f(ii) . Eve n 1* ca n be locally expressed as a rational function , and hence as a polynomial function in (x, xn+d (by th e general fact we recalled before the proof of Theorem 8.4.1). So th ere is some polynomial s(x , xn+I} E 1 lover a number field F , embed X = Xo(C) int o its J acobian A = J(X ) a nd apply t he Mordell-Weil T heore m ensuring th at t he group 9 of F-ration al points in A is finitely generate d . Accordingly decomp ose X o = X n G as a finite union of cosets a + H where a E G and H is a subgroup of g. Take a coset a + H . Its closure a + H is includ ed in X o - an irr edu cible set of di mension 1 -. Conse quent ly, if a + H is infinit e, then X o ju st equa ls a + H an d so inherit s a group st ructure , and genus S; 1. T his means t hat, if t he gen us of X o is > 1, t hen every coset a + H must be finite, whence X o itself is finite .
306 CHAPTER 8. MODEL THEORY AND ALGEBRAIC GEOMETRY
These are the questions we wish to deal with. Now let us report about their solution. Mordell's Conjecture was proved by Faltings in 1983. The echoes of this result spread far and wide, also because it implied an asymptotic solution of Fermat's Last Theorem: in fact, by applying Mordell's Conjecture (or , more precisely, Falting's Theorem) to the projective curve over Q
for n ;::: 3, one gets only finitely many zeros for every n. Also the Manin-Mumford Conjecture was positively answered by Raynaud in 1983. The Mordell-Lang Conjecture (as stated before) was solved just a few years ago: first Faltings handled the particular case when the group 9 is finitely generated , and then McQuillan provided a general positive solution , using Falting's work and other contributions of Hindry. So far we have limited our analysis essentially to the characteristic 0, and to number fields. What can we say when passing to function fields, or prime characteristics? First let us deal with function fields . Still working in characteristic 0, Manin had proved in the sixties the following analogue of Mordell's Conjecture in this setting: if K is a function field over an algebraically closed field Ko (of characteristic 0) and X is a curve of genus> 1 over K, then either X does not descend to Ko (in which case X(K) is finite), or X is isomorphic to a curve X o defined over Ko (and all but finitely many points of X(K) come from elements of X(K o)). When considering prime characteristics p, even the notion of group 9 of finite rank must be rearranged. In fact, what we have to require now is that 9 has some finitely generated subgroup 5 such that, for every g E G, there exists a positive integer m prime to p satisfying mg E S . However 8.7.5 - as it was stated before - does not hold any more. In fact A itself is a torsion group without elements of period p; but there may exist some curves of A which are not finite unions of cosets of subgroups of A. A reasonable restatement of 8.7.5 in the general setting, for an arbitrary characteristic (0 or prime), is the following. Conjecture 8.7.6 Let Ko -< K be algebraically closed fields, A be an abelian variety over K having trace 0 over Ko (this means that A has no non-zero abelian subvarieties isomorphic to abelian varieties over Ko). If X is a Zariski closed subset of A and 9 is a subgroup of A of finite rank, then X n G is a (possibly empty) finite union of cosets of subgroups of g.
In 1994 A. Buium proved this form of the Mordell-Lang Conjecture in characteristic O.
8.7. T HE M ORDELL-LANG CONJECTURE
307
Theorem 8.7.7 (Buium) 8. 7.6 is a true statement when /Co -< /C are algebraically closed fields of characteristic o. What is not eworth y for our purposes in Buium 's line of proof is his use of Differenti al Algebraic Geometry; ind eed Differenti al Alge bra promptl y recalls Mo de l T heory a nd its treatment of differenti ally closed fields. So it is righ t t o spe nd a few words to desc ri be Buium 's strategy : one equips /C wit h a deri vation D whose constant fi eld is ju st /Co, one embeds 9 in a differenti al algebra ic gro up 91 and, finally, one shows by ana lytic arguments th at X n C l is a finite union of cosets of 91 a nd one t ra nsfers this proper t y to 9. But it was Ehud Hru shovski who first proved t he Mo rdell-Lang Conjecture in it s more general form , in any cha racterist ic, following the initial Buium a pproac h and then using mod el th eor etic methods an d , above all, Zari ski geom etries , differentially closed fields in cha rac te rist ic 0 and separably closed fields in prime cha racteristic, in addit ion t o Morley ra nk, elimin ation of im aginari es and t he defina bilit y resul ts of t his chapte r. It should be emphasized t hat no alte rnative genera l proof of the conjecture is known ; and ind eed Hrushovski proposed , so me time lat er , a new mod el t heoretic proof of t he Manin-M um ford Co njecture, based on a cruc ial use of existentially closed fields with an aut om orphism (in pa rticula r Zilber 's Trichotomy in ACFAa) , a nd getting in t his way nice effective bo und s of the number of involved cosets in a decomposition of X n C. Coming back t o t he Mordell-La ng conjecture , we ca n say
Theorem 8 .7 .8 (Hr ushovs ki) 8. 7.6 is a true statem ent in any charact eristic. T his concludes our short and lacu nose history of Mo rdell-Lan g, Morde ll and Manin-Mumford Conjectures. Which is our purpose now? Ce rtainly we do not aim at providing a complete report of Hru shovski 's proof: t his would requ ire man y pages and serious efforts; moreover th ere do exist sever al nice ex pos itory pap ers a nd books wholly devoted to a det ailed ex posit ion (some of t hem are mentioned a mong the references at t he end of th is chapter) . On t he oth er hand , we would like t o spe nd a few words abo ut Hrushovski 's a pproach, ju st to explain where Mo del T heory int ervenes an d why it plays a decisive role. W ith t his in mind , we will sketch Hru shovski 's proof in t he characterist ic 0 case, where so me old frie nds of ours - differenti ally closed fi elds - a re involved. Then we will shortly comment the prime characteristic case, where differenti ally closed fields a re profitably replaced by the separa bly closed ones.
308 CHAPTER 8. MOD EL T HE ORY AND ALGEBRAIC GEOMETRY
So take t wo alge braically closed fields 1(0 -< I( of charact eristic o. Let A be an ab elian variety over I( wit h trace 0 over 1(0 , X be a Zari ski closed set in A , y be a su bgroup of A of finit e rank. Our claim is that X n G is a (possibly empt y) finit e union of cosets of subgroups of y. (a) W ithout loss of generality one ass umes that I( has infinite tra nscende nce degree over 1(0. Then one equips I( with a derivation D making I( a differential field , and even a differenti ally closed field, whose constant subfield C (I( ) coincides with 1(0. J ust to fix our symbols, let L deno te from now on th e usua l language for fields, and L' = L U {D} that of differential fields. So 1(0 is strongly minimal both as a str uctu re of Land L': in fact D is identically 0 on 1(0 and so adds no definable objects to the pure field 1(0 . On t he cont rary, I( is a st rongly minimal struct ure in L , as an algebraically closed field, but it is not any more as a differenti ally closed field; indeed 1( , although w-stable, has Morley rank w in L'. Not ice also t hat, owing to what we saw in t he past sect ions, the a belian variety A is definable (even in L) in 1(. (b) At this poin t one recalls a general resu lt of Manin on differenti al fields: the derivation D yields a group homomorph ism p (definabl e in L') from A onto (I(+) d , where 1(+ is the addit ive group of I( and d is t he dim ension of A. The kernel of p is definabl e in L' a nd has a finite Morley ra nk. Now we deal with y . As y has finite rank and 1(+ has no non zero torsion elements, t he grou p p( G) is finitely generated and t here are go, . . . , gm E K such that
p(G) ~
L
i<m -
Q . s. ~
L
](0 ·
gi·
i<m -
Let II denote L i<m ](0 . gi. H is definabl e (in L' ) and has finite Morley ra nk. Hence p-l (H) is a subgroup of A exte nding y; moreover p-l(H) is definabl e (in L') and has finite Morley rank because both H a nd t he kern el of p are definabl e of finite Morley rank. Without loss of gener alit y for ou r purposes , we ca n replace Y by p-l(H). In fact , if X n p-l (H) is a finit e union of cosets of subgroups of p-l (H) , t hen t he same can be said about X n G a nd G . So we can ass ume t hat y itself is definabl e a nd has a finit e Morley rank. Now, ju st to explain Hr ushov ski's ap proac h in a mor e accessible way, let us restrict a little mo re ou r framework to the particular case when X is an irreducible curve (the setting of the original Mordell Conjecture). If X n G is finite , then we are don e. Otherwise X n G - as a definabl e
8.7. T HE M ORDEL L-LANG CONJECTURE
309
set of finit e Mo rley rank - contains a defina ble strongly minimal subset 5 . As usual, 5 can be viewed as a strong ly minimal str ucture. (c) Now we use t he result of Hr ushovski a nd Sokolovic saying that Zilber's Tri chot om y Conjecture holds for strongly min imal sets definabl e in differenti ally closed fields of characteristic O. In fact , t hese strongly minim al sets a re Zariski struct ures , a nd so obey t he Hru sho vski-Zilber Theor em. T his a pplies to 5, of course . Bu t what Hru sho vski also poin ts out is t hat 5, as a Zariski st ruct ure , is locally modular. In fact, as 5 is st rongly minimal, it suffices to exclude finitely many points fro m 5 to get an ind ecomposable set . So t here is no loss of genera lity in ass uming th at 5 itself is indecomposable, a nd conseque nt ly each t ranslate bS with bE G is also ind ecomposable. Up to replacing 5 by b- 1 5 for a suitable bE G we ca n even assum e t hat t he identi ty element 1G of G is in 5 . Hence we are ju st in a positi on to a pply Zilber 's Ind ecomposability T heorem; accordingly, one ded uces t hat the subgrou p generated by 5 in g is definab le, and ind eed every element in t his subgrou p ca n be ex presse d as a ·c- 1 wit h a and c in 5. Hence 5 interprets an infinite gro up an d so, as a Za riski struct ure , it ca nnot be t rivial. T his means t hat eit her 5 is locally mod ula r or 5 interprets an infinite (algebraically closed) field. We have to exclude t he lat t er opt ion . To obtain t his, one uses a result of Sokolovic already ment ioned in 7.10 and saying what follows. (d) An infinit e field definable in a differenti ally closed field of characte rist ic 0 a nd having finit e Morley rank is isomorphi c to t he constant subfield by a definable fu nction . Recall t hat, owing to t he elimination of imagin aries , t here is no difference between definabl e or interpretable wit hin different ially closed fields. So, if 5 int erpret s a ny infinite field, t hen it defines even C (K ) = Ko up to an £I-definabl e isomorphism. Consequent ly the subgroup t hat 5 generates is isomorphi c t o some group go £I-definable in Ko by a function I also definabl e in L'. As D = 0 in [(0 , go is definable even in L (just as I and 1-1 ). T hen we can a pply t he Hru shov ski-Weil T heorem and dedu ce t hat go is an algebraic gro up over Ko up t o a n L-definabl e isomorphi sm. At t his point one checks t hat go defines a n abelia n subva riety of A in K , which cont radict s the hyp oth esis t hat A has t race 0 over Ko. In concl usion, 5 must be locally modu la r. Now 5 is of t he form (a + L ) - {bo, .. . , btl for some strongly minimal subgroup L of g and suitable a, bo, . . . , b, E G . X is Za riski
310 CHAPTER 8. MODEL THEORY AND ALGEBRAIC GEOMETRY
closed , a nd so contains t he closure a + L of S as well. Hence X equals a + L and consequently inherits a group structure, as Mordell 's Conjecture requires. This concludes our outline of Hru shovski 's proof in cha ract erist ic 0 (when X is a curve). Bu t , as already said , the real novelty of Hru shov ski 's T heorem concerns prime characterist ics p. So it is worth spending some word s also on this case. The pla n of the proof is simila r, but requires some necessary rearrangements. In particular, one can avoid to refer to differentially closed fields and directly handle sepa rably closed fields (with no derivation) . (a) F irst one replaces with no loss of gene rality K by a separably closed non algeb raically closed extension having finit e degree over KP. O bserve that now the theory of K is not w-stable, although it is stable. (b) T he role of the kernel of J-L is now played by n npn A , which is not a definable set, but is the intersection of infinitely many definable sets. The other crucial points in th e proof a re: (c) any strongly minimal struct ure definabl e in K is st ill a Zari ski st ructure (so Zilber's Trichotomy holds);
(cl) a field definable in K and having Morl ey rank 1 is isomorphic to Ko by a definabl e function (a result of M. Messmer). For more details, look at the referen ces quoted below.
8.8
References
The con nection between Mod el T heory a nd Algebraic Geometry is clearl y explained by Poiz at's book [131] (recentl y translated in English [135]): this is a very rich reference on t his matter. In particular it includes a proof of Hru shovski-Weil Theorem 8.6.2 . Cherlin's Conjectu re was raised in [24]. The classification of w-stable groups was given by Macintyr e in [89]. Grou ps of finit e Morl ey rank a re exam ined in [15]. Now let us deal with Mordell-Lang Conjecture and Manin-Mumford Conjecture. A geom etrical int roduction ca n be found in Lang's book [80]; a short but reson abl y ex ha ust ive history of these two questions is also in the recent Pillay paper [128]. Pillay's book [126] explains the main model theoretic techniques involved in Hru shovski's approach. Hrushovski 's original proof
8.8. REFERENCES
311
of t he Mo rdell-Lang Conjecture is in [60]. A det ailed exposit ion is in th e book [16]. [50] and [127] a re shorter surveys , both very read abl e. As already said , Hrushovski 's proof in cha racteristic 0 case is based on some pr elimin ari es concern ing differenti ally closed fields: t hey ca n be found in [155] and [62]. For a prime characterist ic, separa bly closed fields are enough: th e basic prelimin ari es are desc ribed in [115]. Finally, let us deal with Manin-M umford Conjecture: Hru sho vski 's proof is now in [61] . It is based on t he mod el t heory of ACFA, as develop ed in [20] a nd [21] .
Chapter 9
O-minimality 9.1
Introduction
The last part of this book is devoted to o-minimal st ructures . As we saw in t he past chapt ers , t hey a re the infini t e expansion s M = (M , :S; , . . .) of linear orderings such that th e subsets of M definabl e in M are as t rivia l as possibl e, and restrict to th e finit e unions of singlet ons and open int ervals, possibly with infinite end points ± oo (eq uivalent ly to t he finit e unions of op en , closed , ... int ervals in th e broad sen se including half-lin es and the whole M ). a-minimal str uct ures are not sim ple according to t he definition provided in Ch ap ter 7, ju st because t hey define and even expand infinite linear orders; conseq uent ly no good ind ep endence system can be develop ed inside th em , and t hey are not clas sifiable in She lah 's sense. Despi t e thi s, and just owing th e relative triviality of th eir 1-ary definable sets , one ca n see that t hey enjoy several relevant model th eor etic properties and , among th em , a sat isfact or y notion of independence with a related dimension partly resembling Morley rank . Furthermore, they include a lot of noteworthy algebraic exampies. Ind eed we hav e already seen that th e ordered field of real s (R , +, " - , 0, 1, :S;), as well as any real closed field , is o-rninimal; discrete or dense infinit e linear orders, like (N , :S; ), or (Z , :S;), or (Q, :S;), or (R , :S; ), ar e o-minimal as well; and one ca n show t ha t even divi sib le ordered abelian groups, such as (Q, +, 0, :S; ) and (R , +, 0, :S; ), are o-rninimal. In particular , consid ering t he order of th e reals, or expa nding it by addition , or addit ion and multiplication together , yield o-rninimal st ruct ures. On the oth er side, t here do exist some ex pansion of (R, :S;) which are not o-rninimal.
313
314
CHA PTER 9. O-MINIM ALITY
For inst ance, ext end the ord er of reals by t he sinus fun ction si n; in t his enlarged structure, Z -a denumer abl e set of isolated points- get s defin abl e (by sin 1r V = 0) , and so o-mi nimality get s lost. Of course the same argument applies to cos . T his cha pter has a two fold aim. On t he one hand , we will provide a n a bstract structure t heory for o-minimal models M. The sta rt ing poi nt will be ju st the definit ion of o-mi nimality, and t he consequent class ification of t he defina ble subsets of M . But we will see t hat, on t hese seemingly poor grounds , one ca n develop a sign ificant t heory, including a nice characterization of definable subsets of M " for every positi ve int eger n, as well as of definabl e function s from M" into M . This will also lead t o the alr ead y recalled notion of dim ension, satisfying several rem arkable properties resembling those of Morley rank in w-st abl e t heories. Act ually th ere is a good deal of similarity bet ween t he o-mini mal framewor k and t he w-stable set t ing, ju st in t hese di men sions, but also in definabl e groups and in other matters. We will em phasize t hese connections in Sections 9.2-9.6. In part icular Section 9.4 and Section 9.5 will prove, among other th ings, t he already ment ioned a nd not ewor th y fact t hat o-minimalit y, unlik e minimality, is preserved by elementary eq uivalence : if M is a n o-minimal st ructure, t hen every mod el of the t heory of M is o-minimal as well. Accordingly a complete th eory T is said to be o-minimal whe n some (equivalently every) model of T is o-minimal. T he subseq uent section 9.6 will treat definabl e groups, definabl e manifolds , and so on, in o-minim al st ruct ures. On t he other side , we will prop ose ot her relevan t exam ples of o-minim al struct ures (t o which t he previous gene ral t heorems apply) . T his will be t he t heme of Section 9.7, where we will see t hat certain ex pa nsions of t he rea l field by fam ilia r functions , such as ex ponent iation, or suitably restrict ed a nalytic functions, are o-minimal. Here o-minimality largely overla ps real algebraic geomet ry a nd real a nalytic geomet ry, both in acquiring tec hniques and constructi ons fro m t he geomet ric framework towards a general and lar ger spect re of applications, and in providing a new light and in op ening new per sp ecti ves wit hin t he geometrical setting itself. The subseq uent section 9.8 will deal with some variations on t he o-minimal t heme, most not ably wit h a notion of wea k o-m inimality , enla rging t he 0 minimal setting an d intensively st udied in t he latest years. At las t , t he final section 9.9 will int rodu ce very shortly t he qui te recent and attracti ve work of A. On shuus a bo ut a notion of ind ependence enla rging both forkin g ind epend ence in simple t heor ies and algebra ic ind ependence in o-minim al th eories t owards a common general fram ework. Now a few historical not es. O-minimality began its life in th e eight ies; its
9.1. INTRODUCTION
315
origin refers to a classical problem of Tarski, asking whether the real field, expanded by the exponential function x t--+ eX, still has a decidable theory. As we know, decidability closely overlaps definability, so a deeply related question is what is definable in (R, +, " -, 0, 1, :S, exp): is the theory of this structure quantifier eliminable, or model complete? This was the scenery where 1. Van Den Dries introduced o-minimality at the beginning of the eighties. But who gave a considerable impulse to this notion were A. Pillay and C. Steinhorn, who proved the basic structure theorems on o-rninimal structures and greatly developed their abstract theory. In 1991 A. Wilkie partly solved Tarski's Conjecture, showing that the theory of (R, +, " -, 0, 1, :S, exp) is model complete, and even o-minimal (as we will see in 9.7, decidability is still an open question, involving a deep number theoretic problem, usually known as Schanuel's Conjecture, while quantifier elimination fails). This emphasized t he connection with analytic geometry, mentioned some lines ago. And indeed o-minimality became, and still is, a matter of interest not only to model theorists, but to geometers and analysts as well. To conclude this section, we give the proof that any o-rninimal ordered field is real closed. This is the converse of a result we already know, ensuring that every real closed field is o-minimal, and can be viewed as the o-rninimal analogue of Macintyre's theorem saying that any w-stable field must be algebraically closed. The proof requires a very basic machinery from 0minimality -just the definition itself- in addition to the necessary algebraic grounds. Let us preliminarily examine o-minimal groups. Here (and later in this chapter) intervals possibly admit infinite endpoints, and so include half-lines, and the whole line, in case. Lemma 9.1.1 Let A = (A, 0, +, -, :S , . ..) be an o-minimal structure expanding an ordered group (A, 0, +, -,:S) and let H be a subgroup of (A, 0, :S) definable in A. Then H = {U} oppure H = A.
+, -,
Proof. Suppose towards a contradiction that there exists some subgroup H i- {O}, A of (A, 0, +, -, :S) definable in A. Owing to o-minimality, H decomposes as a finite union of pairwise disjoint intervals (possibly closed, or with infinite endpoints). Accordingly write
(0)
H =
U t, j :::; s
where 10, ... , Is are intervals and s is minimal. Notice that H is infinite, because it must contain all the multiples nh of any nonzero element h E H
CHAPTER 9. O-MINIMALITY
316
when n ranges over int egers. Hence t here is j ::; s for which I j is infinite. Without loss of generality pu t j = O. Moreover we ca n even ass ume th at l a contains 0 a nd is sy mmetric with respect to 0 (in t he sense t hat -c E l a for every c E la). This is becau se H is a subgro up, and so includes 0 and is preser ved under + a nd -. As H =f:. A, la is [-a , a] or] - a, a[ for some a > 0 in A. In t he form er case , eve ry b in A satisfying a < b ::; 2a decomposes as b = a + (b - a) where a E la ~ Hand 0 < b - a ::; a , so even b - a is in 10 a nd conseq uently in H ; hence b E H. Therefor e [-2a , 2a] is an interval in H properly including 10' T his impli es t hat [-2a , 2a] shares at least one element with some interval Ij where 0 < j ::; s , say with I s. Put I~ = [-2a , 2a]Uls .
So
Jb
is an interval including 10 U I s and contain ed in H . Co nseq uent ly
H = I~ U
U
O<j<s
i.,
a nd t his cont ra dicts t he choice of s in (0). In t he lat t er case, fix b E A such t hat 0 < b < a, t he n 0 < a - b < a , and consequent ly even a - b is in l a. It follows a = b + (a - b) E H. We get in t his way an interval [- a, a] properly including l a and contained in H . But t his contradicts as before t he minim alit y of s in (0). ., As a conseq uence, one ca n give a full characterization of o-minima l ordered groups. They ar e exa ctl y t hose listed before, namely t he ord er ed div isibl e a belia n groups. Theor em 9 .1.2 An o-m in imal ordered group A = (A , 0, and divi sibl e.
+, - , ::;) is abelian
Proof. For every a EA , th e cent ralizer C (a) of a is a definable subgrou p of A , a nd consequently equ als eit her {O} or A. If a = 0, then clearly C(a) = A. On t he other side, when a =f:. 0, C (a) = A as well, becau se a E C (a) and t his excludes C (a) = {a}. Hence C (a) = A for every a E A, a nd A is abelian . Now take a positive int eger n : n A is a definabl e subgro up of A , clearl y eq ualling A when A = {a}; on t he ot her side, if A =f:. {a}, t hen nA =f:. {a} an d consequent ly nA = A . Then n A = A for every positi ve int eger n , III other words A is divisibl e. .,
Coming back t o ordered fields, we ca n eventua lly pro ve
9.1. INTRODUCTION
317
Theorem 9.1.3 Let /C = (K, 0, 1, +, " -,::;) be an a-minimal ordered ring with identity 1 (in particular let /C be an a-minimal ordered field). Then /C is a real closed field.
Proof. First we claim that /C is an ordered field, in other words the set /C* of the nonzero elements in K is an abelian group with respect to -, It suffices to show that the set «> of the elements > 0 of K is so. In fact, for every a E «>, aK is a definable subgroup of (K , 0, +, -, ::;), and aK -I {O} because a > O. So, owing to Lemma 9.1.1, aK = K, and in particular there exists some b E K satisfying ab = 1. As a > 0, b is positive as well. This proves that x> is a(n ordered) group with respect to -, It remains to check that «> is also abelian. To show this, it suffi ces to observe that «> is o-minimal , and then to use Theorem 9.1.2. In fact «> is definable (as a group) in /C, and consequently every subset X of «> definable in «> is also definable in /C, and hence is a finite union of non-empty intervals of K. All the end points of these intervals lie in «> U {+oo}, with the only possible exception of the left most end point in the first interval, that might equal 0, but can be replaced in this case by -00 in «>. In conclusion, X is actually a finite union of intervals in «>. Hence «> is o-minimal and consequent ly abelian, as claimed. Now let us prove that /C is real closed. Accordingly take a polynomial f(x) E K[x] and two elements a < b in K satisfying f(a) . f(b) < 0, for example f(a) < 0 < f(b). We have to show t hat there is some c E K such th at a < c < band f(c) = O. Recall that /C is an ordered field, and hence ::; is dense in K and in la , b[. The polynomial function that f defines is continuous (with respect to the order topology) , so both la, b[+= {d E K : a < d < b, f(d) > O} and
la, b[-= {d E K : a < d < b, f(d) < O} are open sets. If la, b[-= 0, then the continuity of f is contradicted in a. Hence la, b[- and similarly la, b[+ are not empty. Moreover both la, b[+ and la, b[- are definable, and accordingly decompose as finite unions of intervals, indeed of open intervals. As la, b[+ and la, b[- are disjoint , t here is some c E]a, b[ out of both la, b[+ and la, b[-: so c is a root of [, f( c) = O. '"
318
9.2
CHA PTER 9. O-MINIMALITY
The Monotonicity Theorem
Let M = (M, ~ , . ..) be an a-minimal struct ure . First obser ve t hat M is a t o pological space with resp ect to t he ord er t opology a nd so, for every positi ve int eger n , M " is a topological space as well, with resp ect t o th e product t opology. As already recalled , t he o-minimalit y of M ju st requires t hat t he only definabl e subsets are th e finit e union of singlet ons a nd op en inter vals (possibly including half-lin es and t he whole M) . Bu t what ca n we say abou t th e defina ble subsets of M " when n > I ? We give here a very partial and pr eliminary an swer , dealing with l-ary definable functions f. We show t hat, if the domain of f is an open interval la , b[ with a < b in M U {±oo} , then one can partition la , b[ into finitely many intervals such that, in each of t hem, f is either const ant , or strict ly incr easing , or strict ly decreasing, and a nyhow cont inuous according to t he orde r topology. T his is t he so-called Monotonicity Theorem , say ing in detail what follows. Theorem 9.2.1 Let A1 be an o- m inimal structure, X ~ M, a, b E M U {±oo}, a < b, a an d b be X -defi nabl e when belonging to M . Let f be an X -defi nable f unction of la , b[ in to M . Th en there are a posit ive in teger n and aD, al , .. . , an E MU {±oo} suc h that
1. a = aD < al < ... < an = b, and aI , ... , an- l are X -defi nable;
2. fo r eve ry i < n , f is eithe r constant or strictly m on otonic in ]ai , ai+d; 3. fo r every i < n , if f is strictlu mono to nic in uu, ai+l[, then f( ]ai, ai+l [) is also an in terval, and f con tains a biject ion preserving or reversing ~ bet ween ]ai , ai+d and f (]ai' ai+d ) · In particular f is conti n uous in every inter val ]ai , ai+d for i
< n.
Notic e t ha t, when mor e genera lly f is an arbitrary definable function with both domain an d image in M , t hen the domain of f is definable as well, a nd conseq uent ly is a finit e union of singletons and op en int erval s. Each of t hese intervals satisfies t he ass umpt ions of Theorem 9.1.2 , and hence inh erits its conclusions. Notice also t hat Theor em 9.1.2 implies, as a simple conseq uence, t hat, if M is a n a-minimal structure, a, b E M U {±oo} , a < b a nd f is a definabl e function from la, b[ into M , t hen f (x) has a limit in MU {±oo} when x -T a+ and x -T b>. A full proof of Theorem 9.1.2, as stated before, would require seve ral tec hnical det ails a nd would be quite long. We prefer to propose here a simpler
9.2. THE MONOTONICITY THEOREM
319
argument showing only the continuity result when M expands (R, :S;) (actually this is more or less what we will need later). Proposition 9.2.2 Let M be an o-minimal structure expanding (R, :S;) , X ~ M, f be an X -definable function from ]a, b[ into R . Then th ere are aI, ... , an E]a, b[ such that al < ... < an, aI, ... , an are X -definable and f is continuous in ]a, al [, ]an, b[ and each interval ]ai, ai+l [ for 1 :s; i < n. Proof. Let S denote the set of the points in ]a, b[ where f is not continuous. It is an easy exercise to prove that S is X -definable. If S is finite , then we are done: for, aI, .. . , an are just the elements of S. Hence suppose S infinite. So S contains an infinite open interval I. For every natural n, build two infinite open intervals In and I n such that, for every n,
(i) In
~
I,
(ii) the (topological) closure I n+l of I n+l is included in In'
(iv) the length of I n is smaller than
nil'
Let us see how to define these intervals. First put ID = I. Then take any natural n , suppose In given and form I n and I n+l as follows. If f(In) is finite , then for some d E f(1n) the preimage {c E In : f(c) = d} is infinite. But {c E In : f( c) = d} is definable, and hence includes some infinite interval; f is const a nt , hence continuous, on this interval, which contradicts In ~ I ~ S. Accordingly f(In) must be infinite. But f(In) is definable, too, and hence includes in its turn an infinite interval; let I n be such an interval, notice that we can assume with no loss of generality that the length of I n is < The preimage of I n in In is also definable, and contains some infinite interval. Let I n+l denote such an interval; we can assume In+! ~ In. This determines the In's and the In's for every n. Now put I' = nnEN In. Clearly I' = nnEN In' whence I' =1= 0 because R is compact. Pick d E 1', we claim that f is continuous in d (this contradicts dES and so accomplishes our proof). Take any interval U containing f(d). Owing to (iv), there is some natural n for which U ;2 I n. But this implies U;2 f(In+l) where I n+l is an open neighbourhood of d. ...
nil'
A final remark. When M expands the field of reals, we can say even more, and state a smooth version of the theorem: in fact, one can partition ]a, b[ into finitely many intervals where f is of class cm for every positive integer m.
320
9 .3
CHAPTER 9. O-MINIMALITY
Cells
Now we move to characterize in an o-minimal struct ure M = (M, :S , . . .) th e definable subsets of M": in particular t he definable n-ary function s, for every positive int eger n . To make our life easier , we assume all throughou t t his sect ion (and also in th e following ones) t hat :S is dense without end points: in particular every interval ]a, b[ with - 00 < a < b < + 00 in M must be infinite. As alr ead y said , the reason of t his restriction is ju st to make our treatment and our proofs sim pler; in fact , all th e results we will show below ca n be extend ed -by the appropriate a rr ange ments- to any o-rninimal st ruct ure M. On the other side , t he ord er of reals is j ust dense without end point s, so our fram ework include all the exp an sions of (R , :S ), in particular all the st ruct ures enlarging t he real field; as we said befo re, the notab le o-minimal exam ples we will propose in Section 9.7 lie in t his set t ing. We know that the basic definable subsets of M are t he int ervals and t he singlet ons. More gener ally, th e basic definable subsets of M " are t he cells. So let us define what a cell is, mor e precisely what a k-cell in M " is.
Definition 9 .3 .1 First suppose n = 1. A sub set C of M is a O-cell if an d only if C is a si ngleton, and a 1-cell if and only if C is a non- empty open in ter val, possibly with infi nite endpoints . N ow let n > 1, and let k a natu ral number :S n. A subse t C of M " is called a k -cell if and only on e of the follo wing conditions hold: 1. there are a k- cell D of M n - 1 and a contin uous and defi nable fun ct ion f of D into M such that C is the graph of I, namely
C = {(11, b) E M " : 11 E D , b = f(I1)} ; 2. k ~ 1 and th ere are a (k - I )-ce ll D of M n-l and two fun ctions f and g with domain D such that
(i) eithe r the image of f is a subset of M and f is both continuous and definable, or f(l1) =
-00
VI1 E D ,
(ii) either th e im age of g is a subset of M and g is both continuous and definable, or g(l1) = + 00 VI1 E D , (iii) f(l1) < g (l1) VI1 E D , (iv) C
=
{(11, b) E M n : 11 E D , f (l1)
< b < g (I1)}.
On e eas ily sees t hat every k-cell of Mn is defin able, and t hat a k-cell of jU n is op en in M " if and only if k = n . One can also observe
9.3. CELLS
321
Proposition 9.3.2 Let M be an o-minimal str ucture, k S; n be posit ive integers. For every k-cell C of M" , there is a definable homeomorphism of C onto a k- cell of M k • Proof. It suffices to see , for n > k a nd n > 1, how to det ermine, for every k-cell C of M n , a definable homeomorphism 1re onto some k-cell C' of Mr:" , and then t o iterate th is procedure as long as one needs. First ass ume that C is t he gr aph of some cont inuos definable func tio n fro m a k-cell D of M n-l into M. In this case it suffices to put C' = D , and choose as 1re th e proj ection of C onto C' . Now ass ume
C = {(a , b) E M n
:
a E D , f (a)
< b < 9 (a)}
where D is a (k - I)-cell of M n-l , and I, 9 sat isfy the cond it ions in 9.3.1 , 2. We pro ceed by ind uction on n , If n = 2, then k - 1 = 0, and D red uces t o a single point a of M. P ut C' =JJ(a) , g(a)[ , 1rc(a , b) = b for every b E C '. Now let n > 2. We know th at there is some definabl e homeomorphism 1rD of D onto a (k - 1)-cell D' of M n-2. Consider the two fu nctions one gets by composing l , 9 respectively a nd the inver se of 1rD . T hey have dom ain D' a nd sat isfy t he ass um pt ions in 9.3.1,2. Accordingly they define a k-cell C ' of M n - l ; further mo re 1rc(a , b) = (1rD(a) , b)
V(a , b) E C
det ermines a definab le homeomorphism of C onto C'.
...
When M expands a real closed field (for instance, when M is the real field , or even a n expansion of it) , the cells in M ca n be charact erized as follows. Proposition 9.3.3 Let M be an o-minimal expansion of a real closed fiel d, k , n be two natural nu mbers satisf ying n 2:: k , 1. If C is a k -cell of M n, th en th ere exis ts a definable homeomorphism of C onto ]0, I [k. Proof. When k = 0, our claim is trivial, becau se a O-cell reduces t o a singlet on. So t ake k > 0. Owing to the previous proposition, we can assume n = k. We proceed by induction on n . If n = 1, t hen C = ]a, b[ where a , b EMU {±oo} a nd a < b. So the required homeomorphism between C and ]0, I[ is easily obtain ed: for instance, if both a an d b are in M , th en it suffices to map every x in C = ]a, b[ into x - a b- a'
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CHAPTER 9. O-MINIMALITY
in t he remaining cases one proceeds in a simila r way. Now s uppose n > 1 (a nd our claim true for n - 1). Let C be an n-cell of M n. There do exist an (n -I )-cell D of M n-l and two functions f and 9 as in 9.3.1 ,2 such t hat C is t he set of t hose t uples (ii, b) in M " for which ii E D a nd f (ii) < b < g(ii). By induction , t here is a defina ble homeomorphism h of D ont o ]0, l[n-l. If both f and 9 take th eir values in 1\1, t hen
( a,~ b) ~
(h ( ~)
b - f (ii) ) V(ii, b) E C
a , g(ii) _ f(ii)
is th e required hom eomorphism. All t he other cases ca n be handled simila r way, ju st as when for n = 1. •
III
a
In particular, when M expands th e real field Rand k > 0, every k-cell of M is connecte d in th e ord er topology ; for, ]0, l[k is. This do es not hold any longer when R is replaced by any real closed field. For instance, if Ra is t he orde red field of real algebraic numbers, th en Ra is real closed , and hence o-minimal; Ra is a Lcell of itself, but is not connected becau se, for every real t rascende ntal t , Ra pa rti tion s as R a = {r E R a : r < t } U {r E R a : r > t}. Hence connect ion gets lost. However ever y cell in an o-minimal st ruct ure M sa t isfies a wea k form of con nection, wit h resp ect to ope n definable sets . In fact , conside r t he following notion. Definition 9.3.4 Let M be an o-m inimal structure, n be a posit ive integer. A defi nable set X ~ M " is said to be definably connected if and only if X cannot part it ion as the disjo int un ion of two non-empty open definable subsets . Wh at we will see now is t hat every cell in an o-minimal struct ure M is definably connected . First we give an equivalent charact erization of defin abl y connecte d sets. An open box of M n is t he ca rtes ia n product of n op en int erval s of M . Hence op en boxes form a basis of op en neighbourhoods of t he product topology of M n. Furthermore, for Y ~ X ~ u», an element s of X is called a boundary point of Y in X if a nd only if every op en box of M n containing ii overlaps both Y a nd X - Y. Lemma 9.3.5 Let A1 be an o-m inimal structure, n be a posit ive in teger, X be a definable su bset of M": T hen X is defi nably connected if and only if, fo r every proper non-em pty defi nable subset Y di X , X contains at least one bounda ry point of Y in X.
9.3. CE LLS
323
Proof. X is definably connected if a nd only if, for every definable Y ~ X such t hat Y =1= 0, X , eit he r Y or X - Y is not op en , in other words ther e is eit her a point a E Y such that every open box cont aining a overl aps X - Y , or a point E X - Y such that every open box cont aining overl aps Y . Accordingly X is defin ably connected if a nd only if, for every Y as a bove, t here is a E X which is a bound ary poin t of Y in X . ..
a
a
Now we can show, as promised, T h e orem 9.3.6 Let M be an a-minimal structure, n be a posit ive integer. T hen every cell C of M n is defi nably connec ted.
Proof. We proceed by induction on n . If n = 1, t he n t he claim is trivial becau se the cells of M red uce to singletons a nd open intervals. Hen ce assume n > 1. Let C be a cell of J\;f n . If C is a O-cell, so a singleton, t he n C is definably connected . If C is a k-ce ll for some positi ve int eger k < n, t hen, owin g t o Corollary 9.3.3 , C is defin ably hom eomor phi c t o a cell C' of M k ; by t he induction hypothesis , C' is defin ably co nnected, wh en ce C is defin ably connected , t oo . Fin ally suppose t hat C is a n n-cell of M n. Then ther e exist a cell D of M n-l a nd two fun ctions 1 and 9 as in 9.3.1, 2 such th a t C = {( a, b) E M n : a E D , 1(a) < b < 9 (a) } . D is definably connected by t he induction hyp oth esis. To deduce that even C is defin ably connected we use Lem ma 9.3.5 . Accordingly take a definabl e subset Y of C such t hat Y =1= 0, C . F irst sup pose t hat, for some a E D , t here a re t wo eleme nts b an d b' in M suc h t hat (a, b) E Y a nd (a, b') E C - Y. Then t he int erval ]1(a), g(a)[ contains at leas t one boundary poin t bo of t he definable set {b E M : (a, b) E Y}; t his implies t hat (a, bo) is a bo und ary po int of Y in C . Now suppose t hat , for every a E D , either
{ (a, b) E C : bE M} or
{ (a, b) E C : s « M }
~
~
Y
C - y.
Let Z den ot e t he set of t he eleme nts a of D satisfying t he for mer cond it ion. Y =1= 0, C implies Z =1= 0, D . As D is defin abl y connected, t here is some boundary point ao of Z in D. Let b E M sat isfy (aa , b) E C, we claim th at (aa, b) is a boundary point of Y in C . Let B a n op en box of M n containing (aa, b). As C is open , we can suppose B ~ C . Let B ' denote the projection
CHAPTER 9. O-MINIMALITY
324
of B in M n - 1. Then ao E B ' a nd B ' ~ D. Since ao is a boundary point of Z in D, there are a1 and a2 in B ' such t hat a1 E Z a nd a2 tj Z . Con sequently, if b1, b2 E M satisfy (a1 ' b1), (a2 ' b2) E C respectively (in particular (a1 ' b1), (a2 ' b2) E B) , then (a1' b1) E Y, (a2' b2) tj Y . Hence (ao, b) is a bou ndary poin t of Y in C . In conclusi on , C is defina bly connect ed . ..
9.4
Cell decomposition and other theorems
The aim of this section is to introduce and to state t he basic general theorems for o-rninimal struct ures M = (M , ~ , . . .): we want to characterize all the defin a ble sets and funct ions in M , a nd we wan t also to emphas ize some relevant consequences a nd, am ong t hem, the already ment ioned fact t hat 0minimality is preserved und er elementary equivalence. As th e proofs of t hese fundamental results a re qui te long and intricate, we will defer part of t hem, a nd t he cor responding de tails, to t he next section ; here we provide ju st a basic outline, illustrating these cent ral core s of the theory of o-minimalit y. So a reader simply int erested in a general view may limit her , or his attent ion t o t his section, and to skip the next one. Let us remind once again that , for simplicity 's sake, we a re assuming that (M , ~) is dense without endpoints: this is tacitly acce pted all throughout these sections, unless otherwise stated. The first result we prop ose j ust describes definabl e sets (and functi ons) in o- rnini mal struct ur es. It is a beau ti ful a nd powerful cha racterization, called Cell Decomposition Theorem. In fact , it says t hat every definabl e set decomposes as a fi nite union of cells. Theorem 9.4.1 Let A1 be an o-m inimal structure, n be a positive integer. 1. Eve ry definable set X uni on of cells in M n.
~
M " can be expressed as a fi nit e (disjo int)
2. Furthermore, if X is the domain of a definable fun ction f with values in M , then on e can decompose X as a fin it e disjo int union of cells, such that f is contin uous on each of them . Not ice that t his generalizes what we know when n = 1; in fact, in that case every definable X ~ M is a finit e union of point s and open intervals (in other words, of O-cells a nd I-cells respectively), a nd, when X is th e do main of some definable fu nction I, one can also suppose that f is continuous on eac h of these pieces, owing to the Monotonicity Theorem. But now we can ext end t hese results t o any n .
9.4. CELL DECOMPOSITION AND OTHER THEOREMS
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As already said , t he proof of the Cell Decomposition Theorem will be given in detail in t he next sect ion. But Cell Decomposition , and it s a nalysis of definable sets, is also t he key t ool in showing another basic fact in 0minimality, that is it s preserving under elementary equivalence. Theorem 9.4.2 If M is an a-minimal structure, then the theory of M a-minimal.
is
In fact , which is t he trouble in this claim ? Actually we do know th at, for a n o-minimal M, for every formula O( v, w) in the lan gu age L of M a nd t uple d in M , O(M , il) is a finite union of (possibly closed) intervals . But suppose th at , when ii ranges over M, the minimal number of t he intervals involved in these decompositions of O(M , il) cannot be upperly bounded , and so, for every natural N, on e find s some tuple il( N) in M such that any decomposition of O(M , il( N )) requires at least N intervals. If this is t he case , then it suffices a st raightforward application of Compactness Theorem to provide some M' == M and some t uple a' in M' for which O(M' , a') can not be expressed as a finite union of int ervals , and con sequently to conclude t hat the t heory of M is not o-rninimal. Hence the cr ucial point in showing that the o-rninimality of A1 is preserved by element ary equivalence is to uniformly bound , for every form ula O( v, w) as befor e, the minimal number of intervals necessary to decompose O(M , il) wh en il ranges over M. T his is a definability que stion concerning formulas in arbitrarily many free vari ables, and so directly refers to Cell Decomposition. On the other hand, boundi ng t he number of the invol ved int ervals in O(M , il) is the same as bounding the total number of their end points (forming a defina ble, and finite set). So the key ste p towards the proof of Theor em 9.4 .2 is Theorem 9.4.3 Let M be an a-minimal structure in L , Ifl(V, w) be an Lformula such tha t, for all il in M , Ifl(A1, il) is finit e. Th en there is a positive integer N such that, for eve ry il in M , IIfl(M , il) I ::; N.
The proof of Theorem 9.4.3 will be deferr ed until the next section. But , as we have just pointed ou t , T heorem 9.4.2 is an almost immedi ate conseq uence of Theorem 9.4.3. Let us see in detail why.
Proof. (Theorem 9.4 .2) Let L be the language of M , and let 7)( v , w) be an L-formula ; in pa rticular, let n deno te the lengt h of w. For every d E M"; O(M, il) is a finit e union of intervals. Let Ifl(v, w) be t he L-formula saying v is an end point of 0(... , w).
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CHAPTER 9. O-MINIMALITY
T hen, for every ii E M."; cp(M, ii) is finite; hence, by Theor em 9.4.3 , there exists a positive integer N such that , for every ii E M" , cp(M, ii) cont ains at most N elements, whence B(M , ii) has at most N end poin ts. Bu t t he sentence ViiEj ~ N v cp( v, w) remain s t rue in every struct ure M' == M. Accordingly, for every A1' == M and bE B(M' , b) has at most N end poin ts, and hence is a finit e unio n of intervals. In conclusion t he t heory of M is o-minimal. ..
u»,
The bounds given by Theor em 9.4.3 on formul as B( v, w) ca n be extended to formulas B(V, w) with an a r bit rarily long v. In fact the following result holds . Theore m 9 .4.4 Let M be an o-minimal structure in L , B( V, w) an L formula (n be the length of v and m be that of w) . Th en there exists a posit ive integer N such that , for every M' == M and tuple ii in M ' , B(M, n, ii) is the un ion of at most N cells in M' . Before beginning t he proof, a nd ju st for pr eparing it , let us premi t a sim ple exa mple. Ass ume n = 2. Let B(V I, V2, w) be a n L-formul a , ii be a t uple in M. Suppose t hat t he definabl e set B(M 2 , ii) decomposes as a disjoint union of 2 cells in NI : the form er is a singlet on, so a O-cell, while t he lat ter is a I-cell, a nd more precisely is t he graph of a cont inuous definable func tion f whose dom ain is t he open int erval ]a, b[ wit h a < bin M . Notice t hat t here a re a n L-formula 1]( V I, V2 , Z) a nd a sequence e in J\1 such t hat, for every Cl and C2 in M with a < Cl < b,
Now consider the L-formula
1\
"1](', " Z) defines a cont inuous function of dom ain ]u, w[" 1\ 1]( V l, V2,
Z))) 1\ -, (u
.) = 2" V>. > ~o . With respect to th e count a ble framework , it is wort h emphasizing t ha t Vaught 's Conject ure holds, in t he following st ro ng form. Theorem 9 .4.8 (Mayer) Let T be a (com plete) o-m in imal theory. Th en, up to isomorphism, either T has conti nuum many countable mod els, or there are two naturals n and m such that T has 3n . 6m countable models. Of course, one might get curious in reading t he st ate ment of this th eorem: why, and where, do n and m arise? Basically, they depend on a careful a nalysis of types in o-minimal structures. The interested read er may directly consult Laura Mayer 's work , quoted below .
9.5
Their proofs
This section provides t he proofs of Theorems 9.4.1 and 9.4.3 , stat ed in Section 9.4. As said, th ey are long and intrica t e. In spit e of this, we think it right to propose th em for at least two reason s. Th e form er (and th e principal) is t hat we believe t hat T heorem 9.4 .1 (t he Cell Decomposition Theorem) and T heor em 9.4.2 (the one saying t ha t o-rnin imalit y is pres erv ed under element a ry equivalence) a re two beau tiful and fundamental results and deserv e a full report , including a t echni cal preliminary like Theorem 9.4 .3. The la tter reason just concerns t he intricacy of t he proof; actually this is du e t o it s length and ingenui ty, bu t do es not depend on a relevan t a nd deep t heory, ind eed t he premises it needs a re rather element ary and accessible (they ju st include the definition of o-minimality, t he Monotonicity T heor em, some properties of cells and an induction argument). So our ex position should require no pa rticular efforts but a lit tl e attention and pat ience . And anyhow the reader who is not intere sted in th ese details may neglect this section and proceed directly to th e next ones, t hat will not use these proofs. So conside r a n o-minimal st ructu re A1 in a lan gu age L ; for simplicity, we keep our ass umption t hat t he order of A1 is dense withou t end poin ts. W hat we said in the last section in introducing Theorem 9.4.3 s uggests t he followin g definition . Definition 9.5.1 Let
