Unidimensional Theories: An introduction to geometric stability theory

Logic Colloquium '81 H:D. Ebbinghaus et al. (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1989

73

Un id im ens iona 1 Theories An introduction t o geometric s t a b i l i t y theory

Ehud Hrushovsk i Prince ton University I . Totally categorical theories.

This survey is intended t o serve as a possible introduction t o s t a b i l i t y theory, from what i s called the geometric point of view. The unidimensional theories are those involving only one "geometry"; most of the recent achievements in the area are r e f l e c t e d here. We w i l l r e s t r i c t a t t e n t i o n t o t h i s class. The best understood corner, and the basis f o r more general conjectures, i s the subclass of t o t a l l y categorical structures, and we s t a r t there. Let T be a t o t a l l y categorical theory. This means t h a t T i s a complete f i r s t order theory in a countable language, w i t h exactly one model (up t o isomorphism) in every i n f i n i t e cardinality. Let M be a model of T. We are p r i m a r i l y concerned w i t h understanding T up t o bi-interpretabi l i t y . In model-theoretic language, t h i s means that we work not w i t h M , but w i t h "Meq", the many-sorted structure obtained from M by adjoining a s o r t for S/E, whenever E i s a 0-definable equivalence r e l a t i o n on a definable subset S of PIn.

Once w e

have a detailed description of Meq, possibly in terms of other sorts, we have a basis f o r returning t o the study of the structure on M i t s e l f . Work supported by United States National Science Foundation Mathematical Sciences Postdoctoral Fellowship.

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E. Hrushovski

The f i r s t component of the description of M i s the existence of an i n f i n i t e definable subset of Meq, whose internal structure i s either t r i v i a l , or else t h a t of a projective space over a f i n i t e field. D can be chosen t o be definable w i t h o u t parameters; it i s then unique up t o a unique definable isomorphism. This i s among the deepest r e s u l t s in model theory; the theory of t o t a l l y categorical structures i s entirely based on it. It i s important t o explain precisely what it means. Let F be a finite field, and l e t V be an infinite-dimensional vector space over F. Consider the structure Y = ( V ,+, E,f 1,...,fk). where xEy i f f (X=CXYand y=$x, some

o(.$ [F),

( f l,...,fk) i s a subfield of F, and fi(x)=f i-x.

There i s a s l i g h t technical ambiguity here, in that the subfield (f l,...,fk1 i s not specified; f o r the purposes of t h i s exposition t h i s point should be ignored, and each scalar m u l t i p l i e r w i l l be assumed given as p a r t of the structure. y i s an extremely simple structure; it has quantifier-elimination, and the structure o f the definable sets i s transparent. Vn has a f i n i t e number of definable subgroups (they have the form Zwixi=0); the O-definable subsets of Vn are j u s t the Boolean combinations of these subgroups; the definable subsets of Vn are the Boolean combinations of their cosets. For technical reasons, it i s necessary t o consider not V but the corresponding p r o j e c t i v e space P=V-(O)/E.

The structure of P i s what is induced on i t by V: a

O-definable r e l a t i o n on P has the form ((xl/E ,...,xn/E): where R i s a O-definable r e l a t i o n on Vn.

(X

,,....xn)eR1,

It turns out t h a t V cannot quite

be retrieved from P; f o r each a l p , one can define in Peq an Abelian group Va isomorphic t o V. but when a+b, the resulting isomorphism va+vb i s

in general not canonical.

Unidimensional Theories

15

Now l e t D be a definable subset of a model M (over parameters b). To say that the induced structure on D i s th a t of a p r o j e c t i v e space over F means the following. F i rs t, one can choose a collection o f b-definable

subsets of Dn, so that Dn becomes elementarily equivalent t o the structure P described above. (Under some indexing of the relations). Secondly, there i s no further structure on D: every b-definable subset o f Dn i s equivalent t o a first-order formula constructed from relations in t h i s collection. Thirdly, a s ta b i l i ty condition w i t h o u t which t h i s notion i s ill-behaved: every subset of Dn definable w i t h parameters from M i s also definable w i t h parameters from D. and t h i s happens uniformly. In stable structures, the t h i r d condition i s automatic. Next we describe the way in which a given t o t a l l y categorical structure i s controlled by i t s projective space D. Turning the question around, we can ask: what structures can be b u i l t around a p r o j e c t i v e space, that inherit the t o t a l categoricity from i t ? We w i l l describe a process t hat can be iterated, and that gives r i s e t o every t o t a l l y categorical structure after f i n i t e l y many steps. Assume M i s a to ta l l y categorical structure containing D as i t s O-definable projective space. Let SCMn be a definable set, k an integer. The t r i v i a l k-cover F o f M over S i s the model N obtained by adding

-

S=Sx(O. ....k - I ] as a new sort t o M, together w i t h the p r o j e c t i o n ?+S, and

nothing more. Each of

-

R, M

interprets the other, and one sees easily t h a t

M i s also t ot ally categorical.

(However, n,N are not bi-interpretable. For t o t a l l y categorical structures,

interprets M i f and only i f there exists a continuous

surjective map j:Aut(fi)+Aut(M));

in the present case such a map exists

in either direction, but the groups are not isomorphic (as topological groups). See [A21 f o r a discussion of such considerations.)

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E. Hrushovski

Somewhat more interesting i s the t r i v i a l a f f i n e cover of M over S . Assume that for aaS, there i s given Abelian group VaSMeq definable uniformly in a, and isogenous t o V. (1.e. isogenous t o one of the Va's mentioned above. In r e a l i t y a l l connected 1 -dimensional groups definable in M are isogenous. Gl,GZ are isogenous i f G1/F1=G2/F2 f o r some finite normal subgroups F1,F2, and here the isomorphism i s meant t o be definable. One could specify Va uniquely by demanding that Vanacl(a)=O, but we do not need t o be canonical here, and w i l l not do this.) Let X a be a V,-affine-space;

t h i s means that w e are given an

action of (Va.+> on (Xa,+) isomorphic t o t o the action o f Va on i t s e l f by translation. Assume a l l the Xa's are disjoint. Then the t r i v i a l a f f i n e cover N of M ( w i t h respect t o the family Va, aaS) i s obtained from M by adding X = u a X a as a new sort, together w i t h the p r o j e c t i o n x-a (xaXa), and a r e l a t i o n t h a t gives the Va-affine-space structure of Xa, uniformly

in a. Again, each of M,N interprets the other, though they are not bi-interpretable, and N i s t o t a l l y categorical. This i s the well-understood part of the theory. A f t e r taking a f i n i t e or a f f i n e cover, however, there i s a step over which w e have almost no control a t a l l . Let M be t o t a l l y categorical and N a t r i v i a l cover of one o f the t w o types. Then a cover of M w i t h skeleton N i s any expansion of N that induces no new structure on M. Typically, N induces no new structure on any fiber. (In the a f f i n e case, if any new structure i s induced on a generic fiber, than the covering collapses t o a simpler one, covering a set of smaller dimension. The f i n i t e case could also be defined so as t o make a s i m i l a r statement true, by considering t r i v i a l

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77

covers by a given f i n i t e structure.) Nonetheless i t i s possible f o r N t o include new relations among the different fibers. We i l l u s t r a t e t h i s w i t h an example. Let A be the countable free (2/9Z)-module. categorical. From the point of view of our analysis. it

A i s totally

looks as follows.

The single element 0 can be disregarded, since we are concerned w i t h A only up t o biinterpretability. Let B=3A=(xlA: 3ycA. 3y=x}. Then B has naturally the structure of a GF(3)-vector space. Let D=(B-(O))/-

be the

corresponding projective space. One can check that the structure induced on D i s precisely the projective structure. Going backwards, 8 4 0 ) i s a (double) cover of D. Wi th each blB-(O) i s associated the GF(3)-vector space B. A-(0) i s an a ffi n e cover of B-(0):

the p r o j e c t i o n

map i s a-2a. and the a ffi n e structure on a fiber Ab=(y: 2y=b) i s given by (x,y)-x+y

(xaB.ytA,,).

Note that in both the double cover and the a f f i n e

cover, our description misses the group structure on the cover; it cannot be f u l l y recovered fro m the induced structure on the quotient and on each fiber. The problem of classifying the possible structures on a given t o t a l l y categorical skeleton remains open, even fo r the case of a single finite cover o f a project i v e space. We do have a global theorem l i m i t i n g the number of such expansions. The expansions of a given skeleton M are p a r tially ordered by inclusion (of the set of definable relations,) w i t h rl i t s e l f as the minimal element. We know that t h i s p a r t i a l l y ordered set i s well-founded:

there i s no sequence M
expansion of Mn.

It fo l l o w s th a t i f M i s to ta l l y categorical, there e x i s t s

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E. Hrushovski

a f i n i t e set of re l a ti o n symbols L such th a t every r e l a t i o n on M is equivalent t o some f i r s t order combination of the symbols in L. Moreover, upon r e s t r i c t i n g t o the f i n i t e language L. M becomes almost f i n i t e l y axiomatizable in the following sense: there e x i s t s a f i n i t e set of sentences To of L. such th a t i f M’kTo and card(M’)=card(M). then M ’ W .

The well-foundedness of the class of expansions i s also the key idea here. It follows, in particular, that there are only countably many possible expansions of a given skeleton. This suggests attempting t o classify the maximal expansions of a given skeleton. This has been done in the disintegrated case - when the projective space i s in fa c t degenerate, i.e. a structureless set. Even there cert ain problems associated w i t h finite nilpotent groups prevent us from a f u l l cla s s i fi c a ti o n of the intermediate expansions. In addition, a complete solution fo r the case of the double cover of a p r o j e c t i v e space over GF(2) has recently been announced by Ahlbrandt and Ziegler. The result s described in t h i s section can be found in [BLIJZ 1 I,[CHLI,[H21; t h i s sequence can be read without further references,

except f or a def in i ti o n of Morley rank (from [MI; also given in s2). [BL] i s more general, and w i l l be described in the next section. [ Z l l gives the fundamental theorem on the existence of a (possible degenerate) projective space w i t h not further structure in a t o t a l l y categorical theory. To appreciate the impact of t h i s result, and the progress it made possible in the area, i t i s instructive t o look a t [BCMI, where it was nontrivial even t o c l a s s i fy the to ta l l y categorical rings. [CHL] widened the context and proved the co-ordinatization theorem described above. (In the t ot a l l y categorical context some version of t h i s was

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79

known t o Zil’ber.) [A21 proved the well-foundedness, and the quasi-finite- axiomatizability, in the case where only finite covers are involved (no af f in e ones.) The general case is in [H2, s21. It remains t o be seen whether the fi n a l form o f the theory w i l l include the combinatorial device used t o prove well-foundedness, or whether an e x p licit understanding of the possible expansions w i l l eventually make i t unnecessary.

Intermediate generalizations were achieved in [ L I I and

[ C I I . ( S t i l l excluding affine covers, but allowing orthogonal covers and several dist inct projective spaces.)

The c l a s s i f i c a t i o n o f the maximal

structures in the disintegrated case i s in [H2,§41, f o l l o w i n g work in [ L I I . There i s no reference yet t o the c l a s s i fi c a ti o n in the case of the double cover, 2-element field. 2. N,-categorical structures.

There are t w o points of view regarding the r e s u l t s discussed in the previous section. One i s t o regard No-categoricity as the m a i n assumption; then the second assumption, of N1-categoricity,

iS

somewhat too res tri c ti v e , and one can look f o r a wider context in which s i m ilar result s are valid. The main tool then i s reduction t o f i n i t e questions, and the c l a s s i fi c a ti o n of the f i n i t e simple groups. See [El,[KLMl and [C21 f o r a development from t h i s direction.

Here we w i l l take the point of view of s t a b i l i t y theory. Our goal in t h i s context i s a theory of N1-categorical structures: the success in the locally finite (or No-categorical) case i s an encouraging, but extremely special case. The guiding philosophy here i s Zil’ber’s conjecture.

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E, Hrusho vski

Roughly speaking it says t h a t an N,-categorical theory i s a geometric theory; it comes either from linear algebra or from algebraic geometry (over an algebraically closed field.) There i s also a degenerate case; there i s somewhat more t o it than in the t o t a l l y categorical case, but w e w i l l ignore it here. (It would not m a t t e r t o the general picture i f some bigger class o f degenerate structures existed, if it could be isolated and controlled, and one could show that it cannot support any coverings of interest, or otherwise interact w i t h the ambient model.) Let us s t a r t by describing the known structures of non-linear type. If p i s prime or 0, the theory of algebraically closed f i e l d s of

characteristic p i s complete and N,-categorical.

The theory has

quantif ier-elimination, but the structure of the definable subsets o f K n i s no longer simple; much of algebraic geometry is devoted t o describing it. Every f i n i t e cover of an N1-categorical structure i s s t i l l

N1-categorical, as one easily verifies. The analog of a f f i n e coverings i s no longer finite-dimensional, however; the Abelian groups Va of the a f f i n e covering must be replaced w i t h arbitrary algebraic groups. (They may be assumed connected and simple or irreducible Abelian.) To be precise, l e t M be an N1-categorical structure, b u i l t over ( K , + ; , C ) ~ , ~ ~ (KO i s a set of distinguished constants). Let S be a sort in M, l e t Ga (agS) be an algebraic group over K, definable uniformly in a. Let Xa be a Ga-set, isomorphic t o Ga; assume the Xa's are p a i r w i s e d i s j o i n t and d i s j o i n t from M, and let X=uaaSXa. Then the t r i v i a l principal cover of M over S associated w i t h the f a m i l y (Gal i s given as before, by adding X as

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Unidimensiowl Theories

a sort, together w i t h the projection and the group action on each fiber. An arbitrary principal cover i s obtained by adding structure between the fibers in a way that i s invisible t o both PI and t o each fiber. Every known N,-categorical structure i s either degenerate or of linear type (in which case it i s very w e l l understood, as w i l l be shown below), or can be obtained from an algebraically closed f i e l d by taking f i n i t e and principal coverings a f i n i t e number of times. Zil'ber's conjecture states that a l l N,-categorical structures are in fa c t of t h i s form. The usual statement of the conjecture implies another claim. Note that one has a natural class of f i n i t e coverings of the f i e l d internal t o (i.e. definable over) the field, namely the algebraic curves together w i t h a projection t o the field. The "8"conjecture states that every f i n i t e model-theoretic cover of an algebraically closed f i e l d can be realized algebraically, as a reduct of a possibly reducible algebraic curve, possibly w i t h f i n i t e l y many points added or deleted. This i s somewhat reminiscent of Riemann's existence theorem

- that

every topological

cover of the Riemann sphere (punctured in f i n i t e l y many points) can be realized algebraically. It is a very interesting statement, but not central t o our picture of the field, and we shall say no more about it. It would follow from Zilber's (A) conjecture that the above picture

i s valid not only fo r the known structures but for a l l of them. We w i l l now say how much of the picture is known t o be true.

. . . e f i n i t i q n A structure D is strongly minimal i f every definable subset of D i s uniformly f i n i t e or co-finite. The u n i fo r m i t y condition means this:

if R c D ~ x Di s definable, and TtDn, l e t R ( ~ = ( x : ( ~ , x ) ( R ) .Then f o r each R,

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82

some integer m. f o r a l l

i f R ( 2 i s f i n i t e then it has a t most m

elements. This d e f i n i t i o n mentions only definable subsets of D i t s e l f . One deduces from it, however, the possibility o f ' d e f i n i n g the dimension (Morley rank) of a definable subset of Dn. Assume w e have defined the dimension of definable subsets of Dm, (always an integer cm), and we have: (*) For each ism, and each definable RLDnxDm, 6 # D n : dim(R(g)=i) i s a

definable subset of Dn. Then f o r a definable ECDm+ '=DxDm, f o r each ism, (a#D: dim(E(a))=i) i s f i n i t e or cofinite; so it i s c o - f i n i t e f o r exactly one value iosm; and the maximal value i s jo. Define dim(E)=max(io+l,jo). Then one sees easily t h a t (*) holds f o r m + l also. This d e f i n i t i o n depends on the choice of co-ordinates. but it has many good properties that yield invariant characterizations. Morley's was this: dim(E)=O i f f E i s finite (and nonempty), and dim(E)m+l i f f there exists an i n f i n i t e set of p a i r w i s e d i s j o i n t subsets of E, each of dimension a t least n. A definable set

E of i s said t o have m u l t i p l i c i t y

1

if it cannot be s p l i t i n t o t w o d i s j o i n t sets of the same dimension. Any definable subset of Dn i s the d i s j o i n t union of a f i n i t e number of m u l t i p l i c i t y 1 sets of the same rank; t h i s number i s called the m u l t i p l i c i t y o f E. The key point about t h i s notion of dimension i s t h a t it applies t o a l l definable sets. This can be rephrased as a strong homogeneity property. Let VLDn be a definable s e t of m u l t i p l i c i t y 1. Over any set B of parameters, consider the set of a l l elements of V t h a t do not l i e in any

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B-definable set of smaller dimension. These are called the generic elements of V (over 6). Any t w o of them look exactly the same over 6: given a formula Cp(x,b) w i t h baBk, one of ( x c V : Cp(x,b)l and (xaV: -Cp(x.b)) has smaller dimension than V, and so any t w o generics are in the other, and Cp does not distinguish them. It fo l l o w s (it can be shown) t h a t there i s an automorphism of the model carrying one t o the other. In particular, w i t h n = l . any t w o elements of D not algebraic over a set B are conjugate by an automorphism fi x i n g the algebraic closure of B pointw ise. The property used t o define dimension can now be stated thus: (**) Let E,B be definable sets of m u l t i p l i c i t y I , and f:E+B a definable

map. Assume f i s generically surjective, i.e. (f(e): e a generic element of E l contains the set of generic elements of 6. Then d(E)=d(B)+K, where K i s the dimension of a generic fiber of f .

(i.e. K=d(f-'(b)) f o r a generic

b a 6). Note that i f the strongly minimal set in question i s in f a c t an algebraically closed field, then w e can talk about the dimension and the m u l t i p l i c i t y of a definable set, but cannot actually identify the irreducible components. To show how serious t h i s problem i s we give an alternate statement. I f we w i s h t o talk, in the present language, about curves in a plane (say), we must talk about subsets of D2 o f m u l t i p l i c i t y

I , dimension 1 . However, such subsets can be curves w i t h f i n i t e l y many points removed or added. The best we can do i s t o identify a curve w i t h an equivalence class of definable sets, t w o s e t s being equivalent i f they d i ffer by f i n i t e l y many points. So now we have identified the notion of a curve; what we no longer have is the knowledge, f o r a given curve,

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which points l i e on it and which ones do not! The notion of a strongly minimal set i s our approximation t o the idea of an algebraic structure over which geometry can be done. This i s the weakest part of our understanding of N1-categorical theories. Note th a t an algebraically closed f i e l d i s strongly minimal, but so i s any irreducible curve over the field, and even any t r i v i a l f i n i t e cover o f such a curve. Our present definitions are not sharp enough t o make the d is t inct ion. The r e s t of the picture i s much clearer. Given an N,-categorical structure b u i l t over a strongly minimal s e t D, one naturally defines the class of principal coverings of M w i t h respect t o definable groups in Deq, just as in the algebraic case. Then we do know t h a t every N,-categorical structure can be obtained from some strongly minimal set by a finite sequence of such coverings. There i s also a rudimentary theory of the nature of the definable groups over strongly m i n i m a l sets; from the theory of algebraic groups one has the notions of connected component, generic type, and of course dimension, and t h i s can be used t o go a cert ain way. In the case where the fi b e rs are 1-dimensional (of m u l t i p l i c i t y !), the group i s forced t o be 1-dimensional, since we are assuming the action i s regular, and t h i s implies that the group i s Abelian. But even i f the regularity assumption is removed, w e know the precise nature o f the s tr u ct ural group in t h i s case. It i s either Abelian or isomorphic t o one of t w o specif ic m a t r i x groups (of dimension 2 or 3) over an algebraically closed field. (And in particular, the strongly minimal set can be taken t o have a definable structure of an algebraically closed field. Note that

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Unidimensional Theories

t h i s gives only half of the conjecture, though, since one does not know that the f i e l d has no extra structure above the f i e l d relations.) Much less i s known when the fibers are either 2- or O-dimensional. To i l l u s t r a t e the use of the notion of dimension associated w i t h a strongly minimal s e t we describe the "linear" case. Let us f i r s t w r i t e down some equivalent definitions, and then explain them and the equivalence. A) For some (equivalently, every) strongly minimal set D of M: 1 ) Every definable family of k-dimensional subsets of D I has dimension

at most I-k. 2) Let A,B be algebraically closed sets in Deq. Then A,B are independent over their intersect ion. Equivalently, d(A)+d(B)=d(AuB)+d(AnB). B) 1 ) Let F be a family of k-dimensional definable subsets of a definable

set S of M. Then d(F)+k=d(S). 2) Let A,B be algebraically closed sets in Meq. Then A,B are independent over their intersection.

Explanations: If RlCBxDl i s a definable set, l e t R(b)=Iy: (b,y)tRl. (R(b): b t B ) i s called a definable family of k-dimensional subsets of DI i f

R(b) i s k-dimensional for beB. The equivalence relation: b - b ' i f f

(R(b)-R(b'))u(R(b')-R(b))

has dimension
relation, by (*). The dimension of the family i s d(B)-d', where d' i s the dimension of a generic --class on B. Using imaginaries, it i s possible t o normalize the f a m i l y so that

- becomes- the identity:

and l e t R'=((b.a): f o r generic b w i t h b/-=b, (b,a)tRl. this.

l e t B'=B/-,

We w i l l not need

E. HrushovsM

86

When we w r i t e d ( 2 , w e mean the smallest dimension of a 0-def inable set E such t h a t

rt E.

If acl(2=acl(?).

then d(2=d(aa')=d(2)

as one sees easily, so one can define d(A) where A i s an algebraically closed set. To see the equivalence of A(1) and A(2) (or B ( l ) and B(2)), l e t X=acl(i),Y=acl($

be t w o algebraically closed sets; x ' ( x

,....,xI)

may be

chosen independent. Let R be a 0-definable set of least dimension w i t h

(KqtR, and B a definable s e t of

least dimension w i t h r t B . Then i f B i s

chosen appropriately, (R(b): b l B ) i s a definable family of k-dimensional subsets of DI. Hence by A ( l ) , the family has dimension a t most I-k; i.e. d(B)-@II-k. where

8

i s the dimension of a generic --class.

consider the t w o projections from R, and use

(**I.

Now

The projection t o B

shows that d(R)=d(B)+k. The projection t o D l shows d(R)=d(Dl)+oc= I+oc where

O(

i s the dimension of a generic fiber of the projection. So

oc=d(B)+k-II@. Hence f o r a generic a.DI, (b: (b,a)aRl contains only f i n i t e l y many --classes. over

$.

--

Thus F = y / -

is algebraic over x ( a s w e l l as

Now one computes easily that d($+d(Si)=d(D+d(a, giving I(B).

The argument (really a translation) reverses, showing the equivalence. The f a c t t h a t A(2) implies B(2) over a s u f f i c i e n t l y saturated base w i l l not be proved here; it can be deduced immediately from the existence theorem f o r regular types. On the other hand, A( 1 )=A(2), B( 1)=B(2), and the properties A(2),B(2) are clearly independent of the base. This proves their equivalence over any base. (This proof works a l i t t l e more generally, t o show that when every regular type i s internally I-based, the theory i s globally I-based. The original proof in [CHLI did not r e l y on Shelah's theorem. A n intermediate generalization appeared in [Bu t I.)

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87

There are t w o important principles here. One, seen here in the equivalence of A and B, i s the a b i l i t y t o transfer local s t r u c t u r a l information from the strongly minimal sets t o global information about M.

Generalized t o regular types, t h i s i s the basis of Shelah's theory of

supers tab i I it y . The second principle i s the interplay between the t w o geometries: the combinatorial geometry associated w i t h D, and the geometry of the definable subsets of Dn. The former i s by definition, the structure (D.acl)

- one forgets everything except the algebraic closure structure.

One obtains something called a homogeneous pregeometry; in the known examples, it i s s u f f i c i e n t l y r i c h t o retrieve a l l the structure on D (again ignoring questions of parameters.) In the case where D is a vector space over a division r i n g D ( w i t h no further structure), t h i s i s j u s t the fundamental theorem of projective geometry. The second "geometry" i s the structure of the definable sets, under intersections and projections. It i s w o r t h pointing out t h a t A(1) i s also equivalent t o the f o l l o w i n g

special case: (C)

There i s no 2-dimensional family of 1 -dimensional subsets of D2.

This can be better visualized using the f o l l o w i n g lemma o f Zil'ber's. Let F=(C(b): bC61 be a family of I-dimensional subsets of D2. and assume f o r convenience that B has m u l t i p l i c i t y 1 . Let X={xtD2: f o r generic bqB, xeC(b)) be the "generic intersection" of a l l the elements o f F. Then we have: (*)

Let F be d-dimensional. Then d22 i f and only if t w o mutually

generic elements of meet outside the generic intersection of F.

E. Hrusho vski

88

In other words, i f (b,b') i s a generic point of BxB. then C(b)nC(b')OX.

The proof o f

("1 i s another excercise in dimension counting.

Consider ((a,b,b'): atC(b) and a[C(b'). and b t b ' , i.e. C(b)nC(b') i s infinite), and the p r o j e c t i o n of t h i s set t o the second t w o co-ordinates, (a.b.b')*(b.b').

Clearly the kernel of t h i s projection i s an equivalence

r e l a t i o n w i t h f i n i t e classes (a class being of the f o r m ((a,b.b'): aeC(b)nC(b')), w i t h b,b' fixed.) Hence the dimension of t h i s set equals the dimension of i t s image. An easy dimension computation using the projection t o the f i r s t co-ordinate shows that the set i s 2d-dimensional, hence so i s the image, so indeed i t contains the generic points. This then i s the abstract definition of the structures of linear type. We proceed t o give the structural characterization. There i s much

evidence that there i s a real divide here, w i t h a discontinuous jump in complexity as one moves across it. Results of Buechler and Hrushovski show that in any structure f a i l i n g A or 8, there can be no bound on the dimension of a definable family of subsets of D2, nor on the size o f the

finite Galois groups involved in the algebraic closure. For N,-categorical structures satisfying A-8, on the other hand, we have a detailed algebraic structure theorem, not much i n f e r i o r t o those in the t o t a l l y categorical case. Let V be an infinite-dimensional vector space over a division r i n g F. The definable subsets of Vn can be described just as in § I ; the assumption t h a t F i s finite there had no essential role, and served only t o meet the local finiteness requirement. In particular, V i s &!,-categorical.

The infinitude of the division r i n g a l l o w s f o r a slight

variant. Let B be a subgroup of V, such t h a t wBnB has f i n i t e index in B

Unidimensional Theories

89

fo r each at D. Let V'=V/B. Then V' i s "almost" a D-space, in that f o r each a t D , we have a map x-ax, but it i s defined only up t o a f i n i t e ambiguity. Again, the structure of definable sets i s simple, and they can be described quite e x p l i c i tl y . It i s easy t o f i n d the definable groups over V'; the irreducible ones are a l l I-dimensional, and isogenous t o V ' i ts e lf . So principal covers of V ' are again simply a f f i n e covers. Ignoring quest ions of parameters, an arbitrary non-degenerate N1-categorical structure of linear type can be obtained from such an almost-D-space V ' by iterating the taking of f i n i t e and a f f i n e covers. O f course, the remaining problem from the t o t a l l y categorical case

remains: our understanding of covers is very weak. There are t w o additional issues. The f i r s t concerns the parameters needed t o define V ' and the almost-D-space structure on it. One knows that the set of parameters needed has rank a t most I , and there i s much t o say i f it i s 1 . But even when only algebraic paremeters are needed, so one i s

studying the action of the autornorphism group of acl(0), very delicate questions can arise. The second subtlety, interacting w i t h the f i r s t , arises from the fa c t that the arniguity in the expression .%xixi can be smaller than the sum of the ambiguities of the components. But we have reached a very fine level of detail here; we have a completely algebraic description of a class of structures, and the remaining questions are really model-theoretic problems in modules. or Abelian structures. The fa c t t hat t h i s can be achieved s ta rti n g from a rather dry abstract property of the behaviour of definable sets is, perhaps, encouraging. The f i r s t steps in analyzing the structure of N1-categorical theories were taken in [MI and [BLI. It was the second paper that

90

E. Hrushovski

emphasized the importance of the strongly minimal set controlling the structure. The nature of t h i s control was then understood by showing t h a t the model i s prime and minimal over the strongly minimal set, and analyzing t h i s syntactically, but without using s t a b i l i t y . The f a c t t h a t the model i s obtained in a sequence of coverings, phrased in a d i f f e r e n t language, was proved in [Sh V,S51; it i s a special case of the theorem on semi-regular types. in the unpublished [22l,Zil'ber showed t h a t the f i b e r s can be taken t o be homogeneous spaces f o r definable groups. Quite independently, but nine years later, the same thing was shown in [ H l l in the wider context of semi-regular types. The characterization of the group in the case of 1-dimensional f i b e r s i s also in [ H l l ; special cases of the group theoretic work were previously done in [C3l and [CShl. The discovery of the dichotomy between the "module-like" structures and the others i s also due t o Z i I'ber [Z1 I;he also proved the lemma

("1.

Important work on t h i s class was done by Pillay and Buechler, under d i f f e r e r t names (weakly normal, 1-based). The proof t h a t a n o n t r i v i a l module-like structure i s b u i l t around an Abelian group i s in [HI]. The structure of the definable sets under the assumption t h a t the group i s there was analyzed previously in [PHI, generalizing earlier r e s u l t s of Pillay's in the superstable case. For the general theory of stable groups, see [BeLl and [PI. The f a c t t h a t definable f i e l d s in N,-categorical structures are algebraically closed comes from [Mcl. [PI i s probably the best source for someone wishing t o learn t h i s material, though the locally modular theory i s o m i t t e d there entirely.

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91

3. Unidimensional Theories

A f i r s t order theory T i s called unidimensional i f the class of

I I T I +-saturated

I T +-saturated models of T i s categorical; i.e. T has a unique

model of power

I 2T I

+

(say.) This says t h a t a l l models

of T have the same large-scale structure; i f one l e t s an u l t r a f i l t e r on o inundate the fine details, t w o models become distinguished only by their size. From the point of view of Shelah's theory, t h i s i s the natural class t o deal w i t h , rather than the N1-categorical theories; he shows f o r example that i f a theory is not unidimensional, then it has unboundedly many models of equal power. A theory categorical in some power L N l must be unidimensional, but not necessarily N1-categorical. We w i l l discuss the structure of unidimensional theories, taking Shelah's r e s u l t s as our s t a r t i n g point. The theme here i s that classical modeltheoretical problems, not e x p l i c i t l y mentioning groups or geometries, can require these techniques f o r t h e i r solution. We w i l l i l l u s t r a t e t h i s w i t h the f o l l o w i n g r e s u l t of Laskowski's: a theory i s categorical in uncountable power i f and only i f given t w o models of T. one can be embedded i n t o the other. ([Lasl. The " i f " direction, due t o Shelah, w i l l not be discussed here.)

In general terms, the picture resembles the N1-categorical case, w i t h t w o additional aspects. First, the process of taking f i n i t e coverings need not (a priori) terminate in f i n i t e l y many steps. Secondly, a new topological element i s introduced; a f t e r describing t h e structure on each complete type, one i s s t i l l responsible f o r continuously p u t t i n g

92

E. Hrushovski

these structures together i n t o one model. Shelah proved that every unidimensional theory i s stable. It f o l l o w s t h a t the universal domain o f T (a saturated model in which one works, as a m a t t e r of convenience) contains a minimal (--definable)

set. By

definition, t h i s i s a set DCC satisfying the conditions of strong minimality: i) Every definable subset o f D i s f i n i t e or c o f i n i t e i i) If R i s a definable set of n+1 -tuples, then there i s an integer m such t h a t i f R ( i i h D i s f i n i t e , then it has size a t most m. However, D may be the intersection of definable sets, rather than a definable set i t s e l f . The number of definable sets necessary in t h i s intersection i s a t most the cardinality of the language. A set defined in such a way i s called --definable. The theory of dimension for strongly minimal sets goes through without change f o r minimal sets; a definable subset of Dn i s understood t o be the intersection of Dn w i t h a definable set. Moreover, the technique that shows in the strongly minimal case that the structure is obtained by a number of "covers" from D, the f i b e r s of each of which can be parametrically embedded in Deq, s t i l l works. But the process of constructing M from a minimal set in t h i s way may not terminate in f i n i t e l y many steps. To be precise: s t a r t w i t h a minimal set D. There may be definable groups over D; i.e. groups whose underlying set i s Dn/E f o r some equivalence r e l a t i o n E. such t h a t

(F/E).($E)=(F/E))

{(FEa:

i s a definable subset of D3n. A n =-definable set S i s

said t o be analyzable over D (in o( steps, f o l l o w i n g situation. For each

$
o(

being an ordinal) in the

there i s an =-definable set S 8 ,

Unidimensional Theories analyzable in fewer than

O(

93

steps, and a definable map f8:S+S8.

element ass. l e t n ( a ) = ( f 8 ( a ) : 8 < d .

Then each fiber

For

n-’(g should be

embeddable i n t o Deq by a definable map ( w i t h unrestricted additional parameters.) The above i s the abstract d e f i n i t i o n of analyzability ([HI]). However, it was also shown there (as w i l l be shown below) that one can further

demand the f o l l o w i n g (possibly changing the ordinal but not the notion of analyzability.)

Uniformly in Fan(S). there i s a definable group Gaover

D. and definable regular action of ;G

on the r l ( i i j . This explains the

nature of the parameters needed f o r the definable map of

?-r-’(g into

Deq. Let M be any saturated structure. Then every o r b i t of the automorphism group of M i s --definable; such an --definable set i s called complete. If M i s unidimensional, then every complete --definable

set can be embedded into one analyzable over D. The

d i f f i c u l t y , in addition t o the fact that the “analysis” i s not known t o be finite, i s that each --definable set may be analyzed in a d i f f e r e n t way,

w i t h different

OC’S and

different functions f 8 . Essentially the only

constraint i s that everything be put together continuously; each f 8 must be defined on an open set, and must be continuous. The topology referred t o here i s obtained by taking the definable sets t o be basic open sets,

and identifying conjugate elements of the model. A 2-dimensional structure i s one satisfying exactly the same

conditions, except that there are t w o minimal --definable

sets, and i n

E. Hrushovski

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each of the generalized coverings, the s t r u c t u r a l groups are required t o be over one or the other, but i t i s not specified which. Now there are easy examples of 2-dimensional structures in which the analysis i s in f a c t infinite. However, in attempting t o build a unidimensional structure w i t h an i n f i n i t e covering chain, a new minimal s e t always seems t o appear a t some l i m i t level; i t i s made t o agree w i t h the f i r s t one, a t h i r d arises. The question (raised in [Shl) was whether unidimensional theories always admit a f i n i t e analysis. The f i r s t ingredient of the solution ([H 11, t o be published in [H31) was the description of the covering tower in terms of group actions. The groups can in f a c t be obtained very simply. Suppose f o r example P i s 2-step analyzable, but not l - s t e p analyzable over D. in the abstract sense. This means that there i s a definable function f:P+Q, and g:Dn+Q, ge :Dm++E.

g onto; and f o r each fiber E of f, E=f-'(x), there i s a function (++ signifies onto.) ge depends on E, but it may depend on

more parameters, and in f a c t it must, or else

P would be I - s t e p

analyzable. A t a l l events a fundamental principle of s t a b i l i t y , r e f e r r e d t o in § I , implies that the parameters f o r ge can be chosen in EUD (since

it describes a r e l a t i o n on EUD.)

It f o l l o w s that an automorphism of EuD

also. acts on ge, carrying it t o gel say. Now suppose ff is an automorphism of EUD f i x i n g D. Then 0 i s determined by the information that a(e)=e'; f o r it f o l l o w s that ff(ge(a)=ff(gel(8) f o r each onto. Hence ff may be "coded" by the pair (e,e').

d. and ge i s

One proceeds t o show

that the set of pairs arising in t h i s way, and the r e l a t i o n {(e,e',d,d'): (e,e') codes ff, and b(d)=d') are definable. This shows that Aut(E/D) can

Unidimensional Theories

95

be considered as an =-definable group. (It i s not in general internal t o

D, nor i s the action regular; t h i s can be arranged by easy modifications.) The next step i s the combinatorial heart of the matter. One shows that every =-definable group i s a subgroup of a definable group. The proof i s only a few lines, but I do not understand it, and so w i l l not try t o explain. Then one s t a r t s over, but from inside the definable group. One shows that the original analysis of a unidirnensional structure in

terms of coverings can be done w i t h o u t losing sight of the group theory;

so that equivalence relations are replaced by normal subgroups, functions by homomorphisms, and only sorts carrying a group structure need be considered. This i s done as follows. Given an equivalence r e l a t i o n E on G, consider the equivalence relations Eb defined by xEby i f f xaEya. I f we had an absolute descending chain condition on equivalence relations, then the intersection E* of a l l the Ea’s would be a f i n i t e intersection; clearly E* would be invariant under r i g h t translations by G; and G/E* would be embeddable into a finite products of G/Ea3, and hence i n t o a

finite power of G/E. In f a c t we do not have a descending chain condition of t h i s strength; but i f instead w e l e t E* be the generic intersection of the Ea8s (in the sense of the previous section), then it f o l l o w s f r o m s t a b i l i t y that one can divide G i n t o f i n i t e l y many sets, on each of which E* i s in f a c t a f i n i t e intersection. Then the argument goes through, replacing E by a right-invariant equivalence relation, and s i m i l a r l y we can get l e f t invariance. The price i s that G/E* has higher dimension than G/E. The r e s t of the proof i s quick but requires Poizat’s theory of generic types in stable groups, and we w i l l not do it in detail. The point

E. Hrushovski

96

is that whereas before, each complete --definable

Set may have a

d i f f e r e n t analysis, in the context of a group the analyses of t w o generic types can d i f f e r from each other by translation only. This proof is in [H 1 I. Special cases were solved in [H41,[B2l,and [PSI. The conclusion i s that the analysis i s f i n i t e , as in the N,-categorical case. This also sheds some light on the topological aspect; one concludes from it that the m i n i m a l --definable sets are not iso!ated points, but rather form an open set. In other words, there i s a definable set Do (defined over some algebraically closed base) such that every complete -definable

subset of Do i s minimal.

Such a set i s

called weakly minimal. However, it i s s t i l l far from clear how the various minimal subsets are sewed together. This was understood by Buechler in [BII. He found there a beautiful dichotomy between geometric and topological complexity; they cannot coexist. Either the minimal -definable

set D can be taken t o be

definable, or else it i s of I inear or degenerate type. To prove this, suppose D i s not linear or degenerate, so there exists a 22-dimensional family (C(b):bfB) of 1-dimensional subsets of D2. Take B t o be a complete -definable

set. C(b) i s a definable subset of

D2, i.e. C(b)=Co(b)nD2, where Co(b) is a definable subset of Do2. By c u t t i n g Do and C o down, using compactness, we may get them t o share these properties of D and C: for every be6 and xsDo, there are only f i n i t e l y many yaDo w i t h (x.y)eCo(b); and i f b f b ' c B , then Co(b)nCo(b') i s finite. Let D' be any complete =-definable subset of Do. By assumption, D' i s minimal. Let C'(b)=Co(b)n(D'xDo), and assume Co(b) passes above D',

Unidimensional Theories

97

i.e. C'(b) i s infinite. Then (C'(b): baB) is a family of 1-dimensional subsets of D'xDo. The dimension of t h i s f a m i l y is the dimension of B/-, i.e. i t is the same as the dimension of the original family (C(b): b'B); so it i s 22. In §2 we discussed only f a m i l i e s of subsets of D'xD', but the

same arguments go through for f a m i l i e s of subsets of D'xDo. (In f a c t any such f a m i l y must concentrate on D'x(D1u....UD") for some f i n i t e set of minimal -definable

subsets D1,...,Dn of Do.) So by (=) of s2,

C'(b)nC'(b')z/a f o r generic b,b'sB2. Now t h i s was true for any D' provided only that C'(b) i s infinite; i.e. D'fDl=(xcDo: f o r generic bsB. 3y. (x,y)sCo(b)). So for generic (b,b')eB2. for any complete D'CDI, there is xsD' and yiDo w i t h (x,y)cC(b)nC(b'). However, C(b)nC(b') i s f i n i t e ; so the projection of C(b)nC(b') t o the f i r s t coordinate i s f i n i t e , and thus the set of possible D ' 's i s f i n i t e . Let D2 be a definable subset of D 1 containing D and d i s j o i n t from a l l the other D ' 's; then the only infinite complete =-definable subset of D2 i s D

i t s e l f , so D2 is strongly minimal. This finishes the proof. Let us pass t o the theorem alluded t o in the f i r s t paragraph of t h i s section. (Usually it i s stated in a weaker version, concerning the number of models of power

I T I .)

I f T has a strongly minimal set, then the

theory of N1-categorical theories applies in f u l l , and there i s no problem. So we assume otherwise; and hence by Buechler's dichotomy, T i s of linear type. Let M,,M2 be t w o models. We would l i k e t o show that one can be embedded into the other. We w i l l in f a c t show more: whenever f:X1+X2 i s a maximal elementary map between subsets of M1,M2, then X l = M l or Xl=M2. (This includes a strong homogeneity property.) We may assume

E. Hrushovski

98

Xl=X2=X, and f=id. X must be algebraically closed, else we could extend f t o acl(X).

I f X+Ml and XzM,,

f i n d an element aieMi-X.

The saturated

model M of T can be analyzed by a f i n i t e sequence of covers, M+Dn+D

,-,-,...+ D, over a weakly minimal set; because of linearity the

f i b e r s of these covers can be taken t o be 1-dimensional. Applied t o a26M2, t h i s gives images a2-a2(n)-...-a2(1);

replacing a2 by the smallest

apti) outside X . we may assume that a2 l i e s in some weakly m i n i m a l set Eo, and s i m i l a r l y al l i e s in a weakly minimal set Co.

We now r e s t r i c t attention t o EoUCo. We may do t h i s because of an idea of Lachlan's (I-isolation. in [L21), that implies that the induced theory i s s t i l l categorical. This i s a general s t a b i l i t y r e s u l t and w i l l not be discussed here. We would l i k e t o increase f t o fU((e,a2)l. i.e. t o f i n d an element e in MnE. where E i s the complete ==-definable set in which a2 lies. Here we

must use the categoricity of T. Let A be any minimal ==-definable set. acl(AMuM1) i s a large model of T, hence i s saturated, so E has an element there. Hence there exist FgAm and cl,...,cn~Ml and e t E w i t h ecacl(zc l,...,cn).

Let C i be the complete --definable

set in which ci

lies, and get r i d of unnecessary ti's; then there i s a definable r e l a t i o n RCArlx(Clx

...x

Cn)xE, such that for generic

(Tc l,...,cn),

there e x i s t

f i n i t e l y many (but a t least one) eaE w i t h ( g c l,...,cn,e)cR.

If we could

get a s i m i l a r r e l a t i o n R ' w i t h m=O, then we would get an e c E algebraic over elements of M 1 , hence in M 1 , and w e are done. The heart of the matter i s thus the geometric question, of the

Unidimensiona I Theories

relations on AmxrxE. (Cdenoting

C l x ....x

Cn).

99

We would l i k e t o analyze

RLAmxFxE, and show that it can be replaced by a non-generic X-definable r e l a t i o n R ' r ( C l x

...x

Cn)xE.

We know that T i s of linear type.

Using this, Buechler showed ( B l l ) that the problem reduces t o the case n = l ,m=l . But t h i s i s not enough, and a closer analysis of A.F,E i s apparently required. Clearly A can be replaced by A ' here i f there i s a r e l a t i o n SLAXA' making elements of A and A ' co-algebraic, and s i m i l a r l y for Ci,E.

The

next step, done largely for t h i s purpose, was t o show that there are such A',Ci',E' each w i t h the structure of an Abelian group; but an extra parameter i s necessary for their definition. This r e s u l t was referred t o earlier, in the context of strongly minimal sets of linear type, but i s really v a l i d much more generally f o r regular types, and in particular applies here. Recall the power of t h i s statement: nct only do A',C',E' have Abelian group structures, but there is nothing but the Abelian structure; every R'CA'xC'xE' i s a Boolean combination of cosets of subgroups. Now one had an unprecedented amount of control about a quite abstractly presented s i t u a t i o n

- essentially

quantifier elimination a f t e r

a single parameter is added - but t h i s was s t i l l not enough, and i t was necessary t o remove t h i s parameter. (Apparently the problem i s that there are, in the abstract, geometries that localize t o p r o j e c t i v e spaces, without being isomorphic t o projective or a f f i n e spaces themselves; i f P* i s such a pathological geometry, and A*,C* are the corresponding

a f f i n e and projective geometries, then there does e x i s t a r e l a t i o n

E. Hrushovski

I00

R*sA*xC*xP* satisfying the correct formal relations.) The parameter was removed, by directly improving Buechler’s dichotomy t o show that a nondegenerate unidimensional theory involves either a strongly minimal set, or an acl(o)-definable Abelian group (of linear type). ( [ H I , p. 961). This does not mean that C,E can be replaced by groups (the r e s u l t i s not t h a t every minimal set has an acl(o)-definable group structure, only that some such set does.) But on A there were no constraints except weak m i n i m a l i t y , so we may take A t o be an Abelian group. Ci.E can then be chosen t o be a f f i n e spaces over A. We r e t u r n t o the analysis of the r e l a t i o n RfAmxcxE. Recall the

strength of the statement that A is an Abelian group: every definable subset of Ak is a Boolean combination of cosets of definable subgroups. After choosing a ”0” in E and each Ci, they each become just a copy

o f A.

so R becomes a Boolean combination of cosets. The notion of a coset

does not actually depend on the particular choice of 0 in a group, but only on the a f f i n e structure, i.e. the operation x-y+z.

So R can be taken

t o be an a f f i n e subspace, i.e. invariant under t h i s operation. We also

--

know that for any TeAm, and C ~ C there , i s eeE w i t h x=(zce)eR. So t = ( ~ . c o , e o ) ~ and R f o r some co,eo, and then y=(zco,el)eR f o r some el. Applying x-y+z. we see t h a t R’=Rn((O)xCxE) has the property: f o r

--

generic ceC, there i s some eeE w i t h (0,Ce)eR’.

O f course there are only

f i n i t e l y many such e’s. We have shown that R can be replaced by a r e l a t i o n not involving A, and so we finish. The above was meant t o be an example of arguments concerning unidimensional theories, and the way one i s forced t o recognize the algebraic structure hidden underneath the abstract description. This

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101

recognition then makes a finer analysis possible. W i t h each weakly minimal Abelian group (or rather i t s theory) i s associated a division r i n g F; the models of the theory are "almost" vector spaces over F. In the

strongly minimal context, w e encountered the possibility t h a t the endomorphisms r a F are not w e l l defined; each r i s defined on A but into A/K, K f i n i t e . There i s a formal dual t o this, in the category of Abelian structures: an "action" of F on an Abelian group A, in which each r c F i s defined i n t o A, but the domain i s a subgroup of A of f i n i t e index. Again the problem i s (in dual form) that the maximal domain of d e f i n i t i o n of

n i x i may be bigger than the intersection of the domains of the r i ' s . In the unidimensional context both of these phenomena occur a t once. In addition, the various Ti's may be defined only over acl(0), and not over 0;

i.e. one may not be able t o t e l l apart over

0

c e r t a i n elements of F,

conjugate under the Galois group of F. These issues make themselves f e l t in some longstanding problems; the second t w o are a t the core of

the problem w i t h Vaught's conjecture f o r unidimensional theories. With regard t o t h i s problem i t has been shown, f o r example, that the division r i n g can be taken t o be commutative and locally f i n i t e , and further algebraic work has been done; but one must go deeper in order t o solve the problem. Buechler's theorem has since been generalized t o the superstable context ([HShl). This allows many previously i Il-understood obstacles t o be recognized as manifestations of c e r t a i n embedded Abelian structures; and one i s again pushed t o analyze these a t deeper levels. It is believed in particular that Vaught's conjecture f o r superstable theories w ! l l

eventually f a l l , assuming t h a t the case of the unidimensional Abelian

E. Hrushovski

102

group i s done.

I t r i e d t o indicate how recognizing the underlying geometry of a structure can open up an entire f i e l d of theorems, solving old problems in the process. This i s true both fo r Zilber's theorem and Buechler's, each applying under the appropriate assumptions. These t w o theorems serve t o recognize linear structures; the other class of known examples algebraically closed fields, s t i l l goes unexplained. One can only hope that the subject w i l l eventually be further enriched, by a s i m i l a r theorem opening a bridge t o algebraic geometry.

Unidimensional Theories

103

0. Ahlbrandt and M. Ziegler, Quasi-finitely axiomatirable totally categorical theories, JSL 1986 W. Baur, 0. Cherlin and A. Macintyre, Totally categorical groups and rings, Journal of Algebra 57(1979) 407-440 Ch. Berline and D. Lascar, Superstable Groups, Ann. Pure Appl. Loglc 30 (1986) 1-43. J.T. Baldwln and A.H. Lachlan, On Strongly Minimal Sets. Journal of Symbolic Logic 36(1971), 79-96 Buechler, S., The geometry of weakly minimal types, JSL v. 50(1985) Buechler, S, locally modular theories of finite rank. G. Cherlin, Quasi-finiteaxiomatkability of structures of modular type, preprint G. Cherlin, paper to appear in proceedingsof Trento conference. G. Cherlin, Groups of small Morley rank, Ann. Math. Logic 17, 1-28 G. Cherlin, L. Harringtonand A. Lachlan, No-categorical, No-stable structures, Annals of Pure and Applied Logic 18 (1980), 227-270 0. Cherlin and S. Shelah, Superstable fields and groups, Annals Math. Logic 18,227-270 D. Evans, doctoral dissertation, Universityof London 1986 E. Hrushovski,doctoral dissertation, Berkeley 1986 E. Hrushovski,Totally categorical theories, to appear in AMS Transactions E. Hrushovski, locally modular regular types, proceedings of US.-Israel Binational Science FoundationChicago model theory conference, Springer Verlag E. Hrushovski, Unidimensionaltheories are superstable, to appear E. Hrushovskiand S. Shelah, a dichotomy for regular types, to appear in APAL H.M. Kantor, M.H. Liebeck, and H.D. Macpherson, No-categorical structures smoothly approximable by finite substructures. Preprint. A.H. Lachlan, Structures co-ordinatired by indiscernible sets. Preprint. A.H. Lachlan, On a property of stable theories, Fund. Mat .77(1972) 9-20 C. Laskowski, doctoral dissertation, Berkeley 1987 M. Morley, Categoricity in power, TransactionsAMS 114 (1965), 514-438 A. Maclntyre, On w 1 -categorical theories of fields, Fund. Math. 71,l-25 A. Pillay and E. Hrushovski, Weakly Normal Groups, ASL Orsay Conference proceedings, 1985 B. Pokat, Groupes Stable, nur-el-mantiqwal ma'arlfah, Paris 1987 (write author in Park Vl.) Shelah, S., Classification Theory, North Holland, Amsterdam 1978 Zirber, B. , Ng-categoricalstrongly minimal sets II 8 111, Siberian Math. J. 25 (1984), pp. 396-412 and 559-571 3nnb6ep. 6.H.. C r p o e H n e MOAeAeft vareropnw I reopnti H n p o b n e ~ a KOHeYHOft aKCUOMaTU3UpyeMOCTU. K e M e p o B o r0'lyAapCTBeHH

ynnsepcnrer 1977