Mathematical Reviews

94h:03059 03C45 (03-02 03C35)

Zilber, Boris

Uncountably categorical theories.

Translated from the Russian by D. Louvish. Translations of

Mathematical Monographs, 117.

American Mathematical Society, Providence, RI, 1993. vi+122 pp. ISBN

0-8218-4586-1 [AMS Book Store] _________________________________________________________________

In this volume the author outlines the foundations of "geometric stability theory". Early work on $\aleph\sb 1$-categorical theories established the importance of studying strongly minimal sets---sets on which algebraic closure determines a dependence relation. But the author discovered that for more delicate questions, specifically the determination of whether $\aleph\sb 1$-categorical theories could be finitely axiomatized, a more detailed analysis of the geometry determined by this dependence relation is necessary. The study of $\aleph\sb 1$-categorical structures thus comes in two pieces: (a) classification of the strongly minimal sets and (b) analysis of the construction of an arbitrary model from a strongly minimal set. This monograph contains the first major results on both of these projects and their corollaries for the finite axiomatizability problem. As in much of stability theory, problems such as finite axiomatizability have turned out to be much less important than the structural results which arise in their solution. Nevertheless, for historical reasons, this review is organized around the problem which directly motivated the original work.

The geometry of a dependence structure is called (i) trivial if ${\rm cl}(XY)={\rm cl}(X)\cup{\rm cl}(Y)$ for any $X,Y$, (ii) modular if the lattice of closed sets is modular, (iii) locally modular if the relativization of the geometry to any single point is modular. The author shows that if the geometry is locally finite then one of these three conditions holds. This result was also obtained separately by Cherlin, Mills, and P. Neumann, but their three arguments depended upon the classification of the doubly transitive groups, while the author's is self-contained. (Subsequently, Evans reversed the process and obtained the classification of finite doubly transitive groups using the author's methods.) If the strongly minimal set is $\aleph\sb 0$-categorical, then the geometry is locally finite. This analysis then plays a critical role in the study of the finite axiomatizability problem for totally categorical theories.

If the assumption that the geometry is locally finite is dropped, the author shows that every strongly minimal set falls into one of the above categories or contains a definable pseudoplane, a combinatorial configuration introduced by Lachlan in a different model-theoretic context. (Usually, classes (ii) and (iii) are combined in this context to form the class of module-like structures, so this is called the trichotomy theorem.) The author conjectured that those strongly minimal sets which interpret pseudoplanes are, roughly speaking, bi-interpretable with an algebraically closed field. This conjecture was refuted by Hrushovski. Hrushovski and the author have developed the class of Zariski structures---geometries with sufficient structure to recover the conjecture. This development displays intensive connections between model theory and algebraic geometry.

The author's "ladder theorem" links the global structure of an uncountably categorical theory to that of a strongly minimal set. Let $M\sb 0$, strongly minimal, and $M$ be $\emptyset$-definable in an $\aleph\sb 1$-categorical structure. Then there is a finite sequence $M\sb 0 \subseteq M\sb 1 \subseteq \cdots M\sb n =M$ such that for each $M\sb i$-atom $A \subseteq M\sb {i+1}$, the group $G(A,M\sb i)$ of all automorphisms of $A$ fixing $M\sb i$, and its action on $A$, are definable. Further, if for each $A$, $G(A,M\sb i)$ is abelian then the ladder sequence can be chosen so that each atom $A$ has Morley rank at most 1 over $M\sb i$. The author shows that if an uncountably categorical structure $M$ is module-like or discrete and a connected group $G$ is interpretable in $M$ then $G$ is abelian. Thus, in the module-like (or discrete) case, one may assume that the atoms in the ladder theorem have rank 1. The proof that no totally categorical complete theory is finitely axiomatizable requires one further ingredient. For any definable set $A$ and an arbitrary set $X$, $A[X]$ denotes ${\rm acl }(X) \cap A$. Auxiliary Lemma V.4.: In a totally categorical structure with a fixed $\emptyset$-definable strongly minimal set $A$, for each $\emptyset$-definable set $B$ (of rank 1) there is a natural number $s\sb B$ such that if $Z \subseteq A$ and $\vert Z\vert > s\sb B$ then, for any set $Y$, $B[Z\cup Y]\not = \emptyset$ iff $B[A\cup Y] \not = \emptyset$.

Using this result and the ladder theorems, the author constructs for each finite set $X$ a finite "envelope" ${\rm en}\sb A(X)$ which, for sufficiently large $X$, is an elementary submodel with respect to sentences of a given complexity. This shows that the theory is not finitely axiomatizable.

The author presents two arguments for the construction of envelopes. One relies on the ladder theorem for theories of modular type and uses the auxillary lemma as stated for rank 1 atoms. The second drops the rank 1 hypothesis and relies on a number-theoretic result of Siegel (1921).

A key tool in the author's work is the ability to prove that various groups of automorphisms of the structures are definable. Another chapter of this monograph uses this idea to characterize the strongly minimal quasialgebras (those for which algebraic closure equals definable closure). This analysis generalizes a result of Urbanik on $v\sp *$-algebras and sheds an interesting light on the relation between model theory and universal algebra.

\{Reviewer's remarks: This monograph contains a number of extremely deep results that are well worth study today. However, it is also important as an historic document. In this respect, it is unfortunate that the editors failed to determine when the thesis was actually presented and to what extent the arguments here use later results to simplify the original argument.\}

Reviewed by John T. Baldwin

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Mathematical Reviews on the Web

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Hrushovski, Ehud(IL-HEBR); Zilber, Boris(RS-KEME)

Zariski geometries. (English. English summary)

J. Amer. Math. Soc. 9 (1996), no. 1, 1--56. [ORIGINAL ARTICLE]

FEATURED REVIEW.

The authors show that the Zariski topology associated to an algebraically closed field can be characterized, as a combinatorial geometry, by a handful of familiar and elementary properties. Naturally the axioms are somewhat more elaborate than those used to characterize affine or projective (linear) geometries, but they are sufficiently simple to be quoted in their entirety below; for this reason they can also be verified in nontrivial situations, a fact which has been exploited subsequently in differential algebra [E. Hrushovski and Z. Sokolovic, "Minimal subsets of differentially closed fields", Trans. Amer. Math. Soc., to appear] and in number theory [E. Hrushovski, "The Mordell-Lang conjecture for function fields", J. Amer. Math. Soc., to appear].

The axiomatic framework is as follows. One has a set $X$ and a Noetherian topology on each Cartesian power $X\sp n$. In particular, certain rudimentary notions of algebraic geometry, such as irreducibility and dimension, can be defined in this context. There are four basic axioms, and a nondegeneracy condition. The axioms state that (0) maps from $X\sp m$ to $X\sp n$ defined coordinatewise using constant maps or coordinate maps are continuous, and diagonals are closed; (1) any projection of a closed irreducible subset of $X\sp m$ to $X\sp n$ contains a nonempty open subset of its closure; (2) $X$ is uniformly one-dimensional and irreducible; and (3) the dimension theorem holds for the intersection of a closed irreducible subset of $X\sp n$ with a diagonal $\Delta$. The last two axioms require some elucidation. To say that $X$ is one-dimensional and irreducible means of course that its proper closed subsets are finite. The uniformity condition bounds the size of the closed finite subsets of $X$ occurring in any algebraic family (such a family would be given by a closed subset of $Y\times X$ where the parameter space $Y$ is $X\sp n$ for some $n$). The dimension theorem states that each component of the intersection $U\cap \Delta$ has dimension at least $\dim(U)-1$; the diagonal $\Delta\subseteq X\sp n$ is defined by the equation $x\sb i=x\sb j$ for some $i These are the axioms, apart from nondegeneracy conditions. Two such conditions are considered. A Zariski geometry is ample if there is an algebraic family $\scr C$ of plane curves such that any generic point of $X\sp 2$ lies on one of the curves; it is very ample if, in addition, some such family separates points in the sense that for any two points of $X\sp 2$, one of the curves in the family meets exactly one of the specified points. The intuitive language used here is easily translated into the language of Zariski geometries; for example, something is said to happen generically if it happens outside a proper closed subset.

The results are as follows. Theorem A. Any very ample Zariski geometry is isomorphic to the Zariski geometry induced naturally on some smooth curve over an algebraically closed field. Theorem B. Any ample Zariski geometry is a finite cover of the canonical Zariski geometry on the projective line over some algebraically closed field.

In the proofs of these results the field itself is recovered explicitly (and efficiently) from the geometry. These two results are complemented by an example showing that the latitude left in Theorem B is essential, in the sense that there are ample Zariski geometries not associated to any algebraic set. More specifically, the groups of algebraic automorphisms of algebraic curves are considerably more limited than the groups of definable automorphisms of Zariski geometries built from curves by taking suitable coverings.

In applications it is also important that one can say a good deal about Zariski geometries which are not ample. However, this has been well understood in a more general context for some time, and is reminiscent of standard linear coordinatization theorems. Remarkably, the same model-theoretic techniques that work in the linear case are largely sufficient for the analysis of the ample case.

There are two main steps in the proof. The first is the recovery of the base field from the geometry. Here one uses a notion of tangency which can be defined in a Zariski geometry, starting from any sufficiently rich collection of plane curves, the idea being that two curves in a given family are tangent at a point $p$ if they are the specializations of a pair of curves meeting at distinct points which specialize to $p$. If classes of tangent curves at a point represent slopes, then composition of curves as correspondences should give a multivalued multiplication map, from which a suitable single-valued group operation can be decoded by known model-theoretic methods. In practice the notion of tangency depends on the given family of plane curves, and the group decoded may turn out to be the additive group rather than the multiplicative group, but one gets in any case a constructible (hence smooth) one-dimensional group to which the Zariski geometry can be transferred, and repetition of this argument leads to a two-dimensional group acting on a one-dimensional set, which can be seen to be the semidirect product of the additive and multiplicative groups of the desired field.

In the second main step, one knows that the Zariski geometry contains the canonical Zariski topology coming from a field, and it is necessary to prove that no other sets are closed. The proof amounts to a reversal of Cramer's paradox: given a curve $C$ of degree $d$, the number of points needed to determine an algebraic curve of much higher degree is more than the maximum number of points of intersection of this curve with $C$; so $C$ must be a component of some algebraic curve.

These results provide an effective bridge between the model theory of stable algebraic structures and algebraic geometry, which can fruitfully be crossed in either direction. The fact that any ample Zariski geometry interpretable in a differentially closed field of characteristic zero or a separably closed field of finite degree of imperfection comes from a field, and (as one may also show) that this field is necessarily either the field of constants in the former case, or the maximal perfect subfield in the latter case, has implications both for model theory and for the study of abelian varieties, giving in particular a proof of the geometric Mordell-Lang conjecture in positive characteristic.

A theorem of this type was first conjectured by Zilber in the constructible category; this was eventually refuted by Hrushovski, and the correct form then emerged quite quickly, first for complete Zariski geometries, where there are considerable technical simplifications, and then in the present more general form; its proof relies on a body of machinery which was developed in the constructible category, including the means for decoding multivalued groups. The most striking technical innovation in the paper is the use of two-dimensional arrays of indiscernibles to generate group configurations for which the underlying symmetry prevents obnoxious degeneracies.

A more comprehensive introduction to the subject of Zariski geometries can be found in another paper by Hrushovski and Zilber [Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 315--323; MR 93j:14003].

Reviewed by G. Cherlin

Cited in reviews: 97h:11154