Notes on totally categorical theories

Martin Ziegler

1991

0 Introduction

Cherlin, Harrington and Lachlan’s paper on ω_{0}-categorical, ω_{0}-stable theories ([CHL]) was the starting point of geometrical stability theory. The progress made since then allows us better to understand what they did in modern terms (see [PI]) and also to push the description of totally categorical theories further, (see [HR1, AZ1, AZ2]).

The first two sections of what follows give an exposition of the results of [CHL]. Then I explain how a totally categorical theory can be decomposed by a sequence of covers and in the last section I discuss the problem how covers can look like.

I thank the parisian stabilists for their invation to lecture on these matters, and also for their help during the talks.

1 Fundamental Properties

Let T be a totally categorical theory (i.e. T is a complete countable theory, without finite models which is categorical in all infinite cardinalities).

Since T is ω_{1}-categorical we know that

a) T has finite Morley-rank, which coincides with the Lascar rank U.

b) T is unidimensional : All non-algebraic types are non-orthogonal.

The main result of [CHL] is that

c) T is locally modular.

A pregeometry X (i.e. a matroid) is modular if two subspaces are always independent over their intersection. X is called locally modular if two subspaces are independent over their intersection provided this intersection has positive dimension. A pregeometry is a geometry if the closure of the empty set is empty and the one-dimensional subspaces are singletons. The one-dimensional subspaces of X form a geometry in a natural way, the associated geometry X̃. X is (locally) modular iff X̃ is.

The realization set of a regular type p ∈ S(A) carries the structure of a pregeometry, where

“c is in the closure of C” is interpreted as “c \begin{matrix}/| \\ ⌣ \\ A \end{matrix} C ”, which for U-rank 1 types is the same as

“c ∈ acl_{A}(C)”. A theory T is locally modular if all stationary types of U-rank 1 are locally modular i.e. have locally modular geometry.

By a) every stationary p ∈ S(A) of U-rank 1 is strongly minimal. Whence p contains an Adefinable strongly minimal set D. “acl_{A}” defines on D its pregeometry over A which is the same as the pregeometry of p except that D may contain elements from acl(A).

If A is finite the closure of the empty set in D is finite and the one-dimensional subspaces of D form a uniformly definable family of finite sets. Whence the associated geometry of D is the pregeometry over A of a certain strongly minimal set D_{A}, which lives in C^{eq}. D_{A} is called the associated strictly minimal set over A. It has the same geometry as p. The local modularity of T (at least for types defined over finite sets) follows therefore from

Theorem 1.1 ([CHL]) Let D be a strictly minimal set over ∅. Then the structure of D is one of the following:

a) trivial

b) a projective space over a finite field

c) an affine space over a finite field

Some explanations:

The induced structure on a 0-definable set P tells us what the 0-definable relations on P are. If we are in a countable ω_{o}-categorical model M the induced structure on P is given by the group of all permutations of P which are induced by automorphisms of M: The 0-definable relations on P are the relations which are invariant under all automorphisms of M. If A is a set of parameters and P is A-definable one can also speak about the induced structure of P over A, giving the A-definable relations on P.

Let X be an infinite-dimensional projective (or affine) space over the field F_{p^{n}}. Let G be a group of collineations which contains all linear collineations of X. (There are exactly ϕ(n) possible choices for G). X with the structure given by G is what we mean by a projective (affine) space over F_{p^{n}}. The acl-closed subsets of X are just the subspaces of X. All these structures (and the trivial structure) are totally categorical and strictly minimal. The projective spaces are (as the trivial structure) modular, the affine spaces locally modular.

Many properties of totally categorical theories are shared by all locally modular theories of finite U-Rank - the LMFR-theories. See [B].

The following Lemma is the main use of modularity. If A is a subset of D it specializes just to the definition.

Lemma 1.2 Let T be stable and D be a 0-definable strongly minimal set which is modular over ∅. Let B be a subset of D and A be a set of parameters. Then B and A are independent over acl(B) ∩ acl(A) ∩ D.

Proof: We can assume that acl(B) ∩ acl(A) ∩ D is in the definable closure of the empty set. Using the following easy fact

Lemma Let T be a stable theory. Then for all sets A and elements b there is a set C of

realizations of tp(b/A) such that b \begin{matrix}| \\ ⌣ \\ C \end{matrix} A and b \begin{matrix}| \\ ⌣ \\ A \end{matrix} C.

we obtain a subset C of D such that

B \begin{matrix}| \\ ⌣ \\ C \end{matrix} A (1)

(2)  B \begin{matrix}| \\ ⌣ \\ A \end{matrix} C  .

If B and A are dependent over ∅ then by (1) also B and C are dependent over ∅. Whence by modularity there must be an element d ∈ D which is algebraic over B and over C but not algebraic over ∅. It follows from (2) that d is algebraic over A, whence d belongs to acl(B) ∩ acl(A) ∩ D, a contradiction.□

Lemma 1.2 answers the following question which arises from theorem 1.1. Let D be an Adefinable strongly minimal modular set and B a finite set extending A. The two geometries of D over A and B are given by theorem 1.1, where we added the elements of A and B respectively. What is the relationship between these two geometries ? By 1.2 the answer is that the pregeometry of D over B (also for infinite B) is the pregeometry of D over A relativized by acl(B) ∩ D. We conclude that also the U-rank 1 types which are defined over infinite sets are locally modular.

Corollary 1.3 (see [C.L] for LMFR-theories) All stationary types p ∈ S(A) of U-rank 1 which are not modular are sound i.e. if q ∈ S(B) is a non-forking modular extension of p then p is realized in acl(B).

Proof: We can assume that A is empty. Let D be a strictly minimal set over ∅ which is not modular. By theorem 1.1 D is an affine geometry. Let D’ be the set of the parallel-classes of D, which is strictly minimal and modular over ∅. D ∪ D’ is a set of Morleyrank 1 (and degree 2) which has a modular geometry over ∅ and we can apply 1.2 to it (!). Now assume that D and acl(B) have empty intersection. Then the pregeometry of D over B is the pregeometry of D relativized by acl(B) ∩ D’. Since an affine geometry relativized by a set of parallel-classes remains affine D remains be non-modular over B. □

A stable theory is 1-based if the canonical base of every strong type st(A/B) is contained in

acl(A), which is equivalent to A \begin{matrix}| \\ ⌣ \\ acl(A)∩acl(B) \end{matrix} B. Thus 1-basedness means that A and B can only

be dependent over C for the simple reason that there is an element which is algebraic over A and algebraic over B but not algebraic over C.

Corollary 1.4 T is 1-based

Proof: Assume that a and b are dependent over C. We want to find an element c algebraic over a and b but not over C. In a superstable theory st(a/C) and st(b/C) are domination equivalent to products of regular types p_{i}. This means that one can find an extension M of C, independent from ab over C, and for a an M-independent tuple \overline{a} of realizations of regular types over M which is domination equivalent to a over M and similarly for b a tuple \overline{b}. Since a and b are dependent over M also \overline{a} and \overline{b} are dependent over M. We can assume that all the occuring regular types are equal or orthogonal. Whence there are subsequences \overline{a′} and \overline{b′} of \overline{a} and \overline{b} whose elements all have the same type p and are still dependent over M. Since a and b have finite U-rank over C, we can assume that p is of U-rank 1 (use [D.L]). p is the non-forking extension to M of a stationary type q, which is defined over a small subset of M. If we assume M to be enough saturated q is realized in M. Whence the local modularity of q implies that p is modular. Now by 1.2 we find a c′ realizing p which is algebraic over M\overline{a′} and over M\overline{b′}. c′ is dominated by a and by b over M, whence algebraic over Ma and over Mb. Let E be the canonical base of stp(Mc′/Cab).

Since Mc′ \begin{matrix}| \\ ⌣ \\ Ca \end{matrix} b, Mc′ \begin{matrix}| \\ ⌣ \\ Ca \end{matrix} b and Mc′ \begin{matrix}/| \\ ⌣ \\ C \end{matrix} a , E is a subset of acl(Ca), acl(Cb) but not of acl(C) .

Now choose c as an element from E which is not algebraic over C. □

Corollary 1.5 (“Coordinatization”) If a is non-algebraic over A there is a b which has U-rank 1 over A and is algebraic over Aa

Proof: If n = U(a/A) = 1 there is nothing to show. If n > 1 we choose a c such that U(a/Ac) = n − 1.

Since a and c are dependent over A 1-basedness gives a b′ which is algebraic over Aa and over Ac but not over A. a is not algebraic over Ac and therefore not algebraic over Ab′. The equation

U(a/A) = U(ab′/A) = U(a/Ab′) + U(b′/A)

shows U(b′/A) < n. By induction we find b in the algebraic closure of Ab′ which has U-rank 1 over A. □

Lemma 1.6 For every a there is a sequence a_{1}, ...a_{n} = a such that all a_{i} are in acl(a) and all types tp(a_{i+1}/a_{1}...a_{i}) are algebraic or strictly minimal.

Proof: If a is algebraic over ∅ we set a_{1} = a. Otherwise 1.5 gives us an a_{2} in acl(a) which has U-rank 1 over ∅. By the arguments preceding 1.1 there is an a_{1} such that tp(a_{1}/∅) is strictly minimal and a_{2} is algebraic over a_{1}. Since U(a/a_{1}a_{2}) < U(a/∅) we can use induction to conclude that there is a decomposition a_{3}, ..a_{n} = a of a over the parameters a_{1}a_{2}. □

Note that n can be bounded by two times the Morleyrank of a over ∅.

Example 1 Let p be a prime, n a natural number and M be the totally categorical abelian group ⨁_{ω} (Z/p^{n}Z). For every element a of order p^{m} < p^{n}

D_{a} = {b^{∗} | pb = a, b ≠ 0},

where b^{∗} = b + Zp^{m}b, is strictly minimal over a. A decomposing sequence of a is

(p^{m−1}a)^{∗}, p^{m−1}a, … a^{∗}, a.

Note that p^{i}a is algebraic over (p^{i}a)^{∗} and that (p^{i−1}a)^{∗} belongs to D_{p^{i}a}.

Lemma 1.7 Let D_{1} and D_{2} be 0-definable strictly minimal modular sets. Then there is a unique 0-definable bijection between them.

Proof: If f is a definable bijection between D_{1} and D_{2}, then f(b_{1}) must be the (unique) element b_{2} of D_{2} which is algebraic over b_{1}. Whence f is unique. On the other hand it is enough to find two elements b_{i} of D_{i} (i=1,2) which are interalgebraic. Then the formula f(x, y) = tp(b_{1}, b_{2}) defines the desired bijection:

Since D_{1} and D_{2} are not orthogonal there are subsets B_{i} of D_{i} (i=1,2) which are dependent over ∅. Since D_{2} is modular, there is an element b_{2} of acl(B_{1}) ∩ acl(B_{2}) ∩ D_{2} which is not algebraic over ∅. Since B_{1} and b_{2} are dependent, and D_{1} is modular, there is an element b_{1} of acl(B_{1}) ∩ acl(b_{2}) ∩ D_{1} which is not algebraic over ∅. Now b_{1} and b_{2} are interalgebraic. □

Corollary 1.8 There is a strictly minimal modular set D over ∅.

Proof: By 1.5 there is a 0-definable set D of Morleyrank 1. We can assume that D does not contain elements of acl(∅). The finite equivalence relation theorem gives a 0-definable equivalence relation E which partitions D into finitely many strongly minimal sets D_{1}, ...D_{n}. The D_{i} have over ∅ the same pregeometry as over acl(∅) over which set they are defined. Let D’_{i} be the strictly minimal set of parallel classes of D_{i} and E the 0-definable disjoint union of the D’_{i}. By 1.7 there are unique acl(∅)-definable bijections between the D’_{i}. This shows that in the pregeometry of E every 1-dimensional subspace meets each of the D’_{i} in exactly one point. The set of 1-dimensional subspaces of E is therefore 0-definable and strictly minimal. □

Corollary 1.9 If one strictly minimal set over some parameter set has trivial geometry then all strictly minimal sets over arbitrary parameter sets have trivial geometry.

Proof: If D is strongly minimal over a finite set A it is of one of the types a), b) or c) of 1.1. If B extends A then by 1.2 D_{B} has the relativized geometry (where in the affine case one has not only to relativized by a subset but also by a set of parallel classes, see the proof of 1.3.) This shows that D_{B} is of the same type, except that an affine geometry could have become projective. Now the corollary follows from 1.7. □

2 Envelopes

In this section we fix a totally categorical theory T and a strictly minimal modular set D over ∅ as given by 1.8.

Definition 1 Let B be a set of parameters. A set E is called an envelope of B if E \begin{matrix}| \\ ⌣ \\ B \end{matrix} D and E

is maximal with this property.

If P is a B-definable class in a stable theory then A \begin{matrix}| \\ ⌣ \\ B \end{matrix} P is equivalent to tp(A/B) being almost

orthogonal to every type tp(\overline{c}/B) of a tuple \overline{c} from P. If P is strongly minimal we have to check this only for tuples of B-independent elements of P. If furthermore P is modular over B 1.2 tells us

that it is enough to test single elements of P i.e. A \begin{matrix}| \\ ⌣ \\ B \end{matrix} P is equivalent to acl(AB)∩P = acl(B)∩P.

Example 2 In the group M of example 1 the envelopes are -up to an automorphism- the subgroups ⨁_{i<m} (Z/p^{n}Z).

The following Lemma shows that envelopes are in our situation what for non-totally transcendental unidimensional theories was called “maximal non-modular models” in [PR]. But of course envelopes are not models in general.

Lemma 2.1

1. Let A be algebraically closed over B. Then A \begin{matrix}| \\ ⌣ \\ B \end{matrix} D iff no modular type of U-rank 1 over any

subset of A containing B is realized in A

2. E is an envelope of B iff

(a) E \begin{matrix}| \\ ⌣ \\ B \end{matrix} D

(b) E is algebraically closed over B

(c) All non-modular stationary types of rank 1 over subsets of E are realized in E. It is enough to know this for types over E i.e. that all U-rank 1 types over E are modular.

Proof: 1: Assume A \begin{matrix}| \\ ⌣ \\ B \end{matrix} D , let C be a subset of A which contains B and let p ∈ S(C) be a modular

stationary type of U-rank 1.The proof of 1.7 shows that one (and therefore every) realization a

of p is interalgebraic over C with some element d of D. a \begin{matrix}/| \\ ⌣ \\ C \end{matrix} d implies then Ca \begin{matrix}/| \\ ⌣ \\ B \end{matrix} d . Whence a cannot lie in A. For the converse assume that A \begin{matrix}/| \\ ⌣ \\ B \end{matrix} D . Since A is algebraically closed in C^{eq}

there is an element a of A with a \begin{matrix}/| \\ ⌣ \\ B \end{matrix} D . By 1.6 there is a seqence a_{1}, ...a_{n} = a of elements of

A such that the types tp(a_{i+1}/a_{1}...a_{i}) have at most rank 1. For some i we must have a_{i+1} \begin{matrix}/| \\ ⌣ \\ C \end{matrix} D

where C = Ba_{1}...a_{i}. Since D is modular over C, a_{i+1} is interalgebraic over C with an element of D. But then the geometries of p = tp(a_{i+1}/acl(C)) and D must be the same (see 1.7) and p ist modular. 2: Assume first that E is an envelope of B. (a) and (b) are clearly true. So let C be a subset of E and p a non-modular stationary type of rank 1 over C. Let a be a realization of the nonforking

extension p|E of p to E. By maximality of E we have a \begin{matrix}/| \\ ⌣ \\ E \end{matrix} D. The proof of 1 shows that p|E must

be modular. By 1.3 p is realized in E. This shows (c). For the converse assume (a) and (b) are true but that E is not maximal with (a). Then there is an a which is not algebraic over E such

that a \begin{matrix}| \\ ⌣ \\ E \end{matrix} D. By 1.5 we can assume that p = tp(a/E) has rank 1. By 1 p is not modular. Whence

(c) is false. □

Corollary 2.2 If D is trivial T is almost strongly minimal i.e. C is the algebraic closure of a strongly minimal set.

Proof: Let a be any element and B = acl(a) ∩ D. We show that a ∈ acl(B). By 1.2 aB \begin{matrix}| \\ ⌣ \\ B \end{matrix} D. If a is not

algebraic over B we find a subset C of acl(a) which contains B such that tp(a/C) is stationary and of U-rank 1. Then part 1 of the lemma shows that tp(a/C) is not modular. But tp(a/C) is trivial by 1.9. A contradiction. □.

Corollary 2.3 Envelopes of finite sets are finite in every sort of C^{eq}.

Proof: Let B be finite and E be an envelope of B. We show first that

A = {e ∈ E|U(e/B) ≤ 1}

is finite: Since in every sort there are only finitely many types over acl(B) we can fix a type p ∈ S(acl(B)) of rank at most 1 and show that p has only finitely many realizations in E. If p is algebraic this is clear. If p has rank 1 it is locally modular. If e ∈ E is a realization of p the nonforking extension p|acl(B)e of p to acl(B)e is modular. By 2.1 it cannot be realized in E . We conclude that all other realizations of p in E must be algebraic over Be. Whence there are only finitely many.

Every element e of E of a fixed sort has a decomposition e_{1}, ...e_{n} = e as in 1.6 for a fixed n where all the e_{i} belong to E. We show that E contains only finitely many elements in this sort: The case n = 1 is already clear. If we add the elements of A to the language E remains an envelope of B and all the e’s have a decomposition of length n − 1. Now we are finished by induction. □

The proof shows also that the envelope of an infinite set B has the same cardinality as B.

Corollary 2.4 Any two envelopes of B are isomorphic over B.

Proof: Let E and F be two envelopes of B. Let f : C → F be a maximal elementary map over B from a subset C of E to F. Clearly C is algebraically closed over B. We want to show that C = E. If not, there is an element e in E \ C. By 1.5 we can assume that p = tp(e/C) has rank 1. Now by 2.1(1) p is not modular and by 2.1(2) f(p) is realized in F. This gives an extension of f to C ∪ {e}, a contradiction. Whence f is defined on E. Since E is maximal f is surjectiv. □

Definition 2 A set E is called strongly homogeneous if every isomorphism between two subsets of E can be extended to an automorphism of E.

Corollary 2.5 An envelope of a finite set is strongly homogeneous.

Proof: Let E be an envelope of a finite set. Then E envelopes the finite subspace U = E ∩ D of D, whose dimension we denote by n. Let f be an isomorphism between the subsets A_{1} and A_{2} which we can assume to be algebraically closed. Let m be the dimension of the two subspaces V_{i} = A_{i} ∩ D (i = 1, 2) of U. First extend f to a partial map g defined on E. U and g(U) both have dimension n−m over V_{2}. The remark after 1.2 shows that also in the geometry of D over A_{2} U and g(U) are subspaces of dimension n − m. Whence there is an elementary map which fixes the elements of A_{2} and maps g(U) onto U. Composing g with this map shows that we can assume that g(U) = U.

E is an envelope of both sets U ∪ A_{i} (i = 1, 2). Therefore g(E) is an envelope of U ∪ A_{2} = g(U ∪ A_{1}). By 2.4 there is an elementary map which fixes the elements of U ∪ A_{2} and maps g(E) onto E. If we compose g with this map we obtain the desired extension of f to an automorphism of E. □

Theorem 2.6 Every finite set of axioms of T has a finite model.

Proof: We can assume that the language of T is relational. Let B be a finite set which realizes all ntypes over ∅. Let E be an envelope of B. Since E is strongly homogeneous every 1-type over every (n − 1)-element subset of E is realized in E. Now an induction as in the proof of Tarski’s criterion shows that for all k ≤ n , all formulas ϕ(x_{1}, ...x_{n−k}) of quantifier depth at most k and all e_{1}, ...e_{n−k} ∈ E

E ⊧ ϕ(e_{1}, ...e_{n−k}) ⟺ C ⊧ ϕ(e_{1}, ...e_{n−k})

. Whence all true axioms of quantifier depth at most n are true in E. □

3 Decomposition into covers

In this section let T be an arbitrary complete theory. Let E and F be A-definable classes. We assume E to be non-empty and F to be infinite.

Definition 1 E is F-internal if there is a set B such that E ⊂ dcl(F ∪ B).

Example 3 Let G be a group acting transitively on a class H in a definable way. Then H is G-internal.

Lemma 3.1 E is F-internal iff for some n there is a definable surjection π : F^{n} → E.

Proof: If π is B-definable then each π(\overline{f}) is definable over B\overline{f}. If conversely E ⊂ dcl(F ∪ B), for every e ∈ E there is a B-definable partial function π_{e} and elements \overline{f} of F such that e = π_{e}(\overline{f}). By compactness we find the π_{e} among a finite family π_{0}, ...π_{m} of k-place functions and if we enlarge B to include A we can assume that they map into E. Choose an element e_{0} from E. We define now the n = m + 1 + k-place function π defined on F as follows: If f_{0}, ...f_{m}, \overline{f} are given let i be maximal such that f_{0} = f_{i}. If π_{i}(\overline{f}) = e is defined set π(\overline{f}) = e. Otherwise set π(\overline{f}) = e_{0}. π is surjectiv and Be_{0}-definable. □

Remark 3.2 Let D_{i} (i = 1, 2) be two strictly minimal sets over parameter sets A_{i} in a totally categorical theory. If A_{1} ⊂ A_{2} then D_{2} is D_{1}-internal.

Proof: Since there is a A_{2}-definable surjection from D_{1} onto (D_{1})_{A_{2}} we can assume that A_{1} = A_{2}. Furthermore there is an obvious definable surjection from D_{1} onto D’_{1}, the strictly minimal set of parallel classes of D_{1}. We may therefore assume that D_{1} is modular. If D_{2} is also modular the two sets are isomorphic (1.7). Otherwise D_{2} is an affine space and D_{1} is - up to isomorphy - the set of parallel classes of D_{2}. We fix a line B in D_{2}. Then every point c in D_{2} outside of B can be defined from to points a, b ∈ B and the two directions from c to a and from c to b. Whence D_{2} ⊂ dcl(D_{1} ∪ B). □

A type p ∈ S(A) is F-internal if p(C) ⊂ dcl(F ∪ B) for some B, which is by compactness equivalent to the existence of a formula E(x) in p which is F-internal.

Lemma 3.3 p is F-internal iff there is a realization e of p and an element b such that e \begin{matrix}| \\ ⌣ \\ A \end{matrix} b and

e ∈ dcl(Fb).

Proof: If p is F-internal there is an E(x) ∈ p and a definable surjection π : F^{n} → E. Choose a parameter b over which π is defined and a realization e of p which is independent from b over A. Then

e ∈ dcl(Fb). If conversely e and b with e ⊧ p, e \begin{matrix}| \\ ⌣ \\ A \end{matrix} b and e ∈ dcl(Fb) are given take for B the

elements of a long Morley sequence (b_{i})_{i∈I} of st(b/A). If e′ is any realization of st(e/A) some b_{i} is independent from e′ over A. Then also e′ ∈ dcl(Fb_{i}). Whence st(e/A) ⊂ dcl(F ∪ B) and st(e/A) is F-internal. Then also all the other extensions of p to acl(A) are F-internal, which implies that p is F-internal. □

We call a tuple \overline{e} of elements of E a fundamental solution if E ⊂ dcl(F ∪ A\overline{e}).

Lemma 3.4 If T is stable and E is F-internal there is a fundamental solution \overline{e}. If T is totally transcendental tp(\overline{e}/F ∪ A) is isolated.

Proof: There is a definable surjection π : F^{n} → E. By stability π is definable from parameters in A∪F∪E. The parameters used from E form a fundamental solution. If π : F^{n} → E is an A\overline{e}-definable surjection we can write π(x) = π(x, \overline{e}) for an A-definable partial function π(x, \overline{y}). That π is a surjection F^{n} → E is an elementary property of \overline{e} which can be expressed by a formula with parameters in A. If T is totally transcendental we can therefore find an \overline{e} which is isolated over F ∪ A. □

Let G be a group of permutations of the definable class P. We call G A-definable if there is an isomorphism of G with an A-definable group G such that the induced action of G on P is A-definable. If G is a subclass of Q we say that G lives on Q. For example a definable group which acts regularly on P lives on P^{eq}.

If P is A-definable and G is an A-definable regular permutation group on P then the opposite group G^{opp} of all G-invariant permutations of P is again an A-definable permutation group.

Theorem 3.5 (Groupe de liaison) Let E and F be 0-definable classes. Assume that E is F-internal and that h_{0} = \overline{e}_{0} is a fundamental solution which is isolated over F. Let H be the F-definable class of all fundamental solutions which have the same type over F as h_{0}. Then

1. The natural map Aut(E/F) → Aut(H/F) is an isomorphism.

2. Aut(H/F) acts regularly on H and is F-definable. (This was called the “binding group” in [PO].)

3. Aut(H/F)^{opp} lives on F^{eq}.

Proof: The 0-definable map π(x, y) in the proof of 3.4 maps F^{n} × H to E. π(x, h) is surjective for every h ∈ H. (1): The map is surjective since by stability every automorphism of H over F extends to an automorphism of the monster model C over F. Fix an element h_{1} ∈ H. Then

α(π(\overline{f}, h_{1})) = π(\overline{f}, α(h_{1}))

shows that every α ∈ Aut(E/F) is determined by its value on h_{1}. Whence our map ist also injective. The formula shows also that Aut(E/F) is F-definable: Any pair (h_{1}, h_{2}) of elements of H defines an α by

α(π(\overline{f}, h_{1})) = π(\overline{f}, h_{2}),

which is in Aut(E/F) since h_{1} can be mapped to h_{2} by an element of Aut(E/F), which must be α. If H is defined over F_{0} then also Aut(E/F) is defined over F_{0} . (2): Is clear by the proof of (1). (3): Let m be the length of h_{0}. The m-th power of π gives a F_{0}-definable map σ : F^{nm} × H → E^{m}. σ(x, h) is surjective for every h ∈ H. Since H is a complete type over F the set

S = {x ∈ F^{nm} | σ(x, h) ∈ H}

does not depend on h. We have therefore an F_{0}-definable map

τ : S × H → H.

τ(x, h) and (by completeness) τ(s, y) are surjective for all h ∈ H and s ∈ S. Now for any pair (s, t) ∈ S

Φ_{st}(τ(h, s)) = τ(h, t) (h ∈ H)

defines a map Φ_{st} : H → H which commutes with every β ∈ Aut(H/F). Since Φ_{ts} is invers to Φ_{st} it belongs to Aut(H/F)^{opp}. Since (τ(h, s), τ(h, t)) can be any pair in H every element of Aut(H/F)^{opp} has the form Φ_{st}. This shows that Aut(H/F)^{opp} lives on S^{eq}. □

Definition 2 Let P ⊂ Q be 0-definable classes. We call Q a cover of P if the following entities exist 0-definably:

1. A partition of Q \ P into a family (H\overline{a})_{\overline{a}∈P}

2. A family (G\overline{a})_{\overline{a}∈P} of groups living on P^{eq} (the structure groups)

3. A regular action of each G\overline{a} on H\overline{a}

If all the (H\overline{a})_{\overline{a}∈P} live on D^{eq} for a 0-definable subclass D of P we call Q a D-cover.

Theorem 3.6 ([HR2]) Let T be ω_{1}-categorical and D a 0-definable strongly minimal set. Then there is a sequence

D = P_{0} ⊂ ... ⊂ P_{n}

of 0-definable classes such that each P_{i+1} is a D-cover of P_{i} and the monster model C is in the definable closure of P_{n}. The length n can be bounded by the Morleyrank of C.

The groups we will find are opposite binding groups as constructed in 3.5. We have therefore to show that there are enough D-internal sets. This is done by the following lemma.

Lemma 3.7 Let T and D be as in 3.6. Then for every a of Morleyrank n over ∅ there is a sequence a_{0}, ...a_{n} = a such that all a_{i} are in dcl(a) and all types tp(a_{i+1}/a_{1}...a_{i}) are D-internal.

Proof: If n = 0 then tp(a/∅) is obviously D-internal. Otherwise tp(a/∅) is not orthogonal to D and there must be a b and an element d of D such that a is independent from b over ∅ but not independent from bd. Choose an element c in the canonical base of q = st(bd/a) such that a \begin{matrix}/| \\ ⌣ \end{matrix} c. Since q is definable over any infinite Morley sequence of q c is definable from an a-independent sequence b_{1}d_{1}, ...b_{m}d_{m} of realizations of q. We have a | ⌣ b_{i} for all i. This implies that a | ⌣ b_{1}...b_{m} and - because c ∈ acl(a) - c | ⌣ b_{1}...b_{m}. By 3.3 we know that tp(c/∅) is D-internal. If c_{1}, ...c_{k} are the conjugates of c over a the types tp(c_{i}/∅) are also D-internal. We set a_{1} = {c_{1}, ...c_{k}}. Then tp(a_{1}/∅) is D-internal and a_{1} ∈ dcl(a). Since a is dependent from a_{1} the Morleyrank of tp(a/a_{1}) is smaller that n. After adding the constant a_{1} to the language we can use induction to obtain the desired sequence. □

Proof of 3.6: We express the content of 3.7 by saying that any a of rank n is n-analysable. It is easy to see from the definition that “tp(a/b) is D-internal” is a ⋀-definable property of ab i.e. definable by a conjunction of a set of formulas. Whence also “tp(a/∅) is n-analysable” is ⋀-definable.

Let R be a 0-definable class of elements which are all n-analysable. We show by induction on n that R is in the definable closure of a “covering” sequence of length n. By the above we can apply compactness to

⊧ ∀a ∈ R ∃b ((tp(a/b) D − internal) ∧ (tp(b/∅) (n − 1) − analysable))

to obtain a 0-definable class Q with

For every element a of R there is a b in Q for which tp(a/b) is D-internal.

All elements of Q are (n − 1)-analysable.

Using induction we find a covering sequence D = P_{0} ⊂ … ⊂ P_{n−1} such that Q is in dcl(P_{n−1}). If tp(a/b) is D-internal and b ∈ dcl(\overline{c}) then also tp(a/\overline{c}) is D-internal. Whence for all a ∈ R there is a \overline{c} ∈ P_{n−1} for which tp(a/\overline{c}) is D-internal. We have to find a cover P_{n} of P_{n−1} such that R ⊂ dcl(P_{n}).

By 3.1 every a ∈ R belongs to a D-internal class E which is definable over a subset A of P_{n−1}. Let H be the class of fundamental solutions defined in 3.5 and G = Aut(H/D ∪ A)^{opp}. Then for some parameter \overline{c} ∈ P_{n−1}:

H,G and the - regular - action of G on H are definable from \overline{c}.

a ∈ dcl(H\overline{c}).

We can assume that H = H′ ; \overline{c} and G = G′ ; \overline{c} for 0-definable families

(H′ ; \overline{b}) , (G′ ; \overline{b}) (\overline{b} ∈ P_{n−1} )

of regular acting groups. The families still depend on a. But by compactness there is a finite set of families

(Hi ; \overline{b}) , (Gi ; \overline{b}) (\overline{b} ∈ P_{n−1} , i = 1, … m)

which can be used for every a ∈ R.

Finally define

H\overline{c} = H1 ; \overline{c} × … × Hm ; \overline{c}

G\overline{c} = G1 ; \overline{c} × … × Gm ; \overline{c} (\overline{c} ∈ P_{n−1}).

with the obvious action. We can assume that the H\overline{c} are pairwise disjoint and disjoint to P_{n−1} and that still R ⊂ dcl(P_{n−1} ∪ ⋃_{\overline{c}∈P_{n−1}} H\overline{c}). Set

P_{n} = (P_{n−1} ∪ ⋃_{\overline{c}∈P_{n−1}} H\overline{c}) .

This proves 3.6. □

In totally categorical theories we have a better result. Also the theory of internality is not used at all.

Theorem 3.8 If T is totally categorical and D is a projective space over the field F_{q} we can find a covering sequence

D = P_{0} ⊂ ... ⊂ P_{m}

such that the monster model C is in the definable closure of P_{m} and each P_{i+1} is a D-cover of P_{i} with structure groups G\overline{a} which are either finite or F_{q}-vectorspaces (and with no other structure over \overline{a}).

Proof: By a modification of the proof of 3.6. Instead of 3.7 we use 1.6: Also by ω-categoricity there is no need to deal with a finite family of groups at the same time and use the direct product. The price we pay of course is that we cannot compute a bound for the length of the covering sequence. Furthermore no use is made of internality and the groupe de liason.

Thus the only thing we have to prove is the following: Suppose that \overline{c} is a tuple from P_{n−1} and that tp(a/\overline{c}) is algebraic or strictly minimal. Then there is a two-step cover P_{n+1} of P_{n−1} of the desired kind such that a is definable over P_{n+1}. For this it is enough to find a group G which lives on D^{eq} and acts regularly on a set H, everything definable over P_{n}, such that a is definable over P_{n} ∪ H.

If a is algebraic over \overline{c} only one cover is needed: set P_{n} = P_{n−1}. Let b_{1} … b_{m} be the sequence of conjugates of a over \overline{c} and define G = Sym(1 … m) and H = \{b_{σ(1)} … b_{σ(m)}|σ ∈ G\} with the obvious action.

If tp(a/\overline{c}) is strictly minimal, it is modular or affine. If it is modular it is by 1.7 algebraic over D\overline{c} and we are reduced to the first case. If it is affine the realization set D_{1} of tp(a/\overline{c}) is an affine geometry with no other structure over \overline{c}. The corresponding F_{q}-vectorspace V which acts regularly on D_{1} is definable from \overline{c}. Let D_{2} = D\overline{c} be the strictly minimal set over \overline{c}, associated to D. Then by 1.7 there is a \overline{c}-definable bijection between D_{2} and the projective space of 1dimensional subspaces of V. Whence V is algebraic over P_{n−1} and we can build a cover P_{n} with finite structure groups (a finite cover) such that all elements of V are definable over P_{n} i.e. V lives in P_{n}^{eq}. If we now set H = D_{1} and G = V we are done, besides that G is only to be known to live on P_{n}^{eq} and we want it to live on D^{eq}.

But this is dealt with by finding a P_{n}-definable injection π : V → D_{2}^{3}. We choose three independent elements v_{1}, v_{2}, v_{3} from V and define

π(v) = (F_{q}(v + v_{1}), F_{q}(v + v_{2}), F_{q}(v + v_{3})).

Of course the copy of V living on D^{eq} is now defined using the parameters \overline{c}v_{1}v_{2}v_{3}. □

4 The structure of covers

Let M be an arbitrary structure and let a 0-definable family of groups (G\overline{b})_{\overline{b}∈M} be given. A cover of M with structure groups (G\overline{b}) is a two-sorted structure N = (M, H; R_{1}, R_{2}, …) in which H is 0-definably partitioned into sets (H\overline{b})_{\overline{b}∈M} on which the G\overline{b} act regularly in a 0-definable way. Also we want N not to add new structure to M.

This means that

1. every in N 0-definable relation on M is 0-definable in M

2. every in N definable relation on M is definable in M

The second clause means that M is stably embedded in N in the sense of the next definition. Only by this condition it is that a cover of a totally categorical model is again totally categorical (see 4.4).

Definition 1 Let T be arbitrary and P be a 0-definable class. P is called stably embedded if every definable relation on P is definable with parameters in P . A 0-definable set P in a structure M is called stably embedded with finite support if all types tp(\overline{a}/P) of elements of M over P are definable over a finite subset of P.

Remark 4.1 Pis stably embedded iff all types over Pare definable over P. Let B be a subset of P. Then tp(\overline{a}/P) is definable over B iff tp(\overline{a}/B) ⊢ tp(\overline{a}/P) iff tp(\overline{b}/B) ⊢ tp(\overline{b}/B\overline{a}) for all \overline{b} ∈ P.

If T is stable all P are stably embedded. If in some model M P is stably embedded with finite support then P is stably embedded. If T is ω-stable then all P are stably embedded with finite support. The following theorem was explained to me by Hrushovski and is also contained in [HP]

Theorem 4.2 Let T be a countable complete theory and P be a 0-definable predicate.

1. Let M be an uncountable saturated model. Then P is stably embedded iff every automorphism of P extends to an automorphism of M.

2. Let M be a countable saturated model. Then P is stably embedded with finite support iff every automorphism of P extends to an automorphism of M.

Proof: (1): Assume first that P is stably embedded and that α is an automorphism of P which is already extended to an elementary map β : (P ∪ A) → M for a subset A of small cardinality. We want to extend β to an arbitrary element a of M. Let p = tp(a/P ∪ A). By 4.1 there is a small subset B of P with q = tp(a/B ∪ A) ⊢ p. Then also β(q) ⊢ β(p). Since we can realize β(q) we can also realize β(p) in M and extend β.

For the converse assume that Q is a definable subset of P. The structure (P, Q) - P with its induced structure and with the predicate Q- is saturated. In Aut(M) Q has at most |M|-many conjugates. If one can extend every automorphism of P to an automorphism of M Q has also at most |M|-many conjugates in Aut(P). Kueker’s theorem implies now that Q is definable in P.

(2):Same as (1). But one needs a generalized Kueker theorem: If (P, Q_{i})_{i∈I} is a saturated structure such that (Q_{i})_{i∈I} hat at most |M|-many conjugates in Aut(P) then there is a finite set of parameters over which all the Q_{i} are definable in P. □

Remark 4.3 If P is countable and ω-categorical one has not to assume that (P, Q) is saturated for Kueker’s theorem to be true. For the ”if”-direction of (2) one has therefore not to assume that M is saturated if P is ω-categorical.

Lemma 4.4 Let N = (M, H) be a two-sorted structure where in a 0-definable way H is partitioned into a family of sets H\overline{b} (\overline{b} ∈ M) on which groups G\overline{b} living in M act regularly.

1. Assume that M is stably embedded in N. If then M (with its induced structure) is ω-stable or ω_{1}−categorical then also N is ω-stable or ω_{1}-categorical respectively.

2. Assume that M is stably embedded in N with finite support. Then N is ω-categorical if M is.

Proof: (We write the \overline{b} as 1-tuples) (1): The Morleyrank of H is bounded if the ranks of the H_{b} are uniformly bounded and the rank of the index set M is bounded. Because of the regular action the H_{b} and the G_{b} have the same rank. But since M is stably embedded the (H, M)-Morleyrank of definable subsets of M is the same as their M-Morleyrank. Thus if M has Morleyrank as a structure of its own also H has Morleyrank. Concerning ω_{1}-categoricity we prove that T h(N) has no Vaughtian pairs if T h(M) has no Vaughtian pairs. Suppose that T h(M) has no Vaughtian pairs and let N ≺ N′ be a proper elementary extension and Q = ⋃_{s∈S} Q_{s} , where Q_{s} ⊂ H_{s}, a definable subset of H which does not increase in (M′, H′). Since each H_{b} is in definable bijection with G_{b} M′ is a proper extension of M. That Q does not increase means that S does not increase and that all the Q_{s} do not increase. Since S is definable in M it must be finite. There are definable bijections of the Q_{s} onto definable subsets of G_{s}, which are by assumption definable in M. Whence with these sets the Q_{s} also must be finite. We conclude that Q is finite and that T h(N) has no Vaughtian pairs. (2): We can assume that N is countable. Since Aut(M) has only finitely many orbits on M^{n} by 4.2 also Aut(N) has only finitely many orbits on M^{n}. To count the orbits of Aut(N) on H^{n} it is therefore enough to count for any fixed sequence \overline{s} = s_{0} … s_{m} ∈ M the types of n-tuples from H\overline{s} = ⋃_{i≤m} H_{s_{i}} over \overline{s}. We choose a representative b_{i} from every H_{s_{i}}. Then there is a \overline{b}-definable bijection of H\overline{s} onto the disjoint union G\overline{s} of the G_{s_{i}}. Whence we can bound the number of types of n-tuples from H\overline{s} over \overline{s} by the number of types of n-tuples of elements of M over \overline{s}\overline{b}. Now let tp(\overline{s}\overline{b}/M) be definable over the finite B ⊂ M. Then we have as many types of elements of M over B as we have types over B\overline{s}\overline{b}: Finitely many! □

Corollary 4.5 Covers of totally categorical structures are totally categorical.

□

We fix now a countable totally categorical structure M and a 0-definable family of groups (G\overline{b})_{\overline{b}∈M} living in M^{eq}. For notational simplicity we assume the b = \overline{b} be 1-tupels.

All covers of M with structure groups (G\overline{b}) are (up to isomorphy) expansions of the principal cover N_{0} = (M, H) where the only relations added to M are a map H → M to partition H into sets H_{b} (b ∈ M) and a family of regular actions of the G_{b} on the H_{b}.

Corollary 4.6 The covers of M correspond up to interdefinability to the closed subgroups of Aut(N_{0}) which induce Aut(M) on M.

Proof: The automorphism groups of covers are closed subgroups of Aut(N_{0}) which induce Aut(M) by 4.2. Since covers are ω-categorical they are uniquely determined (as an expansion of N_{0}) by their automorphism groups. If a closed subgroup G of Aut(N_{0}) is given it is the automorphism group of an expansion N of N_{0}. If G induces Aut(M) by 4.2 M is stably embedded in N and N is a cover of M. □

We distinguish covers N by their kernel K = Aut(N/M), which is a closed subgroup of K_{0} = Aut(N_{0}/M) ≅ ∏_{b∈M} G_{b}. The last isomorphism is not natural if the G_{b} are not abelian. The natural isomorphism is between K_{0} and ∏_{b∈M}(G_{b})^{opp}.

By [HHLS] countable totally categorical models M have the small index property: The open subgroups of Aut(M) coincide with the subgroups of countable index. This is the main ingredient of the proof of the following lemma:

Lemma 4.7 ([HP]) Let G be a subgroup of Aut(N_{0}) which induces Aut(M). Then G is closed iff the kernel K = G ∩ K_{0} is closed.

Proof: Assume the kernel of Gto be closed. The small index properties implies that the restriction map from G onto Aut(M) is open. Thus if O is an open neighbourhood of 1 in Aut(N_{0}) then (O∩G)|M is an open neighourhood of 1 in Aut(M). It follows that O′ = K_{0}(O ∩ G) is open in Aut(N_{0}). Let h be an automorphism of N_{0} which does not belong to G . We want to show that h is not in the closure of G . Since G contains an element which has the same restriction to M as h we can divide by this element and assume that h belongs to K_{0}. Now choose an open subgroup O of Aut(N_{0}) such that hO is disjoint from K . Then O′ ∩ hO is a neighourhood of h which is disjoint from G. □.

We assume from now on that all G_{b} are abelian. This is a harmless assumption by 3.8. Then in Aut(N_{0}) conjugation with h depends only on hK_{0}. We obtain therefore a natural action of Aut(M) on Aut(N_{0}). The kernels of subgroups which project onto Aut(M) are obviously invariant under this action.

Lemma 4.8 For every closed subgroup K of K_{0} which is invariant under the action of Aut(M) there is a cover of M with kernel K .

Proof: Choose a transversal t = (t_{b})_{b∈M}, where t_{b} ∈ H_{b}. Then every element of H_{b} has the form t_{b}g for a unique g ∈ G_{b}. We lift now every automorphism α of M to an automorphism α̃ of N_{0} by setting α̃(t_{b}g) = t_{α(b)}α(g). Then the automorphism group of the desired cover is G = {hα̃ |h ∈ K, α ∈ Aut(M)}. □

We call the covers constructed in the proof of 4.8 trivial extensions of K . They depend on the choice of the transversal t, but they are all conjugated by an element of K_{0} i.e. their automorphism groups are in Aut(N_{0}) conjugated by an element of K_{0}.

Theorem 4.9 Let K be a closed, Aut(M)-invariant subgroup of K_{0}. Then there is is a natural bijection between the conjugacy classes of covers of M with structure groups (G_{b})_{b∈M} and H^{1}(Aut(M), K_{0}/K). The trivial covers correspond to 0.

Proof: Let N be a cover with kernel K .Choose a transversal t and define for every α ∈ Aut(M) an element dα of K_{0} by \overline{α}(t) = tdα, where \overline{α} is any extension of α to N. Modulo K this defines a derivation d : Aut(M) → (K_{0}/K) whose cohomology is independent of the choice of t and \overline{α}. This defines the desired bijection. More details and the computation of H^{1}(Aut(M), K_{0}/K) in some special cases can be found in [AZ2]. □

(revision 25-07-1991)

References

[AZ1] G.Ahlbrandt and M.Ziegler, Quasi-finitely axiomatizable totally categorical theories, Annals Pure Appl. Logic, 30 /1986) 63-83

[AZ2] G.Ahlbrandt and M.Ziegler,What’s so special about (Z/4Z)^{ω}?,(1990) to appear in Archive of M.Logic

[HP] W.Hodges and A.Pillay, Cohomology of structures and some problems of Ahlbrandt-Ziegler, preprint(1991)

[HR1] E.Hrushovski, Totally categorical structures , Transactions of the AMS 313,1 (1989)

[HR2] E.Hrushovski, Unidimensional theories, Logic Colloquium 87

[B] S.Buechler, Locally modular theories of finite rank, Annals Pure Appl. Logic 30 (1986) 83-94

[CHL] G.Cherlin, L.Harrington and H.Lachlan, ω_{0}-categorical, ω_{0}-stable theories, Annals Pure Appl. Logic 28 (1985) 103-135

[HHLS] Hodges,Hodkinson,Lascar,Shelah,....,in preparation [D.L] D.Lascar, Relation entre le rang U et le poids, Fundamenta Mathematicae, 121 (1984)

117-123 [C.L] M.C.Laskowski, Uncountable theories that are categorical in a higher power, JSL 53 (1988)

512-530

[PI] A.Pillay, preprint of some chapters of a forthcoming book

[PR] A.Pillay and P.Rothmaler, Non-totally transcendental unidimensional theories (1989), to appear in Archive Math.Logic

[PO] B.Poizat, Groupes stables (1987)