On the canonical base property

Abstract

Abstract. We give an example of a finite rank, in fact ℵ 1 -categorical, theory where the canonical base property (CBP ) fails. In fact we give a "group-like" example in a sense that we will describe below. We also prove, in a finite Morley rank context, that if all definable Galois groups are "rigid" then T has the CBP .

Rigidity, internality and analysability

“…Namely, in a group of finite SU-rank with the CBP, every type with finite stabilizer is almost internal to the family of types of rank 1. This was noted in [16] and elaborated in [12].…”

“…More generally, Chatzidakis proves the following two facts (she works in a simple theory with elimination of hyperimaginaries and the family Σ of all non locally modular (Lascar) strong types of SU-rank 1, but the generalization is straightforward). When stab(g/A) is bounded, Fact 4.1 yields the CBP version of the Corollary above, see for example [12,Fact 1.3]. We shall now prove a rigidity version in the spirit of the original Socle Lemma.…”

On definable Galois groups and the strong canonical base property

“…In the joint paper with Hrushovski [3] an example appears of a finite rank, in fact ℵ 1 -categorical theory, where the CBP fails. Now if T is an ℵ 1categorical theory, then the obstruction to T being almost strongly minimal is given by infinite definable Galois groups (using the theory of analyzability and definable automorphism groups), of which precise definitions will be given below.…”

“…Similarly for finite rank nonmultidimensional theories. In [3] we worked, for convenience, with finite rank theories T which are "coordinatised" by strongly minimal formulas over ∅, and proved that if all relevant definable Galois groups are rigid then T has the CBP in the strong form that whenever c = Cb(stp(b/c)), then c is contained in the algebraic closure of b and realizations of strongly minimal formulas over ∅. (This was generalized in [6] using results in [1]).…”

“…

In [3], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that T has the canonical base property in a strong form; " internality to" being replaced by "algebraicity in". In the current paper we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than [3], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that 1-based groups are rigid.

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