Ample thoughts

Abstract

Non-n-ampleness as defined by Pillay and Evans is preserved under analysability. Generalizing this to a more general notion of Sigma-ampleness, we obtain an immediate proof for all simple theories of CHatzidakis weak Canonical Base Property (CBP) for types of finite SU-rank. This is then applied to the special case of groups

On definable Galois groups and the strong canonical base property

“…In order to prove the main result of this section we need to introduce the following from [5]. First, recall that a stationary type p over B is analyzable in Q if for any realization a of p there are (a i : i < α) ∈ dcl(B, a) such that tp(a i /B, a j : j < i) is internal to Q for all i < α, and a ∈ acl(B, a i : i < α).…”

Rigidity, internality and analysability

“…Another question concerns nilpotent groups: In [15,Remark 6.7] it is shown that if G is Σ-analysable type-definable or supersimple, then there is a nilpotent normal subgroup N such that G/N is almost Σinternal. Is this related to Proposition 3.9 stating that a superstable rigid group is virtually nilpotent?…”

“…This connection was noticed by Kowalski and Pillay in [13], and used to describe the structure of type-definable groups in stable theories satisfying the CBP. The relation between one-basedness and the weak CBP was then used by the authors [15] to generalize and study the weak CBP outside the finite SU-rank setting by replacing the family of types of rank 1 by an arbitrary family of partial types. These ideas also appear in [4] where Blossier, Martín-Pizarro and the second author study a generalization of the CBP in a different direction.…”

On the canonical base property

“…The canonical base property or CBP is a property of finite rank theories, the formulation of which was motivated by results of Campana in bimeromorphic geometry and analogous results by Pillay and Ziegler in differential and difference algebraic geometry in characteristic 0. The notion has been studied by Chatzidakis [1], Moosa and Pillay [3] (where the expression CBP was introduced) and in a somewhat more general framework by Palacín and Wagner [4]. The notion makes sense for arbitrary supersimple theories of finite SU-rank.…”

“…Note also that the canonical base property can be seen as a generalization of 1-basedness: T is 1-based if whenever b = Cb(tp(a/b)) then b ∈ acl(a). This point of view is profitably pursued in [4].…”