Géométries relatives

Abstract

Une analyse des propriétés géométriques d'une structure relatives à un réduit est entamée. En particulier la définissabilité des groupes et des corps dans ce cadre est étudiée. Dans le cas relativement monobasé, tout groupe définissable est isogène à un sous-groupe d'un produit de groupes définissables dans les réduits. Dans le cas relativement CM-trivial, cas qui englobe certains amalgames de Hrushovski (la fusion de deux théories fortement minimales, les expansions d'un corps par un prédicat), tout groupe définissable s'envoie par un homomorphisme à noyau central dans un produit de groupes définissables dans les réduits. English SummaryIn this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expansions of algebraically closed fields by operators. We prove the following (Theorem 3.1) :Theorem. Let T be simple and T 0 be a stable reduct of T . If G is a type-definable group in T , there are a * -interpretable group H in T 0 and a definable homomorphism φ : G 0 → H such that for independent generic elements g, g ′ of G we can name a set D independent of g, g ′ withwhere | 0 ⌣ and acl 0 denote independence and algebraic closure respectively in the reduct T 0 over D.Thus φ captures all possible 0-dependence relations resulting from the group operation. In order to ensure that the morphism is non-trivial, we need to add further hypotheses on the geometric complexity of T over T 0 : Relative one-basedness and relative CM-triviality. The kernel of φ is finite in the first case (Theorem 4.9) and virtually central in the latter (Theorem 5.7).Examples for relatively one-based theories are differentially closed fields in characteristic 0 and fields with a generic automorphism. We therefore recover, up to isogeny, the characterisation of definable groups in these structures from [20,24].Date: 8 novembre 2018. 1991 Mathematics Subject Classification. 03C45.

Tame Expansions of ω-Stable Theories and Definable Groups

“…For the following lemma see [3] and this lemma will be crucial in order to characterize algebraic closure in the pair (K, G).…”

De beaux groupes

“…La caractérisation de l'indépendance ci-dessus permet d'obtenir l'analogue à la propriété ( †) dans [4]. …”

Model Theory of Fields With Operators – a Survey

“…In a recent paper [7], Blossier, Martin-Pizarro and Wagner introduce the notion of a theory being one-based over another, and describe definable groups in models of the first one as almost embeddable in definable groups in the smaller language of the second one. An application of their result gives then that ( * ) holds in all the theories we have so far considered, provided that -the theory eliminates imaginaries, -independence is given by ACF-independence of the algebraic closures in the sense of the theory.…”