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- Title
Géométries relatives.
- Authors
Blossier, Thomas; Martin-Pizarro, Amador; Wagner, Frank O.
- Abstract
In 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 T0 be a stable reduct of T. If G is a type-definable group in T, then there are a *-interpretable group H in T0 and a definable homomorphism φ V G0 ! H such that for independent generic elements g, g0 of G we can name a set D independent of g, g0 with acl.g/; acl.g0/j0 acl.φ.gg0//\acl0.acl.g//;acl.g0//acl.gg0/; where j0 and acl0 denote independence and algebraic closure respectively in the reduct T0 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 T0: 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]. Finally, we show that all coloured fields and fusions constructed by Hrushovski amalgamation are relatively CM-trivial (Theorem 6.4).
- Subjects
GROUP theory; ALGEBRAIC fields; HOMOMORPHISMS; MODEL theory; DEFINABILITY theory (Mathematical logic)
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2015, Vol 17, Issue 2, p229
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/502