×

Relative geometries. (Géométries relatives.) (French. English summary) Zbl 1386.03034

Summary: 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 \(T_0\) be a stable reduct of \(T\). If \(G\) is a type-definable group in \(T\), then there are a \(\ast\)-interpretable group \(H\) in \(T_0\) and a definable homomorphism \(\phi : G^0 \to H\) such that for independent generic elements \(g\), \(g^\prime\) of \(G\) we can name a set \(D\) independent of \(g\), \(g^\prime\) with \[ \begin{aligned} \mathrm{acl}(g), \mathrm{acl}(g') \underset{{\mathrm {acl}(\phi (gg')) \cap \mathrm{acl}_0 (\mathrm{acl}(\phi(g)), \mathrm{acl}(\phi (g')))} } {\perp^0}\mathrm{acl} (gg'), \end{aligned} \] where \(\perp^0\) and \(\mathrm{acl}_0\) denote independence and algebraic closure respectively in the reduct \(T_0\) over \(D\). Thus \(\phi\) 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 \(\phi\) 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 [P. Kowalski and A. Pillay, Proc. Am. Math. Soc. 130, No. 1, 205–212 (2002; Zbl 0983.03033); A. Pillay, Pac. J. Math. 179, No. 1, 179–200 (1997; Zbl 0999.12009)]. Finally, we show that all coloured fields and fusions constructed by Hrushovski amalgamation are relatively CM-trivial (Theorem 6.4).

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C60 Model-theoretic algebra
20A15 Applications of logic to group theory
PDFBibTeX XMLCite
Full Text: DOI arXiv