Logic Seminar
Martin Hils
Equipe de Logique, Universite Paris 7
Generic Automorphisms of Green and Bad Fields
Abstract: The greed field of Poizat is an algebraically closed field $K_\omega$ together with a proper infinite divisible torsion free subgroups $U$ of the multiplicative group. It is of Morley rank $\omega \cdot 2$ and obtained by Hrushovski's amalgamation method. It may be collapsed into a ``bad field'', i.e. a finite Morley rank analogue $(K_\mu, 0, 1, +, \times, U)$. Recall that if T is a model-complete $L$-theory, the generic automorphism is said to be axiomatizable in T is the class of existentially closed models of the theory $T_\sigma = T + (\sigma \mbox{ is an automorphism})$ is elementary. In the talk we show that the generic automorphism is axiomatisable in the theory of green fields of Poizat (once this theory is Morleyised) as well as in the theory of the bad field. As a corollary, we obtain ``bad pseudofinite fields'' in characterisitc $0$. In both cases, we give geometric axioms. In fact, there is a general framework aloowing this kind of axiomatisation. There are serious definability issues related to the necessary ``choice of (unique) gree roots'', and Kummer thoery comes into play. We overcome these difficulties using weak CIT and an effective version of an argument by Zilber, showing that being Kummer-generic is a defiable property for algebraic varieties in characteristic $0$. Finally, we will discuss similar results in various other theories obtained by Hrushovski amalgamation -- both non-cllapsed and collapsed: the fusion of two strongly minimal thoeries, black fields in all characteristics, red fields in positive characteristic etc. The proofs are less involved since the above definability problems do not arise in these contexts.
seminar begins with tea.
Monday February 22, 2010 at 4:00 PM in SEO 612