The geometry of 1-based minimal types

de Piro, Tristram; Kim, Byunghan

doi:10.1090/S0002-9947-03-03327-0

The geometry of 1-based minimal types
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by Tristram de Piro and Byunghan Kim PDF

Trans. Amer. Math. Soc. 355 (2003), 4241-4263 Request permission

Abstract:

In this paper, we study the geometry of a (nontrivial) 1-based $SU$ rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type that is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any $\omega$-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.

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