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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The geometry of 1-based minimal types


Authors: Tristram de Piro and Byunghan Kim
Journal: Trans. Amer. Math. Soc. 355 (2003), 4241-4263
MSC (2000): Primary 03C45
Published electronically: June 18, 2003
MathSciNet review: 1990585
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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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Additional Information

Tristram de Piro
Affiliation: Mathematics Department, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139
Email: tdpdp@math.mit.edu

Byunghan Kim
Affiliation: Mathematics Department, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139
Email: bkim@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03327-0
PII: S 0002-9947(03)03327-0
Received by editor(s): January 4, 2002
Received by editor(s) in revised form: March 11, 2003
Published electronically: June 18, 2003
Additional Notes: The second author was supported by an NSF grant
Article copyright: © Copyright 2003 American Mathematical Society