Transactions of the American Mathematical Society

Author(s): Tristram de Piro; Byunghan Kim
Journal: Trans. Amer. Math. Soc. 355 (2003), 4241-4263.
MSC (2000): Primary 03C45
Posted: June 18, 2003
Retrieve article in: PDF DVI PostScript

Abstract: In this paper, we study the geometry of a (nontrivial) 1-based

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.

1.: E. Artin, Geometric algebra, Interscience Publishers, New York (1957). MR 18:553e
2.: I. Ben-Yaacov, I. Tomasic and F. O. Wagner, Constructing an almost hyperdefinable group, preprint, 2001.
3.: S. Buechler, The geometry of weakly minimal types, J. of Symbolic Logic 50 (1985) 1044-1053. MR 87k:03029
4.: S. Buechler, A. Pillay and F. O. Wagner Supersimple theories, J. Amer. Math. Soc. 14 (2001) 109-124. MR 2001j:03065
5.: Z. Chatzidakis and A. Pillay, Generic structures and simple theories, Ann. of Pure and Applied Logic 95 (1998) 71-92. MR 2000c:03028
6.: G. Cherlin, L. Harrington, and A. H. Lachlan, $\aleph_0$ -categorical, $\aleph_0$ -stable structures, Ann. Pure and Applied Logic 28 (1985) 103-135. MR 86g:03054
7.: J. Doyen and X. Hubaut, Finite regular locally projective spaces, Math. Z. 119 (1971) 83-88. MR 43:3897
8.: D. Evans and F. O. Wagner, Supersimple $\omega$ -categorical groups, J. of Symbolic Logic 65 (2000) 767-776. MR 2001f:03067
9.: B. Hart, B. Kim and A. Pillay, Coordinatisation and canonical bases in simple theories, Journal of Symbolic Logic, 65 (2000) 293-309. MR 2001j:03066
10.: E. Hrushovski, Unimodular minimal theories, Journal of London Math. Soc., 46 (1992) 385-396. MR 94b:03062
11.: E. Hrushovski, Locally modular regular types, in Classification theory, Lecture Notes in Math. 1292, Springer-Verlag, Berlin (1987) 132-164. MR 90n:03064
12.: E. Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996) 667-690. MR 97h:11154
13.: E. Hrushovski, Simplicity and the Lascar group, preprint (1998).
14.: E. Hrushovski and S. Shelah, A dichotomy theorem for regular types, Ann. of Pure and Applied Logic 45 (1989) 157-169. MR 91g:03068
15.: B. Kim, Forking in simple unstable theories, Journal of London Math. Soc. (2) 57 (1998) 257-267. MR 2000a:03052
16.: B. Kim and A. Pillay, Simple theories, Ann. of Pure and Applied Logic 88 (1997) 149-164. MR 99b:03049
17.: A. Pillay, Geometric stability theory, Oxford University Press, Oxford (1996). MR 98a:03049
18.: S. Shelah, Simple unstable theories, Annals of Math. Logic 19 (1980), 177-203. MR 82g:03055
19.: S. Shelah, Classification theory, 2nd edition, North-Holland, Amsterdam (1990). MR 91k:03085
20.: I. Tomasic and F. O. Wagner, Applications of the group configuration theorems in simple theories, preprint, 2001.
21.: E. Vassiliev, Expansions of rank structures, Ph. D. Thesis, Univ. of Notre Dame (2001).
22.: F. O. Wagner, Simple theories, Kluwer Academic Publishers, Dordrecht (2000). MR 2001b:03035
23.: B. I. Zilber, Uncountably categorical theories, Amer. Math. Soc. Translations of Mathematical Monographs, Vol. 117 (1993). MR 94h:03059

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C45

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

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS