### 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 ω-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.

Original language | English |
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Pages (from-to) | 4241-4263 |

Number of pages | 23 |

Journal | Transactions of the American Mathematical Society |

Volume | 355 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2003 Oct 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*355*(10), 4241-4263. https://doi.org/10.1090/S0002-9947-03-03327-0

}

*Transactions of the American Mathematical Society*, vol. 355, no. 10, pp. 4241-4263. https://doi.org/10.1090/S0002-9947-03-03327-0

**The geometry of 1-based minimal types.** / De Piro, Tristram; Kim, Byunghan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The geometry of 1-based minimal types

AU - De Piro, Tristram

AU - Kim, Byunghan

PY - 2003/10/1

Y1 - 2003/10/1

N2 - 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 ω-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.

AB - 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 ω-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.

UR - http://www.scopus.com/inward/record.url?scp=0242382097&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242382097&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-03-03327-0

DO - 10.1090/S0002-9947-03-03327-0

M3 - Article

AN - SCOPUS:0242382097

VL - 355

SP - 4241

EP - 4263

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 10

ER -