The geometry of 1-based minimal types

Tristram De Piro, Byunghan Kim

Research output: Contribution to journalArticle

12 Citations (Scopus)

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 languageEnglish
Pages (from-to)4241-4263
Number of pages23
JournalTransactions of the American Mathematical Society
Volume355
Issue number10
DOIs
Publication statusPublished - 2003 Oct 1

Fingerprint

Geometry
Division ring or skew field
Fold
Vector spaces
Categorical
Vector space
Galois field

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

De Piro, Tristram ; Kim, Byunghan. / The geometry of 1-based minimal types. In: Transactions of the American Mathematical Society. 2003 ; Vol. 355, No. 10. pp. 4241-4263.
@article{921040c1a57d491080aca234668bbcf5,
title = "The geometry of 1-based minimal types",
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.",
author = "{De Piro}, Tristram and Byunghan Kim",
year = "2003",
month = "10",
day = "1",
doi = "10.1090/S0002-9947-03-03327-0",
language = "English",
volume = "355",
pages = "4241--4263",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "10",

}

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

In: Transactions of the American Mathematical Society, Vol. 355, No. 10, 01.10.2003, p. 4241-4263.

Research output: Contribution to journalArticle

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 -