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Isolated types of finite rank: an abstract Dixmier–Moeglin equivalence

  • Omar León SánchezEmail author
  • Rahim Moosa
Article
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Abstract

Suppose T is a totally transcendental first-order theory and every minimal non-locally-modular type is nonorthogonal to a nonisolated minimal type over the empty set. It is shown that a finite rank type \(p={\text {tp}}(a/A)\) is isolated if and only if Open image in new window for every \(b\in {\text {acl}}(Aa)\) and \(q\in S(Ab)\) nonisolated and minimal. This applies to the theory of differentially closed fields—where it is motivated by the differential Dixmier–Moeglin equivalence problem—and the theory of compact complex manifolds.

Keywords

Model theory Totally transcendental theories Differential fields 

Mathematics Subject Classification

03C95 03C98 12H05 

Notes

References

  1. 1.
    Aschenbrenner, M., Moosa, R., Scanlon, T.: Strongly minimal groups in the theory of compact complex spaces. J. Symb. Log. 71(2), 529–552 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bell, J., Launois, S., León Sánchez, O., Moosa, R.: Poisson algebras via model theory and differential-algebraic geometry. J. Eur. Math. Soc. 19(7), 2019–2049 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bell, J., León Sánchez, O., Moosa, R.: \({D}\)-groups and the Dixmier-Moeglin equivalence. Algebra Number Theory 12(2), 343–378 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chatzidakis, Z., Hrushovski, E.: Model theory of difference fields. Trans. Am. Math. Soc. 351(8), 2997–3071 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Freitag, J., Scanlon, T.: Strong minimality and the \(j\)-function. J. Eur. Math. Soc. 20(1), 119–136 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hrushovski, E.: Locally modular regular types. In: Classification Theory (Chicago, IL, 1985), Lecture Notes in Mathematics, vol. 1292, Springer, Berlin, pp. 132–164 (1987)Google Scholar
  7. 7.
    Hrushovski, E., Sokolovic, Z.: Minimal sets in differentially closed fields (Unpublished)Google Scholar
  8. 8.
    Marker, D.: Manin kernels. In: Connections Between Model Theory and Algebraic and Analytic Geometry, vol. 6. Dept. Math., Seconda Univ. Napoli, Caserta, pp. 1–21. http://homepages.math.uic.edu/~marker/manin.ps (2000)
  9. 9.
    Marker, D., Messmer, M., Pillay, A.: Model Theory of Fields. Lecture Notes in Logic, vol. 5, 2nd edn. Association for Symbolic Logic, La Jolla, CA; A K Peters, Ltd., Wellesley, MA (2006)Google Scholar
  10. 10.
    Moosa, R.: On saturation and the model theory of compact Kähler manifolds. J. Reine Angew. Math. 586, 1–20 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Moosa, R., Pillay, A.: \(\aleph _0\)-Categorical strongly minimal compact complex manifolds. Proc. Am. Math. Soc. 140(5), 1785–1801 (2012)CrossRefGoogle Scholar
  12. 12.
    Moosa, R., Pillay, A.: Some model theory of fibrations and algebraic reductions. Sel. Math. (NS) 20, 1067–1082 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pillay, A.: Geometric Stability Theory, Oxford Logic Guides, vol. 32. The Clarendon Press, New York (1996)Google Scholar
  14. 14.
    Pillay, A., Ziegler, M.: Jet spaces of varieties over differential and difference fields. Sel. Math. (N.S.) 9(4), 579–599 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsUniversity of ManchesterManchesterUK
  2. 2.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

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