Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-17T08:30:16.739Z Has data issue: false hasContentIssue false

Lascar and Morley ranks differ in differentially closed fields

Published online by Cambridge University Press:  12 March 2014

Ehud Hrushovski
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel E-mail: ehud@sunset.ma.haji.ac.il
Thomas Scanlon
Affiliation:
Department of Mathematics, University of California, Evans Hall, Berkeley, California 94720, USA E-mail: scanlon@math.berkeley.edu

Extract

We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families. We will be interested here only in sets of finite Morley rank. Section 1 consists of some general lemmas relating the above issues. Section 2 points out a family of sets of finite Morley rank, whose Morley rank exhibits discontinuous upward jumps. To make the base of the family itself have finite Morley rank, we use a theorem of Buium.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Buium, A., Geometry of differential polynomial functions III: Moduli spaces, American Journal of Mathematics, vol. 117 (1995), no. 1, pp. 173.Google Scholar
[2] Hrushovski, E. and Itai, M., On model complete differential fields, preprint, 27 11 1997.Google Scholar
[3] Hrushovski, E. and Sokolović, Ž., Minimal subsets of differentially closed fields, Transactions of the American Mathematical Society, to appear.Google Scholar
[4] Milne, J. S., Abelian varieties, Arithmetic geometry (Cornell, G. and Silverman, J., editors), Springer-Verlag, New York, 1986.Google Scholar
[5] Mumford, D., Abelian varieties, Oxford University Press, Oxford, 1970.Google Scholar
[6] Mumford, D., Fogarty, J., and Kirwan, F., Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, no. 34, Springer-Verlag, New York, 1994.Google Scholar