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A note on CM-triviality and the geometry of forking

Published online by Cambridge University Press:  12 March 2014

Anand Pillay*
Affiliation:
Department of Mathematics, Altgeld Hall, 1409 W. Green Street, University of Illinoisat Urbana-Champaign, Urbana, IL 61801, USA, E-mail: [email protected]

Extract

CM-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions about CM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” by CM-trivial types of rank 1, is itself CM-trivial. (Actually Wagner worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced by P-closure, for P some family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness and CM-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Hrushovski, E., A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147166.CrossRefGoogle Scholar
[2]Pillay, A., The geometry of forking and groups of finite Morley rank, this Journal, vol. 60 (1995), pp. 12511259.Google Scholar
[3]Pillay, A., Geometric Stability Theory, Oxford University Press, 1996.CrossRefGoogle Scholar
[4]Poizat, B., Groupes stables, Nur al-Mantiq wal ma'Rifah, Villeurbanne, 1987.Google Scholar
[5]Wagner, Frank O., CM-triviality and stable groups, this Journal, to appear.Google Scholar