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Differentially algebraic group chunks

Published online by Cambridge University Press:  12 March 2014

Anand Pillay
Anand Pillay*
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
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Extract

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).

I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].

What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.

Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).

Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.

Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).

Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 3 , September 1990 , pp. 1138 - 1142
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

[B]Bouscaren, E., Model theoretic versions of Weil's theorem on pregroups, The model theory of groups (stable group seminar, Notre Dame, Indiana, 1985–1987; Nesin, A. and Pillay, A., editors), Notre Dame Mathematical Lectures, vol. 11, Notre Dame University Press, Notre Dame, Indiana, 1989, pp. 177–185.Google Scholar
[K]Kolchin, E. R., Differentially algebraic groups, Academic Press, Orlando, Florida, 1985.Google Scholar
[PS]Pillay, A. and Srour, G., Closed sets and chain conditions in stable theories, this Journal, vol. 49 (1984), pp. 1350–1362.Google Scholar
[P]Poizat, B., Groupes stables, avec types génériques réguliers, this Journal, vol. 48 (1983), pp. 339–355.Google Scholar
[S]Sacks, G., Saturated model theory, Benjamin, Reading, Massachusetts, 1972.Google Scholar
[Sr]Srour, G., The notion of independence in categories of algebraic structures. Part I: Basic properties, Annals of Pure and Applied Logic, vol. 38 (1988), pp. 185–211.CrossRefGoogle Scholar
[W]Weil, A., On algebraic groups of transformations, American Journal of Mathematics, vol. 77 (1955), pp. 355–391.CrossRefGoogle Scholar