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On central extensions of algebraic groups

Published online by Cambridge University Press:  12 March 2014

Tuna Altinel  and
Gregory Cherlin
Tuna Altinel
Affiliation:
The Fields Institute, 222 College Street, Toronto, ON M5T 3J1, Canada E-mail: [email protected]
Gregory Cherlin
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA
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Extract

In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.

Theorem 1. Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0 is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0 = 1.

Contrary to an impression which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.

Corollary 1. Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4], Section 27.5).

An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification of tame simple K*-groups of finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption of tameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results from K-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 1 , March 1999 , pp. 68 - 74
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1]Altinel, T., Borovik, A., and Cherlin, G., Groups of mixed type, Journal of Algebra, vol. 192 (1997), pp. 524–571.CrossRefGoogle Scholar
[2]Bass, H. and Tate, J., The Milnor ring of a global field, Algebraic K-theory II, Lecture Notes in Mathematics, no. 342, Springer-Verlag, Berlinn, 1973, pp. 349–446.Google Scholar
[3]Borovik, A. V. and Nesin, A., Groups of finite Morley rank, Oxford University Press, Oxford, 1994.CrossRefGoogle Scholar
[4]Humphreys, J. E., Linear algebraic groups, second ed., Springer-Verlag, Berlin-New York, 1981.Google Scholar
[5]Macintyre, A., On ω1-categorical theories of fields, Fundamenta Mathematicae, vol. 71 (1971), pp. 1–25.CrossRefGoogle Scholar
[6]Pillay, A., An introduction to stability theory, Oxford Logic Guides, no. 8, Oxford University Press, 1983.Google Scholar
[7]Poizat, B., Groupes stables, Nur Al-Mantiq Wal-Ma’rifah, Villeurbanne, France, 1987.Google Scholar
[8]Rosenberg, J., Algebraic K-theory and its application, Graduate Texts in Mathematics, no. 147, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar
[9]Steinberg, R., Lectures on Chevalley groups, Yale University, 1967.Google Scholar