Borel fixed-point theorem
Statement of the theorem
A more general version of the theorem holds over a field k that is not necessarily algebraically closed. A solvable algebraic group G is split over k or k-split if G admits a composition series whose composition factors are isomorphic (over k) to the additive group or the multiplicative group . If G is a connected, k-split solvable algebraic group acting regularly on a complete variety V having a k-rational point, then there is a G fixed-point of V.
- Borel (1991), Proposition 15.2
- Borel, Armand (1956). "Groupes linéaires algébriques". Ann. Math. 2. Annals of Mathematics. 64 (1): 20&ndash, 82. doi:10.2307/1969949. JSTOR 1969949. MR 0093006.
- Borel, Armand (1991) , Linear Algebraic Groups (2nd ed.), New York: Springer-Verlag, ISBN 0-387-97370-2, MR 1102012
- V.P. Platonov (2001) , "Borel fixed-point theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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