Difference between revisions of "Borel fixed-point theorem"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Groupes linéaires algébriques" ''Ann. of Math. (2)'' , '''64''' : 1 (1956) pp. 20–82</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Morozov, ''Dokl. Akad. Nauk SSSR'' , '''36''' : 3 (1942) pp. 91–94</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Groupes linéaires algébriques" ''Ann. of Math. (2)'' , '''64''' : 1 (1956) pp. 20–82 {{MR|0093006}} {{ZBL|0070.26104}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Morozov, ''Dokl. Akad. Nauk SSSR'' , '''36''' : 3 (1942) pp. 91–94 {{MR|}} {{ZBL|}} </TD></TR></table> |
Revision as of 10:02, 24 March 2012
A connected solvable algebraic group acting regularly (cf. Algebraic group of transformations) on a non-empty complete algebraic variety
over an algebraically-closed field
has a fixed point in
. It follows from this theorem that Borel subgroups (cf. Borel subgroup) of algebraic groups are conjugate (The Borel–Morozov theorem). The theorem was demonstrated by A. Borel [1]. Borel's theorem can be generalized to an arbitrary (not necessarily algebraically-closed) field
: Let
be a complete variety defined over a field
on which a connected solvable
-split group
acts regularly, then the set of rational
-points
is either empty or it contains a point which is fixed with respect to
. Hence the generalization of the theorem of conjugation of Borel subgroup is: If the field
is perfect, the maximal connected solvable
-split subgroups of a connected
-defined algebraic group
are mutually conjugate by elements of the group of
-points of
[2].
References
[1] | A. Borel, "Groupes linéaires algébriques" Ann. of Math. (2) , 64 : 1 (1956) pp. 20–82 MR0093006 Zbl 0070.26104 |
[2] | A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 |
[3] | V.V. Morozov, Dokl. Akad. Nauk SSSR , 36 : 3 (1942) pp. 91–94 |
Borel fixed-point theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_fixed-point_theorem&oldid=21817