User:Maximilian Janisch/latexlist/Algebraic Groups/Homogeneous space of an algebraic group

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An algebraic variety $N$ together with a regular transitive action of an algebraic group $k$ given on it. If $x \in M$, then the isotropy group $G$ is closed in $k$. Conversely, if $H$ is a closed subgroup of an algebraic group $k$, then the set of left cosets $G / H$ has the structure of an algebraic variety, making it into a homogeneous space of the algebraic group $k$, where the natural mapping $\pi : G \rightarrow G / H$ is regular, separable and has the following universal property: For any morphism $\phi : G \rightarrow X$ constant on cosets, there is a morphism $\psi : G / H \rightarrow X$ such that $\psi \pi = \phi$. If $N$ is any homogeneous space of the algebraic group $k$ and $H = G _ { X }$ for some $x \in M$, then the natural bijection $\psi : G / H \rightarrow M$ is regular, and if the ground field $K$ has characteristic 0, then $25$ is biregular (see [1], [3]).

Suppose that the connected group $k$, the homogeneous space $N$ and the action of $k$ on $N$ are defined over some subfield $k \subset K$. Then the group of $k$-rational points $G ( k )$ takes $M ( k )$ into itself and $G ( k ) _ { x } = G _ { X } ( k )$ for $x \in M ( k )$. If $k$ is finite, then $M ( k ) \neq \emptyset$, and if moreover the isotropy group $G$ is connected, then $G ( k )$ acts transitively on $M ( k )$. In the general case, the study of the $k$-rational points in $N$ reduces to problems in the theory of Galois cohomology (see [2]).

A homogeneous space of an algebraic group $k$ is always a smooth quasi-projective variety (see [5]). If $k$ is an affine algebraic group, then the variety $G / H$ is projective if and only if $H$ is a parabolic subgroup in $k$ (see [3]). If $k$ is reductive, then the variety $G / H$ is affine if and only if the subgroup $H$ is reductive (see Matsushima criterion). A description is also known of the closed subgroups $H$ of a linear algebraic group $k$ over an algebraically closed field of characteristic 0 for which $G / H$ is quasi-affine (see [4], [6]).

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303
[3]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
[4]	A.A. Sukhanov, "A description of observable subgroups of linear algebraic groups" Math. USSR-Sb. , 68 (Forthcoming) Mat. Sb. , 137 : 1 (1988) pp. 90–102
[5]	W. Chow, "On the projective embedding of homogeneous varieties" , Algebraic topology; symposium in honour of S. Lefschetz , Princeton Univ. Press (1957) pp. 122–128 MR0084851 Zbl 0091.33302
[6]	G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) MR0620024 Zbl 0589.20025