Let $G$ be a complex, linear algebraic group and $H\subseteq G$ a closed and normal subgroup. Then, the quotient $G/H$ has the structure of a affine variety. I am looking for the most "modern" reference in English language for this statement. I would prefer a graduate level textbook most. Anything I find, however, restricts to the reductive case, which is not general enough for me.
$\begingroup$
$\endgroup$
8
-
2$\begingroup$ The statement isn't true. Take $G=SL_2$ and $H = \left( \begin{smallmatrix} 1 & \ast \\ 0 & 1 \end{smallmatrix} \right)$. The quotient $G/H$ is isomorphic to $\mathbb{A}^2 \setminus \{ (0,0) \}$, which is not affine. $\endgroup$– David E SpeyerCommented Feb 7, 2014 at 15:43
-
2$\begingroup$ I think you want to claim that $G/H$ must be quasi-projective. $\endgroup$– Peter CrooksCommented Feb 7, 2014 at 16:17
-
2$\begingroup$ @Jesko: Note that working over the complex field is unimpotant here. In general, the quotient of a reductive group by a closed subgroup is affine if and only if the subgroup is also reductive. $\endgroup$– Jim HumphreysCommented Feb 7, 2014 at 16:45
-
2$\begingroup$ And for general $G$ it's true if $H$ is normal ;) $\endgroup$– Piotr AchingerCommented Feb 7, 2014 at 17:43
-
1$\begingroup$ @Daniel: I basically copied Chevalley/Borel. The idea is elementary (once Chevalley thought of it): realize $H$ as isotropy group of a line in a representation $G \rightarrow \mathrm{GL}(V)$, then pass to the projective space $P(V)$ and embed $G/H$ there. $\endgroup$– Jim HumphreysCommented Feb 7, 2014 at 19:14
|
Show 3 more comments
2 Answers
$\begingroup$
$\endgroup$
4
You can prove that $G/H$ is quasi-projective, and a reference is Theorem 4.4.1 of Algebraic Quotients, Torus Actions, and Cohomology by A. Bialynicki-Birula, J. Carrell, and W.M. McGovern.
answered Feb 7, 2014 at 16:20
-
1$\begingroup$ Note that this is a standard textbook fact, which I guess goes back to the Chevalley seminar. An early reference is Theorem 6.8 in Borel's 1969 lecture notes Linear Algebraic Groups, published in an expanded edition as GTM 126 by Springer-Verlag. (Borel's algebraic geometry language is modern enough for this purpose.) $\endgroup$ Commented Feb 7, 2014 at 16:56
-
1$\begingroup$ Just to clarify, Jim Humphreys' comment is referring not to the general coset space construction as in this answer, but rather to the finer assertion of the affineness property when $H$ is normal in $G$. (His reference is to Chapter II of Borel's book.) $\endgroup$ Commented Feb 8, 2014 at 0:47
-
1$\begingroup$ No, my reference to Borel's Theorem 6.8 involves the general case of a quotient $G/H$ for any closed subgroup $H$. Of course, when $H$ is normal it's also shown that $G/H$ has the structure of an affine algebraic group. But the quasi-projective proeprty of the quotient is general. $\endgroup$ Commented Feb 8, 2014 at 20:59
-
$\begingroup$ @JimHumphreys Is the quotient quasi-projective even if the algebraic group is not necessarily affine? $\endgroup$ Commented Oct 19, 2017 at 12:57
$\begingroup$
$\endgroup$
Given an affine (not necessarily algebraic) group $G$ over an arbitrary field and its closed normal subgroup $H$, the fpqc quotient sheaf $G/H$ is affine. A simple proof of this fact, which uses the language of Hopf algebras, was given by M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras. Manuscripta Math. 7 (1972), 251–270. The proof of the theorem in Section 16.3, of Waterhouse, Introduction to affine group schemes, Springer GTM 66, seems based on Takeuchi's idea.
