# Quotient of an algebraic group by a closed algebraic subgroup

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.

• 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. Feb 7, 2014 at 15:43
• I think you want to claim that $G/H$ must be quasi-projective. Feb 7, 2014 at 16:17
• @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. Feb 7, 2014 at 16:45
• And for general $G$ it's true if $H$ is normal ;) Feb 7, 2014 at 17:43
• @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. Feb 7, 2014 at 19:14

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.
• 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.) Feb 8, 2014 at 0:47
• 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. Feb 8, 2014 at 20:59
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.