# Quotient of a (non-linear) algebraic group by a closed subgroup

Let G be a (not necessarily linear) algebraic group and H a closed algebraic subgroup. Does the, say categorical, quotient G/H exist?
If yes, where do I find a proof?
If no, do you have a counterexample?

Remark: I am aware of the following MO discussion, but it does not answer my question: Quotient of an algebraic group

• If I recall correctly, this type of quotient construction was first carried out in an older language of algebraic geometry by people such as Rosenlicht and Chevalley. The transition to later group scheme language (Grothendieck, Demazure) is not trivial. Sources for your question are therefore dependent on exactly how you define the notion of "algebraic group" (and with it the notion of "categorical quotient"). Dec 9 '14 at 14:46
• See SGA3, Exp. VI$_{\rm{A}}$, Theorem 3.2(iv): for any group scheme $G$ locally of finite type over a field $k$ and any closed subgroup scheme $H$ there exists a locally finite type $k$-scheme $G/H$ equipped with a faithfully flat $k$-morphism $G \rightarrow G/H$ invariant under the right $H$-action such that $G \times H \rightarrow G \times_{G/H} G$ is an isomorphism. Thus, this has all "good" quotient properties you could ever want (for maps out, maps in, geometric points, etc.), by descent theory. (The result also holds over any artinian local ring, assuming $H$ is flat.) Dec 9 '14 at 15:03
• Milne's notes jmilne.org/math/CourseNotes/iAG.pdf state this without proof as Aside 6.6 (page 58) and give two references -- the SGA reference of user74230 and also Demazure and Gabriele, Tome II. Dec 9 '14 at 16:58
• Also Perrin, Approximation des schémas en groupes quasi-compacts sur un corps, Bull. SMF 1976, shows the following: if G is quasi-compact and H is either a normal subgroup, or defined by a finitely generated ideal, then the fpqc sheaf G/H is representable by a scheme. Dec 9 '14 at 19:56
• Thanks everybody! We had already searched SGA, but apparently in the wrong place. Dec 10 '14 at 7:54

See SGA3, Exp. VI$_{\rm{A}}$, Theorem 3.2(iv): for any group scheme $G$ locally of finite type over a field $k$ and any closed subgroup scheme $H$ there exists a locally finite type $k$-scheme $G/H$ equipped with a faithfully flat $k$-morphism $G\rightarrow G/H$ invariant under the right $H$-action such that $G×H→G\times_{G/H}G$ is an isomorphism. Thus, this has all "good" quotient properties you could ever want (for maps out, maps in, geometric points, etc.), by descent theory. (The result also holds over any artinian local ring, assuming $H$ is flat.)