I know the fact that for algebraic groups (= connected smooth group scheme over a field) $H \subseteq G$, the fppf sheafification of $G/H$ is a scheme. Is there some short proof of it? (If possible, I want online available one.)
In Milne's note, the author only shows it for quasi-projective $G$. (And he works on a little general situation. So I wish that we can prove the existence of quotient more easily, considering only the quotient of a group by a subgroup...)
But I want to know the general case.
In Conrad's "Modern proof of Chevalley...", the author shows that every algebraic group is quasi-projective. But in its proof, it seems for me that he uses the existence of quotients for general algebraic groups.
So are there some easy proofs for the existence of the quotient of general algebraic groups?
Thank you very much!
If $H$ is normal, it's fairly easy to prove the existence of a quotient, but otherwise it is difficult. In B.38 of his book (Algebraic Groups), Milne explains how to remove the quasi-projectivity hypothesis in his proof there (with a reference to SGA 3). He also indicates a more direct proof that all algebraic group schemes over a field are quasi-projective. Working "only with a subgroup" doesn't help.
Apart from the appendix to Milne book, the only references I know of for this are SGA 3 and Brochard, Sylvain. Topologies de Grothendieck, descente, quotients. (French) [[Grothendieck topology, descent, quotients]] Autour des schémas en groupes. Vol. I, 1--62, Panor. Synthèses, 42/43, Soc. Math. France, Paris, 2014. MR3362639