Homogeneous spaces of smooth affine groups are separated schemes

By Chevalley's theorem [F1] there are a finite-dimensional rational representation $r$ of $G$ on $V$ and a line $L\subseteq V$ such that $H(R)=\{g\in G(R):r(g)L_R=L_R\}$ for every $k$-algebra $R$.

Since $G$ is smooth, the orbit lemma [F3] applies to the action of [F2] on ${\mathbf{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(V)$: the orbit $O_{[L]}$ is locally closed and stable under $G$, it is smooth over $k$, and ${\varrho }_{[L]}:G\to O_{[L]}$ is faithfully flat and locally of finite presentation.

The projective action of [F2] has scheme-theoretic stabilizer $G_{[L]}$ with $G_{[L]}(R)=\{g:r(g)L_R=L_R\}$ for every $k$-algebra $R$, which is $H(R)$ by step 1.1; hence $G_{[L]}=H$ as closed subgroup schemes of $G$.

The orbit $O_{[L]}$ is a locally closed subscheme of the separated finite-type $k$-scheme ${\mathbf{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(V)$, so it is separated and finite type over $k$ by [F5], and the morphism $O_{[L]}\hookrightarrow {\mathbf{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(V)$ is an immersion.

Applying the coset-quotient proposition [F4] with $x=[L]$, $H=G_{[L]}$ and the faithfully flat orbit map of step 1.2 shows that $O_{[L]}$ represents the fppf quotient sheaf $G\mathrm{/}H$, with quotient morphism ${\varrho }_{[L]}$ faithfully flat and locally of finite presentation, and that $G{\times }_{k}H\cong G{\times }_{O_{[L]}}G$.

By steps 2.2 and 3.1 the quotient $G\mathrm{/}H$ is represented by the separated finite-type $k$-scheme $O_{[L]}$, with quotient morphism ${\varrho }_{[L]}$, and the morphism $G\mathrm{/}H\cong O_{[L]}\hookrightarrow {\mathbf{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(V)$ is an immersion; the representation and line of step 1.1 satisfy the stated requirement that the scheme-theoretic line stabilizer is $H$. Any other representing scheme is uniquely isomorphic by the Yoneda lemma applied to a natural isomorphism of the represented functors. No smoothness of $H$ was used.