Quotients of affine group schemes by normal subgroup schemes are affine

Choose by [F1] a representation $\rho :G\to \mathrm{GL}(E)$ with exact kernel $N$. By [F2], the quotient $Q=G\mathrm{/}N$ and projection $q:G\to Q$ exist with all the stated torsor and flatness properties. Its universal property gives a homomorphism $\bar{\rho }:Q\to \mathrm{GL}(E)$ with $\rho =\bar{\rho }q$. For every test scheme $T$ and every $x\in Q(T)$ in the kernel of $\bar{\rho }$, there is an fppf cover $T^{\mathrm{\prime }}\to T$ on which $x$ lifts to $g\in G(T^{\mathrm{\prime }})$: use the pullback of the quotient torsor itself as that cover. The equality $\bar{\rho }(x)=1$ gives $\rho (g)=1$, so $g\in N(T^{\mathrm{\prime }})$ by the exact kernel assertion of [F1]. Therefore $x{\mathrm{\mid }}_{T^{\mathrm{\prime }}}=q(g)=1$. Since a represented fppf sheaf detects equality on covers, $x=1$. Thus $\bar{\rho }$ has trivial scheme-theoretic kernel.

Apply [F3] to $\bar{\rho }$: $Q$ and $\mathrm{GL}(E)$ are separated finite-type group schemes, so $Q\hookrightarrow \mathrm{GL}(E)$ is a closed immersion. In a basis of $E$, the target is the affine scheme with ring $k[t_{ij},1\mathrm{/}\det (t_{ij})]$. A closed subscheme of an affine scheme is affine, with coordinate ring the corresponding quotient ring, proving that $G\mathrm{/}N$ is affine. The remaining claims were obtained from [F2] in step 1.1. This proof uses the represented fppf quotient and an exact representation kernel; it makes no faithful-flatness assumption about an inclusion of Hopf algebras. AC is inherited from [F1]–[F3].