# Is the quotient of two linear group schemes linear?

Let $S$ be an affine scheme. Call a group scheme $G\to S$ linear if there exists an $S$-group morphism $G\to \mathrm{GL}_{n,S}$ with trivial kernel. Assuming this, suppose $H\to S$ is a central closed subgroup of $G$ and that the quotient fpqc sheaf $G/H$ happens to be representable by a flat group $S$-scheme, also denoted $G/H$. My question is whether $G/H$ linear in the previous sense?

I am willing to impose various assumptions such as regularity of $S$, smoothness of $G\to S$, flatness of $H\to S$, and that $G$ embeds as a closed subgroup of $\mathrm{GL}_{n,S}$, but I want to avoid assuming that $G$ or $H$ are reductive or limiting the dimension of $S$.

Note: The question of linearity of arbitrary smooth affine group schemes had been discussed here; it is open in general. I am asking in the hope that perhaps something more concrete can be said about quotients of linear group schemes.

• Do you suppose that $G$ and $H$ are linear? (the title suggests it but it is not stated in the question). – Damian Rössler Apr 17 '18 at 7:06
• I do. (Of course, if $G$ is linear, then so is $H$.) I slightly rephrased the question to remove this ambiguity. – Uriya First Apr 18 '18 at 11:34