# Quotient of a reductive group by a non-smooth subgroup

This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.

Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic $p>0$. Let $H\subset G$ be a $k$-subgroup, not necessarily smooth. Question 1: Does the quotient $G/H$ exist as a $k$-variety?

I am interested in the following special case. Let $H^{\rm mult}$ denote the largest quotient of $H$ which is a $k$-group ($k$-group scheme) of multiplicative type. Set $H_1=\ker[H\to H^{\rm mult}]$. I assume that $H_1$ is smooth, connected and semisimple. Question 2: Does the quotient $G/H$ exist as a $k$-variety under this assumption? (I do not assume that $H^{\rm mult}$ is smooth.)

All comments and references are welcome!

• For any group scheme $G$ of finite type over any field $k$ and any closed $k$-subgroup scheme $H$ of $G$, the fppf quotient sheaf $G/H$ is represented by a separated finite type $k$-scheme and the map $q:G \rightarrow G/H$ is faithfully flat (so $q$ inherits "any" property of $H \rightarrow {\rm{Spec}}(k)$ that is fppf-local on the base). In particular, if $G$ is smooth then $G/H$ is smooth. See SGA3, Exp. VI$_{\rm{A}}$, section 3 for the construction (doesn't use quasi-projective methods, but see footnote 35). For affineness when $G$ is affine and $H$ is normal, see VI$_{\rm{B}}$, 11.17. – user29283 Mar 28 '13 at 14:36
• "footnote 35" in the preceding comment refers to the new SMF edition. – user29283 Mar 28 '13 at 14:37

Let $G$ be an affine algebraic group scheme over a field, and let $N$ be an affine normal subgroup scheme. Then the quotient $Q=G/N$ certainly always exists as an affine algebraic group scheme, and it is smooth if $G$ is smoooth (the coordinate ring of $Q$ is contained in that of $G$, and hence geometrically reduced).