Quotient of group schemes

Question

Is the following sequence of affine group schemes over $k \in \mathsf{CRing}$ exact, or even for that matter, is there a meaningful notion of quotient of (group) schemes: $$0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0 $$ Additionally, under which condition is the sequence still exact in $\mathsf{Grp}$ ? $$0 \rightarrow H(A) \rightarrow G(A) \rightarrow G(A)/H(A) \rightarrow 0 $$ (for any $k$-algebra $A$).

Answer 1

In general there is not a good of quotients for general bases. Namely, let's fix a ring $R$ and assume that $H\unlhd G$ is a normal containment of flat and finite presentation group schemes over $R$. Then, by standard theory we know that since $G$ and $H$ are representable (by assumption) that $G$ and $H$ are sheaves on the fppf site of $\mathrm{Spec}(R)$. One can form the fppf sheaf quotient $Q$ (i.e. the $Q$ is the fppf sheafification of $R\mapsto G(R)/H(R)$). One then gets a short exact sequence of group fppf sheaves

$$1\to H\to G\to Q\to 1\qquad (1)$$

on $R$. The issue is that, in general, $Q$ will only be an algebraic space (either in the etale sense or the fppf sense thanks to Artin's theorem).

If $\dim \mathrm{Spec}(k)\leqslant 1$ then, in fact, $Q$ is a scheme (e.g. see [1]) but for higher-dimensional bases this is false (e.g. see this).

Now even in the case when $Q$ is representable one may not have that $(1)$ is exact as a sequence of presheaves. Namely, for any $k$-algebra $A$ one gets an exact sequence of pointed sets

$$1\to H(A)\to G(A)\to Q(A)\to H^1_\text{fppf}(A,H)\to H^1_\text{fppf}(A,G)\to H^1_\text{fppf}(A,Q)$$

(which one can extend further if, for instance, $Q$ is central in $A$). One can define $H^1_\text{fppf}(A,X$) (for a group scheme $X$) to either be

• The set of isomorphism class of $X$-torsors over $A$
• The Cech cohomology of $X$ over $A$

(e.g. see [2, Theorem 6.5.10] for the equivalence). Note that if $X$ is a smooth group scheme over $A$ then one can replace 'fppf torsors' with 'etale torsors' in the case when you assume that $\dim(\mathrm{Spec}(k))\leqslant 1$--for then any $X$-torsor is representable (see loc.cit.) and then one uses the fact that a smooth $X$-group scheme has a section etale locally.

Note that in the torsor language one associates in $(1)$ to any $\alpha\in Q(A)$ the fiber in $G_A$ which, as one can easily check, is a $G_A$-torsor.

[1] S. Anantharaman. Schemas en groupes, espaces homogenes et espaces algebriques sur une base de dimension 1 Bull. Soc. Math. France, M´emoire 33 (1973), 5–79.

[2] Poonen, B., 2017. Rational points on varieties (Vol. 186). American Mathematical Soc.

Answer 2

The answer for your first question.

Suppose that $G$ and $H$ are group schemes over $k$, $H\subseteq G$. Consider $G, H$ as fpqc sheaves on $\textbf{Alg}_k$. Then the quotient $G/H$ exists in the category of fpqc sheaves and your sequence is (by definition) exact as a sequence of pointed objects in this category.

If in addition $H$ and $G$ are affine, of finite type over $k$ and $k$ is a field, then the fpqc sheaf $G/H$ is representable by a $k$-scheme of finite type.

This can be found in Demazure's and Gabriel's book Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. You can also find this fact in Jantzen's book Representations of Algebraic Groups and in SGA III.