# Group scheme counterexample

Could someone give me an example of a finite group scheme $G$ (over some base $S$) so that $G$ minus a point is still a group scheme over $S$, but not affine over $S$?

Oort mentions that there are examples of this kind in his book "Commutative Group Schemes", but doesn't give an example.

• Let $S = {\rm{Spec}}(R)$ for a local normal noetherian domain $R$ of dimension $\ge 2$, with $s \in S$ the closed point, and let $H = (\mathbf{Z}/(2))_S$ be the constant finite etale $S$-group associated to $\mathbf{Z}/(2)$. The open subscheme $G$ complementary to the non-identity point in the closed fiber $H_s$ is an $S$-subgroup, and as a scheme it is a disjoint union of $S$ and $S-\{s\}$. Thus, it is not $S$-affine since it is not affine (as $S - \{s\}$ is not affine, due to the hypotheses on $R$). – Marguax Sep 23 '13 at 3:02
• @Marguax: I think it's better to post this as an answer rather than a comment; this makes your post more visible and makes it clear that the question has been answered. – Tom De Medts Sep 23 '13 at 15:21

Here's a minor variant on Margaux's comment from a year ago. Let $S = \mathbb{A}^2$ (say, over your favorite field), and let $\pi: G \to S$ be the constant $S$-group scheme $(\mathbb{Z}/2\mathbb{Z})_S$. As a scheme, it is a disjoint union of two copies of $S$.
If we remove a point $g$ from the non-identity section, we get an $S$-group scheme $G - g$, whose underlying scheme is a disjoint union of $S$ and $S - \pi(g)$ (note that $(G-g) \times_S (G-g)$ is a disjoint union of an identity section $S$ and 3 copies of $S - \pi(g)$). This is not affine over $S$, since $S$ is affine and $G - g$ is not.