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.

  • 10
    $\begingroup$ 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 $R$). $\endgroup$ – Marguax Sep 23 '13 at 3:02
  • 4
    $\begingroup$ @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. $\endgroup$ – 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.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.