purity for finite flat group schemes - MathOverflow

purity for finite flat group schemes

2013-02-04T16:17:54Z

Let $X$ be a "nice" scheme and $Z \hookrightarrow X$ closed of codimension $\geq 2$. Let $Y$ over $X \setminus Z$ be a torsor for a finite flat group scheme $G/X$.

Does $Y$ spread out to a $G$-torsor over the whole of $X$?

For $G/X$ finite étale one can use Zariski-Nagata purity to spread $Y$ out to a scheme and then [Szamuely, Galois Groups and Fundamental Groups], p. 171, Lemma 5.3.13 to show that the extension is in fact a $G$-torsor.

Answer by Angelo for purity for finite flat group schemes

2013-02-04T16:57:07Z

Here is a counterexample for $G =\alpha_p$. Suppose that $k$ is a field of characteristic $p > 0$, and set $X = \mathbb A^2_k$, $Z= \{0\}$ . Set $U := X \smallsetminus Z$. Then $\mathrm H^1(U, \mathbb{G}_{\rm a}) = \mathrm{H}^1(U, \mathcal{O})$ is an infinite dimensional vector space over $k$; hence it is $p$-torsion, and from the long exact sequence associated with the exact sequence $$ 0 \longrightarrow \alpha_p\longrightarrow \mathbb{G}_{\rm a}\longrightarrow \mathbb{G}_{\rm a}\longrightarrow 0 $$ we see that $\mathrm H^1(U, \alpha_p)$ surjects onto $\mathrm H^1(U, \mathbb{G}_{\rm a})$ . If we take any class in $\mathrm H^1(U, \alpha_p)$ such that its image in $\mathrm H^1(U, \mathbb{G}_{\rm a})$ is $\neq 0$, this represents a torsor that does not extend to $X$, since $\mathrm H^1(X, \mathbb{G}_{\rm a}) = 0$.

[Edit:] Anon is right, my construction is nonsense. I apologize.

Answer by Laurent Moret-Bailly for purity for finite flat group schemes

2013-02-05T08:04:30Z

It is true if $X$ is regular. This is stated (as Lemme 2) in my CRAS 1985 note on purity for families of curves:
http://gallica.bnf.fr/ark:/12148/bpt6k5495813c/f45.image
where, unfortumately, the proof is missing (I only claim that it extends Auslander's proof of the étale case).