purity for finite flat group schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:30:20Z http://mathoverflow.net/feeds/question/120778 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120778/purity-for-finite-flat-group-schemes purity for finite flat group schemes Timo Keller 2013-02-04T16:17:54Z 2013-02-05T08:12:04Z <p>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 $G/X$.</p> <p>Does $Y$ spread out to a $G$-torsor over the whole of $X$?</p> <p>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.</p> http://mathoverflow.net/questions/120778/purity-for-finite-flat-group-schemes/120784#120784 Answer by Angelo for purity for finite flat group schemes Angelo 2013-02-04T16:57:07Z 2013-02-05T08:12:04Z <p>Here is a counterexample for $G =\alpha_p$. Suppose that $k$ is a field of characteristic $p > 0$, and set $X = \mathbb A^2_k$, <code>$Z= \{0\}$</code>. Set $U := X \smallsetminus Z$. Then <code>$\mathrm H^1(U, \mathbb{G}_{\rm a}) = \mathrm{H}^1(U, \mathcal{O})$</code> is an infinite dimensional vector space over $k$; hence it is $p$-torsion, and from the long exact sequence associated with the exact sequence <code>$$0 \longrightarrow \alpha_p\longrightarrow \mathbb{G}_{\rm a}\longrightarrow \mathbb{G}_{\rm a}\longrightarrow 0$$</code> we see that <code>$\mathrm H^1(U, \alpha_p)$</code> surjects onto <code>$\mathrm H^1(U, \mathbb{G}_{\rm a})$</code>. If we take any class in <code>$\mathrm H^1(U, \alpha_p)$</code> such that its image in <code>$\mathrm H^1(U, \mathbb{G}_{\rm a})$</code> is $\neq 0$, this represents a torsor that does not extend to $X$, since $\mathrm H^1(X, \mathbb{G}_{\rm a}) = 0$.</p> <p>[Edit:] Anon is right, my construction is nonsense. I apologize.</p> http://mathoverflow.net/questions/120778/purity-for-finite-flat-group-schemes/120841#120841 Answer by Laurent Moret-Bailly for purity for finite flat group schemes Laurent Moret-Bailly 2013-02-05T08:04:30Z 2013-02-05T08:04:30Z <p>It is true if $X$ is regular. This is stated (as Lemme 2) in my CRAS 1985 note on purity for families of curves:<br> <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5495813c/f45.image" rel="nofollow">http://gallica.bnf.fr/ark:/12148/bpt6k5495813c/f45.image</a><br> where, unfortumately, the proof is missing (I only claim that it extends Auslander's proof of the étale case). </p>