Quotient by p-th roots of unity in characteristic p - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:02:38Z http://mathoverflow.net/feeds/question/48096 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48096/quotient-by-p-th-roots-of-unity-in-characteristic-p Quotient by p-th roots of unity in characteristic p Piotr Achinger 2010-12-02T21:14:44Z 2010-12-03T08:54:00Z <p>Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ k[\varepsilon]/(\varepsilon^p - 1)$. Denote the quotient $X/\mu_p$ by $Y$ and the quotient morphism $X\to Y$ by $\pi$. </p> <blockquote> <p>Let $D$ be a Weil divisor on $X$. Is the sheaf $\pi_* (\mathscr{O}_X(D))$ a direct sum of sheaves of the form $\mathscr{O}_Y(E)$ for some Weil divisors $E$ on $X$?</p> </blockquote> <p>For an action of $\mu_n$, $n$ not divisible by $p$, we get the above statement because the push-forward decomposes according to the characters of $\mu_n$, and if we assume that there is an orbit on which $\mu_n$ acts freely, then all these summands are nonzero - hence all have rank one and are reflexive. So we need a form of ,,diagonalizability'' for $\mu_p$. I don't see how the proof for $\mu_n$ above could be translated to a proof for $\mu_p$. </p> http://mathoverflow.net/questions/48096/quotient-by-p-th-roots-of-unity-in-characteristic-p/48153#48153 Answer by Bugs Bunny for Quotient by p-th roots of unity in characteristic p Bugs Bunny 2010-12-03T08:54:00Z 2010-12-03T08:54:00Z <p>If you cannot generalize, you may try to simplify:-)) Representations of $\mu_p$ are still completely reducible: they are just graded vector spaces by the cyclic group of order $p$. The first part of your argument goes through.</p> <p>For the second part, you have a simplification as $\mu_p$ has no subgroup schemes! So your stabilisers are either trivial of full $\mu_p$. If the action is nontrivial, there is a point with trivial stabiliser, and there you know that each simple character of $\mu_p$ appears.</p> <p>Did I miss anything?</p>