Timeline for finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme
Current License: CC BY-SA 3.0
8 events
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| Feb 25, 2017 at 6:21 | vote | accept | CommunityBot | moved from User.Id=19475 by developer User.Id=69903 | |
| Dec 24, 2016 at 11:22 | comment | added | darx | 1) If you have a closed subscheme of the generic fibre, then the closure is a closed subscheme of $G$ over $X$, but may not be flat. Then after a blowup the strict transform is flat and still a closed subscheme. 2) Just look at the equations for the group schemes in Oort-Tate paper. As you say one gets $\mu_p$, $Z/pZ$, of $\alpha_p$ in the generic point. In the last two cases looking at the invertible sheaf you mention gives a group scheme homomorphism $G \to L$ and then you just show the quotient is another line bundle $L'$ but the map $L \to L'$ is not linear, just additive. | |
| Dec 24, 2016 at 7:40 | comment | added | user19475 | This is clear to me. My question was 1) why there exists such a filtration after an alteration, and 2) why the Tate-Oort classification gives us the two cases. | |
| Dec 23, 2016 at 23:15 | comment | added | darx | If $Y \to X$ is an alteration with $Y$ normal and $H^1(Y, G)$ is finite, then Lemma 3 tells us $H^1(X, G)$ is finite. Hence we may replace $X$ by $Y$. | |
| Dec 23, 2016 at 20:13 | comment | added | user19475 | Regarding 1): The category of finite flat commutative group schemes of $p$-power order is Abelian, and over an algebraically closed field, its simple objects are $\mu_p,\alpha_p$ and $\mathbf{Z}/p$. Perhaps this helps; regarding 2): Since $X$ has characteristic $p$, the element $w_p$ of [Tate-Oort] is $= 0$, and if $a=0$, the $p$-Lie algebra of $G$ is $\mathscr{L}$ with the $p$-power morphism. | |
| Dec 23, 2016 at 18:03 | comment | added | user19475 | Can you please give more details for: "By Lemma 3 we may replace $X$ by an alteration. Hence we may assume that over the function field of $X$ we have a filtration of $G$ by closed subgroup schemes such that the successive quotients have order $p$." and why the Tate-Oort classification gives exactly the two cases you mention? | |
| Dec 22, 2016 at 5:55 | comment | added | user19475 | Thank you very much! I will read this in the next weeks! | |
| Dec 22, 2016 at 1:54 | history | answered | darx | CC BY-SA 3.0 |