1st-flat cohomology group for elliptic curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:57:07Z http://mathoverflow.net/feeds/question/105473 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105473/1st-flat-cohomology-group-for-elliptic-curves 1st-flat cohomology group for elliptic curves unknown 2012-08-25T17:20:22Z 2013-05-20T08:19:47Z <p>Let $E$ be an elliptic curve over an algebraically closed field $k$ of characteristic $p$. Is there any nice computation for the group $H^1(E,\alpha_p)$ and $H^1(E,\mathbb{G}_a)$? The cohomology is taken in the flat topology.</p> <p>When these groups are trivial? and is there any way to describe them? I am not asking for the description of $H^1$ using flat torsors on $E$, I looking for a description using which I could (at least) determine the groups are trivial or not.</p> http://mathoverflow.net/questions/105473/1st-flat-cohomology-group-for-elliptic-curves/105476#105476 Answer by Will Sawin for 1st-flat cohomology group for elliptic curves Will Sawin 2012-08-25T17:44:20Z 2012-08-25T17:44:20Z <p>$\mathbb G_a$ is a smooth group scheme, so the flat cohomology is the same as the etale cohomology. It is also a quasicoherent sheaf, so the etale cohomology is the same as the Zariski cohomology, which is a $1$-dimensional vector space over $k$.</p> <p>For $\alpha_p$, you can use the exact sequence $0 \to \alpha_p \to \mathbb G_a \to \mathbb G_a \to 0$ and take cohomology:</p> <p>$H^0(E,\mathbb G_a)\to H^0(E,\mathbb G_a) \to H^1(E,\alpha_p) \to H^1(E,\mathbb G_a)\to H^1(E,\mathbb G_a)$</p> <p>The map $H^0(E,\mathbb G_a)\to H^0(E,\mathbb G_a)$ given by taking $p$th powers is surjective since $k$ is algebraically closed. If it is only separably closed and not perfect than this is false. The map $H^1(E,\mathbb G_a)\to H^1(E,\mathbb G_a)$ given by taking $p$th powers is injective because it's a field, which has no nilpotent elements. So $H^1(E,\alpha_p)$ is trivial.</p>