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Given a double cover $\pi: C \to \mathbb P^1$, where $C$ is a genus $g$ curve over algebraically closed field, I want to compute the group $\mathrm H^1(\mathbb P^1, \pi_*\mathbb G_m)$ in flat topology.

The only way to attack this that comes to my mind is to use Leray spectral sequence, which gives:

$0 \to \mathrm H^1(\mathbb P^1, \pi_*\mathbb G_m) \to \text{Pic}C \to \Gamma (\text{R}^1\pi_*\mathbb G_m) \to 0$

However, I don't know how to compute higher direct image $\text{R}^1$ either. Would be happy to any hint or a reference.

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    $\begingroup$ The étale and flat cohomology groups with values in smooth group schemes coincide, so you can as well use étale cohomology. Then $R^q\pi _*(F)=0$ for $q>0$, any finite morphism $\pi $ and any abelian sheaf $F$ (see SGA 4$\frac{1}{2} $, II, Prop. 3.6). $\endgroup$
    – abx
    Commented Feb 16, 2015 at 7:19
  • $\begingroup$ I completely ignored the fact the $\pi$ was finite. Thank you! $\endgroup$
    – Den
    Commented Mar 6, 2015 at 2:36

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