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First, I am by no means well-versed on cohomology so I apologize if this is too elementary.

I have been going through some basics of etale cohomology, with my ultimate goal being an understanding of some basic applications. I have gone through the Kummer and Artin-Schreier sequences, and wanted to get an idea for how these sequences can help us classify $\mathbb{Z}/n$ and $\mathbb{Z}/p$ torsors.

I found a short exposition by Artin that said $H^1_{et}(X,\mathbb{G}_m)=Pic(X)$, and this was was labelled as Hilbert 90. This presumably has something to do with $\mathbb{G}_m$ being related to $\mathcal{O}_X^*$. Can someone tell me what the additive version of this is, ie what $H^1_{et}(X,\mathbb{G}_a)$ is equal to? Also if anyone has a reference for this being worked out that would be great as well.

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up vote 6 down vote accepted

$H^1_{ét}(X, \mathbf{G}_a) = H^1_{Zar}(X, \mathcal{O}_X)$, see Milne, Étale Cohomology III.§3. (as for the étale cohomology of any coherent sheaf)

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Ah thank you. I did not know about that comparison theorem for coherent sheaves. – Randall Jun 16 '11 at 18:16

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