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.



Additive form of Hilbert 90 for schemes?
$H^1\_{ét}(X, \mathbf{G}\_a) = H^1\_{Zar}(X, \mathcal{O}\_X)$, see Milne, Étale Cohomology III.§3. (as for the étale cohomology of any coherent sheaf)


