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The algebraic cohomology of $\mathbb G_{\rm m}$ is very different from the analytic cohomology of $\mathcal O^*$. For a regular scheme the former is always torsion in degree at least 2 (this is a well-known result of Grothendieck), whereas the analytic cohomology of a complex manifold tends to contain positive-dimensional $\mathbb Q$-vector spaces.
I guess that "Galois with group $G$" should not be interpreted as being necessarily étale. Anyway, it seems to me that (2) and (3) follows immediately from (1), personally I would not worry with a reference. Of course, I am not very good with references.
Can't you work out an example before asking a question? How can you think that the intersection of all curves with a given tangent vector is 1-dimensional?
It follows easily from the fact that a finitely dimensional algebra is Jacobson; in particular, the closed points are dense. So, if there is only one closed point, the spectrum consists of that point, so the algebra is 0-dimensional, hence artinian.
It seems to the me that the étale case should reduced to the Zariski case, because the pushforward from the étale site to the Zariski site is also flabby. Am I missing something?
