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What is known about the absolute Galois cohomology of function fields of varieties of dimension 2 or larger? Specifically, I am interested in multiplicative coefficients $\mathbb G_m$. I have seen ...

Let $K/\mathbb F_q(x)$ be a finite Galois extension with Galois group $G$. Let $Aut(K)$ be the group of $\mathbb F_q$-automorphisms of $K$. Obviously, $G\subseteq Aut(K)$. It is well known that $H^1(G,...