Let $K$ be a field and $K^s$ a separable closure of $K$, and let $\mathcal{F}$ be a sheaf on $\mathrm{Spec}(K)$ (in the étale topology).
By Grothendieck's Galois Theory, we have the isomorphism
$$H_{\text{et}}^k(\mathrm{Spec}(K),\mathcal{F}) \cong H^k(\mathrm{Gal}(K^s/K),\mathcal{F}_{\mathrm{Spec}(K^s)})$$
i.e. the étale cohomology groups of $\mathrm{Spec}(K)$ with coefficients in $\mathcal{F}$ correspond to the Galois cohomology groups of the absolute Galois group of $K$ acting on the stalk of $\mathcal{F}$ at the (separably closed) geometric point $\mathrm{Spec}(K^s) \to \mathrm{Spec}(K)$.
Question: Is there any similar interpretation of more general Galois cohomology groups in terms of étale cohomology groups? For instance, can we write the cohomology groups $H^k(\mathrm{Gal}(L/K),L^\times)$ directly in terms of étale cohomology groups?
I do not see how this would be possible, given that topos-theoretically we can just restrict our attention to points with separably closed residue field for the geometric points of the relevant étale topos.
Is there any chance we can describe more Galois cohomology groups if we consider other topologies on a category of schemes (Edit: preferably to obtain a uniform description using a single site)?
Finally, if this is not possible, are there any "higher dimensional" appearances of this discrepancy? This is deliberately vague, but I am thinking about some sort of cohomology groups that restrict to étale cohomology groups when considering just separably closed geometric points (via some sort of higher dimensional version of Grothendieck's Galois Theory).
Best Answer
If $L/K$ is a Galois extension, then one can define a site in which the objects are intermediate fields $K \subseteq E \subseteq L$ which are finite over $K$. Or, you can take finite étale algebras $A$ over $K$ with a homorphism of $K$-algebras $A \to L$. The topology is the étale topology. These two sites define the same topos. There is an equivalence between sheaves on either of these sites and discrete $\mathrm{Gal}(L/K)$-modules, hence cohomology coincides with Galois cohomology.
Does this answer your question? I am not sure I understand what you have in mind.