Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “etale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see short discussion here). For now I will assume $G$ is abelian. In this setting an open cover of $X$ will be a set of intermediate extensions $\{E/E_i/F\}$ which together generate $E$.

My question is about how much this viewpoint transfers the properties of sheaves on topological spaces, e. g. coherent sheaves on schemes. I think the correct analogy is that $G$ is the structure sheaf, and discrete $G$-modules $M$ are coherent sheaves on $X$. If this is correct, is the correct notion of restriction to a subfield $L$ the quotient of $M$ by the invariants of Gal($E/L$)?

If the answer is yes, it is clear how to define cochains and coboundary operator via restriction maps, as in the construction of Cech cohomology. Does this give a cochain complex? If so we obtain an associated cohomology $\grave{H}^\bullet(G, M)$. What is the relation, if any, between this and the typical group cohomology?

To what extent do these constructions “globalize”? For instance, can we somehow patch together these objects, in the same way affine schemes patch into general schemes? Are there interesting analogies of moduli spaces of Galois groups? In general I am interested in whatever geometric aspects of Galois theory (or class field theory more specifically) that this perspective may elucidate.

If anyone can shed light themselves or suggest a reference I would greatly appreciate it. Please excuse me if these questions are not research level. Thanks