“Sheaf cohomology” of Galois groups

Question

Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “étale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see short discussion at Galois Group as a Sheaf). 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 $\operatorname{Gal}(E/L)$?

If the answer is yes, it is clear how to define cochains and coboundary operator via restriction maps, as in the construction of Čech cohomology. Does this give a cochain complex? If so we obtain an associated cohomology $\grave{H}^\bullet(G, M)$. As with Čech cohomology, a priori this is defined with respect to a choice of cover. Is there some choice of cover which will give a canonical result? What is the relation, if any, between this and the typical group cohomology? What is the correct analogy of a short exact sequence of sheaves? Is there a better definition of cohomology in terms of derived functors?

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 analogues of moduli spaces of Galois groups? In general I am interested in whatever geometric aspects of Galois theory (or class field theory more specifically) 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.

Answer 1

The usual geometric point of view on the Galois group of $F$ is that the Galois group is the etale fundamental group of $\operatorname{Spec} F$. It's possible that you're already aware of this point of view and are looking for something different, but let me say a few words about it anyway.

This analogy comes not from thinking of the fundamental group as loops, but as the automorphism group of the universal cover. The starting point is the observation that if $X$ is any reasonable topological space, say connected for simplicity, then the fundamental group $\pi_1(X,x)$ is isomorphic to the group of deck transformations of the universal cover $\tilde{X} \to X$, via monodromy. Working in the etale site over $\operatorname{Spec} F$ (which is now our $X$), the "universal cover" is $\operatorname{Spec}F^{\text{sep}}$ and the Galois group $G_F = \operatorname{Gal}(F^{\text{sep}} / F)$ of $F$ becomes the group of deck transformations. Strictly speaking, $\operatorname{Spec} F^{\text{sep}}$ isn't a finite etale cover of $\operatorname{Spec} F$, which is usually dealt with by viewing $\operatorname{Spec} F^{\text{sep}}$ as a limit of connected etale covers $\operatorname{Spec} E \to \operatorname{Spec} F$, where $E$ is a finite (Galois) extension of $F$. So the etale fundamental group of $\operatorname{Spec} F$ is identified with $G_F$ as a profinite group.

If $\mathcal{F}$ is an abelian sheaf on the etale site of $\operatorname{Spec} F$, then $M := \mathcal{F}(\operatorname{Spec} F^{\text{sep}})$ has a natural $G_F$-action. This construction gives an equivalence between abelian sheaves on the etale site and discrete $G_F$-modules. Moreover, $M^{G_F} \cong \mathcal{F}(\operatorname{Spec} F)$, which implies that $H^*(G_F,M) \cong H^*_{et}(\operatorname{Spec} F, \mathcal{F})$.

In sum, I think that any reference about algebraic geometry containing the word "etale" may be relevant. As for globalizing these constructions, the keyword you're looking for may be "descent". For example, the Stacks Project contains a lot of material about these topics.