I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois group $\mathrm{Gal}(L/K)$ of a field extension $L/K$ is the sections over $L/K$. This makes sense because we have a natural restriction map $\mathrm{Gal}(M/K) \to \mathrm{Gal}(L/K)$ when $M/L/K$. Furthermore, we can define an open covering of an extension $L/K$ as a collection of subextensions whose compositum is $L/K$, and then a collection of elements of the Galois group in each subextension which match up under restriction gives an element of $\mathrm{Gal}(L/K)$. This is a key ingredient in proving that the Galois group of an infinite extension is an inverse limit of finite Galois groups. Is there a way to formalize this?


It also has interesting ramifications for understanding the topology of the Galois groups. Two elements of an infinite Galois group of $L/K$ are close if they agree on larger and larger subextensions of $L/K$, i.e. if they agree on larger "open sets" in this topology. So there's some kind of interesting inverse relation between the topology on the infinite Galois group and the "topology" of the subextensions.



Galois Group as a Sheaf
You might introduce a Grothendieck topology on the Galois subextensions of $L/K$; then the Galois group is indeed a sheaf.


I just want to remark the following: There is a natural homeomorphism between $Gal(L/K)$ and $Spec(L \otimes\_K \overline{K})$. Namely, if $\sigma \in Gal(L/K)$, then the kernel of $L \otimes\_K \overline{K} \to L \otimes\_K \overline{K} \subseteq \overline{K} \otimes\_K \overline{K} \to \overline{K}$ is a prime ideal of $L \otimes\_K \overline{K}$. This is easily checked to be a homeomorphism if $L/K$ is finite, and then also in general. In particular, you can endow the profinite space $Gal(L/K)$ with a sheaf so that we get an affine scheme. If $L'/K$ is a Galois subextension of $L/K$, then the restriction $Gal(L/K) \to Gal(L'/K)$ comes from the inclusion $L \otimes\_K \overline{K} \subseteq L' \otimes\_K \overline{K}$. This provides another reason why $Gal(-/K)$ is a "sheaf", namely because $Spec$ commutes with filtered direct limits.


