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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.

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I think you just rediscovered the notion of an etale sheaf. – Kevin Buzzard Jun 30 '10 at 11:18
@Kevin: well, at least for the "finite etale" topology. :) – Boyarsky Jun 30 '10 at 13:20
up vote 11 down vote accepted

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.

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That is really nice - that the Galois group is not just in bijection with spec (which is not too surprising given, but is actually homeomorphic! This raises the question: If one wishes to motivate or even define the topology on the Galois group, could one do it using $Spec(L \otimes_K \bar{K})$? – David Corwin Jul 28 '10 at 11:36
Yes, but I would choose - in any case - the profinite topology. – Martin Brandenburg Jul 28 '10 at 16:24
In the case of normal inseparable extensions, might one study $Spec(L \otimes_K \bar{K}$ in place of the Galois group? Also, I believe the inclusion in $L \otimes_K \overline{K} \subseteq L' \otimes_K \overline{K}$ is backward. – David Corwin Jul 28 '10 at 20:40
In case anyone is curious, Kirsten Wickelgren and I discuss some related issues and constructions in ("Universal covering spaces and fundamental groups in algebraic geometry as schemes"). – Ravi Vakil Oct 31 '10 at 3:48
Thanks - that's a great article. – David Corwin Oct 29 '12 at 6:47

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