Why is the fundamental group a sheaf in the etale topology?

Question

In this paper by Minhyong Kim on p5, there is a variety $X$ defined over $\mathbb{Q}$, $G = \pi_1(X(\mathbb{C}),b)$ the topological fundamental group of the associated complex algebraic variety, and $G$^ the profinite completion of $G$.

Kim states that $G$^ is a sheaf of groups for the etale topology on Spec($\mathbb{Q}$). Why is this? A sheaf should be an assignment of groups to the etale covers of $\mathbb{Q}$ with morphisms in the opposite direction. How does this assignment work? The ways I've tried to think about it have the functoriality going in the wrong direction.

He then says that such a sheaf is just a set with a continuous action of Gal($\bar{\mathbb{Q}}/\mathbb{Q}$). I suppose this is just the outer action coming from the homotopy exact sequence, but I'm more interested in understanding the concrete description of a sheaf. How do they relate?

Answer 1

You're right that the action of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on $G$ is just the outer action from the homotopy exact sequence, though technically you really need a legit action (not just an outer action), so perhaps he's assuming that the homotopy exact sequence is split (possibly by a $\mathbb{Q}$-rational point of $X$?)

Now, noting that $Gal(\overline{\mathbb{Q}}/\mathbb{Q})\cong\pi_1(\text{Spec }\mathbb{Q})$, recall that the Galois correspondence says that the category $FEt_\mathbb{Q}$ of finite etale covers of Spec $\mathbb{Q}$ is equivalent to the category of finite sets equipped with an action of $\pi_1(\text{Spec }\mathbb{Q})$. There's actually a refinement of the Galois correspondence which you can find in SGA 1 (Expose V, Prop 5.2), which says that the pro-category Pro-$FEt_\mathbb{Q}$ is equivalent to the category of compact hausdorff totally disconnected topological spaces equipped with a continuous action of $\pi_1(\text{Spec }\mathbb{Q})$. Now, clearly $\hat{G}$ is such a compact hausdorff totally disconnected space with an action of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, so by this equivalence this corresponds to an object $\underline{G}\in$ Pro-$FEt_\mathbb{Q}$.

Now, by definition, objects of Pro-$FEt_\mathbb{Q}$ are just formal inverse systems of objects in $FEt_\mathbb{Q}$ (they are not their limits!), but by taking limits, in this case we can identify $\underline{G}$ with an actual scheme (which in this case is a group scheme). To view it as a sheaf, you simply define its sections over any etale $T\rightarrow\text{Spec }\mathbb{Q}$ as $Hom_\mathbb{Q}(T,\underline{G})$.

Since $\underline{G}$ is a group scheme, this set is a group, though in general any scheme $X$ determines a sheaf of sets of the form $Hom(*,X)$.

Answer 2

In this case, it is indeed an action rather than an outer action, since a splitting has been chosen using the base point. It is a classical fact that a sheaf is the same as a set with Galois action. One way to describe the sheaf starting from the set goes like this: It suffices to describe the sections on connected open sets. These then are just the spectra of finite field extensions. It then suffices to describe the sections on the full subcategory of finite field extensions that are in a fixed algebraic closure. Each such extensions corresponds to an open subgroup, and you take the subset fixed by the subgroup.

A small point of care: This describes the sheaf associated to each finite quotient of $\pi_1$. For the whole $\pi_1$, what you have is actually a "pro-sheaf" consisting of the sequences of sheaves associated to the finite quotients.

Thanks for the answer. So the description of $G\text{^}$ as an étale sheaf only holds in the case the action is well defined, e.g. the sequence splits due to the existence of a rational point? By fixed subset you mean subset of the geometric fundamental group fixed under the action? In the case of an outer action, could we define the sections of the sheaf just to consist of conjugacy classes of fixed subsets?

I always get a bit confused with this formalism, but if you don't have splitting, I think the corresponding object is the gerbe of splittings.