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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?

Simone Weil
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Somatic Custard
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2 Answers2

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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 $Hom(*,X)$.

oxeimon
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    Thanks for the answer. As Simone pointed out, Kim does assume that the base point is rational, though Kim didn't make it too clear that the result depended on that. Are the Homs $T\to\underline{G}$ just a collection of morphisms $T\to G_i$to each finite etale scheme in the inverse system, compatible with the morphisms $G_i \to G_j$? Wouldn't that collection be empty though? No finite etale cover $T$ of Spec$\mathbb{Q}$ would have a morphism to all finite etale covers of Spec$\mathbb{Q}$. – Somatic Custard Aug 28 '16 at 23:28
  • @festering Yes it follows from the universal property of inverse limits that a map $T\rightarrow\underline{G}$ is just a compatible family of morphisms ${T\rightarrow G_i}$. This set is actually not empty. For one, since $\underline{G}$ is a group scheme over $\mathbb{Q}$, it must have a rational point (namely, the identity!). – oxeimon Aug 29 '16 at 01:35
  • I don't really understand the group scheme perspective well. So $G$ starts out as the geometric fundamental group for a variety $X/\mathbb{Q}$, and then it corresponds to a pro-etale cover of Spec($\mathbb{Q}$) rather than of $X$? And why does that sheaf you defined actually have sections? For any etale $G_i$ that is a nontrivial connected etale cover of $T$, there will be no $T \to G_i$. Doesn't a section need a compatible collection of morphisms to all elements in the inverse system? – Somatic Custard Aug 29 '16 at 12:41
  • @festering By the Galois correspondence, given a finite etale cover $f : C\rightarrow D$ and a geometric point $d\in D$, the fiber $f^{-1}(d)$ is a finite set with an action of $\pi_1(D)$, and this defines a functor which is an equivalence of categories between finite etale covers of $D$ and finite sets equipped with an action of $\pi_1(D)$. Given any category, one can talk about "group objects" of that category. For the category of finite $\pi_1(D)$-sets, group objects are just finite groups equipped with an action of $\pi_1(D)$ acting via group automorphisms... – oxeimon Aug 29 '16 at 19:11
  • via the Galois correspondence, finite groups equipped with an action of $\pi_1(D)$ via group automorphisms correspond to group objects in the category of finite etale covers of $D$. In this latter category, the group objects are just "group schemes over $D$, which are finite etale over $D$". Now, given a group scheme $G$ over $D$, a section is a morphism $D\rightarrow G$ as schemes over $D$. Since both $D$ and $G$ are finite etale over $D$ we can again apply the equivalence of categories, and see that this diagram corresponds to the inclusion of a 1-point set into the fiber of $G$ over $d$... – oxeimon Aug 29 '16 at 19:15
  • which is $\pi_1(D)$-equivariant. This translates into the following statement: Given a finite etale cover $f : C\rightarrow D$ and a geometric point $d\in D$, the set of sections of $C$ over $D$ is in bijection with the set of elements of $f^{-1}(d)$ fixed by the action of $\pi_1(D)$.

    Why does this mean that $G$ has a section? Well, since $\pi_1(D)$ acts on the fiber of $G$ via group automorphisms, and since any group automorphism fixes the identity, this implies that $G$ has a section, namely the identity section.

     – oxeimon Aug 29 '16 at 19:17
  • The Galois correspondence also extends to pro-finite etale covers, and pro-finite sets/groups equipped with a $\pi$-action. Thus, via this extension of the Galois correspondence, the geometric fundamental group $\pi := \pi_1(\overline{X})$ of a variety $X/\mathbb{Q}$ can be viewed as a pro-etale group scheme over $\mathbb{Q}$ because the split exact sequence $1\rightarrow \pi_1(\overline{X})\rightarrow\pi_1(X)\rightarrow\pi_1(\text{Spec }\mathbb{Q})\rightarrow 1$ gives you an action of $\pi_1(\text{Spec }\mathbb{Q})$ on $\pi_1(\overline{X})$, which via the Galois corresp gives you a pro-etale – oxeimon Aug 29 '16 at 19:22
  • group scheme over Spec $\mathbb{Q}$. If you have a $\mathbb{Q}$-rational point of $X$, then the corresponding map Spec $\mathbb{Q}\rightarrow X$ gives you a map on fundamental groups $\pi_1(X)\leftarrow\pi_1(\text{Spec }\mathbb{Q})$ which is precisely the "section" splitting the exact sequence. Via this section, for any element of $\pi_1(\mathbb{Q})$, one can lift it to an element of $\pi_1(X)$, which then via conjugation acts as an automorphism of $\pi_1(\overline{X})$ (since it's a normal subgroup) – oxeimon Aug 29 '16 at 19:24
  • Note that one can also view $\pi_1(\overline{X})$ as a group equipped with an action of $\pi_1(X)$ (again acting via conjugation). Viewing it this way would give you a pro-etale group scheme over $X$, whose fiber over the given $\mathbb{Q}$-rational point of $X$ would be precisely the pro-etale group scheme over $\mathbb{Q}$ described above. – oxeimon Aug 29 '16 at 19:32
  • you may also be interested in my answer here: http://mathoverflow.net/questions/197874/the-difference-between-an-étale-finite-group-scheme-and-a-finite-group/197876#197876 – oxeimon Aug 29 '16 at 19:36
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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.

Simone Weil
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  • Thanks for the answer. So the description of $G$^ as an etale 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? – Somatic Custard Aug 28 '16 at 19:15