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In section 5.5 of Deligne + Mumford's paper "Irreducibility of the Moduli Space of Curves", they introduce the notion of Teichmuller level structures.

You can find the paper here: http://publications.ias.edu/sites/default/files/Number7.pdf

In short, fix a finite group $G$, and let $P$ be the set of primes dividing $|G|$. If $S$ is any conneccted scheme whose residue characteristics are not in $P$, then for any smooth stable curve $f : X\rightarrow S$ which has a section $g : S\rightarrow X$, and any geometric point $s\in S$, they define a 'relative fundamental group' $\pi_1(X/S,g,s)$, which is a projective system of finite etale group schemes over $S$ (where each member of the system has only prime divisors from $P$), which for our purposes we may identify with its limit. They define the sheaf $\text{Hom}_S^\text{ext}(\pi_1(X/S),G)$ on the big etale site $(\text{Sch}/S)_{etale}$ sending $$(T\rightarrow S)\mapsto\text{Hom}_T(\pi_1(X/S,g,s)_T,G_T)/\pi_1(X/S,g,s)$$ where $G_T$ is the constant group scheme over $T$ associated to $G$, and $\pi_1(X/S,g,s)$ acts by inner automorphisms on the domain. This is actually finite locally constant, hence representable, since $\pi_1(X/S,g,s)$ is finitely generated. Since changing $g$ or $s$ results in fundamental groups which are canonically isomorphic up to inner automorphism, this sheaf is indepedent of the choice of $g,s$.

My question: Right before definition 5.6, they say that since sections (ie, the $g$) exist locally in the etale topology, the sheaf $\text{Hom}_S^\text{ext}(\pi_1(X/S),G)$ makes sense without assuming there to be exist a section over $S$. My question is, if there is no section over $S$, but only one over an etale covering $S'\rightarrow S$, then sure you can define the sheaf $\text{Hom}_{S'}^\text{ext}(\pi_1(X/S),G)$ over $S'$, but in general giving a scheme/sheaf over $S'$ doesn't uniquely define a scheme/sheaf over $S$ (ie, there are many sheaves over $S$ whose pullback is the given sheaf over $S'$), so if $X/S$ doesn't admit a section over $S$, then how are they defining the sheaf over $S$?

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You have a sheaf over $S'$ and, by independence of the choice of section, an isomorphism between the two different pullbacks of the sheaf to $S' \times_S S'$. This gives descent data to descend the sheaf down to $S$, no?

Maybe a a different characterization would be: The sheafification of the presheaf which associates to an etale open set S' the set of isomorphism classes of $G$-bundles over $X \times_S S'$, where a $G$ bundle is a finite \'{e}tale cover with a left action of $G$. I'm not sure this is right.

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  • $\begingroup$ Hmm, nice, that makes a lot of sense. $\endgroup$ – Will Chen Jul 10 '14 at 17:18

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