$\newcommand{\LL}{\mathbb{L}}$
I'm reading SGA 1, obtained from http://arxiv.org/abs/math/0206203
My questions regard 4.5 and 4.5.1 (page 309) of Expose XIII.
Following "Exemples 4.4" in Expose XIII, let $f : X\rightarrow S$ be such that $S$ is connected, and $X$ is the complement of a normal crossings divisor in a smooth scheme $Z$ over $S$, such that $Z/S$ has "fibres geometriques séparables connexes", which I assume means "separated connected geometric fibers". Let $\LL$ be a set of primes which are not the residue characteristics of any point of $S$. Further, assume we have a section $g : S\rightarrow X$.
Let $s$ be a geometric point of $S$, then we have a split exact sequence $$1\rightarrow\pi_1^\LL(X_s,g(s))\rightarrow\pi_1'(X,g(s))\rightarrow\pi_1(S,s)\rightarrow 1$$ which is split by the section $g_* : \pi_1(S,s)\rightarrow \pi_1'(X,g(s))$. Here, $\pi_1'(X,g(s))$ is the quotient of $\pi_1(X,g(s))$ by the image of the kernel of the natural map $\pi_1(X_s,g(s))\rightarrow \pi_1^\LL(X_s,g(s))$ in $\pi_1(X,g(s))$.
The section $g_*$ gives an action of $\pi_1(S,s)$ on $\pi_1^\LL(X,g(s))$ by conjugation, which corresponds to a Pro-object in the category of finite etale group schemes over $S$, or equivalently a Pro-object in the category of finite locally constant sheaves on $S_{et}$. Let $\pi_1^\LL(X/S,g)$ denote this pro-object.
My first question is this: I've read that for even a complete categories $\mathcal{C}$, in general it is not equivalent to Pro-$\mathcal{C}$. However, it seems to me that for any object $c\in\mathcal{C}$, and a Pro-object $p\in$ Pro-$\mathcal{C}$ (ie, an inverse system of objects of $\mathcal{C}$), it is correct to say that $\text{Hom}_\mathcal{C}(\lim p,c) = \text{Hom}_{\text{Pro}-\mathcal{C}}(p,c)$?
In 4.5.1, she says that for any finite etale group scheme $G/S$ with rank divisible only by primes in $\LL$, the set $\text{Hom}_S(\pi_1^\LL(X/S,g),G)/\{\text{inner automorphisms of $G$}\}$ is canonically (functorially?) isomorphic to the set of isomorphism classes of torsors $P$ over $X$ under the group $G_X := G\times_S X$, equipped with an isomorphism $g^*P\cong G$.
Firstly, am I correct in saying that here by "torsor" she's referring to a principal $G$-bundle as defined in Expose XI, section 4 (pages 226-227)? Often I see torsors in this sort of context refer to sheaves of sets with a freely transitive action of a sheaf of groups, so that a principal $G$-bundle would be a representable sheaf of sets with a free transitive action of a representable sheaf of groups. Certainly here she's assuming that the sheaf of groups is representable (ie, by $G$), but is she also restricting herself to representable torsors?
Secondly, am I correct in assuming that the additional data of the isomorphism $g^*P\cong G$ does not change the notion of isomorphism class of torsors? I.e., $\text{Hom}_S(\pi_1^\LL(X/S),g),G)/\{\text{inner autos}\}$ is in bijection with the subset of isomorphism classes of $G_X$-torsors over $X$ which are isomorphic over $S$ to $G$ (via $g$)?
Third, I was hoping that maps from this pro object $\pi_1^\LL(X/S,g)$ to, say, a constant group scheme $G$ associated to an abstract group $\textbf{G}$ would classify $\textbf{G}$-galois covers of $X/S$. However, 4.5.1 makes this somewhat less clear. However, here's my last question, which I hope might remedy the situation: For any $X\rightarrow S$ satisfying the above conditions and $\textbf{G}$ an abstract finite group of size divisible only by primes in $\LL$, can you always find a scheme $S'$ surjective etale over $S$ such that every $\textbf{G}$-galois cover $P'$ of $X' := X\times_S S'$ (ie, torsor over $X'$ by the constant group scheme $G'/S'$) has the property that $(g')^*P'\cong G'$ (ie, is completely decomposed?)
If $S = \text{Spec } k$ for a field $k$, then you could take $S'$ to be the compositum of all $\textbf{G}$-galois extensions of $k$, which will be etale over $k$. However, if you tried taking the inverse limit of all the $\textbf{G}$-galois covers of $S$, I feel like you might get something which is no longer quasi-finite over $S$ (hence not etale over $S$).
I'm particularly interested in the case where $Z/S$ is a proper smooth family with all geometric fibers irreducible nonsingular projective curves of some genus, and $X/S$ is the complement of the image of a section $S\rightarrow Z$.
EDIT: If I knew that there were only finitely many isomorphism classes of $\textbf{G}$-galois covers of $S$, then taking $S'$ to be the inverse limit of representatives of these isomorphism classes would work right?
thanks,
- will
