Skip to main content
MathOverflow

Return to Question

added 217 characters in body
Source Link
Will Chen
Will Chen
  • 10.7k
  • 2
  • 32
  • 74

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,

thanks,

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,

The author of XIII is "she", not "he" (there were two students of Grothendieck named M. Raynaud).
Source Link
edit approved Jun 19, 2014 at 5:44
user27920
edit approved Jun 19, 2014 at 5:44
user27920
  • 1.3k
  • 11
  • 11

In 4.5.1, heshe 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" he'sshe's referring to a principal $G$-bundle as he 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 he'sshe's assuming that the sheaf of groups is representable (ie, by $G$), but is heshe also restricting himselfherself 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? IeI.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 $g$)?

In 4.5.1, he 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" he's referring to a principal $G$-bundle as he 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 he's assuming that the sheaf of groups is representable (ie, by $G$), but is he also restricting himself 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? Ie, $\text{Hom}_S(\pi_1^\LL(X/S),g),G)/\{\text{inner autos}\}$ is in bijection with the subset of isomorphism classes of $G_X$-torsors over $X$ which are isomorphic over $S$ to $G$ (via $g$)?

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 $g$)?

Source Link
Will Chen
Will Chen
  • 10.7k
  • 2
  • 32
  • 74

questions about the "relative fundamental group" of SGA 1 Expose XIII

$\newcommand{\LL}{\mathbb{L}}$

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, he 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" he's referring to a principal $G$-bundle as he 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 he's assuming that the sheaf of groups is representable (ie, by $G$), but is he also restricting himself 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? Ie, $\text{Hom}_S(\pi_1^\LL(X/S),g),G)/\{\text{inner autos}\}$ 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 $S\rightarrow Z$.

thanks,

  • will