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Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:

$$g_Y \colon Y \times_X Y \cong Y \times_X G$$

Two $G$-torsors $Y$ and $Y'$ are isomorphic to each other if and only if there is an isomorphism $Y \cong Y'$ which is compatible with $g_Y$ and $g_{Y'}$ keeping G identity.

I heard that there is a one-to-one correspondence between isomorphism classes of $G$-torsors and $\mathrm{Hom}(\pi_1(X), G)$.

I tried to see this by associating $\rho_Y \colon \pi_1(X) \to G$ to the given $G$-torsor $Y$. Assuming that $Y$ is an etale Galois covering of $X$, I suppose that I can have a correspondence

$$g_Y(y,y) = (y, \theta)$$

with $\theta \in G$. I choose an element $\sigma \in \pi_1(X)$ and let it act on $y$ getting

$$g_Y(\sigma(y), \sigma(y)) = (\sigma(y), \lambda)$$

with $\lambda \in G$ and I will define $\rho_Y \colon \pi_1(X) \to G$ by

$$\rho_Y(\sigma) \colon\!= \theta^{-1}\lambda$$

Question: With this definition of $\rho_Y$, is it easy to see that $\rho_Y(\sigma)$ is well-defined in the sense that it does $not$ depend on the chosen element $y$?

Then if so, is it easy also to see that $\rho_Y$ is a group homomorphism?

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  • $\begingroup$ Have you tried working this out first when X is the spectrum of a field? $\endgroup$
    – Martin Bright
    Commented Feb 11, 2015 at 16:49
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    $\begingroup$ Also the equivalence is with Hom$(\pi_1(X,G)/Inn(G)$. $\endgroup$
    – Will Chen
    Commented Feb 11, 2015 at 19:14
  • $\begingroup$ Perhaps one way to intuitively see this is to notice that any such $G$-torsor is going to be a twisted form of $G$, and we can produce such twisted forms by "twisting" by an element of $\pi_1(X)$. It's a standard fact to associate such twisted forms to a sort of Cech cohomology with coefficients in $G$, but I'm not sure how that compares with $\pi_1(X)$. $\endgroup$
    – Jonathan Beardsley
    Commented Feb 11, 2015 at 21:13
  • $\begingroup$ You should make $X$ into a pointed connected scheme to define $\pi_1(X)$, and your definition of torsor is missing the condition that $Y$ admits a local section. You have defined the notion of pseudo-torsor, and the empty scheme is a standard example of a pseudo-torsor. $\endgroup$
    – S. Carnahan ♦
    Commented Feb 12, 2015 at 11:42

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