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Edit: the original question was imprecise, I'm sorry about that. I hope this is better.

Let $k$ be a field, $S/k$ any scheme, $G/k$ be an algebraic group and $X$ an $S$-torsor under $G$ (so $G$ acts simply transitively on the fibers of $G \to S$). Under what (sufficient/necessary/whatever) conditions on the schemes $X$, $G$, $S$, ... do we get an isomorphism $\overline{X} \cong \overline{G} \times_{\overline{k}} \overline{S}$, where the bar denotes base change to the algebraic closure of $\overline{k}$? For example, do we always have this isomorphism if $\text{Pic}\ \overline{S}$ is trivial?

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How is the right-hand side different from $\overline{G}$? –  Michael Thaddeus Aug 29 '11 at 22:59
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Is $S$ proper over $k$? Otherwise there are counterexamples where $S$ equals the multiplicative group scheme $\mathbb{G}_{m,k}$. –  Jason Starr Aug 29 '11 at 23:58
    
Minor nitpick: you probably want the isomorphism $\bar{X} \to \bar{G}$ to be $\bar{G}$-equivariant. –  S. Carnahan Aug 30 '11 at 0:20
    
Maybe you mean $Pic^0 \overline{S}$, rather than $Pic \overline{S}$, in your final sentence? –  Emerton Aug 30 '11 at 2:44
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@Moret-Bailly: That was also the counterexample I had in mind. –  Jason Starr Aug 30 '11 at 17:53
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1 Answer 1

up vote 1 down vote accepted

-The following is a general condition for trivialization:

Let $\pi:X\to S$ be the structure map and let $T_G$ be the sheaf of trivialization maps where for $U\subset X$ open we have $T_G(U) := \lbrace \psi: \pi^{-1}(U)\to U\times G, \mbox{an isomorphism such that } \pi = p_1\circ\psi\rbrace$ (you can replace $G$ with any scheme you want) The collection of such trivialization is an $\underline{\mathrm{Aut}}(G)$ torsor. If there exists some $\lbrace U_i\rbrace_{i\in I}$ an open cover of $S$ with $t_i\in T_G(U_i)$ let's define $t_{ij}=t_i\circ t_j^{-1}\in \underline{\mathrm{Aut}}(G)(U_{ij})$ so that we get a (well-defined) cohomology class $$\eta\in \Check{H}^1(S,\underline{\mathrm{Aut}}(G))$$ One can show that $\eta$ is trivial if and only if $X\cong S\times G$.

-The statement as you have it will never work since the dimensions don't match if $\dim_{\bar{k}}(S_{\bar k})>0$ since the relative dimension of a torsor and its group scheme are the same. Am I missing something here?

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