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I am working on Fabien Morel's paper : "A1-Algebraic topology over a field" and I am a bit confused about certains properties of classifying spaces.

For example : How to show that $BSL_r \longrightarrow BGL_r$ is a $G_m$-torsor?

Here $BGL_r$ and $BSL_r$ are living in the $A^1$-homotopy category of schemes over a field.

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  • $\begingroup$ hint : this is the pull-back of the universal $BG_m$ torsor via $Bdet: BGL_r\to BG_m$. $\endgroup$
    – Niels
    Nov 11, 2013 at 21:45
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    $\begingroup$ More concretely, given a scheme $S$ and a map $S\to BGL_r$--equivalently, a vector bundle $V$ over $S$ of constant rank $r$--the fiber-product $S\times_{BGL_r}BSL_r$ is precisely the scheme over $S$ that parameterizes trivializations of the determinant line bundle $\wedge^{r}V$. This latter space is clearly a $\mathbb{G}_m$-torsor over $S$. $\endgroup$ Nov 12, 2013 at 2:41

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