I have just started studying stacks. Trying to understand the theory I was thinking about a (very interesting) toy example: $ BG $ the classifying stack of a smooth (over a base scheme $ S $) group G. It is well known that a (smooth) cover is given by its canonical point $ s_0: S \rightarrow BG $ (the trivial torsor over $ S $ seen through Yoneda). Is it true that a map from a scheme $ X $ to $ BG $ factor through $ s_0 $?

In particular I had in mind the concrete example where $ S=\mathrm{Spec}(\mathbb C) $ and where $ G=GL_n $ (so it is true that every torsor is locally trivial in the étale topology).

  • $\begingroup$ When $S=\operatorname{Spec}(\mathbb{C})$ and If $X\rightarrow BG$ factors through $s_0$, then I think the corresponding family is necessarily trivial $\endgroup$ – loch Sep 24 '18 at 14:39
  • $\begingroup$ @loch: If you take $ X \rightarrow \mathrm{Spec}(\mathbb C) $ as the structure morphism then you're certainly right. But I was thinking about something like an endomorphism of $ X $ composed with the structure morphism $ x:X \rightarrow \mathrm{Spec}(\mathbb C) $ and then $s_0$. But I am not comfortable with stacks yet, so maybe I am making huge mistakes. $\endgroup$ – user192820 Sep 24 '18 at 15:47
  • $\begingroup$ @loch: as stated by Sasha you were right, thank you both. $\endgroup$ – user192820 Sep 24 '18 at 15:59

No, a map $X \to BG$ is determined by a map $f:X \to S$ and a $f^*G$-torsor over $X$, while those map that factor through $s_0$ correspond to trivial torsors. So, as soon as there are nontrivial torsors over $X$, there are maps that do not factor through $s_0$.


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