Let $S$ be a scheme and $\mathscr{X}$ an Artin stack over $S$. Let $X$ be a scheme and $P : X \to \mathscr{X}$ a representable morphism, which is smooth and surjective. Then this $P$ is an epimorphism.
(i.e., for any $S$-schemes $U$ and $x \in \mathscr{X}(U)$, there exists an etale surjective morphism $U' \to U$ (of schemes), such that $x|_{U'}$ comes from $X(U')$)

I want to know it in order to show that $\mathscr{X} = [X/ R]$, the quotient stack. ($R = X \times_\mathscr{X} X$)

In 4.3. of Laumon, Moret-Bailly's Champs algébriques, the authors say that this is obvious. But I don't understand.

I've heard that every Artin stack is a stack over fppf topology. If we use it, then trivially $\mathscr{X} = [X/ R]$. But in the proof of this proposition (10.7. of LMB's Champ algebriques), they use the highlighted statement, so it is circular reasoning.

How can I show the highlighted statement without using the stack-ness over fppf topology?

  • 3
    $\begingroup$ Take the fiber product $X \times_{\mathcal X} U$, which is a smooth scheme over $U$, then use that every smooth surjective morphism of schemes has an etale section. $\endgroup$ – Will Sawin Aug 1 at 14:48
  • $\begingroup$ Has étale-locally a section, no? $\endgroup$ – David Roberts Aug 1 at 20:47
  • $\begingroup$ @WillSawin I did not notice that. Thank you! $\endgroup$ – k.j. 2 days ago
  • 1
    $\begingroup$ @DavidRoberts Yes, that's what I should have said. $\endgroup$ – Will Sawin 2 days ago

As Will Sawin pointed out in comments, every smooth surjective morphism of schemes admits étale-local sections (Tag 055U in the Stacks Project). See Tag 055S more generally for details and discussion. As mentioned at Tag 021Y the étale topology and the smooth topology give rise to the same topos, but it is useful to deal with the extra flexibility that smooth morphisms give, as evidenced by examples of Artin stacks that are not Deligne–Mumford stacks (for instance, IIRC, the classifying stack of $\mathbb{G}_m$).


A "similar" result along with proof can be found as Lemma 2.14 of Differentiable Stacks and Gerbes.

I would like to give more details if you want.

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