My question is about the passage (11.1) in the book of Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$, an algebraic stack $\mathscr{X}$ over $S$, and a point $\xi$ of $\mathscr{X}$. The residual gerbe $\mathscr{G}_\xi$ at $\xi$ is defined as follows: choose an $S$-field $K$ and an $S$-morphism $x\colon \mathrm{Spec}(K) \rightarrow \mathscr{X}$ representing $\xi$, and let $\mathscr{G}_\xi \subset \mathscr{X}$ be the smallest substack (for the fppf topology) through which $x$ factors. The claim is that this $\mathscr{G}_\xi$ does not depend on the choices of $K$ and $x$. How does one verify this?

I am especially having trouble in the case when $K'$ is a big overfield of $K$, when I cannot figure out how to make contact with the fppf topology that is used in defining $\mathscr{G}_\xi$. It seems to me that $\mathscr{G}_\xi$ could shrink after replacing $K$ by this $K'$ in the defining procedure.

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    $\begingroup$ The definition should require $\xi$ admits a representative $(x,K)$ with $K$ essentially of finite type over the residue field at a point in a smooth scheme cover $X \rightarrow \mathscr{X}$ ("ess. finite type" defined via Rem. 5.3), and only use such $K$ in 11.1. Then for such $\xi$ the same gerbe is obtained using any representative $(x,K)$ without finiteness conditions on $K$ (because ${\rm{Spec}}(K)\times_{\mathscr{X}} X$ is a non-empty smooth algebraic space, so admits a $K'$-point for $K'/K$ finite separable)! For the appeal to 3.7 near the start of 11.1, they should also point out 10.7. $\endgroup$ – grghxy Sep 10 '15 at 0:49
  • $\begingroup$ @grghxy: Thank you. Could you clarify your first sentence? Any $\xi$ admits a representative $(x, K)$ with $K$ even of finite type over the residue field at a point of a smooth scheme cover (because already the classes of these residue fields exhaust $\xi$). Do you mean that one should use only such fields $K$ in 11.1 (for arbitrary $\xi$)? $\endgroup$ – O-Ren Ishii Sep 10 '15 at 2:43
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    $\begingroup$ Sorry yes, I meant that in 11.1 one should only consider such $K$. It is more robust to impose "essentially finite type" (i.e., finitely generated as field extension) rather than "finite type" (= "finite" by Nullstellensatz) because a typical point of the algebraic space $x \times_{\mathscr{X}} X$ has residue field ess. finite type over $K$. $\endgroup$ – grghxy Sep 10 '15 at 4:53
  • $\begingroup$ @grghxy: Thanks for clarifying. I am still puzzled: consider the case $\mathscr{X} = \mathrm{Spec}(F)$ for some field $F$. Then choosing $x = \mathrm{id}_{F}$ I get that the residual gerbe is $\mathscr{X}$ itself. However, I could choose a smooth cover $X \rightarrow \mathrm{Spec}(F)$, choose a generic point $\mathrm{Spec}(K)$ of $X$, and look at $x' \colon \mathrm{Spec}(K) \rightarrow \mathrm{Spec}(F)$. Now I should be getting that $x'$ is surjective for the fppf topology, but it is not: there is no fppf cover of $\mathrm{Spec}(F)$ that would factor through $\mathrm{Spec}(K)$. $\endgroup$ – O-Ren Ishii Sep 10 '15 at 15:50
  • $\begingroup$ Sorry, you are absolutely correct. So focusing on finite extensions of residue fields at points of a smooth scheme cover is the right thing to do. Looking back at my copy of L-MB, in the margin I had made the incorrect note to myself that $K'/K$ is an fppf cover for a finitely generated extension field, which as you say is false; that was where my error crept in. I have now fixed my margin comment to say "finite extension" (in 11.1, after which one bootstraps to the general case). $\endgroup$ – grghxy Sep 11 '15 at 19:29

IIRC a reference for a discussion is David Rydh's paper \'Etale d\'evissage, descent and pushouts of stacks, Appendix B. (Of course grghxy also already answered your question too, my apologies to grghxy.)

You can look at Tag 06ML for an answer to your question if you already know the thing as defined in Laumon and Moret-Bailly is algebraic. This is not hard.

To see that it is algebraic (following Rydh and in the quasi-separated case, i.e., in the case of algebraic stacks as discussed in Laumon and Moret-Bailly), you can look Section Tag 06UH. Essentially the idea is to reduce to the case where $x$ is the generic point and $\mathcal{X}$ is an integral algebraic stack, then reduce to the case where $\mathcal{X}$ is a gerbe (flattening stratification for inertia), then pull back from the algebraic space that the algebraic stack is a gerbe over. Cheers!

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  • $\begingroup$ Thanks! These references are very helpful and clarify my confusion. The second paragraph in Appendix B of Rydh's paper is exactly what I was looking for. $\endgroup$ – O-Ren Ishii Sep 10 '15 at 20:45

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