Proposition 66.14.1 (07S6)—The Stacks project

Proposition 66.14.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume

$s, t : R \to U$ finite locally free,
$j = (t, s)$ is an equivalence, and
for a dense set of points $u \in U$ the $R$-equivalence class $t(s^{-1}(\{ u\} ))$ is contained in an affine open of $U$.

Then there exists a finite locally free morphism $U \to M$ of schemes over $S$ such that $R = U \times _ M U$ and such that $M$ represents the quotient sheaf $U/R$ in the fppf topology.

Proof. By assumption (3) and Groupoids, Lemma 39.24.1 we can find an open covering $U = \bigcup U_ i$ such that each $U_ i$ is an $R$-invariant affine open of $U$. Set $R_ i = R|_{U_ i}$. Consider the fppf sheaves $F = U/R$ and $F_ i = U_ i/R_ i$. By Spaces, Lemma 65.10.2 the morphisms $F_ i \to F$ are representable and open immersions. By Groupoids, Proposition 39.23.9 the sheaves $F_ i$ are representable by affine schemes. If $T$ is a scheme and $T \to F$ is a morphism, then $V_ i = F_ i \times _ F T$ is open in $T$ and we claim that $T = \bigcup V_ i$. Namely, fppf locally on $T$ we can lift $T \to F$ to a morphism $f : T \to U$ and in that case $f^{-1}(U_ i) \subset V_ i$. Hence we conclude that $F$ is representable by a scheme, see Schemes, Lemma 26.15.4. $\square$

Comments (3)

Comment #1521 by jojo on June 20, 2015 at 18:27

Is it really ovious that the sheaves $U_i/R_i$ cover $R/U$ ? I think that copying the arguments used in 02WU I can see why it is true but if that's the way to do it then maybe it would deserve a proof (or at least a sentence indicating how to proceed). But it wouldn't surprise me if I were missing something :)

Comment #1525 by jojo on June 21, 2015 at 19:09

In fact I wonder ... can we not bypass the use of 01JJ and thus of 02WU by invoking 03C5 ? I mean I think that the conditions of 03C5 can be tested locally for a cover that's $R$ -invariant? Sadly I don't have enough self-confidence in algebraic geometry to be really sure but it feels like it might be true.

I would be really nice because the proof of 02WU is kind of hard while 03C5 is easy.

Also maybe it would be nice to state proposition 3.4.6 in http://arxiv.org/pdf/1210.0431.pdf which says when 03C5 is an equivalence.

Comment #1532 by Johan on June 22, 2015 at 15:53

OK, I sort of agree with you that $F = \bigcup F_i$ being a covering needs a argument. I added a small argument. I can make it longer by choosing coverings and writing everything out, but if we do that, then we should move this propositions to the chapter on groupoids. Here is the commit.

If you want to write out your suggestion, you are welcome to do so and submit it by email. Thanks!

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