The Stacks project

Lemma 39.23.3. 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. Then

\[ U = \coprod \nolimits _{r \geq 1} U_ r \]

is a disjoint union of $R$-invariant opens such that the restriction $R_ r$ of $R$ to $U_ r$ has the property that $s, t : R_ r \to U_ r$ are finite locally free of rank $r$.

Proof. By Morphisms, Lemma 29.48.5 there exists a decomposition $U = \coprod \nolimits _{r \geq 0} U_ r$ such that $s : s^{-1}(U_ r) \to U_ r$ is finite locally free of rank $r$. As $s$ is surjective we see that $U_0 = \emptyset $. Note that $u \in U_ r \Leftrightarrow $ if and only if the scheme theoretic fibre $s^{-1}(u)$ has degree $r$ over $\kappa (u)$. Now, if $z \in R$ with $s(z) = u$ and $t(z) = u'$ then using notation as in Lemma 39.13.4

\[ \text{pr}_1^{-1}(z) \to \mathop{\mathrm{Spec}}(\kappa (z)) \]

is the base change of both $s^{-1}(u) \to \mathop{\mathrm{Spec}}(\kappa (u))$ and $s^{-1}(u') \to \mathop{\mathrm{Spec}}(\kappa (u'))$ by the lemma cited. Hence $u \in U_ r \Leftrightarrow u' \in U_ r$, in other words, the open subsets $U_ r$ are $R$-invariant. In particular the restriction of $R$ to $U_ r$ is just $s^{-1}(U_ r)$ and $s : R_ r \to U_ r$ is finite locally free of rank $r$. As $t : R_ r \to U_ r$ is isomorphic to $s$ by the inverse of $R_ r$ we see that it has also rank $r$. $\square$

Comments (3)

Comment #1540 by jojo on

I think that the conclusion of this lemma should be that are finite locally free of rank not .

Comment #1541 by jojo on

You explain that is a base change of

$$ s^{-1}(u) \to \text{Spec}(\kappa(u)) \;\;\text{ by }\;\; $ and of

$$ s^{-1}(u') \to \text{Spec}(\kappa(u')) \;\;\text{ by }\;\; $

Instead of concluding directly that it means that is -stable I think that it might be nice to add the following short argument :

The first base change implies that is of rank over because is of rank above . Now the second base change implies that is of rank over because is faithfully flat and affine (because surjective flat and finite). So this means that is in which in turn implies that is -stable.

Comment #1568 by on

Thanks for the typo. I do not agree entirely with your second comment because I think faithfully flat descent isn't needed here because we already have the opens . In fact, in some cases one can obtain invariant loci like this without assuming that and are flat; take a look at the (unfortunately rather hard to parse) Section 40.6.

For the edits corresponding to your comments, see here. Thanks very much!

There are also:

  • 6 comment(s) on Section 39.23: Finite flat groupoids, affine case

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