The Stacks project

Lemma 39.25.1. Let $X \to Y$ be a surjective finite locally free morphism. Let $V$ be a scheme over $X$ such that for all $(y, v_1, \ldots , v_ d)$ where $y \in Y$ and $v_1, \ldots , v_ d \in V_ y$ there exists an affine open $U \subset V$ with $v_1, \ldots , v_ d \in U$. Then any descent datum on $V/X/Y$ is effective.

Proof. Let $\varphi $ be a descent datum as in Descent, Definition 35.34.1. Recall that the functor from schemes over $Y$ to descent data relative to $\{ X \to Y\} $ is fully faithful, see Descent, Lemma 35.35.11. Thus using Constructions, Lemma 27.2.1 it suffices to prove the lemma in the case that $Y$ is affine. Some details omitted (this argument can be avoided if $Y$ is separated or has affine diagonal, because then every morphism from an affine scheme to $X$ is affine).

Assume $Y$ is affine. If $V$ is also affine, then we have effectivity by Descent, Lemma 35.37.1. Hence by Descent, Lemma 35.35.13 it suffices to prove that every point $v$ of $V$ has a $\varphi $-invariant affine open neighbourhood. Consider the groupoid $(X, X \times _ Y X, \text{pr}_1, \text{pr}_0, \text{pr}_{02})$. By Lemma 39.21.3 the descent datum $\varphi $ determines and is determined by a cartesian morphism of groupoid schemes

\[ (V, R, s, t, c) \longrightarrow (X, X \times _ Y X, \text{pr}_1, \text{pr}_0, \text{pr}_{02}) \]

over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Since $X \to Y$ is finite locally free, we see that $\text{pr}_ i : X \times _ Y X \to X$ and hence $s$ and $t$ are finite locally free. In particular the $R$-orbit $t(s^{-1}(\{ v\} ))$ of our point $v \in V$ is finite. Using the equivalence of categories of Lemma 39.21.3 once more we see that $\varphi $-invariant opens of $V$ are the same thing as $R$-invariant opens of $V$. Our assumption shows there exists an affine open of $V$ containing the orbit $t(s^{-1}(\{ v\} ))$ as all the points in this orbit map to the same point of $Y$. Thus Lemma 39.24.1 provides an $R$-invariant affine open containing $v$. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CCI. Beware of the difference between the letter 'O' and the digit '0'.