The Stacks project

39.25 Descending quasi-projective schemes

We can use Lemma 39.24.1 to show that a certain type of descent datum is effective.

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 35.31.1. Recall that the functor from schemes over $Y$ to descent data relative to $\{ X \to Y\} $ is fully faithful, see Descent, Lemma 35.32.11. 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 35.34.1. Hence by Descent, Lemma 35.32.13 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 $v$. $\square$

Lemma 39.25.2. Let $X \to Y$ be a surjective finite locally free morphism. Let $V$ be a scheme over $X$ such that one of the following holds

  1. $V \to X$ is projective,

  2. $V \to X$ is quasi-projective,

  3. there exists an ample invertible sheaf on $V$,

  4. there exists an $X$-ample invertible sheaf on $V$,

  5. there exists an $X$-very ample invertible sheaf on $V$.

Then any descent datum on $V/X/Y$ is effective.

Proof. We check the condition in Lemma 39.25.1. Let $y \in Y$ and $v_1, \ldots , v_ d \in V$ points over $y$. Case (1) is a special case of (2), see Morphisms, Lemma 29.43.10. Case (2) is a special case of (4), see Morphisms, Definition 29.40.1. If there exists an ample invertible sheaf on $V$, then there exists an affine open containing $v_1, \ldots , v_ d$ by Properties, Lemma 28.29.5. Thus (3) is true. In cases (4) and (5) it is harmless to replace $Y$ by an affine open neighbourhood of $y$. Then $X$ is affine too. In case (4) we see that $V$ has an ample invertible sheaf by Morphisms, Definition 29.37.1 and the result follows from case (3). In case (5) we can replace $V$ by a quasi-compact open containing $v_1, \ldots , v_ d$ and we reduce to case (4) by Morphisms, Lemma 29.38.2. $\square$


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