Lemma 39.25.1. Let $X \to Y$ be a surjective finite locally free morphism. Let $d$ be a positive integer. Assume that for every geometric point $\bar{y} : \mathrm{Spec}(k) \to Y$ the fiber $X_{\bar{y}}$ has at most $d$ points. 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.
39.25 Descending quasi-projective schemes
We can use Lemma 39.24.1 to show that a certain type of descent datum 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, and their geometric fibers have at most $d$ points. In particular the $R$-orbit $t(s^{-1}(\{ v\} ))$ of our point $v \in V$ has at most $d$ points. 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 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. As in the proof of Lemma 39.25.1, we may assume that $Y$ is affine, in particular quasi-compact. Thus there exists an integer $d$ such that all geometric fibers of $X \to Y$ have at most $d$ points, see the discussion in Morphisms, Section 29.57. Our assumption on $V$ trivially implies that we can apply Lemma 39.25.1, which completes the proof. $\square$
Lemma 39.25.3. 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
$V \to X$ is projective,
$V \to X$ is quasi-projective,
there exists an ample invertible sheaf on $V$,
there exists an $X$-ample invertible sheaf on $V$,
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 39.25.2. 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. $\square$
Lemma 39.25.4. Let $X \to Y$ be a surjective finite locally free morphism which is radicial. Let $V$ be a scheme over $X$. Then any descent datum on $V/X/Y$ is effective.
Proof. Apply Lemma 39.25.1 with $d = 1$. $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)