The Stacks project

Proof. Suppose that the lemma holds whenever $Y$ is a scheme. Let $U$ be a scheme and let $U \to Y$ be a surjective étale morphism. Let $R = U \times _ Y U$ with projections $t, s : R \to U$. Denote $X_ U = U \times _ Y X$ and $\mathcal{L}_ U$ the pullback. Then we get an open subscheme $V' \subset U$ as in the lemma for $(X_ U \to U, \mathcal{L}_ U)$. By the functorial characterization we see that $s^{-1}(V') = t^{-1}(V')$. Thus there is an open subspace $V \subset Y$ such that $V'$ is the inverse image of $V$ in $U$. In particular $V' \to V$ is surjective étale and we conclude that $\mathcal{L}_ V$ is ample on $X_ V/V$ (Divisors on Spaces, Lemma 71.14.6). Now, if $Y' \to Y$ is a morphism such that $\mathcal{L}'$ is ample on $X'/Y'$, then $U \times _ Y Y' \to Y'$ must factor through $V'$ and we conclude that $Y' \to Y$ factors through $V$. Hence $V \subset Y$ is as in the statement of the lemma. In this way we reduce to the case dealt with in the next paragraph.

Assume $Y$ is a scheme. Since the question is local on $Y$ we may assume $Y$ is an affine scheme. We will show the following:

1. If $\mathop{\mathrm{Spec}}(k) \to Y$ is a morphism such that $\mathcal{L}_ k$ is ample on $X_ k/k$, then there is an open neighbourhood $V \subset Y$ of the image of $\mathop{\mathrm{Spec}}(k) \to Y$ such that $\mathcal{L}_ V$ is ample on $X_ V/V$.

It is clear that (A) implies the truth of the lemma.

Let $X \to Y$, $\mathcal{L}$, $\mathop{\mathrm{Spec}}(k) \to Y$ be as in (A). By Lemma 74.13.1 we may assume that $k = \kappa (y)$ is the residue field of a point $y$ of $Y$.

As $Y$ is affine we can find a directed set $I$ and an inverse system of morphisms $X_ i \to Y_ i$ of algebraic spaces with $Y_ i$ of finite presentation over $\mathbf{Z}$, with affine transition morphisms $X_ i \to X_{i'}$ and $Y_ i \to Y_{i'}$, with $X_ i \to Y_ i$ proper and of finite presentation, and such that $X \to Y = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$. See Limits of Spaces, Lemma 70.12.2. After shrinking $I$ we may assume $Y_ i$ is an (affine) scheme for all $i$, see Limits of Spaces, Lemma 70.5.10. After shrinking $I$ we can assume we have a compatible system of invertible $\mathcal{O}_{X_ i}$-modules $\mathcal{L}_ i$ pulling back to $\mathcal{L}$, see Limits of Spaces, Lemma 70.7.3. Let $y_ i \in Y_ i$ be the image of $y$. Then $\kappa (y) = \mathop{\mathrm{colim}}\nolimits \kappa (y_ i)$. Hence $X_ y = \mathop{\mathrm{lim}}\nolimits X_{i, y_ i}$ and after shrinking $I$ we may assume $X_{i, y_ i}$ is a scheme for all $i$, see Limits of Spaces, Lemma 70.5.11. Hence for some $i$ we have $\mathcal{L}_{i, y_ i}$ is ample on $X_{i, y_ i}$ by Limits, Lemma 32.4.15. By Divisors on Spaces, Lemma 71.15.3 we find an open neighbourhood $V_ i \subset Y_ i$ of $y_ i$ such that $\mathcal{L}_ i$ restricted to $f_ i^{-1}(V_ i)$ is ample relative to $V_ i$. Letting $V \subset Y$ be the inverse image of $V_ i$ finishes the proof (hints: use Morphisms, Lemma 29.37.9 and the fact that $X \to Y \times _{Y_ i} X_ i$ is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma 29.37.7). $\square$