The Stacks project

Proof. The implication (1) $\Rightarrow $ (2) follows from Divisors on Spaces, Lemma 71.14.3. Assume (2). To check $\mathcal{L}$ is ample on $X/Y$ we may work étale localy on $Y$, see Divisors on Spaces, Lemma 71.14.6. Thus we may assume that $Y$ is a scheme and then we may in turn assume each $Y_ i$ is a scheme too, see Topologies on Spaces, Lemma 73.9.5. In other words, we may assume that $\{ Y_ i \to Y\} $ is an fpqc covering of schemes.

By Divisors on Spaces, Lemma 71.14.4 we see that $X_ i \to Y_ i$ is representable (i.e., $X_ i$ is a scheme), quasi-compact, and separated. Hence $f$ is quasi-compact and separated by Lemmas 74.11.1 and 74.11.18. This means that $\mathcal{A} = \bigoplus _{d \geq 0} f_*\mathcal{L}^{\otimes d}$ is a quasi-coherent graded $\mathcal{O}_ Y$-algebra (Morphisms of Spaces, Lemma 67.11.2). Moreover, the formation of $\mathcal{A}$ commutes with flat base change by Cohomology of Spaces, Lemma 69.11.2. In particular, if we set $\mathcal{A}_ i = \bigoplus _{d \geq 0} f_{i, *}\mathcal{L}_ i^{\otimes d}$ then we have $\mathcal{A}_ i = g_ i^*\mathcal{A}$. It follows that the natural maps $\psi _ d : f^*\mathcal{A}_ d \to \mathcal{L}^{\otimes d}$ of $\mathcal{O}_ X$ pullback to give the natural maps $\psi _{i, d} : f_ i^*(\mathcal{A}_ i)_ d \to \mathcal{L}_ i^{\otimes d}$ of $\mathcal{O}_{X_ i}$-modules. Since $\mathcal{L}_ i$ is ample on $X_ i/Y_ i$ we see that for any point $x_ i \in X_ i$, there exists a $d \geq 1$ such that $f_ i^*(\mathcal{A}_ i)_ d \to \mathcal{L}_ i^{\otimes d}$ is surjective on stalks at $x_ i$. This follows either directly from the definition of a relatively ample module or from Morphisms, Lemma 29.37.4. If $x \in |X|$, then we can choose an $i$ and an $x_ i \in X_ i$ mapping to $x$. Since $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X_ i, \overline{x}_ i}$ is flat hence faithfully flat, we conclude that for every $x \in |X|$ there exists a $d \geq 1$ such that $f^*\mathcal{A}_ d \to \mathcal{L}^{\otimes d}$ is surjective on stalks at $x$. This implies that the open subset $U(\psi ) \subset X$ of Divisors on Spaces, Lemma 71.13.1 corresponding to the map $\psi : f^*\mathcal{A} \to \bigoplus _{d \geq 0} \mathcal{L}^{\otimes d}$ of graded $\mathcal{O}_ X$-algebras is equal to $X$. Consider the corresponding morphism

It is clear from the above that the base change of $r_{\mathcal{L}, \psi }$ to $Y_ i$ is the morphism $r_{\mathcal{L}_ i, \psi _ i}$ which is an open immersion by Morphisms, Lemma 29.37.4. Hence $r_{\mathcal{L}, \psi }$ is an open immersion by Lemma 74.11.14. Hence $X$ is a scheme and we conclude $\mathcal{L}$ is ample on $X/Y$ by Morphisms, Lemma 29.37.4. $\square$

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 neigbourhood $V_ i \subset Y_ i$ 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). $\square$