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 locally 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. $\square$