The Stacks project

Proof. Let $I$ be a directed set and let $(A_ i, \varphi _{ii'})$ be a system of rings over $I$. Set $S = \mathop{\mathrm{Spec}}(A)$ and $S_ i = \mathop{\mathrm{Spec}}(A_ i)$. We have to show that on fibre categories we have

We know that the category of schemes of finite presentation over $S$ is the colimit of the category of schemes of finite presentation over $S_ i$, see Limits, Lemma 32.10.1. Moreover, given $X_ i \to S_ i$ of finite presentation, with limit $X \to S$, then the category of invertible $\mathcal{O}_ X$-modules $\mathcal{L}$ is the colimit of the categories of invertible $\mathcal{O}_{X_ i}$-modules $\mathcal{L}_ i$, see Limits, Lemma 32.10.2 and 32.10.3. If $X \to S$ is proper and flat, then for sufficiently large $i$ the morphism $X_ i \to S_ i$ is proper and flat too, see Limits, Lemmas 32.13.1 and 32.8.7. Finally, if $\mathcal{L}$ is ample on $X$ then $\mathcal{L}_ i$ is ample on $X_ i$ for $i$ sufficiently large, see Limits, Lemma 32.4.15. Putting everything together finishes the proof. $\square$