The Stacks project

Proof. We may write $X_\lambda = \mathop{\mathrm{colim}}\nolimits _{\omega \in \Omega _\lambda } X_{\lambda , \omega }$ as the colimit of affine schemes over a directed set $\Omega _\lambda$ such that the transition morphisms $X_{\lambda , \omega } \to X_{\lambda , \omega '}$ are thickenings. For each $\lambda , \mu \in \Lambda$ and $\omega \in \Omega _\lambda$, with $\mu \geq \lambda$ there exists an $\omega ' \in \Omega _\mu$ such that the morphism $X_{\lambda , \omega } \to X_\mu$ factors through $X_{\mu , \omega '}$, see Lemma 87.9.4. Then the morphism $X_{\lambda , \omega } \to X_{\mu , \omega '}$ is a closed immersion inducing an isomorphism on reductions and hence a thickening. Set $\Omega = \coprod _{\lambda \in \Lambda } \Omega _\lambda$ and say $(\lambda , \omega ) \leq (\mu , \omega ')$ if and only if $\lambda \leq \mu$ and $X_{\lambda , \omega } \to X_\mu$ factors through $X_{\mu , \omega '}$. It follows from the above that $\Omega$ is a directed set and that $X = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } X_\lambda = \mathop{\mathrm{colim}}\nolimits _{(\lambda , \omega ) \in \Omega } X_{\lambda , \omega }$. This finishes the proof. $\square$

Proof. Since we take the colimit in the category of fppf sheaves, we see that $X$ is a sheaf. Choose and fix $\lambda \in \Lambda$. Choose a covering $\{ X_{i, \lambda } \to X_\lambda \}$ as in Definition 87.11.1. In particular, we see that $\{ X_{i, \lambda , red} \to X_{\lambda , red}\}$ is an étale covering by affine schemes. For each $\mu \geq \lambda$ there exists a cartesian diagram

with étale vertical arrows. Namely, the étale morphism $X_{i, \lambda , red} \to X_{\lambda , red} = X_{\mu , red}$ corresponds to an étale morphism $X_{i, \mu } \to X_\mu$ of formal algebraic spaces with $X_{i, \mu }$ an affine formal algebraic space, see Lemma 87.34.4. The same lemma implies the base change of $X_{i, \mu }$ to $X_\lambda$ agrees with $X_{i, \lambda }$. It also follows that $X_{i, \mu } = X_\mu \times _{X_{\mu '}} X_{i, \mu '}$ for $\mu ' \geq \mu \geq \lambda$. Set $X_ i = \mathop{\mathrm{colim}}\nolimits X_{i, \mu }$. Then $X_{i, \mu } = X_ i \times _ X X_\mu$ (as functors). Since any morphism $T \to X = \mathop{\mathrm{colim}}\nolimits X_\mu$ from an affine (or quasi-compact) scheme $T$ maps into $X_\mu$ for some $\mu$, we see conclude that $\mathop{\mathrm{colim}}\nolimits X_{i, \mu } \to \mathop{\mathrm{colim}}\nolimits X_\mu$ is étale. Thus, if we can show that $\mathop{\mathrm{colim}}\nolimits X_{i, \mu }$ is an affine formal algebraic space, then the lemma holds. Note that the morphisms $X_{i, \mu } \to X_{i, \mu '}$ are closed immersions as a base change of the closed immersion $X_\mu \to X_{\mu '}$. Finally, the morphism $X_{i, \mu , red} \to X_{i, \mu ', red}$ is an isomorphism as $X_{\mu , red} \to X_{\mu ', red}$ is an isomorphism. Hence this reduces us to the case discussed in Lemma 87.36.1. $\square$