Lemma 87.36.1. Let $S$ be a scheme. Suppose given a directed set $\Lambda $ and a system of affine formal algebraic spaces $(X_\lambda , f_{\lambda \mu })$ over $\Lambda $ where each $f_{\lambda \mu } : X_\lambda \to X_\mu $ is a closed immersion inducing an isomorphism $X_{\lambda , red} \to X_{\mu , red}$. Then $X = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } X_\lambda $ is an affine formal algebraic space over $S$.
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. $\square$
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