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The Stacks project

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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