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The Stacks project

Lemma 87.13.1. Let S be a scheme. Suppose given a directed set \Lambda and a system of algebraic spaces (X_\lambda , f_{\lambda \mu }) over \Lambda where each f_{\lambda \mu } : X_\lambda \to X_\mu is a thickening. Then X = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } X_\lambda is a formal algebraic space over S.

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 an étale covering \{ X_{i, \lambda } \to X_\lambda \} where X_ i is an affine scheme over S, see Properties of Spaces, Lemma 66.6.1. For each \mu \geq \lambda there exists a cartesian diagram

\xymatrix{ X_{i, \lambda } \ar[r] \ar[d] & X_{i, \mu } \ar[d] \\ X_\lambda \ar[r] & X_\mu }

with étale vertical arrows, see More on Morphisms of Spaces, Theorem 76.8.1 (this also uses that a thickening is a surjective closed immersion which satisfies the conditions of the theorem). Moreover, these diagrams are unique up to unique isomorphism and hence X_{i, \mu } = X_\mu \times _{X_{\mu '}} X_{i, \mu '} for \mu ' \geq \mu . The morphisms X_{i, \mu } \to X_{i, \mu '} is a thickening as a base change of a thickening. Each X_{i, \mu } is an affine scheme by Limits of Spaces, Proposition 70.15.2 and the fact that X_{i, \lambda } is affine. Set X_ i = \mathop{\mathrm{colim}}\nolimits _{\mu \geq \lambda } X_{i, \mu }. Then X_ i is an affine formal algebraic space. The morphism X_ i \to X is étale because given an affine scheme U any U \to X factors through X_\mu for some \mu \geq \lambda (details omitted). In this way we see that X is a formal algebraic space. \square


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