The Stacks project

Lemma 85.9.1. 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 64.6.1. 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 74.8.1 (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 68.15.2 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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