The Stacks project

Theorem 87.27.4. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Let $U \subset X$ be the open subspace with $|U| = |X| \setminus T$. The completion functor (

\[ \left\{ \begin{matrix} \text{morphisms of algebraic spaces} \\ f : X' \to X\text{ which are locally} \\ \text{of finite type and such that} \\ f^{-1}U \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{morphisms }g : W \to X_{/T} \\ \text{of formal algebraic spaces} \\ \text{with }W\text{ locally Noetherian} \\ \text{and }g\text{ rig-étale} \end{matrix} \right\} \]

sending $f : X' \to X$ to $f_{/T} : X'_{/T'} \to X_{/T}$ is an equivalence.

Proof. The functor is fully faithful by Lemma 87.27.2. Let $g : W \to X_{/T}$ be a morphism of formal algebraic spaces with $W$ locally Noetherian and $g$ rig-étale. We will prove $W$ is in the essential image to finish the proof.

Choose an étale covering $\{ X_ i \to X\} $ with $X_ i$ affine for all $i$. Denote $U_ i \subset X_ i$ the inverse image of $U$ and denote $T_ i \subset X_ i$ the inverse image of $T$. Recall that $(X_ i)_{/T_ i} = (X_ i)_{/T} = (X_ i \times _ X X)_{/T}$ and $W_ i = X_ i \times _ X W = (X_ i)_{/T} \times _{X_{/T}} W$, see Lemma 87.27.1. Observe that we obtain isomorphisms

\[ \alpha _{ij} : W_ i \times _{X_{/T}} (X_ j)_{/T} \longrightarrow (X_ i)_{/T} \times _{X_{/T}} W_ j \]

satisfying a suitable cocycle condition. By Lemma 87.27.3 applied to $X_ i, T_ i, U_ i, W_ i \to (X_ i)_{/T}$ there exists a morphism $X'_ i \to X_ i$ of algebraic spaces which is locally of finite type and an isomorphism over $U_ i$ and an isomorphism $\beta _ i : (X'_ i)_{/T} \cong W_ i$ over $(X_ i)_{/T}$. By fully faithfullness we find an isomorphism

\[ a_{ij} : X'_ i \times _ X X_ j \longrightarrow X_ i \times _ X X'_ j \]

over $X_ i \times _ X X_ j$ such that $\alpha _{ij} = \beta _ j|_{X_ i \times _ X X_ j} \circ (a_{ij})_{/T} \circ \beta _ i^{-1}|_{X_ i \times _ X X_ j}$. By fully faithfulness again (this time over $X_ i \times _ X X_ j \times _ X X_ k$) we see that these morphisms $a_{ij}$ satisfy the same cocycle condition as satisfied by the $\alpha _{ij}$. In other words, we obtain a descent datum (as in Descent on Spaces, Definition 73.22.3) $(X'_ i, a_{ij})$ relative to the family $\{ X_ i \to X\} $. By Bootstrap, Lemma 79.11.3, this descent datum is effective. Thus we find a morphism $f : X' \to X$ of algebraic spaces and isomorphisms $h_ i : X' \times _ X X_ i \to X'_ i$ over $X_ i$ such that $a_{ij} = h_ j|_{X_ i \times _ X X_ j} \circ h_ i^{-1}|_{X_ i \times _ X X_ j}$. The reader can check that the ensuing isomorphisms

\[ (X' \times _ X X_ i)_{/T} \xrightarrow {\beta _ i \circ (h_ i)_{/T}} W_ i \]

over $X_ i$ glue to an isomorphism $X'_{/T} \to W$ over $X_{/T}$; some details omitted. $\square$

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