The Stacks project

Theorem 86.27.4. 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 (86.27.0.1)

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

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

Proof. The functor is fully faithful by Lemma 86.27.2. 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 86.27.1. 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 86.27.3 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 72.21.3) $(X'_ i, a_{ij})$ relative to the family $\{ X_ i \to X\} $. By Bootstrap, Lemma 78.11.3, 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ARB. Beware of the difference between the letter 'O' and the digit '0'.