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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 85.9: Colimits of algebraic spaces along thickenings

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Your email address will not be published.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 0AIU. Beware of the difference between the letter 'O' and the digit '0'.