Lemma 66.20.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a subsheaf of the final object of the étale topos of $X$ (see Sites, Example 7.10.2). Then there exists a unique open $W \subset X$ such that $\mathcal{F} = h_ W$.
Proof. The condition means that $\mathcal{F}(U)$ is a singleton or empty for all $\varphi : U \to X$ in $\mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$. In particular local sections always glue. If $\mathcal{F}(U) \not= \emptyset $, then $\mathcal{F}(\varphi (U)) \not= \emptyset $ because $\varphi (U) \subset X$ is an open subspace (Lemma 66.16.7) and $\{ \varphi : U \to \varphi (U)\} $ is a covering in $X_{spaces, {\acute{e}tale}}$. Take $W = \bigcup _{\varphi : U \to S, \mathcal{F}(U) \not= \emptyset } \varphi (U)$ to conclude. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)