Lemma 66.20.1 (04K8)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.20: Supports of abelian sheaves
Lemma 66.20.1 (cite)

next lemma

numberstags

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. $\square$

Comments (0)

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).

Comments (0)

Post a comment