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 $\mathcal{F} = h_ W$.
66.20 Supports of abelian sheaves
First we talk about supports of local sections.
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. $\square$
Lemma 66.20.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an abelian sheaf on $X_{spaces, {\acute{e}tale}}$. Let $\sigma \in \mathcal{F}(U)$ be a local section. There exists an open subspace $W \subset U$ such that
$W \subset U$ is the largest open subspace of $U$ such that $\sigma |_ W = 0$,
for every $\varphi : V \to U$ in $X_{spaces, {\acute{e}tale}}$ we have
for every geometric point $\overline{u}$ of $U$ we have
where $\overline{x} = (U \to X) \circ \overline{u}$.
\[ \sigma |_ V = 0 \Leftrightarrow \varphi (V) \subset W, \]
\[ (U, \overline{u}, \sigma ) = 0\text{ in }\mathcal{F}_{\overline{x}} \Leftrightarrow \overline{u} \in W \]
Proof. Since $\mathcal{F}$ is a sheaf in the étale topology the restriction of $\mathcal{F}$ to $U_{Zar}$ is a sheaf on $U$ in the Zariski topology. Hence there exists a Zariski open $W$ having property (1), see Modules, Lemma 17.5.2. Let $\varphi : V \to U$ be an arrow of $X_{spaces, {\acute{e}tale}}$. Note that $\varphi (V) \subset U$ is an open subspace (Lemma 66.16.7) and that $\{ V \to \varphi (V)\} $ is an étale covering. Hence if $\sigma |_ V = 0$, then by the sheaf condition for $\mathcal{F}$ we see that $\sigma |_{\varphi (V)} = 0$. This proves (2). To prove (3) we have to show that if $(U, \overline{u}, \sigma )$ defines the zero element of $\mathcal{F}_{\overline{x}}$, then $\overline{u} \in W$. This is true because the assumption means there exists a morphism of étale neighbourhoods $(V, \overline{v}) \to (U, \overline{u})$ such that $\sigma |_ V = 0$. Hence by (2) we see that $V \to U$ maps into $W$, and hence $\overline{u} \in W$. $\square$
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. By Remark 66.19.11 the isomorphism class of the stalk of the sheaf $\mathcal{F}$ at a geometric points lying over $x$ is well defined.
Definition 66.20.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$.
The support of $\mathcal{F}$ is the set of points $x \in |X|$ such that $\mathcal{F}_{\overline{x}} \not= 0$ for any (some) geometric point $\overline{x}$ lying over $x$.
Let $\sigma \in \mathcal{F}(U)$ be a section. The support of $\sigma $ is the closed subset $U \setminus W$, where $W \subset U$ is the largest open subset of $U$ on which $\sigma $ restricts to zero (see Lemma 66.20.2).
Lemma 66.20.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ and $\sigma \in \mathcal{F}(U)$.
The support of $\sigma $ is closed in $|X|$.
The support of $\sigma + \sigma '$ is contained in the union of the supports of $\sigma , \sigma ' \in \mathcal{F}(X)$.
If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of abelian sheaves on $X_{\acute{e}tale}$, then the support of $\varphi (\sigma )$ is contained in the support of $\sigma \in \mathcal{F}(U)$.
The support of $\mathcal{F}$ is the union of the images of the supports of all local sections of $\mathcal{F}$.
If $\mathcal{F} \to \mathcal{G}$ is surjective then the support of $\mathcal{G}$ is a subset of the support of $\mathcal{F}$.
If $\mathcal{F} \to \mathcal{G}$ is injective then the support of $\mathcal{F}$ is a subset of the support of $\mathcal{G}$.
Proof. Part (1) holds by definition. Parts (2) and (3) hold because they holds for the restriction of $\mathcal{F}$ and $\mathcal{G}$ to $U_{Zar}$, see Modules, Lemma 17.5.2. Part (4) is a direct consequence of Lemma 66.20.2 part (3). Parts (5) and (6) follow from the other parts. $\square$
Lemma 66.20.5. The support of a sheaf of rings on the small étale site of an algebraic space is closed.
Proof. This is true because (according to our conventions) a ring is $0$ if and only if $1 = 0$, and hence the support of a sheaf of rings is the support of the unit section. $\square$
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