Lemma 59.31.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a subsheaf of the final object of the étale topos of $S$ (see Sites, Example 7.10.2). Then there exists a unique open $W \subset S$ such that $\mathcal{F} = h_ W$.
59.31 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 $\varphi : U \to S$ in $\mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$. In particular local sections always glue. If $\mathcal{F}(U) \not= \emptyset $, then $\mathcal{F}(\varphi (U)) \not= \emptyset $ because $\{ \varphi : U \to \varphi (U)\} $ is a covering. Hence we can take $W = \bigcup _{\varphi : U \to S, \mathcal{F}(U) \not= \emptyset } \varphi (U)$. $\square$
Lemma 59.31.2. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_{\acute{e}tale}$. Let $\sigma \in \mathcal{F}(U)$ be a local section. There exists an open subset $W \subset U$ such that
$W \subset U$ is the largest Zariski open subset of $U$ such that $\sigma |_ W = 0$,
for every $\varphi : V \to U$ in $S_{\acute{e}tale}$ we have
for every geometric point $\overline{u}$ of $U$ we have
where $\overline{s} = (U \to S) \circ \overline{u}$.
\[ \sigma |_ V = 0 \Leftrightarrow \varphi (V) \subset W, \]
\[ (U, \overline{u}, \sigma ) = 0\text{ in }\mathcal{F}_{\overline{s}} \Leftrightarrow \overline{u} \in W \]
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 $S_{\acute{e}tale}$. Note that $\varphi (V) \subset U$ is an open subset 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 $\mathcal{F}_{\overline{s}}$, 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 $s \in S$. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. By Remark 59.29.8 the isomorphism class of the stalk of the sheaf $\mathcal{F}$ at a geometric points lying over $s$ is well defined.
Definition 59.31.3. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_{\acute{e}tale}$.
The support of $\mathcal{F}$ is the set of points $s \in S$ such that $\mathcal{F}_{\overline{s}} \not= 0$ for any (some) geometric point $\overline{s}$ lying over $s$.
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 59.31.2).
In general the support of an abelian sheaf is not closed. For example, suppose that $S = \mathop{\mathrm{Spec}}(\mathbf{A}^1_{\mathbf{C}})$. Let $i_ t : \mathop{\mathrm{Spec}}(\mathbf{C}) \to S$ be the inclusion of the point $t \in \mathbf{C}$. We will see later that $\mathcal{F}_ t = i_{t, *}(\mathbf{Z}/2\mathbf{Z})$ is an abelian sheaf whose support is exactly $\{ t\} $, see Section 59.46. Then
\[ \bigoplus \nolimits _{n \in \mathbf{N}} \mathcal{F}_ n \]
is an abelian sheaf with support $\{ 1, 2, 3, \ldots \} \subset S$. This is true because taking stalks commutes with colimits, see Lemma 59.29.9. Thus an example of an abelian sheaf whose support is not closed. Here are some basic facts on supports of sheaves and sections.
Lemma 59.31.4. Let $S$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $S_{\acute{e}tale}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ and $\sigma \in \mathcal{F}(U)$.
The support of $\sigma $ is closed in $U$.
The support of $\sigma + \sigma '$ is contained in the union of the supports of $\sigma , \sigma ' \in \mathcal{F}(U)$.
If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of abelian sheaves on $S_{\acute{e}tale}$, 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 59.31.2 part (3). Parts (5) and (6) follow from the other parts. $\square$
Lemma 59.31.5. The support of a sheaf of rings on $S_{\acute{e}tale}$ 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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