The Stacks project

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 $\sigma \in \mathcal{F}(U)$.

  1. The support of $\sigma $ is closed in $U$.

  2. The support of $\sigma + \sigma '$ is contained in the union of the supports of $\sigma , \sigma ' \in \mathcal{F}(U)$.

  3. 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 $\sigma \in \mathcal{F}(U)$.

  4. The support of $\mathcal{F}$ is the union of the images of the supports of all local sections of $\mathcal{F}$.

  5. If $\mathcal{F} \to \mathcal{G}$ is surjective then the support of $\mathcal{G}$ is a subset of the support of $\mathcal{F}$.

  6. 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. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 59.31: Supports of abelian sheaves

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Your email address will not be published.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 04FT. Beware of the difference between the letter 'O' and the digit '0'.