Lemma 17.23.2. Let $(X, \mathcal{O}_ X)$ be a ringed space and let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. If $\mathcal{F}$ is of finite type, then $(\text{Ann}_{\mathcal{O}_ X}(\mathcal{F}))_ x = \text{Ann}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x)$.

**Proof.**
By Lemma 17.22.4 the map

is injective. Thus any section $f$ of $\mathcal{O}_ X$ over an open neighbourhood $U$ of $x$ which acts as zero on $\mathcal{F}_ x$ will act as zero on $\mathcal{F}|_ V$ for some $U \supset V \ni x$ open. Hence the inclusion (17.23.1.1) is an equality. $\square$

## 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Your email address will not be published.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).

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.

## Comments (0)

There are also: