Lemma 81.6.2. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module with $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k$. Let $Z$ be an integral closed subspace of $X$ with $\dim _\delta (Z) = k$. Let $\xi \in |Z|$ be the generic point. Then the coefficient of $Z$ in $[\mathcal{F}]_ k$ is the length of $\mathcal{F}$ at $\xi $.

**Proof.**
Observe that $|Z|$ is an irreducible component of $\text{Supp}(\mathcal{F})$ if and only if $\xi \in \text{Supp}(\mathcal{F})$, see Lemma 81.4.5. Moreover, the length of $\mathcal{F}$ at $\xi $ is zero if $\xi \not\in \text{Supp}(\mathcal{F})$. Combining this with Definition 81.6.1 we conclude.
$\square$

## 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Your email address will not be published.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)