The Stacks project

Lemma 82.6.2. In Situation 82.2.1 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 82.4.5. Moreover, the length of $\mathcal{F}$ at $\xi $ is zero if $\xi \not\in \text{Supp}(\mathcal{F})$. Combining this with Definition 82.6.1 we conclude. $\square$

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