The Stacks project

Lemma 72.18.6. In the situation of Definition 72.18.3 let $Z_ i \subset Z$ be the irreducible components of dimension $d$. Let $m_ i = \text{length}_{\mathcal{O}_{X, \overline{x}_ i}} (\mathcal{O}_{Z, \overline{x}_ i})$ where $\overline{x}_ i$ is a geometric generic point of $Z_ i$. Then

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = \sum m_ i(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z_ i) \]

Proof. Immediate from Lemma 72.18.2 and the definitions. $\square$

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