The Stacks project

Lemma 43.16.4. In Lemma 43.16.3 assume that $c = 1$, i.e., $V$ is an effective Cartier divisor. Then

\[ e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/f_1\mathcal{O}_{W, \xi }). \]

Proof. In this case the image of $f_1$ in $\mathcal{O}_{W, \xi }$ is nonzero by properness of intersection, hence a nonzerodivisor divisor. Moreover, $\mathcal{O}_{W, \xi }$ is a Noetherian local domain of dimension $1$. Thus

\[ \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/f_1^ t\mathcal{O}_{W, \xi }) = t \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/f_1\mathcal{O}_{W, \xi }) \]

for all $t \geq 1$, see Algebra, Lemma 10.121.1. This proves the lemma. $\square$

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