Lemma 43.16.4 (0B05)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 43: Intersection Theory
Section 43.16: Computing intersection multiplicities
Lemma 43.16.4 (cite)

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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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