Lemma 43.16.5. In Lemma 43.16.3 assume that the local ring $\mathcal{O}_{W, \xi }$ is Cohen-Macaulay. Then we have

$e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/ f_1\mathcal{O}_{W, \xi } + \ldots + f_ c\mathcal{O}_{W, \xi }).$

Proof. This follows immediately from Lemma 43.16.1. Alternatively, we can deduce it from Lemma 43.16.3. Namely, by Algebra, Lemma 10.104.2 we see that $f_1, \ldots , f_ c$ is a regular sequence in $\mathcal{O}_{W, \xi }$. Then Algebra, Lemma 10.69.2 shows that $f_1, \ldots , f_ c$ is a quasi-regular sequence. This easily implies the length of $\mathcal{O}_{W, \xi }/(f_1, \ldots , f_ c)^ t$ is

${c + t \choose c} \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/ f_1\mathcal{O}_{W, \xi } + \ldots + f_ c\mathcal{O}_{W, \xi }).$

Looking at the leading coefficient we conclude. $\square$

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