The Stacks project

Proof. Let $Z \subset Y$ be an integral closed subscheme of $\delta $-dimension $k$. Let $\xi \in Z$ be its generic point. We want to compare the coefficient $n$ of $[Z]$ in the expression $\sum n_ i[Z(s_ i)]_ k$ with the coefficient $m$ of $[Z]$ in the expression $[Z(s)]_ k$. Choose a generator $s_\xi \in \mathcal{L}_\xi $. Write $A = \mathcal{O}_{Y, \xi }$, $L = \mathcal{L}_\xi $. Then $L = As_\xi $. Write $s = f s_\xi $ for some (unique) $f \in A$. Hypothesis (3) means that $f : A \to A$ is injective. Since $\dim _\delta (Y) \leq k + 1$ and $\dim _\delta (Z) = k$ we have $\dim (A) = 0$ or $1$. We have

which is finite in either case.

If $\dim (A) = 0$, then $f : A \to A$ being injective implies that $f \in A^*$. Hence in this case $m$ is zero. Moreover, the condition $\dim (A) = 0$ means that $\xi $ does not lie on any irreducible component of $\delta $-dimension $k + 1$, i.e., $n = 0$ as well.

Now, let $\dim (A) = 1$. Since $A$ is a Noetherian local ring it has finitely many minimal primes $\mathfrak q_1, \ldots , \mathfrak q_ t$. These correspond 1-1 with the $Y_ i$ passing through $\xi '$. Moreover $n_ i = \text{length}_{A_{\mathfrak q_ i}}(A_{\mathfrak q_ i})$. Also, the multiplicity of $[Z]$ in $[Z(s_ i)]_ k$ is $\text{length}_ A(A/(f, \mathfrak q_ i))$. Hence the equation to prove in this case is

which follows from Lemma 42.3.2. $\square$