The Stacks project

Proof. Assume $Z \subset X$ as in (1). Then set $\mathcal{F} = \mathcal{O}_ Z$. The assumption that $D \cap Z$ is an effective Cartier divisor is equivalent to the assumption that $s : \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}$ is injective. Moreover $[Z]_{k + 1} = [\mathcal{F}]_{k + 1}]$ and $[D \cap Z]_ k = [\mathcal{O}_{D \cap Z}]_ k = [i^*\mathcal{F}]_ k$. See Lemma 42.10.3. Hence part (1) follows from part (2).

Write $[\mathcal{F}]_{k + 1} = \sum m_ j[W_ j]$ with $m_ j > 0$ and pairwise distinct integral closed subschemes $W_ j \subset X$ of $\delta $-dimension $k + 1$. The assumption that $s : \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}$ is injective implies that $W_ j \not\subset D$ for all $j$. By definition we see that

We claim that

as cycles. Let $Z \subset D$ be an integral closed subscheme of $\delta $-dimension $k$. Let $\xi \in Z$ be its generic point. Let $A = \mathcal{O}_{X, \xi }$. Let $M = \mathcal{F}_\xi $. Let $f \in A$ be an element generating the ideal of $D$, i.e., such that $\mathcal{O}_{D, \xi } = A/fA$. By assumption $\dim (\text{Supp}(M)) = 1$, the map $f : M \to M$ is injective, and $\text{length}_ A(M/fM) < \infty $. Moreover, $\text{length}_ A(M/fM)$ is the coefficient of $[Z]$ in $[i^*\mathcal{F}]_ k$. On the other hand, let $\mathfrak q_1, \ldots , \mathfrak q_ t$ be the minimal primes in the support of $M$. Then

is the coefficient of $[Z]$ in $\sum [D \cap W_ j]_ k$. Hence we see the equality by Lemma 42.3.2. $\square$