Lemma 42.29.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1.

1. Let $Z \subset X$ be a closed subscheme such that $\dim _\delta (Z) \leq k + 1$ and such that $D \cap Z$ is an effective Cartier divisor on $Z$. Then $i^*[Z]_{k + 1} = [D \cap Z]_ k$.

2. Let $\mathcal{F}$ be a coherent sheaf on $X$ such that $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k + 1$ and $s : \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}$ is injective. Then

$i^*[\mathcal{F}]_{k + 1} = [i^*\mathcal{F}]_ k$

in $\mathop{\mathrm{CH}}\nolimits _ k(D)$.

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

$i^*[\mathcal{F}]_{k + 1} = \sum m_ j [D \cap W_ j]_ k.$

We claim that

$\sum [D \cap W_ j]_ k = [i^*\mathcal{F}]_ k$

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

$\sum \text{length}_{A_{\mathfrak q_ i}}(M_{\mathfrak q_ i}) \text{ord}_{A/\mathfrak q_ i}(f)$

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

Comment #6637 by WhatJiaranEatsTonight on

I think the formula for $i^*[F]_{k+1}$ missed coefficients $m_j$ before $[D\cap W_j]$.

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