Lemma 81.22.7. In Situation 81.2.1 let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 81.22.1. 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 $i^*([Z]_{k + 1}) = [D \cap Z]_ k$.

Proof. The assumption means that $s|_ Z$ is a regular section of $\mathcal{L}|_ Z$. Thus $D \cap Z = Z(s)$ and we get

$[D \cap Z]_ k = \sum n_ i [Z(s_ i)]_ k$

as cycles where $s_ i = s|_{Z_ i}$, the $Z_ i$ are the irreducible components of $\delta$-dimension $k + 1$, and $[Z]_{k + 1} = \sum n_ i[Z_ i]$. See Lemma 81.18.3. We have $D \cap Z_ i = Z(s_ i)$. Comparing with the definition of the gysin map we conclude. $\square$

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