Definition 81.22.1. In Situation 81.2.1 let $X/B$ be good. Let $(\mathcal{L}, s)$ be a pair consisting of an invertible sheaf and a global section $s \in \Gamma (X, \mathcal{L})$. Let $D = Z(s)$ be the vanishing locus of $s$, and denote $i : D \to X$ the closed immersion. We define, for every integer $k$, a (refined) Gysin homomorphism

$i^* : Z_{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(D).$

by the following rules:

1. Given a integral closed subspace $W \subset X$ with $\dim _\delta (W) = k + 1$ we define

1. if $W \not\subset D$, then $i^*[W] = [D \cap W]_ k$ as a $k$-cycle on $D$, and

2. if $W \subset D$, then $i^*[W] = i'_*(c_1(\mathcal{L}|_ W) \cap [W])$, where $i' : W \to D$ is the induced closed immersion.

2. For a general $(k + 1)$-cycle $\alpha = \sum n_ j[W_ j]$ we set

$i^*\alpha = \sum n_ j i^*[W_ j]$
3. If $D$ is an effective Cartier divisor, then we denote $D \cdot \alpha = i_*i^*\alpha$ the pushforward of the class to a class on $X$.

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