Definition 82.18.1. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. We define, for every integer $k$, an operation
\[ c_1(\mathcal{L}) \cap - : Z_{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X) \]
called intersection with the first Chern class of $\mathcal{L}$.
Given an integral closed subspace $i : W \to X$ with $\dim _\delta (W) = k + 1$ we define
\[ c_1(\mathcal{L}) \cap [W] = i_*(c_1({i^*\mathcal{L}}) \cap [W]) \]where the right hand side is defined in Definition 82.17.1.
For a general $(k + 1)$-cycle $\alpha = \sum n_ i [W_ i]$ we set
\[ c_1(\mathcal{L}) \cap \alpha = \sum n_ i c_1(\mathcal{L}) \cap [W_ i] \]
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