The Stacks project

Definition 42.25.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. 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 $\mathcal{L}$.

  1. Given an integral closed subscheme $i : W \to X$ 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 42.24.1.

  2. 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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