Definition 81.17.1. In Situation 81.2.1 let $X/B$ be good. Assume $X$ is integral and $n = \dim _\delta (X)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module.

1. For any nonzero meromorphic section $s$ of $\mathcal{L}$ we define the Weil divisor associated to $s$ is the $(n - 1)$-cycle

$\text{div}_\mathcal {L}(s) = \sum \text{ord}_{Z, \mathcal{L}}(s) [Z]$

defined in Spaces over Fields, Definition 71.7.4. This makes sense because Weil divisors have $\delta$-dimension $n - 1$ by Lemma 81.12.1.

2. We define Weil divisor associated to $\mathcal{L}$ as

$c_1(\mathcal{L}) \cap [X] = \text{class of }\text{div}_\mathcal {L}(s) \in \mathop{\mathrm{CH}}\nolimits _{n - 1}(X)$

where $s$ is any nonzero meromorphic section of $\mathcal{L}$ over $X$. This is well defined by Spaces over Fields, Lemma 71.7.3.

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