Definition 42.24.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim _\delta (X)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module.

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 Divisors, Definition 31.27.4. This makes sense because Weil divisors have $\delta $-dimension $n - 1$ by Lemma 42.16.1.

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 Divisors, Lemma 31.27.3.

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