The Stacks project

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.

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

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 Divisors, Lemma 31.27.3.