The Stacks project

Definition 72.7.4. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. 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 $s$ as

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

    where the sum is over prime divisors. This is well defined by Lemma 72.7.2.

  2. We define Weil divisor class associated to $\mathcal{L}$ as the image of $\text{div}_\mathcal {L}(s)$ in $\text{Cl}(X)$ where $s$ is any nonzero meromorphic section of $\mathcal{L}$ over $X$. This is well defined by Lemma 72.7.3.

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