Definition 70.7.4. 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.

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 70.7.2.

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 70.7.3.

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