Definition 31.27.1. Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{K}_ X(\mathcal{L}))$ be a regular meromorphic section of $\mathcal{L}$. For every prime divisor $Z \subset X$ we define the order of vanishing of $s$ along $Z$ as the integer

$\text{ord}_{Z, \mathcal{L}}(s) = \text{ord}_{\mathcal{O}_{X, \xi }}(s/s_\xi )$

where the right hand side is the notion of Algebra, Definition 10.121.2, $\xi \in Z$ is the generic point, and $s_\xi \in \mathcal{L}_\xi$ is a generator.

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