Definition 72.7.1. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic algebraic space over $S$. 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 $Z \subset X$ with generic point $\xi \in |Z|$ we define the order of vanishing of $s$ along $Z$ as the integer
\[ \text{ord}_{Z, \mathcal{L}}(s) = \text{length}_{\mathcal{O}_{X, \xi }^ h} (\mathcal{O}_{X, \xi }^ h/a \mathcal{O}_{X, \xi }^ h) - \text{length}_{\mathcal{O}_{X, \xi }^ h} (\mathcal{O}_{X, \xi }^ h/b \mathcal{O}_{X, \xi }^ h) \]
where $a, b \in \mathcal{O}_{X, \xi }^ h$ are nonzerodivisors such that the element $s/s_\xi $ of $Q(\mathcal{O}_{X, \xi }^ h)$ constructed above is equal to $a/b$. This is well defined by the above and Algebra, Lemma 10.121.1.
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