The Stacks project

Definition 70.6.4. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \in R(X)^*$. For every prime divisor $Z \subset X$ we define the order of vanishing of $f$ along $Z$ as the integer

\[ \text{ord}_ Z(f) = \text{length}_{\mathcal{O}_{X, \xi }^ h} (\mathcal{O}_{X, \xi }^ h/a \mathcal{O}_{X, \xi }^ h) - \text{length}_{\mathcal{O}_{X, \xi }^ h} (\mathcal{O}_{X, \xi }^ h/b \mathcal{O}_{X, \xi }^ h) \]

where $a, b \in \mathcal{O}_{X, \xi }^ h$ are nonzerodivisors such that the image of $f$ in $Q(\mathcal{O}_{X, \xi }^ h)$ (Lemma 70.6.3) is equal to $a/b$. This is well defined by Algebra, Lemma 10.120.1.

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