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Definition 31.26.3. Let $X$ be a locally Noetherian integral scheme. 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

\[ \text{ord}_ Z(f) = \text{ord}_{\mathcal{O}_{X, \xi }}(f) \]

where the right hand side is the notion of Algebra, Definition 10.121.2 and $\xi $ is the generic point of $Z$.

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