The Stacks project

Lemma 70.6.5. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$. Let $f \in R(X)^*$. If the prime divisor $Z \subset X$ meets the schematic locus of $X$, then the order of vanishing $\text{ord}_ Z(f)$ of Definition 70.6.4 agrees with the order of vanishing of Divisors, Definition 31.26.3.

Proof. After shrinking $X$ we may assume $X$ is an integral Noetherian scheme. If $\xi \in Z$ denotes the generic point, then we find that $\mathcal{O}_{X, \xi }^ h$ is the henselization of $\mathcal{O}_{X, \xi }$ (Decent Spaces, Lemma 66.11.8). To prove the lemma it suffices and is necessary to show that

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

This follows immediately from Algebra, Lemma 10.51.13 (and the fact that $\mathcal{O}_{X, \xi } \to \mathcal{O}_{X, \xi }^ h$ is a flat local ring homomorphism of local Noetherian rings). $\square$


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