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The Stacks project

Definition 10.121.2. Suppose that K is a field, and R \subset K is a local1 Noetherian subring of dimension 1 with fraction field K. In this case we define the order of vanishing along R

\text{ord}_ R : K^* \longrightarrow \mathbf{Z}

by the rule

\text{ord}_ R(x) = \text{length}_ R(R/(x))

if x \in R and we set \text{ord}_ R(x/y) = \text{ord}_ R(x) - \text{ord}_ R(y) for x, y \in R both nonzero.

[1] We could also define this when R is only semi-local but this is probably never really what you want!

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View Definition 10.121.2 as pdf