Why does a Weil Divisor $ D $ on a scheme $ X $ induce a Weil Divisor $ D_{x} $ on the local scheme $ \text{Spec}(\mathcal{O}_{X,x})$?

Asked May 08 '19 at 23:10

Active May 27 '19 at 23:51

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Let $ X $ be an integral, separated noetherian scheme, and $ x \in X. $

In Proposition 6.11 of Hartshorne's Algebraic Geometry(Page 141), the author states that if $ D $ is a Weil divisor on $ X, $ then $ D $ induces a Weil divisor $ D_{x} $ on $ \text{Spec}(\mathcal{O}_{X,x}). $

I am not seeing why this follows.

edited May 27 '19 at 23:51

asked May 08 '19 at 23:10

Confused Student

@reuns $ \mathcal{O}{x} $ is a UFD, and so the locally induced divisor $ D{x} $ is principal, yes. – Confused Student May 08 '19 at 23:35
For $D \ge 0$ is it represented locally by a function/ideal $(f)\subset O_X(U)$ ? If so $ (f)\subset O_{X,x}$ represents a divisor $D_x$ – reuns May 08 '19 at 23:37

Why does a Weil Divisor $ D $ on a scheme $ X $ induce a Weil Divisor $ D_{x} $ on the local scheme $ \text{Spec}(\mathcal{O}_{X,x})$?

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