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The Stacks project

Lemma 31.15.7. Let X be a locally Noetherian scheme. Let D \subset X be an integral closed subscheme. Assume that

  1. D has codimension 1 in X, and

  2. \mathcal{O}_{X, x} is a UFD for all x \in D.

Then D is an effective Cartier divisor.

Proof. Let x \in D and set A = \mathcal{O}_{X, x}. Let \mathfrak p \subset A correspond to the generic point of D. Then A_\mathfrak p has dimension 1 by assumption (1). Thus \mathfrak p is a prime ideal of height 1. Since A is a UFD this implies that \mathfrak p = (f) for some f \in A. Of course f is a nonzerodivisor and we conclude by Lemma 31.15.2. \square


Comments (2)

Comment #8518 by Zhenhua Wu on

The argument seems to hold in the case of locally Noetherian scheme.


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