Lemma 31.15.7. Let X be a locally Noetherian scheme. Let D \subset X be an integral closed subscheme. Assume that
D has codimension 1 in X, and
\mathcal{O}_{X, x} is a UFD for all x \in D.
Then D is an effective Cartier divisor.
Lemma 31.15.7. Let X be a locally Noetherian scheme. Let D \subset X be an integral closed subscheme. Assume that
D has codimension 1 in X, and
\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
Comment #9119 by Stacks project on