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The Stacks project

Lemma 31.15.9. Let Z \subset X be a closed subscheme of a Noetherian scheme. Assume

  1. Z has no embedded points,

  2. every irreducible component of Z has codimension 1 in X,

  3. every local ring \mathcal{O}_{X, x}, x \in Z is a UFD or X is regular.

Then Z is an effective Cartier divisor.

Proof. Let D = \sum a_ i D_ i be as in Lemma 31.15.8 where D_ i \subset Z are the irreducible components of Z. If D \to Z is not an isomorphism, then \mathcal{O}_ Z \to \mathcal{O}_ D has a nonzero kernel sitting in codimension \geq 2. This would mean that Z has embedded points, which is forbidden by assumption (1). Hence D \cong Z as desired. \square


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