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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BXH. Beware of the difference between the letter 'O' and the digit '0'.