Lemma 31.13.10. Let $X$ be a scheme. Let $D, D' \subset X$ be effective Cartier divisors such that the scheme theoretic intersection $D \cap D'$ is an effective Cartier divisor on $D'$. Then $D + D'$ is the scheme theoretic union of $D$ and $D'$.

Proof. See Morphisms, Definition 29.4.4 for the definition of scheme theoretic intersection and union. To prove the lemma working locally (using Lemma 31.13.2) we obtain the following algebra problem: Given a ring $A$ and nonzerodivisors $f_1, f_2 \in A$ such that $f_1$ maps to a nonzerodivisor in $A/f_2A$, show that $f_1A \cap f_2A = f_1f_2A$. We omit the straightforward argument. $\square$

## Comments (0)

There are also:

• 1 comment(s) on Section 31.13: Effective Cartier divisors

## 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Your email address will not be published.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 0C4R. Beware of the difference between the letter 'O' and the digit '0'.