Lemma 31.18.4. Let $f : X \to S$ be a morphism of schemes. If $D_1, D_2 \subset X$ are relative effective Cartier divisor on $X/S$ and $D_1 \subset D_2$ as closed subschemes, then the effective Cartier divisor $D$ such that $D_2 = D_1 + D$ (Lemma 31.13.8) is a relative effective Cartier divisor on $X/S$.

Proof. This translates into the following algebra fact: Let $A \to B$ be a ring map and $h_1, h_2 \in B$. Assume the $h_ i$ are nonzerodivisors, that $B/h_ iB$ is flat over $A$, and that $(h_2) \subset (h_1)$. Then we can write $h_2 = h h_1$ where $h \in B$ is a nonzerodivisor. We get a short exact sequence

$0 \to B/hB \to B/h_2B \to B/h_1B \to 0$

where the first arrow is given by multiplication by $h_1$. Since the right two are flat modules over $A$, so is the middle one, see Algebra, Lemma 10.39.13. $\square$

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.

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 0B8V. Beware of the difference between the letter 'O' and the digit '0'.