The Stacks project

Lemma 31.18.3. Let $f : X \to S$ be a morphism of schemes. If $D_1, D_2 \subset X$ are relative effective Cartier divisor on $X/S$ then so is $D_1 + D_2$ (Definition 31.13.6).

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 and that $B/h_ iB$ is flat over $A$. Then $h_1h_2$ is a nonzerodivisor and $B/h_1h_2B$ is flat over $A$. The reason is that we have a short exact sequence

\[ 0 \to B/h_1B \to B/h_1h_2B \to B/h_2B \to 0 \]

where the first arrow is given by multiplication by $h_2$. Since the outer two are flat modules over $A$, so is the middle one, see Algebra, Lemma 10.39.13. $\square$

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