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. $\square$

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