The Stacks project

Proof. In the statement of the lemma and in the proof we use the equivalence of Morphisms, Lemma 29.4.1 to think of quasi-coherent modules on closed subschemes of $X$ as quasi-coherent modules on $X$. Let $\mathcal{I}$ be the ideal sheaf of $D$ in $D'$ and let $\mathcal{I}_ D = \mathcal{O}_ X(-D)$ be the ideal sheaf of $D$ in $X$. Then there is a canonical surjection $\mathcal{I}_ D \to \mathcal{I}$. Thus it suffices to show the kernel of this map is equal to $\mathcal{I}_ C\mathcal{I}_ D$ (with obvious notation). This can be checked locally, hence we may assume $X = \mathop{\mathrm{Spec}}(A)$, $D = \mathop{\mathrm{Spec}}(A/fA)$ and $C = \mathop{\mathrm{Spec}}(A/gA)$ where $f, g \in A$ are nonzerodivisors (Lemma 31.13.2). Then $\mathcal{I}_ D$ corresponds to $fA$, the nonzerodivisor $fg \in A$ generates the ideal of $D'$, and $\mathcal{I}$ corresponds to $I = fA/fgA$. Thus the result is clear. $\square$