The Stacks project

Lemma 31.14.3. Let $X$ be a scheme. Let $D, C \subset X$ be effective Cartier divisors with $C \subset D$ and let $D' = D + C$. Then there is a short exact sequence

\[ 0 \to \mathcal{O}_ X(-D)|_ C \to \mathcal{O}_{D'} \to \mathcal{O}_ D \to 0 \]

of $\mathcal{O}_ X$-modules.

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 $D'$. Then there is a short exact sequence

\[ 0 \to \mathcal{I} \to \mathcal{O}_{D'} \to \mathcal{O}_ D \to 0 \]

because $D \to D'$ is a closed immersion. There is a canonical surjection $\mathcal{I} \to \mathcal{I}/\mathcal{I}^2 = \mathcal{C}_{D/D'}$. We have $\mathcal{C}_{D/X} = \mathcal{O}_ X(-D)|_ D$ by Lemma 31.14.2 and there is a canonical surjective map

\[ \mathcal{C}_{D/X} \longrightarrow \mathcal{C}_{D/D'} \]

see Morphisms, Lemmas 29.31.3 and 29.31.4. Thus it suffices to show: (a) $\mathcal{I}^2 = 0$ and (b) $\mathcal{I}$ is an invertible $\mathcal{O}_ C$-module. Both (a) and (b) 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). Since $C \subset D$ we see that $f \in gA$. Then $I = fA/fgA$ has square zero and is invertible as an $A/gA$-module as desired. $\square$

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