The Stacks project

Remark 31.14.11. Let $X$ be a scheme, $\mathcal{L}$ an invertible $\mathcal{O}_ X$-module, and $s$ a regular section of $\mathcal{L}$. Then the zero scheme $D = Z(s)$ is an effective Cartier divisor on $X$ and there are short exact sequences

\[ 0 \to \mathcal{O}_ X \to \mathcal{L} \to i_*(\mathcal{L}|_ D) \to 0 \quad \text{and}\quad 0 \to \mathcal{L}^{\otimes -1} \to \mathcal{O}_ X \to i_*\mathcal{O}_ D \to 0. \]

Given an effective Cartier divisor $D \subset X$ using Lemmas 31.14.10 and 31.14.2 we get

\[ 0 \to \mathcal{O}_ X \to \mathcal{O}_ X(D) \to i_*(\mathcal{N}_{D/X}) \to 0 \quad \text{and}\quad 0 \to \mathcal{O}_ X(-D) \to \mathcal{O}_ X \to i_*(\mathcal{O}_ D) \to 0 \]

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