Definition 31.14.1. Let $S$ be a scheme. Let $D \subset S$ be an effective Cartier divisor with ideal sheaf $\mathcal{I}_ D$.

1. The invertible sheaf $\mathcal{O}_ S(D)$ associated to $D$ is defined by

$\mathcal{O}_ S(D) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{I}_ D, \mathcal{O}_ S) = \mathcal{I}_ D^{\otimes -1}.$
2. The canonical section, usually denoted $1$ or $1_ D$, is the global section of $\mathcal{O}_ S(D)$ corresponding to the inclusion mapping $\mathcal{I}_ D \to \mathcal{O}_ S$.

3. We write $\mathcal{O}_ S(-D) = \mathcal{O}_ S(D)^{\otimes -1} = \mathcal{I}_ D$.

4. Given a second effective Cartier divisor $D' \subset S$ we define $\mathcal{O}_ S(D - D') = \mathcal{O}_ S(D) \otimes _{\mathcal{O}_ S} \mathcal{O}_ S(-D')$.

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