Definition 31.14.1. Let S be a scheme. Let D \subset S be an effective Cartier divisor with ideal sheaf \mathcal{I}_ D.
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}.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.
We write \mathcal{O}_ S(-D) = \mathcal{O}_ S(D)^{\otimes -1} = \mathcal{I}_ D.
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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