The Stacks project

Definition 70.7.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I}_ D$.

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

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

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

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

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