Definition 31.13.1. Let $S$ be a scheme.

A

*locally principal closed subscheme*of $S$ is a closed subscheme whose sheaf of ideals is locally generated by a single element.An

*effective Cartier divisor*on $S$ is a closed subscheme $D \subset S$ whose ideal sheaf $\mathcal{I}_ D \subset \mathcal{O}_ S$ is an invertible $\mathcal{O}_ S$-module.

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