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Definition 31.12.1. Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The reflexive hull of $\mathcal{F}$ is the $\mathcal{O}_ X$-module

\[ \mathcal{F}^{**} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}( \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{O}_ X), \mathcal{O}_ X) \]

We say $\mathcal{F}$ is reflexive if the natural map $j : \mathcal{F} \longrightarrow \mathcal{F}^{**}$ is an isomorphism.

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