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

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

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