Remark 31.12.3. If $X$ is a scheme of finite type over a field, then sometimes a different notion of reflexive modules is used (see for example [bottom of page 5 and Definition 1.1.9, HL]). This other notion uses $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ into a dualizing complex $\omega _ X^\bullet $ instead of into $\mathcal{O}_ X$ and should probably have a different name because it can be different when $X$ is not Gorenstein. For example, if $X = \mathop{\mathrm{Spec}}(k[t^3, t^4, t^5])$, then a computation shows the dualizing sheaf $\omega _ X$ is not reflexive in our sense, but it is reflexive in the other sense as $\omega _ X \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\omega _ X, \omega _ X), \omega _ X)$ is an isomorphism.

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