Lemma 48.24.4. Let $X$ be a locally Noetherian scheme.

If $X$ has a dualizing complex $\omega _ X^\bullet $, then

$X$ is Gorenstein $\Leftrightarrow $ $\omega _ X^\bullet $ is an invertible object of $D(\mathcal{O}_ X)$,

$\mathcal{O}_{X, x}$ is Gorenstein $\Leftrightarrow $ $\omega _{X, x}^\bullet $ is an invertible object of $D(\mathcal{O}_{X, x})$,

$U = \{ x \in X \mid \mathcal{O}_{X, x}\text{ is Gorenstein}\} $ is an open Gorenstein subscheme.

If $X$ is Gorenstein, then $X$ has a dualizing complex if and only if $\mathcal{O}_ X[0]$ is a dualizing complex.

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