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

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

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

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

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

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

Proof. Looking affine locally this follows from the corresponding result in algebra, namely Dualizing Complexes, Lemma 47.21.4. $\square$

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