The Stacks project

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.