Example 48.22.2. Say $X$ is an equidimensional scheme of finite type over a field $k$. Then it is customary to take $\omega _ X^\bullet $ the dualizing complex normalized relative to $k[0]$ and to refer to

as the dualizing module of $X$. If $X$ is separated over $k$, then $\omega _ X^\bullet = f^!\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}$ where $f : X \to \mathop{\mathrm{Spec}}(k)$ is the structure morphism by Lemma 48.20.9. If $X$ is proper over $k$, then this is a special case of Example 48.22.1.

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