The Stacks project

Example 48.22.1. Say $S = \mathop{\mathrm{Spec}}(A)$ with $(A, \mathfrak m, \kappa )$ a local Noetherian ring, and $\omega _ S^\bullet $ corresponds to a normalized dualizing complex $\omega _ A^\bullet $. Then if $f : X \to S$ is proper over $S$ and $\omega _ X^\bullet = f^!\omega _ S^\bullet $ the coherent sheaf

\[ \omega _ X = H^{-\dim (X)}(\omega _ X^\bullet ) \]

is a dualizing module and is often called the dualizing module of $X$ (with $S$ and $\omega _ S^\bullet $ being understood). We will see that this has good properties.

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