The Stacks project

Remark 54.7.7. Let $X$ be an integral Noetherian normal scheme of dimension $2$. In this case the following are equivalent

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

  2. there is a coherent $\mathcal{O}_ X$-module $\omega _ X$ such that $\omega _ X[n]$ is a dualizing complex, where $n$ can be any integer.

This follows from the fact that $X$ is Cohen-Macaulay (Properties, Lemma 28.12.7) and Duality for Schemes, Lemma 48.23.1. In this situation we will say that $\omega _ X$ is a dualizing module in accordance with Duality for Schemes, Section 48.22. In particular, when $A$ is a Noetherian normal local domain of dimension $2$, then we say $A$ has a dualizing module $\omega _ A$ if the above is true. In this case, if $X \to \mathop{\mathrm{Spec}}(A)$ is a normal modification, then $X$ has a dualizing module too, see Duality for Schemes, Example 48.22.1. In this situation we always denote $\omega _ X$ the dualizing module normalized with respect to $\omega _ A$, i.e., such that $\omega _ X[2]$ is the dualizing complex normalized relative to $\omega _ A[2]$. See Duality for Schemes, Section 48.20.

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