I have a question on a proposition from Shihoko Ishii's book "introduction to Singularities":
Preliminaries: A Noetherian local ring $R$ with the maximal ideal $m$ is called Cohen-Macaulay ring if $\operatorname{depth} R = \dim R$. An algebraic variety or an analytic space $X$ or a scheme of finite type is called Cohen–Macaulay if for every $x \in X$, the local ring $O_{X,x}$ is a Cohen–Macaulay ring.
The dualizing complex is introduced and is defined in Note 3.5.9 (main source: Hartshorne's "Residues and Duality"; abbrev by Ha1):
Note 3.5.9 (dualizing complex [Ha1, V, §2]). Let $X$ be an algebraic variety, let $\mathcal{C} = (Mod_{O_X} )$ and let $D_c(\mathcal{C} )$ be the subcategory of $D(\mathcal{C})$ (= the derived category of $\mathcal{C}$: see) consisting of complexes with each cohomology coherent. The subcategory $D^+ _c (\mathcal{C})$ of $D^+ (\mathcal{C})$ is defined in the same way. An object $R^• \in D^+ _c (Mod_{O_X} )$ is called a dualizing complex if it has an injective resolution of finite length and for a functor $D = \mathbb{R}Hom( , R^• ) : D_c(\mathcal{C})^0 \to D_c(\mathcal{C})$ the canonical morphism $\mathcal{F}^• \to DD\mathcal{F}^•$ is isomorphic. A dualizing complex is not unique [Ha1, V, 3.1], but a normalized dualizing complex $D^• X$ is locally unique up to isomorphisms in $D^+ _c (\mathcal{C})$ [Ha1, V, §6].
The proposition states
Proposition 3.5.12. For an $n$-dimensional algebraic variety $X$, the following are equivalent:
(i) X is a Cohen–Macaulay variety;
(ii) $D_{X^{\bullet}} \cong D_X^{-n}[n]$ (i.e., $D_{X^{\bullet}}$ is the complex with $D_X^{-n}$ as the $−n$-part and with $0$ as the other parts).
In this case $D_X^{-n} \cong \omega_X $, holds, where $\omega_X$ is the canonical sheaf that will be defined in Sect. 5.3.
Question: Unfortunately, for this Proposition the book neither gives a proof nor a reference. The main source Hartshorne's "Residues and Duality" not contains such statement. Does anybody know where a proof of this Proposition can be found?