Dualizing complexes and the normalized dualizing sheaf on a projective CM scheme

A dualizing complex on a Noetherian scheme $X$ is an object $D_X\in D_{\mathrm{C}\mathrm{o}\mathrm{h}}^b(X)$ such that, locally on affine open neighborhoods $U=\mathrm{Spec}B$, its corresponding complex has finite injective dimension over $B$ and the homothety map $B\to R{\mathrm{Hom}}_B(D_X{\mathrm{\mid }}_U,D_X{\mathrm{\mid }}_U)$ is an isomorphism. Here $D_{\mathrm{C}\mathrm{o}\mathrm{h}}^b$ means bounded complexes with coherent cohomology, $\mathcal{R}\text{\hspace{0.17em}⁣}Hom$ denotes derived internal Hom, and ${\mathrm{Ext}}_X^r(M,N)={\mathrm{Hom}}_{D(X)}(M,N[r])$ is global Ext, rather than a sheaf Ext.

For a projective scheme over a field $k$, a normalization over $k$ consists of a dualizing complex $D_X$ and a trace $t_X:R\mathrm{\Gamma }(X,D_X)\to k$ for which the evaluation map induces natural isomorphisms ${\mathrm{Hom}}_{D(X)}(K,D_X[r])\cong {\mathrm{Hom}}_k(H^{-r}(X,K),k)$ for all $K\in D_{\mathrm{C}\mathrm{o}\mathrm{h}}^b(X)$ and $r\in \mathbb{Z}$. For a pure $d$-dimensional Cohen–Macaulay scheme (every local ring has depth equal to dimension), the normalized dualizing sheaf is ${\omega }_X={\mathcal{H}}^{-d}(D_X)$. The local constructions on this page prove existence and $D_X\cong {\omega }_X[d]$; concentration is a conclusion, rather than an additional definition. The shift convention is $H^a(K[b])=H^{a+b}(K)$. Pure dimension means every irreducible component has dimension $d$. A dualizing sheaf on a singular CM scheme need not be invertible.