Existence and biduality from a projective embedding

Choose the embedding in [F1]. Resolve ${\omega }_P[N]$ by injective module sheaves and apply $i^b$ from [F3]. This constructs $D_i$ with its actual ${\mathcal{O}}_X$-action, rather than treating an ambient locally free resolution as a complex of ${\mathcal{O}}_X$-modules. On a standard chart $U=\mathrm{Spec}A\subset P$, $X\cap U=\mathrm{Spec}(A\mathrm{/}I)$, and the derived complex corresponds to $R{\mathrm{Hom}}_A(A\mathrm{/}I,{\omega }_P(U)[N])$; finite resolutions in [F2] verify this correspondence and its compatibility with localization.

The rings $A$ are regular of dimension $N$, and the restriction of ${\omega }_P$ is invertible. Hence [F3] proves finite injective dimension, finite cohomology and homothety on $X\cap U$. These affine opens cover $X$. The finite ambient resolutions [F2] bound the cohomology degrees uniformly, and their local Hom cohomology sheaves are coherent, so $D_i\in D_{\mathrm{C}\mathrm{o}\mathrm{h}}^b(X)$ and is dualizing in the sense of Dualizing complexes and the normalized dualizing sheaf on a projective CM scheme. For any bounded coherent $K$, the same local argument gives bounded coherent duals and bidual evaluation is an isomorphism on this cover, hence globally. This proves existence and biduality before any use of duality on singular schemes. AC is used by the listed regularity and resolution suppliers.