Concentration of the projective dualizing complex on a pure CM scheme

At a closed point $x\in X$, its residue field is finite algebraic over $k$ by Finite-type maps from Jacobson rings induce finite residue-field extensions at maximal ideals, so [F3] gives $\dim {\mathcal{O}}_{X,x}=d$ and $\dim {\mathcal{O}}_{P,x}=N$. The local ring of $P$ is a localization of a polynomial chart. CM gives depth $d$ for the quotient. Hence [F2], with the local trivialization of ${\omega }_P$, shows that ${\mathcal{H}}^a(D_X{)}_x=0$ unless $a=(N-d)-N=-d$. Every cohomology sheaf is coherent by [F1]. A nonzero coherent sheaf on a finite-type scheme over a field has a closed point in its support: on an affine chart its support is the vanishing locus of its annihilator, and a maximal ideal of this quotient has finite algebraic residue field; such points are closed in the finite-type scheme. Thus the asserted vanishing at closed points proves vanishing everywhere. Canonical truncation gives $D_X\cong {\omega }_X[d]$, and the construction gives the ambient sheaf Ext formula.

At a closed point set $R={\mathcal{O}}_{P,x}$, $B={\mathcal{O}}_{X,x}$ and $c=N-d$. The proof of [F2] gives a finite free resolution of $B$ of length $c$. Dualizing it over $R$ gives an exact sequence $0\to F_0^{\vee }\to \cdots \to F_c^{\vee }\to E\to 0$, where $E={\mathrm{Ext}}_R^c(B,R)$; for $c=0$ this means $E=B^{\vee }$. Thus ${\mathrm{pd}}_{R}E\le c$. The module $E$ is nonzero: $R{\mathrm{Hom}}_R(B,R)$ is nonzero, since if it vanished then the biduality $B\cong R{\mathrm{Hom}}_R(R{\mathrm{Hom}}_R(B,R),R)$ of [F1] would give $B=0$, contrary to $B\mathrm{\ne }0$; and [F2] identifies this complex with $E[-c]$. It is annihilated by $I$, so its support has dimension at most $d$. By [F4], ${\mathrm{depth}}_{R}E=N-{\mathrm{pd}}_{R}E\ge d$; depth is at most support dimension by A finite local module has depth at most its dimension, so both equal $d$. Regularity over $R$ and over $B$ is measured by the same multiplication maps; hence $E$ is CM over $B$. Its localizations are CM by [F4], and every point specializes to a closed point, so ${\omega }_X$ is CM everywhere. Finally at any point $y$, if $({\omega }_X{)}_y$ vanished then $(D_X{)}_y$ would vanish, contradicting the local homothety for nonzero ${\mathcal{O}}_{X,y}$; therefore its support is all of $X$. AC is inherited through the dimension, resolution and CM suppliers.