Serre duality for coherent sheaves on a projective Cohen–Macaulay scheme

Apply [F1] to $F$ with $r=-i$. Since $D_X={\omega }_X[d]$ by [F2], its left side becomes ${\mathrm{Hom}}_{D(X)}(F,{\omega }_X[d-i])={\mathrm{Ext}}_X^{d-i}(F,{\omega }_X)$; its right side is ${\mathrm{Hom}}_k(H^i(X,F),k)$. For $\eta \in H^i(X,F)$, [F3] represents it by $\beta :{\mathcal{O}}_X\to F[i]$. A class $\alpha :F\to {\omega }_X[d-i]$ pairs with it by $t_X(\alpha [i]\circ \beta )$. This is the Yoneda composition of [F3] and is exactly the evaluation comparison proved in [F1]. Thus it is bilinear, natural in $F$, and compatible with shifts and connecting maps.

Apply [F4] to the closed subscheme $X\hookrightarrow {\mathbb{P}}_k^N$ supplied by a projective embedding: its finite-dimensionality clause gives that each $H^i(X,F)$ is a finite-dimensional $k$-vector space. The isomorphism in step 1.1 proves that the adjoint map from Ext to the dual of cohomology is an isomorphism. For a finite-dimensional space, evaluation into its double dual is an isomorphism, so the other adjoint map is an isomorphism as well. Outside $0\le i\le d$, cohomology vanishes by [F4] and the negative-degree convention; step 1.1 then proves the corresponding Ext vanishing. For bounded coherent $K$, retain the actual global derived comparison used in [F1]: closed-immersion adjunction gives $R{\mathrm{Hom}}_X(K,D_X)\cong R{\mathrm{Hom}}_P(i_{\ast }K,{\omega }_P[N])$, and the projective-space evaluation/trace map identifies the latter with $R{\mathrm{Hom}}_k(R\mathrm{\Gamma }(P,i_{\ast }K),k)=R{\mathrm{Hom}}_k(R\mathrm{\Gamma }(X,K),k)$. This comparison is an isomorphism in $D(k)$ induced by the same counit and trace, so its naturality and signs are those of [F1]. The biduality $K\cong \mathcal{R}\text{\hspace{0.17em}⁣}Ho{m}_X(\mathcal{R}\text{\hspace{0.17em}⁣}Ho{m}_X(K,D_X),D_X)$ is the coherent biduality of the embedding construction [F1], valid after pushing into the ambient projective space, without imposing perfectness on $K$ over the singular scheme. AC enters through the cited resolution, derived-composition and cohomology suppliers.