Dualizing complexes and coherent biduality for regular-ring quotients

Every finite $A$-module has a bounded resolution by finite projective modules: finite generation of each kernel follows from Noetherianity, and the $n$th syzygy is projective by the global-dimension bound. Bounded complexes with finite cohomology are perfect as well, by truncation triangles and cones of such resolutions. For a bounded finite-projective complex $P$, the ordinary termwise map $P\to {\mathrm{Hom}}_A({\mathrm{Hom}}_A(P,L[s]),L[s])$ is an isomorphism of complexes with the signed evaluation convention: the two shifts and the two factors of $L$ cancel. Thus $R{\mathrm{Hom}}_A(-,L[s])$ has coherent biduality.

By [F1], $L$ has a bounded injective resolution $I^{\bullet }$. Applying ${\mathrm{Hom}}_A(B,-)$ gives a bounded complex of injective $B$-modules by [F2], representing $D_B$. Its cohomology is finite over $A$, since $B$ has a finite projective resolution, and the $B$-action makes it finite over $B$. For bounded finite $B$-cohomology, the adjunction identifies the underlying $A$-complex of ${\mathbb{D}}_B(M)$ with $R{\mathrm{Hom}}_A(M,L[s])$, so it too has bounded finite cohomology.

Apply that adjunction twice. The underlying $A$-complex of ${\mathbb{D}}_B{\mathbb{D}}_B(M)$ becomes the ambient double dual in step 1.1. Under these adjunctions the canonical $B$-evaluation becomes the ambient evaluation: both send an element $m$ to the functional $\phi \mapsto \phi (m)$, followed by evaluation at $1\in B$; resolving gives the same signed identity for complexes. It is therefore a quasi-isomorphism after forgetting the $B$-action, hence a quasi-isomorphism over $B$. At $M=B$, the identification ${\mathbb{D}}_B(B)=D_B$ shows that $B\to R{\mathrm{Hom}}_B(D_B,D_B)$ is precisely the homothety isomorphism. Together with step 2.1 this proves all conditions in [F3] and biduality. Finite projective resolutions show that derived Hom and evaluation commute with localization, so the construction agrees on intersections of affine charts. AC is used in the ambient global-dimension and resolution suppliers.