External affine derived and perfect comparisons

PDF p. 16: Perfect complexes with support

Assume the Axiom of Choice. Let $A$ be a commutative unital ring and $X=\mathrm{Spec}A$. Degreewise associated sheaf induces an equivalence $D(A)\to D_{\mathrm{q}\mathrm{c}}(X)$, where the target is the full subcategory of $D({\mathcal{O}}_X)$ whose cohomology sheaves are quasi-coherent. The associated object of an $A$-complex is locally perfect if and only if the complex is isomorphic in $D(A)$ to a bounded complex of finitely generated projective modules.

Given: $A$ and $X$ as above, and possibly unbounded complexes.

Goal: Record the classical affine comparison and its perfection criterion.

[F1] The cited Stacks Lemma 36.3.5 gives the derived equivalence.

[F2] The cited Stacks Lemma 36.10.7 gives the perfection criterion; the ring convention is that of Perfect complexes over a ring and its graded version.

Import [F1] and [F2] with the stated reading limits. This item asserts a classical derived equivalence; the enhanced comparison is established separately for the actual induced functor.