Enhanced affine perfect equivalence

Exact degreewise associated sheaf preserves quasi-isomorphisms and cones, so [F1] gives an exact induced functor $L:\mathcal{D}(A)\to \mathcal{D}(X)$. Its cohomology sheaves are associated sheaves of module cohomology, so it lands in ${\mathcal{D}}_{\mathrm{q}\mathrm{c}}(X)$. This full subcategory is stable: on affine opens, quasi-coherent sheaves are closed under kernels, cokernels and extensions, and the cohomology sequence of a cone therefore remains quasi-coherent.

By [F2], the functor induced by this same $L$ on homotopy categories is an equivalence. In a stable infinity-category, the homotopy groups of a mapping space at zero are ${\pi }_m\mathrm{Map}(M,N)\cong {\mathrm{Hom}}_{h\mathcal{D}}(M[m],N)$ for $m\ge 0$. Exactness and the homotopy-category equivalence identify these groups for $L$; translation in the grouplike mapping spaces treats all basepoints. Thus every mapping-space map is an equivalence, and essential surjectivity follows from the homotopy-category equivalence. Hence $L$ is an equivalence.

By [F2], a module complex is ring-perfect exactly when its image is locally perfect. By [F3], every locally perfect object belongs to the target of step 2.1. The equivalence therefore restricts to the stated full subcategories.