K-flat roofs give the full derived localization

The subcategory H is full because between K-flat complexes it keeps all chain maps modulo all chain homotopies. K-flatness is invariant under chain homotopy equivalence: tensoring preserves homotopies, hence preserves contractibility and the acyclicity test defining K-flatness. Zero and shifts are K-flat, and the cone of a map between K-flat complexes is K-flat by [F2]. Distinguished triangles inherited from K therefore make H a full pretriangulated subcategory. By [F1], S is saturated and compatible with the triangulated structure, and S⁻¹$K=D$($O_X$).

Every M∈K receives a quasi-isomorphism R→M with R K-flat by [F2]. This verifies the replacement hypothesis of Stacks Lemma 13.5.8 for every ambient object, without boundedness or global scheme hypotheses. Together with step 1.1 all its hypotheses hold. Thus $S_H$ is a saturated compatible multiplicative system and ($S_H$)⁻¹H→S⁻¹K is an exact equivalence by [F1]. The following two steps make the full-faithfulness content explicit.

For fullness, let P,Q∈H and represent a derived morphism by the right roof $P\stackrel{s}{\leftarrow }M\stackrel{a}{\to }Q$ in K, with s∈S, using [F1]. Choose r:R→M as in [F2]. Then $P\stackrel{sr}{\leftarrow }R\stackrel{ar}{\to }Q$ is a roof in H with denominator sr∈$S_H$ and represents the same ambient derived morphism. Thus the map on each localized Hom set is surjective. Essential surjectivity on objects follows from the same replacement R→M.

For faithfulness, let roofs $P\stackrel{s_i}{\leftarrow }{R}_i\stackrel{a_i}{\to }Q$ in H, for $i=1,2$, have equal images in S⁻¹K. The right-calculus equivalence relation in [F1] provides a common refining roof $P\stackrel{t}{\leftarrow }M\stackrel{b}{\to }Q$ in K and maps $u_i$:M→$R_i$ in K with $s_i$ $u_i$=t∈S and $a_i$ $u_i$=b. Choose a K-flat quasi-isomorphism r:R→M. Then $u_i$ r:R→$R_i$ are morphisms of the full subcategory H, the common denominator tr belongs to $S_H$, and $s_i$ $u_i$ r=tr and $a_i$ $u_i$ r=br as equalities in H. They are equalities of homotopy classes; any witnessing chain homotopies are the same homotopies in the full complex subcategory, so no lifting of homotopies is needed. Hence $P\stackrel{tr}{\leftarrow }R\stackrel{br}{\to }Q$ is a common refining roof in H. The two original roofs are equal in ($S_H$)⁻¹H by the same [F1] equivalence relation. Thus the map on Hom sets is injective.

Let $C_X$=F(X)[$W_X$⁻¹] denote the common model's fiber. K-flat complexes are closed under the finite chain interval because it is built from finite sums, shifts and cones. By [F3], first localizing F(X) at chain homotopy equivalences gives its stable dg nerve; localizing further at quasi-isomorphisms is the stable Verdier localization by acyclic objects. Thus $C_X$ is stable. Actual inclusion into all complexes induces $I_X$:$C_X$→𝒟(X), and $I_X$ is exact: zero and the cofiber cone diagrams are preserved by inclusion and the exact localizations. By [F4], the homotopy category of this localization is precisely ($S_H$)⁻¹H, so hI_X is the functor proved fully faithful and essentially surjective in steps 2.1 through 2.3.

For P,Q∈$C_X$ and m≥0, the homotopy group at zero of its stable mapping space is Hom_{hC_X}(P[m],Q), and similarly in 𝒟(X). Exactness of $I_X$ identifies the shift comparison, so hI_X being fully faithful identifies these homotopy groups and the sets of components. Stable mapping spaces are grouplike; translation by any component reduces its basepoint to zero. Hence every Map(P,Q)→Map($I_{X}P$,$I_{X}Q$) is an equivalence of spaces. Essential surjectivity follows from hI_X, so $I_X$ is an exact equivalence of the actual stable enhancements. This proves exactly the missing fiber comparison for the unchanged coherent-family construction.