Classical and derived smooth Zariski sites over a classical base

For a smooth derived $Y\to S$, choose affine charts $\mathrm{Spec}B\subset Y$ over $\mathrm{Spec}A\subset S$. Since $S$ is classical, ${\pi }_{i}A=0$ for $i>0$. Smoothness and [F2] give ${\pi }_{i}B\cong {\pi }_{i}A{\otimes }_{{\pi }_{0}A}{\pi }_{0}B=0$ for $i>0$, and a classically smooth map $A\to {\pi }_{0}B$. Thus these charts, and hence $Y$, are classical. Conversely a classical smooth morphism satisfies the same flatness criterion: both sides vanish in positive degrees and the degree-zero condition is ordinary flatness. It is therefore derived smooth. The fully faithful embedding in [F2] identifies every mapping space between classical schemes with the discrete set of classical morphisms; passing to the slice over the classical $S$ retains this identification. This proves the categorical equivalence with inverse $t_0$. The underlying spaces and affine opens are unchanged, so qcqs is preserved and reflected; open immersions and their joint surjectivity agree. Thus this is an equivalence of the stated Zariski sites.

Precomposition with this site equivalence and with $t_0$ gives inverse equivalences on presheaves. Covering diagrams correspond, so their descent limit conditions correspond as well, and the equivalences restrict to sheaves. These inverse functors preserve all existing limits and colimits. They preserve pointwise products and the terminal object; after adding basepoints they preserve the smash pushouts. Applied levelwise to ordinary spectra they preserve the loop equivalences defining stabilization; the spectrum tensor is obtained from pointed smash by stabilization and colimits, so it too is preserved. Full faithfulness identifies each representable with the corresponding representable, including the usual ${\mathbb{P}}_S^1$ and its infinity section. For classical $Y$, its derived product with ${\mathbb{P}}_S^1$ over $S$ is classical, since it is smooth over $Y$ by [F1]–[F2], and its truncation is ${\mathbb{P}}_Y^1$; hence this is also the ordinary projective line over $Y$. Under the tensor-preserving equivalence the right adjoints defining projective-line cotensor correspond by their adjunctions. Therefore the explicit extension $B\circ t_0$ transports the actual graded products, pointed maps and projective-line equivalences, with every identity, composite and homotopy retained by functoriality.