Classical-base comparison of the native smooth site and bundle constructions

On an affine chart $\mathrm{Spec}A$ of the classical base, a smooth native affine chart $\mathrm{Spec}B$ over it has discrete $B$: in animated geometry [F1] gives ${\pi }_{i}B=0$ for $i>0$ because ${\pi }_{i}A=0$; in spectral geometry use flatness and the discrete-base assertion of [F2]. Its map on ordinary rings is smooth, by [F1] or the classical criterion in [F2]. Conversely an ordinary smooth map gives a native smooth map, using [F1] in the animated instance and the same classical criterion of [F2] affine-locally in the spectral instance. These assertions glue on affine opens, so every smooth native object over $S$ is classical and conversely. The discrete inclusion is fully faithful on affine charts by [F1] or [F2]; morphisms of schemes glue from these chartwise morphisms, so it is fully faithful globally, including for all $S$-morphisms between smooth objects. By [F3], qcqs is the same property and open restrictions have the same underlying opens. A family covers precisely when these opens cover the underlying space. Thus the inclusion is an equivalence of the specified sites, with their covering sieves.

Precomposition with the site equivalence of step 1.1 identifies sheaves and has inverse precomposition with a quasi-inverse. Their products, limits, colimits and adjunctions correspond, hence so do pointing, smash, disjoint basepoints, cotensors and the ordinary and symmetric projective-line spectrum constructions of [F3]. On a discrete affine chart $\mathrm{Spec}A$, the free-module adjunction of [F2], also applicable to animated modules by [F1], identifies the space of maps $A^r\to A^s$ with the discrete set of $s$-by-$r$ matrices over $A$. Composition and pullback are the usual matrix operations. Thus all bundle maps, as well as isomorphisms, agree locally. Gluing free modules and their maps identifies the bundle groupoids and their morphism sets on every test scheme, naturally under pullback; rank one identifies the pointed Picard stack. Now apply the explicit quotient construction of [F4]. By step 1.1 every test $T$ is classical, so surjectivity on ${\pi }_0$ is ordinary sheaf surjectivity. It is locally a split surjection: trivialize $L$ and lift its frame locally along $q$ to obtain a right inverse. Thus the quotient condition is local and preserved by pullback. Bundles, bundle maps and isomorphisms glue, and surjectivity is local, so these pairs and arrows satisfy Zariski descent. The bundle-map comparison therefore identifies the entire quotient groupoids, not just their isomorphism classes. For a bundle on a test scheme, the representing bijection of Projective bundle represents line quotients identifies their classes with maps to the classical projective bundle. An automorphism $a$ of a pair satisfies $aq=q$, hence $a=1$ by local surjectivity; the groupoid is therefore equivalent to this discrete represented sheaf. The theorem identifies the universal pair with the actual tautological quotient and is natural in test schemes and base change. For a trivial bundle of positive rank $r$ this gives ${\mathbb{P}}^{r-1}$ with its coordinate sections and standard quotient; rank zero gives the empty quotient sheaf, and varying rank is handled on the trivializing opens. In the animated instance this is the rank-one quotient convention of reference 3. In the flat spectral instance, reference 1 identifies the flat projective space over each discrete affine base with the classical projective space and its twisting line. In its monoid charts the degree-one coordinate sections give that same quotient, so it realizes the same quotient sheaf on these tests; no such claim is used over non-discrete spectral rings. In both instances changes of bundle frame act by the same matrices on the actual quotients, so the local identifications glue with their tautological lines. For arbitrary $X$, the construction of [F4] applies over each $x:T\to X$; its compatibility with pullback and paths in $X(T)$ gives an equivalence over $X$, carrying the line map to $\mathrm{P}\mathrm{i}\mathrm{c}$ to the classical one. This also proves compatibility with the test-scheme descent construction of reference 3. The coordinate quotient at infinity sends the selected basis vector to $1$ and the other vector to $0$, so its trivialized line and basepoint are preserved. Finally the transported coefficient functors, products and pointed bonding classes have the same values and maps on the identified tests. Their class-power maps and pullbacks consequently correspond as actual maps under this equivalence, not merely on component groups.