Actual second-stage comparison on every qcqs field-scheme

The map $T\times B\to S$, $(t,b)\mapsto i(t)+\kappa (b)$, is an equivalence over $B$: translation by $\kappa (b)$ identifies the zero fiber with the fiber over each generalized point $b$, using the specified homotopy $d\kappa \simeq \mathrm{i}\mathrm{d}$. Its inverse gives a natural pointed retraction $r:S\to T$, underlying $s\mapsto s-\kappa (d(s))$ with that nullhomotopy; $ri\simeq \mathrm{i}\mathrm{d}$. No additivity of $\kappa$ is required. The section makes ${\pi }_{1}S(Y)\to {\pi }_{1}B(Y)$ surjective on every qcqs $Y$, so ${\pi }_{0}T(Y)\to {\pi }_{0}S(Y)$ is injective. The discrete rank target also makes ${\pi }_{0}S(Y)\to K_0(Y)$ injective.

For a rank-$n$ bundle $V$, $r([V]-n)$ has underlying class $\delta (V)=[V]-[\det V]-(n-1)$. Pull back to a detecting flag tower and let its line quotients be $L_s$. Put $D_s={⨂}_{i\le s}{L}_i$ and $R_s={\sum }_{i\le s}[L_i]-[D_s]-(s-1)$. Exact flag sequences give this equality without splitting, and $R_1=0,R_s=R_{s-1}-([L_s]-1)([D_{s-1}]-1)\mathrm{.}$ The latter product is the augmentation of the actual stage product of the line lifts $\kappa (L_s),\kappa (D_{s-1})$. Injectivity in step 1.1 identifies these equal K-classes with their unique second-stage classes. The actual product square in [F2] and induction give $Q_{2}r([V]-n)=-{\sum }_{i<s}{c}_1(L_i)c_1(L_s)=-c_2(V)\mathrm{.}$ Rank zero/one give zero, and finite clopen decomposition handles varying ranks. Detection in [F3] descends the identity.

Put $a=c_1(V)=c_1(\det V)$. Full-scope Cartan in [F3] gives $c_t(\delta (V))=c_t(V)(1+at{)}^{-1}$ through degree two. Its second coefficient is $c_2(V)-a^2+a^2=c_2(V)$. Thus $(Q_2+C_2{i}_K)r$ vanishes on every actual universal bundle on finite Grassmannians over $X$.

The particular difference $(Q_2+C_2{i}_K)r$ is built from the natural actual stage maps, Chern operations, determinant and line-class section of [F1]–[F3]. Form these same constructions first on the smooth big Zariski site over $X$, where the universal-operation and coefficient constructions in [F3] are defined. This gives a specific big-site difference map; it is not an extension of an arbitrary small-site map. Apply the actual coefficient adjunction and Grass/Vect detection there, then stabilize to $BGL$. In weight two the actual negative-coefficient vanishing supplied in [F3] eliminates every coefficient of monomial weight greater than two in all degrees. Therefore both mapping ${\pi }_0$ and ${\pi }_1$ stabilize at rank at least two, and ${\lim }^1{\pi }_1=0$. The resulting big-site homotopy restricts along the open immersions to the small Zariski site and shows that the difference vanishes after its actual rank-zero free-class map $j_0:BGL\to S$. By the first helper's derived-chain detection it vanishes on $S$. Restrict along $i$ and use $ri\simeq \mathrm{i}\mathrm{d}$ to prove the displayed map identity on $T$. This extends the comparison by a retraction before using plus detection; it never asserts preservation of an arbitrary homotopy fiber by a homology equivalence.