External projective bundle input for the actual motivic complex

PDF p. 8: Computations after the splitting principle

Let $\mathbb{F}=\mathbb{Q}$ or ${\mathbb{F}}_p$, let $X$ be a qcqs $\mathbb{F}$-scheme, and let $E$ be finite locally free of constant rank $r+1>0$. For $j\ge 0$, the Elmanto-Morrow projective-bundle map ${⨁}_{s=0}^r\mathbb{Z}(j-s{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X)[-2s]\stackrel{\sum c_1(\mathcal{O}(1){)}^s{\pi }^{\ast }}{\to }\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}({\mathbb{P}}_{\mathrm{s}\mathrm{u}\mathrm{b},X}(E))$ is an equivalence of derived complexes. The source uses line subbundles, and $\mathcal{O}(1)$ is the dual of its tautological line. Negative-weight terms in this formula are zero by the source's stated convention. In particular, the component $s=0$ is the actual pullback and has a retraction in $D(\mathbb{Z})$.

Given: Exactly the prime field, qcqs scheme, rank and weight in the Statement.

Goal: Record the established projective-bundle input needed for the actual motivic tower, without strengthening the abstract contract or substituting classical smooth cohomology.

[F1] Theorem 5.24(2) is the displayed derived equivalence in the cited PDF. The paragraph preceding it defines its line-subbundle convention.

Import [F1]. Projecting its inverse onto the zero summand gives the stated retraction. The native quotient convention for a bundle $G$ is obtained by applying this result to $E=G^{\vee }$: a locally split invertible quotient of $G$ dualizes to a locally split line subbundle of $G^{\vee }$, and dualizing back is inverse for finite locally free sheaves. The source's $\mathcal{O}(1)$ then becomes exactly the native tautological quotient. The target uses this explicit adaptation and the existing actual Nisnevich-sheaf contract; it does not claim that the abstract projective-bundle axioms alone imply finite-clopen detection.