External EM projective bundle equivalence with its specified components

PDF p. 2: Motivic projective bundle property; PDF p. 8: Orientation for the splitting principle

Let $F$ be a prime field, $X$ a qcqs $F$-scheme, $V$ a finite locally free sheaf of constant rank $r>0$, and $j\ge 0$. For the line projectivization $\pi :P_X^{\mathrm{l}\mathrm{i}\mathrm{n}}(V)\to X$ of the preceding external input, the actual map ${\sum }_{m=0}^{r-1}{c}_1(\mathcal{O}(1){)}^m{\pi }^{\ast }:{⨁}_{m=0}^{r-1}{C}_{j-m}(X)[-2m]⟶C_j(P_X^{\mathrm{l}\mathrm{i}\mathrm{n}}(V))$ is an equivalence. Each component is pullback followed by multiplication by the $m$-th power of the specific Definition5.2 class; $C_{j-m}=0$ when $j-m<0$.

Import EM v1 Theorem5.24(2). The source proves regular-blowup excision, deduces the projective-space formula from its P¹ formula using Annala–Iwasa’s argument, and then obtains the vector-bundle formula by Zariski descent. This imports that established proof; it does not pretend to reconstruct all its prerequisites. The explicitly displayed components and the first-Chern/product convention are part of this input. It is not merely an assertion that the two groups or complexes happen to be isomorphic.