Homotopy invariance for vector bundles

Compactification and localization. If $r=0$, $E=T$ and $p=\mathrm{id}$, so pullback and its inverse are the identity after every base change. Assume $r\ge 1$ for the compactification argument. Let $q:\mathbb{P}(E^{\vee }\oplus 1)\to T$ be the projective completion with $\mathcal{O}(1)$ and $\xi =c_1(\mathcal{O}(1))$, and let $j:\mathbb{P}(E^{\vee })\hookrightarrow \mathbb{P}(E^{\vee }\oplus 1)$ be the infinity divisor, the zero scheme of the section of $\mathcal{O}(1)$ induced by the direct-summand $1\subseteq E^{\vee }\oplus 1$. Its complement is $E$, and the restriction of $q$ to $E$ is $p$; hence the localization sequence of [L2] gives the exact sequence $A_m(\mathbb{P}(E^{\vee }))\stackrel{j_{\ast }}{\to }{A}_m(\mathbb{P}(E^{\vee }\oplus 1))\stackrel{}{\to }{A}_m(E)\to 0$.

The image of the infinity pushforward. Write $\pi =q\circ j$. For an integral cycle $[V]$ on $T$, the trivial-summand section cuts the relative hyperplane $\mathbb{P}(E^{\vee }{\mathrm{\mid }}_V)$ in the projective completion, with multiplicity one. The Cartier formula therefore gives $j_{\ast }{\pi }^{\ast }[V]=\xi \cap q^{\ast }[V]$, and linearity gives this identity for all classes $\beta$ on $T$. Compatibility of the first Chern cap with proper pushforward then yields $j_{\ast }({\xi }^a\cap {\pi }^{\ast }\beta )={\xi }^{a+1}\cap q^{\ast }\beta$. Consequently, on the basis $1,\xi ,\dots ,{\xi }^{r-1}$ over $A_{\ast }(T)$ of the source and $1,\xi ,\dots ,{\xi }^r$ of the target supplied by [L3], the image of $j_{\ast }$ is exactly the span of the positive powers ${\xi }^1,\dots ,{\xi }^r$.

Conclusion. By step 2.1 the quotient of $A_{\ast }(\mathbb{P}(E^{\vee }\oplus 1))$ by the image of $j_{\ast }$ is the direct summand $q^{\ast }{A}_{\ast }(T)$ spanned by $1$, and by step 1.1 this quotient is exactly $A_{\ast }(E)$; since $q{\mathrm{\mid }}_E=p$, the induced map is the flat pullback $p^{\ast }$, which is therefore an isomorphism onto the summand spanned by the classes $q^{\ast }\alpha$ with $m$ shifted by $r$. Both bundles and both bases pull back along any $k$-base change $T^{\mathrm{\prime }}\to T$ with $T^{\mathrm{\prime }}$ locally of finite type over $k$, so the same computation applies verbatim after base change; the inverse of $p^{\ast }$ is by definition $s_E^{!}$.