The projective bundle formula for Chow groups

Pushforward identities. Let $V\subseteq X$ be an integral closed subscheme of dimension $m$, so that ${\pi }^{\ast }[V]=[\mathbb{P}(\mathcal{E}{\mathrm{\mid }}_V)]$ is an $(m+r-1)$-cycle. First suppose $\mathcal{E}{\mathrm{\mid }}_V$ is trivial. The zero schemes of $r-1$ coordinate sections of $\mathcal{O}(1)$ are relative hyperplanes meeting in a section $\sigma :V\to \mathbb{P}(\mathcal{E}{\mathrm{\mid }}_V)$, so ${\xi }^{r-1}\cap {\pi }^{\ast }[V]={\sigma }_{\ast }[V]$ and ${\pi }_{\ast }{\sigma }_{\ast }[V]=[V]$. For any $\mathcal{E}$ and $0\le s<r-1$, the cycle ${\xi }^s\cap {\pi }^{\ast }[V]$ has dimension $m+r-1-s>m$ and is supported on $\mathbb{P}(\mathcal{E}{\mathrm{\mid }}_V)$, whose image has dimension $m$; hence its components push to zero by [L4]. For the top power with general $\mathcal{E}$, choose a dense open $U\subseteq V$ trivializing the bundle. The identity ${\pi }_{\ast }({\xi }^{r-1}\cap {\pi }^{\ast }[V])=[V]$ holds over $U$, and its difference is in $A_m(V)$ supported on $V\setminus U$. This complement has dimension less than $m$, so localization [L3] makes the difference zero. Both identities extend by linearity to all $\alpha \in A_{\ast }(X)$.

Surjectivity over a trivializing open. Let $U\subseteq X$ be an open on which $\mathcal{E}$ is trivial, so that $\mathbb{P}(\mathcal{E}){\mathrm{\mid }}_U\cong U\times {\mathbb{P}}^{r-1}$. The Chow group of $U\times {\mathbb{P}}^{r-1}$ is generated by the classes ${\xi }^i\cap {\pi }^{\ast }\alpha$ with $0\le i\le r-1$ and $\alpha \in A_{\ast }(U)$: stratify ${\mathbb{P}}^{r-1}$ by the affine spaces ${\mathbb{A}}^j$, use the localization sequence and affine homotopy invariance [L3] to reduce every class to classes pulled back from the strata, and represent each stratum class as an $i$-fold cut with $\xi$ by repeated Cartier cutting of coordinate hyperplanes, which realizes it as ${\xi }^i\cap {\pi }^{\ast }$ of a class on $U$; the claim is compatible with the restriction to any smaller open by [L2].

Injectivity. Suppose ${\sum }_{i=0}^{r-1}{\xi }^i\cap {\pi }^{\ast }{\alpha }_i=0$. Applying ${\pi }_{\ast }$ and step 1.1 gives ${\alpha }_{r-1}=0$, since the other summands push to zero. Capping the remaining relation with ${\xi }^j$ and applying ${\pi }_{\ast }$ successively gives ${\alpha }_{r-1-j}=0$ for every $0\le j\le r-2$ by the same computation, using associativity of repeated caps from [L2]. Hence the displayed map is injective.

Global surjectivity. It suffices to express an integral cycle $[V]$ on $\mathbb{P}(\mathcal{E})$, and we induct on the dimension of the reduced image closure $W\subseteq X$. The cycle is supported on $\mathbb{P}(\mathcal{E}{\mathrm{\mid }}_W)$. Choose a dense open $U\subseteq W$ trivializing the bundle. Step 1.2 expresses $[V]{\mathrm{\mid }}_U$ in the required form; lift its coefficient classes from $A_{\ast }(U)$ to $A_{\ast }(W)$ using localization [L3]. The resulting difference is the pushforward of a cycle class on $\mathbb{P}(\mathcal{E}{\mathrm{\mid }}_{W\setminus U})$. Each integral component of that cycle has image closure of dimension strictly less than $\dim W$, so induction expresses its class in the required form over its image. Push these expressions to $W$ and then $X$, using compatibility of flat pullback with a closed immersion and of caps with proper pushforward. The induction terminates because the integral image has finite dimension. For locally finite cycles the constructions remain locally finite: $\pi$ is proper, so the image closures of a locally finite family are locally finite; each residual family is supported within its respective image closure, and the dimension bound for a fixed-degree input cycle is uniform. Thus the same argument sums locally finitely without asserting that the whole base has finitely many components.

Conclusion. Injectivity is step 2.1 and surjectivity is step 2.2; both hold degreewise, so the displayed map is an isomorphism for every $d$. The final paragraph of the statement follows because $\mathcal{E}$ locally free of rank $r$ makes $\pi$ smooth of relative dimension $r-1$ by [L1], and smoothness of $X$ inherits to the total space of a smooth morphism.