Part A: $\Phi$ is surjective.

Step A1: Generation by twists. We claim that every class in $K_0(\mathbb{P}(\mathcal{E}))$ is a $K_0(X)$-linear combination of $[\mathcal{O}], [\mathcal{O}(1)], \ldots, [\mathcal{O}(r-1)]$.

For any coherent sheaf $\mathcal{F}$ on $\mathbb{P}(\mathcal{E})$, Serre's theorem gives: for $n \gg 0$, the sheaf $\mathcal{F}(n) = \mathcal{F} \otimes \mathcal{O}(n)$ is generated by its direct image $\pi_*\mathcal{F}(n)$. That is, the evaluation map

$\pi^*\pi_*\mathcal{F}(n) \to \mathcal{F}(n) \to 0$

is surjective. Thus $[\mathcal{F}(n)] = \pi^*[\pi_*\mathcal{F}(n)] - [\ker]$ where $\ker$ is a coherent sheaf on $\mathbb{P}(\mathcal{E})$ with "lower complexity." By Noetherian induction on the support, we reduce to sheaves of the form $\pi^*\mathcal{G} \otimes \mathcal{O}(m)$.

Step A2: Reducing twists modulo $r$. The tautological sequence on $\mathbb{P}(\mathcal{E})$ is:

$0 \to \mathcal{O}(-1) \to \pi^*\mathcal{E} \to \mathcal{Q} \to 0$

where $\mathcal{Q}$ is the universal quotient of rank $r-1$. Taking the $r$-th exterior power of the dual sequence (Koszul complex):

$0 \to \pi^*(\det \mathcal{E})^\vee \otimes \mathcal{O}(r) \to \pi^*\Lambda^{r-1}\mathcal{E}^\vee \otimes \mathcal{O}(r-1) \to \cdots \to \pi^*\mathcal{E}^\vee \otimes \mathcal{O}(1) \to \mathcal{O} \to 0$

This exact complex (the Koszul complex associated to the tautological section) gives the relation in $K_0$:

$[\mathcal{O}(r)] = \sum_{j=0}^{r-1} (-1)^{r-1-j} \pi^*[\Lambda^{r-1-j}\mathcal{E}^\vee \otimes \det\mathcal{E}] \cdot [\mathcal{O}(j)]$

This expresses $[\mathcal{O}(r)]$ as a $K_0(X)$-linear combination of $[\mathcal{O}], \ldots, [\mathcal{O}(r-1)]$. By induction, all $[\mathcal{O}(m)]$ for $m \geq r$ (and by tensoring with $\mathcal{O}(-1)$ repeatedly, also for $m < 0$) are in the span of $\{[\mathcal{O}(i)]\}_{i=0}^{r-1}$.

Combining Steps A1 and A2, $\Phi$ is surjective.

Part B: $\Phi$ is injective.

Step B1: Construct a left inverse. Define $\Psi_j: K_0(\mathbb{P}(\mathcal{E})) \to K_0(X)$ for $0 \leq j \leq r-1$ by:

$\Psi_j(\alpha) = \sum_{i \geq 0} (-1)^i [R^i\pi_*(\alpha \otimes \mathcal{O}(-j))]$

This is the K-theoretic direct image of $\alpha(-j)$.

Step B2: Compute $\Psi_j \circ \Phi$. For the basis elements $\pi^*(a_k) \cdot [\mathcal{O}(k)]$:

$\Psi_j(\pi^*(a_k) \cdot [\mathcal{O}(k)]) = a_k \cdot \sum_i (-1)^i [R^i\pi_*\mathcal{O}(k-j)]$

by the projection formula $R^i\pi_*(\pi^*\mathcal{G} \otimes \mathcal{O}(m)) = \mathcal{G} \otimes R^i\pi_*\mathcal{O}(m)$.

Step B3: Cohomology of $\mathcal{O}(m)$ on fibers. For the projective bundle $\mathbb{P}(\mathcal{E}) \to X$ with fibers $\mathbb{P}^{r-1}$:

$R^i\pi_*\mathcal{O}(m) = \begin{cases} S^m\mathcal{E} & i = 0, \; m \geq 0 \\ 0 & 0 < i < r-1, \; \text{all } m \\ (\det\mathcal{E}) \otimes S^{-m-r}\mathcal{E}^\vee & i = r-1, \; m \leq -r \\ 0 & i = 0, \; -r < m < 0 \end{cases}$

where $S^m$ denotes the $m$-th symmetric power. In particular, for $0 \leq k, j \leq r-1$:

$R^i\pi_*\mathcal{O}(k-j) = \begin{cases} S^{k-j}\mathcal{E} & \text{if } k \geq j \text{ and } i = 0 \\ 0 & \text{if } k < j \text{ (since } -(r-1) \leq k-j < 0 \text{)} \end{cases}$

Step B4: The matrix is upper triangular. The matrix $M_{jk} = \sum_i (-1)^i [R^i\pi_*\mathcal{O}(k-j)]$ for $0 \leq j, k \leq r-1$ satisfies:

• $M_{jk} = [S^{k-j}\mathcal{E}]$ for $k \geq j$ (in $K_0(X)$)
• $M_{jk} = 0$ for $k < j$
• $M_{jj} = [S^0\mathcal{E}] = [\mathcal{O}_X] = 1$

So $M$ is upper triangular with $1$'s on the diagonal in $K_0(X)$. Such a matrix is invertible over $K_0(X)$, so $\Psi = (\Psi_0, \ldots, \Psi_{r-1})$ provides a left inverse to $\Phi$. Since $\Phi$ is already surjective, $\Phi$ is an isomorphism. $\square$