Lemma 82.27.3. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let
be the associated vector bundle over $X$. Then $p^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k + r}(E)$ is an isomorphism for all $k$.
Proof. (For the case of linebundles, see Lemma 82.25.2.) For surjectivity see Lemma 82.25.1. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective space bundle associated to the finite locally free sheaf $\mathcal{E} \oplus \mathcal{O}_ X$. Let $s \in \Gamma (P, \mathcal{O}_ P(1))$ correspond to the global section $(0, 1) \in \Gamma (X, \mathcal{E} \oplus \mathcal{O}_ X)$. Let $D = Z(s) \subset P$. Note that $(\pi |_ D : D \to X , \mathcal{O}_ P(1)|_ D)$ is the projective space bundle associated to $\mathcal{E}$. We denote $\pi _ D = \pi |_ D$ and $\mathcal{O}_ D(1) = \mathcal{O}_ P(1)|_ D$. Moreover, $D$ is an effective Cartier divisor on $P$. Hence $\mathcal{O}_ P(D) = \mathcal{O}_ P(1)$ (see Divisors on Spaces, Lemma 71.7.8). Also there is an isomorphism $E \cong P \setminus D$. Denote $j : E \to P$ the corresponding open immersion. For injectivity we use that the kernel of
are the cycles supported in the effective Cartier divisor $D$, see Lemma 82.15.2. So if $p^*\alpha = 0$, then $\pi ^*\alpha = i_*\beta $ for some $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + r}(D)$. By Lemma 82.27.2 we may write
for some $\beta _ i \in \mathop{\mathrm{CH}}\nolimits _{k + i}(X)$. By Lemmas 82.24.1 and 82.19.4 this implies
Since the rank of $\mathcal{E} \oplus \mathcal{O}_ X$ is $r + 1$ this contradicts Lemma 82.19.4 unless all $\alpha $ and all $\beta _ i$ are zero. $\square$
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