The Stacks project

Lemma 42.36.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let

\[ p : E = \underline{\mathop{\mathrm{Spec}}}(\text{Sym}^*(\mathcal{E})) \longrightarrow X \]

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 42.32.2.) For surjectivity see Lemma 42.32.1. 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, Lemma 31.14.10). 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

\[ j^* : \mathop{\mathrm{CH}}\nolimits _{k + r}(P) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k + r}(E) \]

are the cycles supported in the effective Cartier divisor $D$, see Lemma 42.19.3. So if $p^*\alpha = 0$, then $\pi ^*\alpha = i_*\beta $ for some $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + r}(D)$. By Lemma 42.36.2 we may write

\[ \beta = \pi _ D^*\beta _0 + \ldots + c_1(\mathcal{O}_ D(1))^{r - 1} \cap \pi _ D^* \beta _{r - 1}. \]

for some $\beta _ i \in \mathop{\mathrm{CH}}\nolimits _{k + i}(X)$. By Lemmas 42.31.1 and 42.26.4 this implies

\[ \pi ^*\alpha = i_*\beta = c_1(\mathcal{O}_ P(1)) \cap \pi ^*\beta _0 + \ldots + c_1(\mathcal{O}_ D(1))^ r \cap \pi ^*\beta _{r - 1}. \]

Since the rank of $\mathcal{E} \oplus \mathcal{O}_ X$ is $r + 1$ this contradicts Lemma 42.26.4 unless all $\alpha $ and all $\beta _ i$ are zero. $\square$

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