The Stacks project

Lemma 42.32.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let

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

be the associated vector bundle over $X$. Then $p^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k + 1}(L)$ is an isomorphism for all $k$.

Proof. For surjectivity see Lemma 42.32.1. Let $o : X \to L$ be the zero section of $L \to X$, i.e., the morphism corresponding to the surjection $\text{Sym}^*(\mathcal{L}) \to \mathcal{O}_ X$ which maps $\mathcal{L}^{\otimes n}$ to zero for all $n > 0$. Then $p \circ o = \text{id}_ X$ and $o(X)$ is an effective Cartier divisor on $L$. Hence by Lemma 42.31.1 we see that $o^* \circ p^* = \text{id}$ and we conclude that $p^*$ is injective too. $\square$

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