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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