Lemma 42.67.8. In Situation 42.67.1 let $X \to S$ be locally of finite type and let $X' \to S'$ be the base change by $S' \to S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module of rank $r$ with base change $\mathcal{E}'$ on $X'$. Then the diagram

$\xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]_{g^*} \ar[d]_{c_ i(\mathcal{E}) \cap -} & \mathop{\mathrm{CH}}\nolimits _{k + c}(X') \ar[d]^{c_ i(\mathcal{E}') \cap -} \\ \mathop{\mathrm{CH}}\nolimits _{k - i}(X) \ar[r]^{g^*} & \mathop{\mathrm{CH}}\nolimits _{k + c - i}(X') }$

of chow groups commutes for all $i$.

Proof. Set $P = \mathbf{P}(\mathcal{E})$. The base change $P'$ of $P$ is equal to $\mathbf{P}(\mathcal{E}')$. Since we already know that flat pullback and cupping with $c_1$ of an invertible module commute with base change (Lemmas 42.67.5 and 42.67.7) the lemma follows from the characterization of capping with $c_ i(\mathcal{E})$ given in Lemma 42.38.2. $\square$

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