Lemma 42.37.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $Y$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\alpha $ be a $k$-cycle on $Y$. Then

**Proof.**
Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.37.2 we have

in the chow group of the projective bundle $(\pi : P \to Y, \mathcal{O}_ P(1))$ associated to $\mathcal{E}$. Consider the fibre product diagram

Note that $\mathcal{O}_{P_ X}(1)$ is the pullback of $\mathcal{O}_ P(1)$. Apply flat pullback $(f')^*$ (Lemma 42.20.2) to the displayed equation above. By Lemmas 42.25.2 and 42.14.3 we see that

holds in the chow group of $P_ X$. By the characterization of Lemma 42.37.2 we conclude. $\square$

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