The Stacks project

Lemma 42.38.3. 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 $X$. If $\alpha \sim _{rat} \beta $ are rationally equivalent $k$-cycles on $X$ then $c_ j(\mathcal{E}) \cap \alpha = c_ j(\mathcal{E}) \cap \beta $ in $\mathop{\mathrm{CH}}\nolimits _{k - j}(X)$.

Proof. By Lemma 42.38.2 the elements $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, $j \geq 1$ and $\beta _ j = c_ j(\mathcal{E}) \cap \beta $, $j \geq 1$ are uniquely determined by the same equation in the chow group of the projective bundle associated to $\mathcal{E}$. (This of course relies on the fact that flat pullback is compatible with rational equivalence, see Lemma 42.20.2.) Hence they are equal. $\square$

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