Lemma 82.28.4. In Situation 82.2.1 let X/B be good. Let \mathcal{E} be a locally free \mathcal{O}_ X-module of rank r. Then c_ j(\mathcal{L}) \in A^ j(X) commutes with every element c \in A^ p(X). In particular, if \mathcal{F} is a second locally free \mathcal{O}_ X-module on X of rank s, then
c_ i(\mathcal{E}) \cap c_ j(\mathcal{F}) \cap \alpha = c_ j(\mathcal{F}) \cap c_ i(\mathcal{E}) \cap \alpha
as elements of \mathop{\mathrm{CH}}\nolimits _{k - i - j}(X) for all \alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X).
Proof.
Let X' \to X be a morphism of good algebraic spaces over B. Let \alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X'). Write \alpha _ j = c_ j(\mathcal{E}) \cap \alpha , so \alpha _0 = \alpha . By Lemma 82.28.1 we have
\sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap (\pi ')^*(\alpha _{r - i}) = 0
in the chow group of the projective bundle (\pi ' : P' \to X', \mathcal{O}_{P'}(1)) associated to (X' \to X)^*\mathcal{E}. Applying c \cap - and using Lemma 82.26.8 and the properties of bivariant classes we obtain
\sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap \pi ^*(c \cap \alpha _{r - i}) = 0
in the Chow group of P'. Hence we see that c \cap \alpha _ j is equal to c_ j(\mathcal{E}) \cap (c \cap \alpha ) by the uniqueness in Lemma 82.28.1. This proves the lemma.
\square
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