Lemma 81.28.4. In Situation 81.2.1 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 81.28.1 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 81.26.8 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 81.28.1. This proves the lemma.
$\square$

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