The Stacks project

Lemma 42.38.9. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Then

  1. $c_ j(\mathcal{E}) \in A^ j(X)$ is in the center of $A^*(X)$ and

  2. if $f : X' \to X$ is locally of finite type and $c \in A^*(X' \to X)$, then $c \circ c_ j(\mathcal{E}) = c_ j(f^*\mathcal{E}) \circ c$.

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. It is immediate that (2) implies (1). Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$. Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.38.2 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 $(\pi : P \to Y, \mathcal{O}_ P(1))$ associated to $\mathcal{E}$. Denote $\pi ' : P' \to X'$ the base change of $\pi $ by $f$. Using Lemma 42.34.5 and the properties of bivariant classes we obtain

\begin{align*} 0 & = c \cap \left(\sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i})\right) \\ & = \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap (\pi ')^*(c \cap \alpha _{r - i}) \end{align*}

in the Chow group of $P'$ (calculation omitted). Hence we see that $c \cap \alpha _ j$ is equal to $c_ j(f^*\mathcal{E}) \cap (c \cap \alpha )$ by the characterization of Lemma 42.38.2. This proves the lemma. $\square$

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