The Stacks project

Lemma 42.46.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be perfect. If the Chern classes of $E$ are defined then

  1. $c_ p(E)$ is in the center of the algebra $A^*(X)$, and

  2. if $g : X' \to X$ is locally of finite type and $c \in A^*(X' \to X)$, then $c \circ c_ p(E) = c_ p(Lg^*E) \circ c$.

Proof. Part (1) follows immediately from part (2). Let $g : X' \to X$ and $c \in A^*(X' \to X)$ be as in (2). To show that $c \circ c_ p(E) - c_ p(Lg^*E) \circ c = 0$ we use the criterion of Lemma 42.35.3. Thus we may assume that $X$ is integral and by Lemma 42.46.5 we may even assume that $E$ is represented by a bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules of constant rank. Then we have to show that

\[ c \cap c_ p(\mathcal{E}^\bullet ) \cap [X] = c_ p(\mathcal{E}^\bullet ) \cap c \cap [X] \]

in $\mathop{\mathrm{CH}}\nolimits _*(X')$. This is immediate from Lemma 42.38.9 and the construction of $c_ p(\mathcal{E}^\bullet )$ as a polynomial in the chern classes of the locally free modules $\mathcal{E}^ n$. $\square$

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