Lemma 42.46.6 (0FAC)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.46: Chern classes and the derived category
Lemma 42.46.6 (cite)

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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

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