The Stacks project

Proof. Let $f : Y \to X$ be an envelope and let $\alpha ^\bullet : \mathcal{E}_1^\bullet \to \mathcal{E}_2^\bullet $ be a map of locally bounded complexes of finite locally free $\mathcal{O}_ Y$-modules representing $Lf^*E_1 \to Lf^*E_2$. Then the cone $C(\alpha )^\bullet $ represents $Lf^*E_3$. Since $C(\alpha )^ n = \mathcal{E}_2^ n \oplus \mathcal{E}_1^{n + 1}$ we see that $C(\alpha )^\bullet $ is a locally bounded complex of finite locally free $\mathcal{O}_ Y$-modules. We conclude that the Chern classes of $E_1$, $E_2$, $E_3$ are defined. Moreover, recall that $c_ p(E_1)$ is defined as the unique element of $A^ p(X)$ which restricts to $c_ p(\mathcal{E}_1^\bullet )$ in $A^ p(Y)$. Similarly for $E_2$ and $E_3$. Hence it suffices to prove $c(\mathcal{E}_2^\bullet ) = c(\mathcal{E}_1^\bullet ) c(C(\alpha )^\bullet )$ in $\prod _{p \geq 0} A^ p(Y)$. In turn, it suffices to prove this after restricting to a connected component of $Y$. Hence we may assume the complexes $\mathcal{E}_1^\bullet $ and $\mathcal{E}_2^\bullet $ are bounded complexes of finite locally free $\mathcal{O}_ Y$-modules of fixed rank. In this case the desired equality follows from the multiplicativity of Lemma 42.40.3. In the case of $ch$ or $P_ p$ we use Lemmas 42.45.2.

In the previous paragraph we have seen that the lemma holds if condition (1) is satisfied. Since (2) implies (1) this deals with the second case. Assume (3). Arguing exactly as in the proof of Lemma 42.46.4 we find an envelope $f : Y \to X$ such that $Y$ is a disjoint union $Y = \coprod Y_ i$ of quasi-compact (and quasi-separated) schemes each having the resolution property. Then we may represent the restriction of $Lf^*E_1 \to Lf^*E_2$ to $Y_ i$ by a map of bounded complexes of finite locally free modules, see Derived Categories of Schemes, Proposition 36.37.5. In this way we see that condition (3) implies condition (1). Of course condition (4) implies condition (3) and the proof is complete. $\square$