Lemma 42.47.10. In Lemma 42.47.1 assume $E_2$ has constant rank $0$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then

$c'_ i(E_2 \otimes \mathcal{L}) = \sum \nolimits _{j = 0}^ i \binom {- i + j}{j} c'_{i - j}(E_2) c_1(\mathcal{L})^ j$

Proof. The assumption on rank implies that $E_2|_{X_1 \cap X_2}$ is zero. Hence $c'_ i(E_2)$ is defined for all $i \geq 1$ and the statement makes sense. The actual equality follows immediately from Lemma 42.46.10 and the characterization of $c'_ i$ in Lemma 42.47.1. $\square$

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