Lemma 42.46.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $i_ j : X_ j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E, F \in D(\mathcal{O}_ X)$ be perfect objects. Assume

chern classes of $E$ and $F$ are defined,

the restrictions $E|_{X_1 \cap X_2}$ and $F|_{X_1 \cap X_2}$ are isomorphic to a finite locally free $\mathcal{O}_{X_1}$-modules of rank $< p$ and $< q$ sitting in cohomological degree $0$.

With notation as in Remark 42.33.7 set

with $c'_ p(E|_{X_2})$ as in Lemma 42.46.1. Similarly for $c^{(q)}(F)$ and $c^{(p + q)}(E \oplus F)$. Then $c^{(p + q)}(E \oplus F) = c^{(p)}(E)c^{(q)}(F)$ in $A^{(p + q)}(X_2 \to X)$.

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