The Stacks project

Lemma 41.42.8. Let $(S, \delta )$ be as in Situation 41.7.1. 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 $E, F \in D(\mathcal{O}_{X_2})$ be perfect objects. Assume

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

  2. the restrictions $E|_{X_1 \cap X_2}$ and $F|_{X_1 \cap X_2}$ are zero,

Denote $P'_ p(E), P'_ p(F), P'_ p(E \oplus F) \in A^ p(X_2 \to X)$ for $p \geq 0$ the classes constructed in Lemma 41.42.1. Then $P'_ p(E \oplus F) = P'_ p(E) + P'_ p(F)$.

Proof. Immediate from the characterization of the classes in Lemma 41.42.1 and the additivity in Lemma 41.41.4. $\square$


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