Lemma 42.49.8. In Lemma 42.49.1 assume $Q|_ T$ is zero. Assume we have another perfect object $Q' \in D(\mathcal{O}_ W)$ whose Chern classes are defined such that the restriction $Q'|_ T$ is zero. In this case the classes $P'_ p(Q), P'_ p(Q'), P'_ p(Q \oplus Q') \in A^ p(Z \to X)$ constructed in Lemma 42.49.1 satisfy $P'_ p(Q \oplus Q') = P'_ p(Q) + P'_ p(Q')$.

**Proof.**
This follows immediately from the construction of these classes and Lemma 42.47.9.
$\square$

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