The Stacks project

Lemma 42.47.5. In Lemma 42.47.1 suppose we have closed subschemes $X'_2 \subset X_2$ and $X_1 \subset X'_1 \subset X$ such that $X = X'_1 \cup X'_2$ set theoretically. Assume $E_2|_{X'_1 \cap X_2}$ is zero, resp. isomorphic to a finite locally free module of rank $< p$ placed in degree $0$. Then we have $(X'_2 \to X_2)_* \circ P'_ p(E_2|_{X'_2}) = P'_ p(E_2)$, resp. $(X'_2 \to X_2)_* \circ c'_ p(E_2|_{X'_2}) = c_ p(E_2)$ (with $\circ $ as in Lemma 42.33.4).

Proof. This follows immediately from the characterization of these classes in Lemma 42.47.1. $\square$

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