Lemma 42.47.4. In Lemma 42.47.1 say $E_2$ is the restriction of a perfect $E \in D(\mathcal{O}_ X)$ such that $E|_{X_1}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$. If Chern classes of $E$ are defined, then $i_{2, *} \circ P'_ p(E_2) = P_ p(E)$, resp. $i_{2, *} \circ c'_ p(E_2) = c_ p(E)$ (with $\circ$ as in Lemma 42.33.4).

Proof. First, assume $E|_{X_1}$ is zero. With notations as in the proof of Lemma 42.47.1 the lemma in this case follows from

\begin{align*} P_ p(E) \cap \alpha ' & = i'_{1, *}(P_ p(E) \cap \alpha '_1) + i'_{2, *}(P_ p(E) \cap \alpha '_2) \\ & = i'_{1, *}(P_ p(E|_{X_1}) \cap \alpha '_1) + i'_{2, *}(P'_ p(E_2) \cap \alpha ') \\ & = i'_{2, *}(P'_ p(E_2) \cap \alpha ') \end{align*}

The case where $E|_{X_1}$ is isomorphic to a finite locally free $\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$ is similar. $\square$

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