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

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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