Lemma 42.47.7. In Lemma 42.47.1 assume $E_2|_{X_1 \cap X_2}$ is zero. Then

and so on with multiplication as in Remark 42.34.7.

Lemma 42.47.7. In Lemma 42.47.1 assume $E_2|_{X_1 \cap X_2}$ is zero. Then

\begin{align*} P'_1(E_2) & = c'_1(E_2), \\ P'_2(E_2) & = c'_1(E_2)^2 - 2c'_2(E_2), \\ P'_3(E_2) & = c'_1(E_2)^3 - 3c'_1(E_2)c'_2(E_2) + 3c'_3(E_2), \\ P'_4(E_2) & = c'_1(E_2)^4 - 4c'_1(E_2)^2c'_2(E_2) + 4c'_1(E_2)c'_3(E_2) + 2c'_2(E_2)^2 - 4c'_4(E_2), \end{align*}

and so on with multiplication as in Remark 42.34.7.

**Proof.**
The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c'_ p(E_2)$ are defined for all $p \geq 1$. The equalities follow immediately from the characterization of the classes produced by Lemma 42.47.1 and the corresponding result for capping with the Chern classes of $E_2$ given in Remark 42.46.8.
$\square$

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