The Stacks project

Proof. The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c'_ p(Q)$ are defined for all $p \geq 1$. In the proof of Lemma 42.49.1 we have constructed the classes $P'_ p(Q)$ and $c'_ p(Q)$ using the bivariant class $C \in A^0(W_\infty \to X)$ of Lemma 42.48.1 and the bivariant classes $P'_ p(Q|_ E)$ and $c'_ p(Q|_ E)$ of Lemma 42.47.1 for the restriction $Q|_ E$ of $Q$ to the inverse image $E$ of $Z$ in $W_\infty $. Observe that by Lemma 42.47.7 we have the desired relationship between $P'_ p(Q|_ E)$ and $c'_ p(Q|_ E)$. Recall that

To finish the proof it suffices to show the multiplications defined in Remark 42.34.7 on the classes $a_ p = c'_ p(Q)$ and on the classes $b_ p = c'_ p(Q|_ E)$ agree:

Some details omitted. If $r = 1$, then this is true. For $r > 1$ note that by Remark 42.34.8 the multiplication in Remark 42.34.7 proceeds by inserting $(Z \to X)_*$, resp. $(E \to W_\infty )_*$ in between the factors of the product $a_{p_1}a_{p_2} \ldots a_{p_ r}$, resp. $b_{p_1}b_{p_2} \ldots b_{p_ r}$ and taking compositions as bivariant classes. Now by Lemma 42.47.1 we have

and by Lemma 42.49.4 we have

for $i = 2, \ldots , r$. A calculation shows that the left and right hand side of the desired equality both simplify to

and the proof is complete. $\square$