The Stacks project

Proof. Recall that the image of $c'_ i(Q)$ in $A^ p(X)$ is equal to $c_ i(Q|_{X \times \{ 0\} })$ for $i \geq p$ and similarly for $Q'$ and $Q \oplus Q'$, see Lemma 42.49.1. Hence the equality in degrees $< p + p'$ follows from the additivity of Lemma 42.46.7.

Let's take $n \geq p + p'$. As in the proof of Lemma 42.49.1 let $E \subset W_\infty$ denote the inverse image of $Z$. Observe that we have the equality

in $A^{(p + p')}(E \to W_\infty )$ by Lemma 42.47.8. Since by construction

we conclude that suffices to show for all $i + j = n$ we have

in $A^ n(Z \to X)$ where the multiplication is the one from Remark 42.34.7 on both sides. There are three cases, depending on whether $i \geq p$, $j \geq p'$, or both.

Assume $i \geq p$ and $j \geq p'$. In this case the products are defined by inserting $(E \to W_\infty )_*$, resp. $(Z \to X)_*$ in between the two factors and taking compositions as bivariant classes, see Remark 42.34.8. In other words, we have to show

By Lemma 42.47.1 the left hand side is equal to

Since $c'_ i(Q) = (E \to Z)_* \circ c'_ i(Q|_ E) \circ C$ the right hand side is equal to

which is immediately seen to be equal to the above by Lemma 42.49.4.

Assume $i \geq p$ and $j < p$. Unwinding the products in this case we have to show

Again using that $c'_ i(Q) = (E \to Z)_* \circ c'_ i(Q|_ E) \circ C$ we see that it suffices to show $c_ j(Q'|_{W_\infty }) \circ C = C \circ c_ j(Q'|_{X \times \{ 0\} })$ which is part of Lemma 42.49.4.

Assume $i < p$ and $j \geq p'$. Unwinding the products in this case we have to show

However, since $c'_ j(Q|_ E)$ and $c'_ j(Q')$ are bivariant classes, they commute with capping with Chern classes (Lemma 42.38.9). Hence it suffices to prove

which we reduces us to the case discussed in the preceding paragraph. $\square$