in $A^*(Y \times _ X Z \to X)$.
Proof. Let $E \subset W_\infty $ be the inverse image of $Z$. Recall that $P'_ p(Q) = (E \to Z)_* \circ P'_ p(Q|_ E) \circ C$, resp. $c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C$ where $C$ is as in Lemma 42.47.1 and $P'_ p(Q|_ E)$, resp. $c'_ p(Q|_ E)$ are as in Lemma 42.46.1. By Lemma 42.47.4 we see that $C$ commutes with $c$ and by Lemma 42.46.5 we see that $P'_ p(Q|_ E)$, resp. $c'_ p(Q|_ E)$ commutes with $c$. Since $c$ is a bivariant class it commutes with proper pushforward by $E \to Z$ by definition. This finishes the proof. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.