Lemma 45.14.9. Assume given data (D0), (D1), and (D2') satisfying axioms (A1) – (A7). Let $p : P \to X$ be as in axiom (A3) with $X$ nonempty equidimensional. Then $\gamma $ commutes with pushforward along $p$.
Proof. It suffices to prove this on generators for $\mathop{\mathrm{CH}}\nolimits _*(P)$. Thus it suffices to prove this for a cycle class of the form $\xi ^ i \cdot p^*\alpha $ where $0 \leq i \leq r - 1$ and $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(X)$. Note that $p_*(\xi ^ i \cdot p^*\alpha ) = 0$ if $i < r - 1$ and $p_*(\xi ^{r - 1} \cdot p^*\alpha ) = \alpha $. On the other hand, we have $\gamma (\xi ^ i \cdot p^*\alpha ) = c^ i \cup p^*\gamma (\alpha )$ and by the projection formula (Lemma 45.9.1) we have
\[ p_*\gamma (\xi ^ i \cdot p^*\alpha ) = p_*(c^ i) \cup \gamma (\alpha ) \]
Thus it suffices to show that $p_*c^ i = 0$ for $i < r - 1$ and $p_*c^{r - 1} = 1$. Equivalently, it suffices to prove that $\lambda _ P : H^{2d + 2r - 2}(P)(d + r - 1) \to F$ defined by the rules
\[ \lambda _ P(c^ i \cup p^*(a)) = \left\{ \begin{matrix} 0
& \text{if}
& i < r - 1
\\ \int _ X(a)
& \text{if}
& i = r - 1
\end{matrix} \right. \]
satisfies the condition of axiom (A5). This follows from the computation of the class of the diagonal of $P$ in Lemma 45.6.2. $\square$
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Your email address will not be published.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.
Comments (0)