Lemma 82.27.1. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ of rank $r$. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}$. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ the element
\[ \pi _*\left( c_1(\mathcal{O}_ P(1))^ s \cap \pi ^*\alpha \right) \in \mathop{\mathrm{CH}}\nolimits _{k + r - 1 - s}(X) \]
is $0$ if $s < r - 1$ and is equal to $\alpha $ when $s = r - 1$.
Proof.
Let $Z \subset X$ be an integral closed subspace of $\delta $-dimension $k$. We will prove the lemma for $\alpha = [Z]$. We omit the argument deducing the general case from this special case; hint: argue as in Remark 82.15.3.
Let $P_ Z = P \times _ X Z$ be the base change; of course $\pi _ Z : P_ Z \to Z$ is the projective bundle associated to $\mathcal{E}|_ Z$ and $\mathcal{O}_ P(1)$ pulls back to the corresponding invertible module on $P_ Z$. Since $c_1(\mathcal{O}_ P(1) \cap -$, and $\pi ^*$ are bivariant classes by Lemmas 82.26.4 and 82.26.5 we see that
\[ \pi _*\left( c_1(\mathcal{O}_ P(1))^ s \cap \pi ^*[Z] \right) = (Z \to X)_*\pi _{Z, *}\left( c_1(\mathcal{O}_{P_ Z}(1))^ s \cap \pi _ Z^*[Z] \right) \]
Hence it suffices to prove the lemma in case $X$ is integral and $\alpha = [X]$.
Assume $X$ is integral, $\dim _\delta (X) = k$, and $\alpha = [X]$. Note that $\pi ^*[X] = [P]$ as $P$ is integral of $\delta $-dimension $r - 1$. If $s < r - 1$, then by construction $c_1(\mathcal{O}_ P(1))^ s \cap [P]$ a $(k + r - 1 - s)$-cycle. Hence the pushforward of this cycle is zero for dimension reasons.
Let $s = r - 1$. By the argument given above we see that $\pi _*(c_1(\mathcal{O}_ P(1))^ s \cap [P]) = n [X]$ for some $n \in \mathbf{Z}$. We want to show that $n = 1$. For the same dimension reasons as above it suffices to prove this result after replacing $X$ by a dense open. Thus we may assume $X$ is a scheme and the result follows from Chow Homology, Lemma 42.36.1.
$\square$
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