Lemma 42.36.2 (Projective space bundle formula). Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. 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 $\mathcal{E}$. The map
\[ \bigoplus \nolimits _{i = 0}^{r - 1} \mathop{\mathrm{CH}}\nolimits _{k + i}(X) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(P), \]
\[ (\alpha _0, \ldots , \alpha _{r-1}) \longmapsto \pi ^*\alpha _0 + c_1(\mathcal{O}_ P(1)) \cap \pi ^*\alpha _1 + \ldots + c_1(\mathcal{O}_ P(1))^{r - 1} \cap \pi ^*\alpha _{r-1} \]
is an isomorphism.
Proof.
Fix $k \in \mathbf{Z}$. We first show the map is injective. Suppose that $(\alpha _0, \ldots , \alpha _{r - 1})$ is an element of the left hand side that maps to zero. By Lemma 42.36.1 we see that
\[ 0 = \pi _*(\pi ^*\alpha _0 + c_1(\mathcal{O}_ P(1)) \cap \pi ^*\alpha _1 + \ldots + c_1(\mathcal{O}_ P(1))^{r - 1} \cap \pi ^*\alpha _{r-1}) = \alpha _{r - 1} \]
Next, we see that
\[ 0 = \pi _*(c_1(\mathcal{O}_ P(1)) \cap (\pi ^*\alpha _0 + c_1(\mathcal{O}_ P(1)) \cap \pi ^*\alpha _1 + \ldots + c_1(\mathcal{O}_ P(1))^{r - 2} \cap \pi ^*\alpha _{r - 2})) = \alpha _{r - 2} \]
and so on. Hence the map is injective.
It remains to show the map is surjective. Let $X_ i$, $i \in I$ be the irreducible components of $X$. Then $P_ i = \mathbf{P}(\mathcal{E}|_{X_ i})$, $i \in I$ are the irreducible components of $P$. Consider the commutative diagram
\[ \xymatrix{ \coprod P_ i \ar[d]_{\coprod \pi _ i} \ar[r]_ p & P \ar[d]^\pi \\ \coprod X_ i \ar[r]^ q & X } \]
Observe that $p_*$ is surjective. If $\beta \in \mathop{\mathrm{CH}}\nolimits _ k(\coprod X_ i)$ then $\pi ^* q_* \beta = p_*(\coprod \pi _ i)^* \beta $, see Lemma 42.15.1. Similarly for capping with $c_1(\mathcal{O}(1))$ by Lemma 42.26.4. Hence, if the map of the lemma is surjective for each of the morphisms $\pi _ i : P_ i \to X_ i$, then the map is surjective for $\pi : P \to X$. Hence we may assume $X$ is irreducible. Thus $\dim _\delta (X) < \infty $ and in particular we may use induction on $\dim _\delta (X)$.
The result is clear if $\dim _\delta (X) < k$. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(P)$. For any locally closed subscheme $T \subset X$ denote $\gamma _ T : \bigoplus \mathop{\mathrm{CH}}\nolimits _{k + i}(T) \to \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(\pi ^{-1}(T))$ the map
\[ \gamma _ T(\alpha _0, \ldots , \alpha _{r - 1}) = \pi ^*\alpha _0 + \ldots + c_1(\mathcal{O}_{\pi ^{-1}(T)}(1))^{r - 1} \cap \pi ^*\alpha _{r - 1}. \]
Suppose for some nonempty open $U \subset X$ we have $\alpha |_{\pi ^{-1}(U)} = \gamma _ U(\alpha _0, \ldots , \alpha _{r - 1})$. Then we may choose lifts $\alpha '_ i \in \mathop{\mathrm{CH}}\nolimits _{k + i}(X)$ and we see that $\alpha - \gamma _ X(\alpha '_0, \ldots , \alpha '_{r - 1})$ is by Lemma 42.19.3 rationally equivalent to a $k$-cycle on $P_ Y = \mathbf{P}(\mathcal{E}|_ Y)$ where $Y = X \setminus U$ as a reduced closed subscheme. Note that $\dim _\delta (Y) < \dim _\delta (X)$. By induction the result holds for $P_ Y \to Y$ and hence the result holds for $\alpha $. Hence we may replace $X$ by any nonempty open of $X$.
In particular we may assume that $\mathcal{E} \cong \mathcal{O}_ X^{\oplus r}$. In this case $\mathbf{P}(\mathcal{E}) = X \times \mathbf{P}^{r - 1}$. Let us use the stratification
\[ \mathbf{P}^{r - 1} = \mathbf{A}^{r - 1} \amalg \mathbf{A}^{r - 2} \amalg \ldots \amalg \mathbf{A}^0 \]
The closure of each stratum is a $\mathbf{P}^{r - 1 - i}$ which is a representative of $c_1(\mathcal{O}(1))^ i \cap [\mathbf{P}^{r - 1}]$. Hence $P$ has a similar stratification
\[ P = U^{r - 1} \amalg U^{r - 2} \amalg \ldots \amalg U^0 \]
Let $P^ i$ be the closure of $U^ i$. Let $\pi ^ i : P^ i \to X$ be the restriction of $\pi $ to $P^ i$. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(P)$. By Lemma 42.32.1 we can write $\alpha |_{U^{r - 1}} = \pi ^*\alpha _0|_{U^{r - 1}}$ for some $\alpha _0 \in \mathop{\mathrm{CH}}\nolimits _ k(X)$. Hence the difference $\alpha - \pi ^*\alpha _0$ is the image of some $\alpha ' \in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(P^{r - 2})$. By Lemma 42.32.1 again we can write $\alpha '|_{U^{r - 2}} = (\pi ^{r - 2})^*\alpha _1|_{U^{r - 2}}$ for some $\alpha _1 \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(X)$. By Lemma 42.31.1 we see that the image of $(\pi ^{r - 2})^*\alpha _1$ represents $c_1(\mathcal{O}_ P(1)) \cap \pi ^*\alpha _1$. We also see that $\alpha - \pi ^*\alpha _0 - c_1(\mathcal{O}_ P(1)) \cap \pi ^*\alpha _1$ is the image of some $\alpha '' \in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(P^{r - 3})$. And so on.
$\square$
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