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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