## 42.36 Projective space bundle formula

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Consider a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ of rank $r$. Our convention is that the projective bundle associated to $\mathcal{E}$ is the morphism

$\xymatrix{ \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_ X(\text{Sym}^*(\mathcal{E})) \ar[r]^-\pi & X }$

over $X$ with $\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$ normalized so that $\pi _*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)) = \mathcal{E}$. In particular there is a surjection $\pi ^*\mathcal{E} \to \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. We will say informally “let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}$” to denote the situation where $P = \mathbf{P}(\mathcal{E})$ and $\mathcal{O}_ P(1) = \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$.

Lemma 42.36.1. 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 $\mathcal{E}$. 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 subscheme of $\delta$-dimension $k$. Note that $\pi ^*[Z] = [\pi ^{-1}(Z)]$ as $\pi ^{-1}(Z)$ is integral of $\delta$-dimension $r - 1$. If $s < r - 1$, then by construction $c_1(\mathcal{O}_ P(1))^ s \cap \pi ^*[Z]$ is represented by a $(k + r - 1 - s)$-cycle supported on $\pi ^{-1}(Z)$. Hence the pushforward of this cycle is zero for dimension reasons.

Let $s = r - 1$. By the argument given above we see that $\pi _*(c_1(\mathcal{O}_ P(1))^ s \cap \pi ^*\alpha ) = n [Z]$ 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 $X \setminus T$ where $T \subset Z$ is a proper closed subset. Let $\xi$ be the generic point of $Z$. We can choose elements $e_1, \ldots , e_{r - 1} \in \mathcal{E}_\xi$ which form part of a basis of $\mathcal{E}_\xi$. These give rational sections $s_1, \ldots , s_{r - 1}$ of $\mathcal{O}_ P(1)|_{\pi ^{-1}(Z)}$ whose common zero set is the closure of the image a rational section of $\mathbf{P}(\mathcal{E}|_ Z) \to Z$ union a closed subset whose support maps to a proper closed subset $T$ of $Z$. After removing $T$ from $X$ (and correspondingly $\pi ^{-1}(T)$ from $P$), we see that $s_1, \ldots , s_ n$ form a sequence of global sections $s_ i \in \Gamma (\pi ^{-1}(Z), \mathcal{O}_{\pi ^{-1}(Z)}(1))$ whose common zero set is the image of a section $Z \to \pi ^{-1}(Z)$. Hence we see successively that

\begin{eqnarray*} \pi ^*[Z] & = & [\pi ^{-1}(Z)] \\ c_1(\mathcal{O}_ P(1)) \cap \pi ^*[Z] & = & [Z(s_1)] \\ c_1(\mathcal{O}_ P(1))^2 \cap \pi ^*[Z] & = & [Z(s_1) \cap Z(s_2)] \\ \ldots & = & \ldots \\ c_1(\mathcal{O}_ P(1))^{r - 1} \cap \pi ^*[Z] & = & [Z(s_1) \cap \ldots \cap Z(s_{r - 1})] \end{eqnarray*}

by repeated applications of Lemma 42.25.4. Since the pushforward by $\pi$ of the image of a section of $\pi$ over $Z$ is clearly $[Z]$ we see the result when $\alpha = [Z]$. We omit the verification that these arguments imply the result for a general cycle $\alpha = \sum n_ j [Z_ j]$. $\square$

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. $\square$

Lemma 42.36.3. 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 sheaf of rank $r$ on $X$. Let

$p : E = \underline{\mathop{\mathrm{Spec}}}(\text{Sym}^*(\mathcal{E})) \longrightarrow X$

be the associated vector bundle over $X$. Then $p^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k + r}(E)$ is an isomorphism for all $k$.

Proof. (For the case of linebundles, see Lemma 42.32.2.) For surjectivity see Lemma 42.32.1. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective space bundle associated to the finite locally free sheaf $\mathcal{E} \oplus \mathcal{O}_ X$. Let $s \in \Gamma (P, \mathcal{O}_ P(1))$ correspond to the global section $(0, 1) \in \Gamma (X, \mathcal{E} \oplus \mathcal{O}_ X)$. Let $D = Z(s) \subset P$. Note that $(\pi |_ D : D \to X , \mathcal{O}_ P(1)|_ D)$ is the projective space bundle associated to $\mathcal{E}$. We denote $\pi _ D = \pi |_ D$ and $\mathcal{O}_ D(1) = \mathcal{O}_ P(1)|_ D$. Moreover, $D$ is an effective Cartier divisor on $P$. Hence $\mathcal{O}_ P(D) = \mathcal{O}_ P(1)$ (see Divisors, Lemma 31.14.10). Also there is an isomorphism $E \cong P \setminus D$. Denote $j : E \to P$ the corresponding open immersion. For injectivity we use that the kernel of

$j^* : \mathop{\mathrm{CH}}\nolimits _{k + r}(P) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k + r}(E)$

are the cycles supported in the effective Cartier divisor $D$, see Lemma 42.19.3. So if $p^*\alpha = 0$, then $\pi ^*\alpha = i_*\beta$ for some $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + r}(D)$. By Lemma 42.36.2 we may write

$\beta = \pi _ D^*\beta _0 + \ldots + c_1(\mathcal{O}_ D(1))^{r - 1} \cap \pi _ D^* \beta _{r - 1}.$

for some $\beta _ i \in \mathop{\mathrm{CH}}\nolimits _{k + i}(X)$. By Lemmas 42.31.1 and 42.26.4 this implies

$\pi ^*\alpha = i_*\beta = c_1(\mathcal{O}_ P(1)) \cap \pi ^*\beta _0 + \ldots + c_1(\mathcal{O}_ D(1))^ r \cap \pi ^*\beta _{r - 1}.$

Since the rank of $\mathcal{E} \oplus \mathcal{O}_ X$ is $r + 1$ this contradicts Lemma 42.26.4 unless all $\alpha$ and all $\beta _ i$ are zero. $\square$

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