## 80.27 Projective space bundle formula

In Situation 80.2.1 let $X/B$ be good. 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 80.27.1. In Situation 80.2.1 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 80.15.3.

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 80.26.4 and 80.26.5 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 42.35.1. $\square$

Lemma 80.27.2 (Projective space bundle formula). Let $(S, \delta )$ be as in Situation 80.2.1. 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 80.27.1 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.

To prove the map is surjective, we will argue exactly as in the proof of Lemma 80.25.1 to reduce to the case of schemes. We urge the reader to skip the proof.

Let $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + r - 1}(P)$. Write $\beta = \sum m_ j[W_ j]$ with $m_ j \not= 0$ and $W_ j$ pairwise distinct integral closed subspaces of $\delta$-dimension $k + r$. Then the family $\{ W_ j\}$ is locally finite in $P$. Let $Z_ j \subset X$ be the “image” of $W_ j$ as in Lemma 80.7.1. For any quasi-compact open $U \subset X$ we see that $\pi ^{-1}(U) \cap W_ j$ is nonempty only for finitely many $j$. Hence the collection $Z_ j$ of images is a locally finite collection of integral closed subspaces of $X$.

Consider the fibre product diagrams

$\xymatrix{ P_ j \ar[r] \ar[d]_{\pi _ j} & P \ar[d]^\pi \\ Z_ j \ar[r] & X }$

Suppose that $[W_ j] \in Z_{k + r - 1}(P_ j)$ is rationally equivalent to

$\pi _ j^*\alpha _{j, 0} + c_1(\mathcal{O}(1)) \cap \pi _ j^*\alpha _{j, 1} + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi _ j^*\alpha _{j, r - 1}$

for some $(k + i)$-cycle $\alpha _{j, i} \in \mathop{\mathrm{CH}}\nolimits _{k + i}(Z_ j)$. Then $\alpha _ i = \sum m_ j \beta _{j, i}$ will be a $(k + i)$-cycle on $X$ and

$\pi ^*\alpha _0 + c_1(\mathcal{O}(1)) \cap \pi ^*\alpha _1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi ^*\alpha _{r - 1}$

will be rationally equivalent to $\beta$ (see Remark 80.15.3). This reduces us to the case $X$ integral, and $\alpha = [W]$ for some integral closed subscheme of $P$ dominating $X$. In particular we may assume that $d = \dim _\delta (X) < \infty$.

Hence we can use induction on $d = \dim _\delta (X)$. If $d < k$, then $\mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X) = 0$ and the lemma holds; this is the base case of the induction. Consider a nonempty open $U \subset X$. Suppose that we can show that

$\beta |_{\pi ^{-1}(U)} = \pi ^*\alpha _0 + c_1(\mathcal{O}(1)) \cap \pi ^*\alpha _1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi ^*\alpha _{r - 1}$

for some $\alpha _ i \in Z_{k + i}(U)$. By Lemma 80.10.2 we see that $\alpha _ i = \alpha '_ i|_ U$ for some $\alpha '_ i \in Z_{k + i}(X)$. By the exact sequences $\mathop{\mathrm{CH}}\nolimits _{k + i}(\pi ^{-1}(X \setminus U)) \to \mathop{\mathrm{CH}}\nolimits _{k + i}(P) \to \mathop{\mathrm{CH}}\nolimits _{k + i}(\pi ^{-1}(U))$ of Lemma 80.15.2 we see that

$\beta - \left(\pi ^*\alpha '_0 + c_1(\mathcal{O}(1)) \cap \pi ^*\alpha '_1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi ^*\alpha '_{r - 1}\right)$

comes from a cycle $\beta ' \in \mathop{\mathrm{CH}}\nolimits _{k + r}(\pi ^{-1}(X \setminus U))$. Since $\dim _\delta (X \setminus U) < d$ we win by induction on $d$.

In particular, by replacing $X$ by a suitable open we may assume $X$ is a scheme and we have reduced our problem to Chow Homology, Lemma 42.35.2. $\square$

Lemma 80.27.3. In Situation 80.2.1 let $X/B$ be good. 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 80.25.2.) For surjectivity see Lemma 80.25.1. 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 on Spaces, Lemma 69.7.8). 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 80.15.2. So if $p^*\alpha = 0$, then $\pi ^*\alpha = i_*\beta$ for some $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + r}(D)$. By Lemma 80.27.2 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 80.24.1 and 80.19.4 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 80.19.4 unless all $\alpha$ and all $\beta _ i$ are zero. $\square$

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