The Stacks project

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

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. $\square$

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 82.27.1 we see that

Next, we see that

and so on. Hence the map is injective.

To prove the map is surjective, we will argue exactly as in the proof of Lemma 82.25.1 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 82.7.1. 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

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

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

will be rationally equivalent to $\beta $ (see Remark 82.15.3). 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

for some $\alpha _ i \in Z_{k + i}(U)$. By Lemma 82.10.2 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 82.15.2 we see that

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 42.36.2. $\square$

Proof. (For the case of linebundles, see Lemma 82.25.2.) For surjectivity see Lemma 82.25.1. 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 71.7.8). 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

are the cycles supported in the effective Cartier divisor $D$, see Lemma 82.15.2. So if $p^*\alpha = 0$, then $\pi ^*\alpha = i_*\beta $ for some $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + r}(D)$. By Lemma 82.27.2 we may write

for some $\beta _ i \in \mathop{\mathrm{CH}}\nolimits _{k + i}(X)$. By Lemmas 82.24.1 and 82.19.4 this implies

Since the rank of $\mathcal{E} \oplus \mathcal{O}_ X$ is $r + 1$ this contradicts Lemma 82.19.4 unless all $\alpha $ and all $\beta _ i$ are zero. $\square$