The Stacks project

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

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 $\alpha = \sum n_ j [Z_ j]$. $\square$

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

Next, we see that

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

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

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

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

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. $\square$

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

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

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

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. $\square$