The Stacks project

Lemma 42.43.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}_ i$ be a finite collection of locally free $\mathcal{O}_ X$-modules of rank $r_ i$. There exists a projective flat morphism $\pi : P \to X$ of relative dimension $d$ such that

  1. for any morphism $f : Y \to X$ the map $\pi _ Y^* : \mathop{\mathrm{CH}}\nolimits _*(Y) \to \mathop{\mathrm{CH}}\nolimits _{* + d}(Y \times _ X P)$ is injective, and

  2. each $\pi ^*\mathcal{E}_ i$ has a filtration whose successive quotients $\mathcal{L}_{i, 1}, \ldots , \mathcal{L}_{i, r_ i}$ are invertible ${\mathcal O}_ P$-modules.

Moreover, when (1) holds the restriction map $A^*(X) \to A^*(P)$ (Remark 42.34.2) is injective.

Proof. We may assume $r_ i \geq 1$ for all $i$. We will prove the lemma by induction on $\sum (r_ i - 1)$. If this integer is $0$, then $\mathcal{E}_ i$ is invertible for all $i$ and we conclude by taking $\pi = \text{id}_ X$. If not, then we can pick an $i$ such that $r_ i > 1$ and consider the morphism $\pi _ i : P_ i = \mathbf{P}(\mathcal{E}_ i) \to X$. We have a short exact sequence

\[ 0 \to \mathcal{F} \to \pi _ i^*\mathcal{E}_ i \to \mathcal{O}_{P_ i}(1) \to 0 \]

of finite locally free $\mathcal{O}_{P_ i}$-modules of ranks $r_ i - 1$, $r_ i$, and $1$. Observe that $\pi _ i^*$ is injective on chow groups after any base change by the projective bundle formula (Lemma 42.36.2). By the induction hypothesis applied to the finite locally free $\mathcal{O}_{P_ i}$-modules $\mathcal{F}$ and $\pi _{i'}^*\mathcal{E}_{i'}$ for $i' \not= i$, we find a morphism $\pi : P \to P_ i$ with properties stated as in the lemma. Then the composition $\pi _ i \circ \pi : P \to X$ does the job. Some details omitted. $\square$

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