The Stacks project

Lemma 81.31.1. In Situation 81.2.1 let $X/B$ be good. 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 $f : Y \to X$ of good algebraic spaces over $B$ 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.

Proof. We prove this by induction on the integer $r = \sum r_ i$. If $r = 0$ we can take $\pi = \text{id}_ X$. If $r_ i = 1$ for all $i$, then we can also take $\pi = \text{id}_ X$. Assume that $r_{i_0} > 1$ for some $i_0$. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}_{i_0}$. The canonical map $\pi ^*\mathcal{E}_{i_0} \to \mathcal{O}_ P(1)$ is surjective and hence its kernel $\mathcal{E}'_{i_0}$ is finite locally free of rank $r_{i_0} - 1$. Observe that $\pi _ Y^*$ is injective for any morphism $f : Y \to X$ of good algebraic spaces over $B$, see Lemma 81.27.2. Thus it suffices to prove the lemma for $P$ and the locally free sheaves $\pi ^*\mathcal{E}_ i$. However, because we have the subbundle $\mathcal{E}_{i_0} \subset \pi ^*\mathcal{E}_{i_0}$ with invertible quotient, it now suffices to prove the lemma for the collection $\{ \mathcal{E}_ i\} _{i \not= i_0} \cup \{ \mathcal{E}'_{i_0}\} $. This decreases $r$ by $1$ and we win by induction hypothesis. $\square$

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