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

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

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.

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