The Stacks project

Remark 42.43.2. The proof of Lemma 42.43.1 shows that the morphism $\pi : P \to X$ has the following additional properties:

1. $\pi $ is a finite composition of projective space bundles associated to locally free modules of finite constant rank, and

2. for every $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ we have $\alpha = \pi _*(\xi _1 \cap \ldots \cap \xi _ d \cap \pi ^*\alpha )$ where $\xi _ i$ is the first Chern class of some invertible $\mathcal{O}_ P$-module.

The second observation follows from the first and Lemma 42.36.1. We will add more observations here as needed.