The Stacks project

Lemma 45.6.1 (Projective space bundle formula). In the situation above, the map

\[ \sum \nolimits _{i = 0, \ldots , r - 1} c_ i : \bigoplus \nolimits _{i = 0, \ldots , r - 1} h(X)(-i) \longrightarrow h(P) \]

is an isomorphism in the category of motives.

Proof. By Lemma 45.5.3 it suffices to show that our map defines an isomorphism on Chow groups of motives after taking the product with any smooth projective scheme $Z$. Observe that $P \times Z \to X \times Z$ is the projective bundle associated to the pullback of $\mathcal{E}$ to $X \times Z$. Hence the statement on Chow groups is true by the projective space bundle formula given in Chow Homology, Lemma 42.36.2. Namely, pushforward of cycles along $[\Gamma _ p]$ is given by pullback of cycles by $p$ according to Lemma 45.3.6 and Chow Homology, Lemma 42.59.5. Hence pushforward along $c_ i$ sends $\alpha $ to $c_1(\mathcal{O}_ P(1))^ i \cap p^*\alpha $. Some details omitted. $\square$

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