Let $k$ be a base field. Let $X$ be a smooth projective scheme over $k$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Our convention is that the projective bundle associated to $\mathcal{E}$ is the morphism
over $X$ with $\mathcal{O}_ P(1)$ normalized so that $p_*(\mathcal{O}_ P(1)) = \mathcal{E}$. Recall that
See Example 45.3.2. For $i = 0, \ldots , r - 1$ consider the correspondences
We may and do think of $c_ i$ as a morphism $h(X)(-i) \to h(P)$.
Lemma 45.6.1 (Projective space bundle formula). In the situation above, the map is an isomorphism in the category of motives.
\[ \sum \nolimits _{i = 0, \ldots , r - 1} c_ i : \bigoplus \nolimits _{i = 0, \ldots , r - 1} h(X)(-i) \longrightarrow h(P) \]
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. $\square$
In the situation above, for $j = 0, \ldots , r - 1$ consider the correspondences
For $i, j \in \{ 0, \ldots , r - 1\} $ we have
The cycles $\text{pr}_{12}^{-1}\Gamma _ p$ and $\text{pr}_{23}^{-1}\Gamma _ p^ t$ intersect transversally and with intersection equal to the image of $(p, 1, p) : P \to X \times P \times X$. Observe that the fibres of $(p, p) = \text{pr}_{13} \circ (p, 1, p) : P \to X \times X$ have dimension $r - 1$. We immediately conclude $c'_ j \circ c_ i = 0$ for $i + r - 1 - j < r - 1$, in other words when $i < j$. On the other hand, by the projective space bundle formula (Chow Homology, Lemma 42.36.2) the cycle $c_1(\mathcal{O}_ P(1))^{r - 1} \cap [P]$ maps to $[X]$ in $X$. Hence for $i = j$ the pushforward above gives the class of the diagonal and hence we see that
for all $i \in \{ 0, \ldots , r - 1\} $. Thus we see that the matrix of the composition
is invertible (upper triangular with $1$s on the diagonal). We conclude from the projective space bundle formula (Lemma 45.6.1) that also the composition the other way around is invertible, but it seems a bit harder to prove this directly.
Lemma 45.6.2. Let $p : P \to X$ be as in Lemma 45.6.1. The class $[\Delta _ P]$ of the diagonal of $P$ in $\mathop{\mathrm{CH}}\nolimits ^*(P \times P)$ can be written as where $\mathcal{S}$ is the kernel of the canonical surjection $p^*\mathcal{E} \to \mathcal{O}_ P(1)$.
\[ [\Delta _ P] = \left(\sum \nolimits _{i = 0, \ldots , r - 1} {r - 1 \choose i} c_{r - 1 - i}(\text{pr}_1^*\mathcal{S}^\vee ) \cap c_1(\text{pr}_2^*\mathcal{O}_ P(1))^ i\right) \cap (p \times p)^*[\Delta _ X] \]
Proof. Observe that $(p \times p)^*[\Delta _ X] = [P \times _ X P]$. Since $\Delta _ P \subset P \times _ X P \subset P \times P$ and since capping with Chern classes commutes with proper pushforward (Chow Homology, Lemma 42.38.4) it suffices to show that the class of $\Delta _ P \subset P \times _ X P$ in $\mathop{\mathrm{CH}}\nolimits ^*(P \times _ X P)$ is equal to
where $q_ i : P \times _ X P \to P$, $i = 1, 2$ are the projections. Set $q = p \circ q_1 = p \circ q_2 : P \times _ X P \to X$. Consider the maps
where the final arrow is the pullback by $q$ of the evaluation map $\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{E}^\vee \to \mathcal{O}_ X$. The source of the composition is a module locally free of rank $r - 1$ and a local calculation shows that this map vanishes exactly along $\Delta _ P$. By Chow Homology, Lemma 42.44.1 the class $[\Delta _ P]$ is the top Chern class of the dual
The desired result follows from Chow Homology, Lemma 42.39.1. $\square$
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