Lemma 82.28.1. In Situation 82.2.1 let X/B be good. Let \mathcal{E} be a finite locally free sheaf of rank r on X. Let (\pi : P \to X, \mathcal{O}_ P(1)) be the projective space bundle associated to \mathcal{E}. For every morphism X' \to X of good algebraic spaces over B there are unique maps
c_ i(\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - i}(X'),\quad i = 0, \ldots , r
such that for \alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X') we have c_0(\mathcal{E}) \cap \alpha = \alpha and
\sum \nolimits _{i = 0, \ldots , r} (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap (\pi ')^*\left(c_{r - i}(\mathcal{E}) \cap \alpha \right) = 0
where \pi ' : P' \to X' is the base change of \pi . Moreover, these maps define a bivariant class c_ i(\mathcal{E}) of degree i on X.
Proof.
Uniqueness and existence of the maps c_ i(\mathcal{E}) \cap - follows immediately from Lemma 82.27.2 and the given description of c_0(\mathcal{E}). For every i \in \mathbf{Z} the rule which to every morphism X' \to X of good algebraic spaces over B assigns the map
t_ i(\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - i}(X'),\quad \alpha \longmapsto \pi '_*(c_1(\mathcal{O}_{P'}(1))^{r - 1 + i} \cap (\pi ')^*\alpha )
is a bivariant class1 by Lemmas 82.26.4, 82.26.5, and 82.26.7. By Lemma 82.27.1 we have t_ i(\mathcal{E}) = 0 for i < 0 and t_0(\mathcal{E}) = 1. Applying pushforward to the equation in the statement of the lemma we find from Lemma 82.27.1 that
(-1)^ r t_1(\mathcal{E}) + (-1)^{r - 1}c_1(\mathcal{E}) = 0
In particular we find that c_1(\mathcal{E}) is a bivariant class. If we multiply the equation in the statement of the lemma by c_1(\mathcal{O}_{P'}(1)) and push the result forward to X' we find
(-1)^ r t_2(\mathcal{E}) + (-1)^{r - 1} t_1(\mathcal{E}) \cap c_1(\mathcal{E}) + (-1)^{r - 2} c_2(\mathcal{E}) = 0
As before we conclude that c_2(\mathcal{E}) is a bivariant class. And so on.
\square
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