Lemma 80.28.1. In Situation 80.2.1 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 80.27.2 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 80.26.4, 80.26.5, and 80.26.7. By Lemma 80.27.1 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 80.27.1 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. $\square$

[1] Up to signs these are the Segre classes of $\mathcal{E}$.

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