Processing math: 100%

The Stacks project

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

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

Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.