Lemma 42.38.5 (02U8)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.38: Intersecting with Chern classes
Lemma 42.38.5 (cite)

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Lemma 42.38.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ , $Y$ be locally of finite type over $S$ . Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $Y$ . Let $f : X \to Y$ be a flat morphism of relative dimension $r$ . Let $\alpha$ be a $k$ -cycle on $Y$ . Then

$f^*(c_ j(\mathcal{E}) \cap \alpha ) = c_ j(f^*\mathcal{E}) \cap f^*\alpha$

Proof. Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha$ , so $\alpha _0 = \alpha$ . By Lemma 42.38.2 we have

$\sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i}) = 0$

in the chow group of the projective bundle $(\pi : P \to Y, \mathcal{O}_ P(1))$ associated to $\mathcal{E}$ . Consider the fibre product diagram

$\xymatrix{ P_ X = \mathbf{P}(f^*\mathcal{E}) \ar[r]_-{f'} \ar[d]_{\pi _ X} & P \ar[d]^\pi \\ X \ar[r]^ f & Y }$

Note that $\mathcal{O}_{P_ X}(1)$ is the pullback of $\mathcal{O}_ P(1)$ . Apply flat pullback $(f')^*$ (Lemma 42.20.2) to the displayed equation above. By Lemmas 42.26.2 and 42.14.3 we see that

$\sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P_ X}(1))^ i \cap \pi _ X^*(f^*\alpha _{r - i}) = 0$

holds in the chow group of $P_ X$ . By the characterization of Lemma 42.38.2 we conclude. $\square$

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