Lemma 42.38.4 (02U9)—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 42.38.4 (cite)

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Lemma 42.38.4. 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 $X$. Let $p : X \to Y$ be a proper morphism. Let $\alpha $ be a $k$-cycle on $X$. Let $\mathcal{E}$ be a finite locally free sheaf on $Y$. Then

\[ p_*(c_ j(p^*\mathcal{E}) \cap \alpha ) = c_ j(\mathcal{E}) \cap p_*\alpha \]

Proof. Let $(\pi : P \to Y, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}$. Then $P_ X = X \times _ Y P$ is the projective bundle associated to $p^*\mathcal{E}$ and $\mathcal{O}_{P_ X}(1)$ is the pullback of $\mathcal{O}_ P(1)$. Write $\alpha _ j = c_ j(p^*\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.38.2 we have

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

in the chow group of $P_ X$. Consider the fibre product diagram

\[ \xymatrix{ P_ X \ar[r]_-{p'} \ar[d]_{\pi _ X} & P \ar[d]^\pi \\ X \ar[r]^ p & Y } \]

Apply proper pushforward $p'_*$ (Lemma 42.20.3) to the displayed equality above. Using Lemmas 42.26.4 and 42.15.1 we obtain

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

in the chow group of $P$. By the characterization of Lemma 42.38.2 we conclude. $\square$

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