Lemma 82.29.1 (0ES4)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 82: Chow Groups of Spaces
Section 82.29: Polynomial relations among Chern classes
Lemma 82.29.1 (cite)

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Lemma 82.29.1. 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 $\mathcal{L}$ be an invertible sheaf on $X$ . Then we have

82.29.1.1

$\begin{equation} \label{spaces-chow-equation-twist} c_ i({\mathcal E} \otimes {\mathcal L}) = \sum \nolimits _{j = 0}^ i \binom {r - i + j}{j} c_{i - j}({\mathcal E}) c_1({\mathcal L})^ j \end{equation}$

in $A^*(X)$ .

Proof. The proof is identical to the proof of Chow Homology, Lemma 42.39.1 replacing the lemmas used there by Lemmas 82.26.9 and 82.28.1. $\square$

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