Lemma 82.30.1 (0ES7)—The Stacks project

Table of contents
Part 5: Topics in Geometry
Chapter 82: Chow Groups of Spaces
Section 82.30: Additivity of Chern classes
Lemma 82.30.1 (cite)

next lemma

Lemma 82.30.1. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{E}$, $\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence

\[ 0 \to \mathcal{O}_ X \to \mathcal{E} \to \mathcal{F} \to 0 \]

Then we have

\[ c_ r(\mathcal{E}) = 0, \quad c_ j(\mathcal{E}) = c_ j(\mathcal{F}), \quad j = 0, \ldots , r - 1 \]

in $A^*(X)$.

Proof. The proof is identical to the proof of Chow Homology, Lemma 42.40.1 replacing the lemmas used there by Lemmas 82.26.9, 82.24.1, 82.19.4, and 82.28.1. $\square$

The Stacks project

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