Lemma 81.30.2. In Situation 81.2.1 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

where $\mathcal{L}$ is an invertible sheaf. Then

in $A^*(X)$.

Lemma 81.30.2. In Situation 81.2.1 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

\[ 0 \to \mathcal{L} \to \mathcal{E} \to \mathcal{F} \to 0 \]

where $\mathcal{L}$ is an invertible sheaf. Then

\[ c(\mathcal{E}) = c(\mathcal{L}) c(\mathcal{F}) \]

in $A^*(X)$.

**Proof.**
The proof is identical to the proof of Chow Homology, Lemma 42.40.2 replacing the lemmas used there by Lemmas 81.30.1 and 81.29.1.
$\square$

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