Lemma 82.30.3. In Situation 82.2.1 let $X/B$ be good. Suppose that $\mathcal{E}$ sits in an exact sequence

of finite locally free sheaves $\mathcal{E}_ i$ of rank $r_ i$. The total Chern classes satisfy

in $A^*(X)$.

Lemma 82.30.3. In Situation 82.2.1 let $X/B$ be good. Suppose that $\mathcal{E}$ sits in an exact sequence

\[ 0 \to \mathcal{E}_1 \to \mathcal{E} \to \mathcal{E}_2 \to 0 \]

of finite locally free sheaves $\mathcal{E}_ i$ of rank $r_ i$. The total Chern classes satisfy

\[ c(\mathcal{E}) = c(\mathcal{E}_1) c(\mathcal{E}_2) \]

in $A^*(X)$.

**Proof.**
The proof is identical to the proof of Chow Homology, Lemma 42.40.3 replacing the lemmas used there by Lemmas 82.26.9, 82.30.2, and 82.28.1.
$\square$

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