Lemma 81.30.1. 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

$0 \to \mathcal{O}_ X \to \mathcal{E} \to \mathcal{F} \to 0$

Then we have

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

in $A^*(X)$.

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