The Stacks project

Lemma 42.40.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let ${\mathcal L}_ i$, $i = 1, \ldots , r$ be invertible $\mathcal{O}_ X$-modules on $X$. Let $\mathcal{E}$ be a locally free rank $\mathcal{O}_ X$-module endowed with a filtration

\[ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_ r = \mathcal{E} \]

such that $\mathcal{E}_ i/\mathcal{E}_{i - 1} \cong \mathcal{L}_ i$. Set $c_1({\mathcal L}_ i) = x_ i$. Then

\[ c(\mathcal{E}) = \prod \nolimits _{i = 1}^ r (1 + x_ i) \]

in $A^*(X)$.

Proof. Apply Lemma 42.40.2 and induction. $\square$

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