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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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