Remark 42.37.11. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. In general we write $X = \coprod X_ r$ as in Remark 42.37.10. If only a finite number of the $X_ r$ are nonempty, then we can set

$c_{top}(\mathcal{E}) = \sum \nolimits _ r c_ r(\mathcal{E}|_{X_ r}) \in A^*(X) = \bigoplus A^*(X_ r)$

where the equality is Lemma 42.34.4. If infinitely many $X_ r$ are nonempty, we will use the same notation to denote

$c_{top}(\mathcal{E}) = \prod c_ r(\mathcal{E}|_{X_ r}) \in \prod A^ r(X_ r) \subset A^*(X)^\wedge$

see Remark 42.34.5 for notation.

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