Definition 42.37.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. For $i = 0, \ldots , r$ the $i$th Chern class of $\mathcal{E}$ is the bivariant class $c_ i(\mathcal{E}) \in A^ i(X)$ of degree $i$ constructed in Lemma 42.37.7. The total Chern class of $\mathcal{E}$ is the formal sum

$c(\mathcal{E}) = c_0(\mathcal{E}) + c_1(\mathcal{E}) + \ldots + c_ r(\mathcal{E})$

which is viewed as a nonhomogeneous bivariant class on $X$.

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