Definition 82.28.2. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. 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 82.28.1. 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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