Definition 42.36.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim _\delta (X)$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective space bundle associated to $\mathcal{E}$.

By Lemma 42.35.2 there are elements $c_ i \in \mathop{\mathrm{CH}}\nolimits _{n - i}(X)$, $i = 0, \ldots , r$ such that $c_0 = [X]$, and

42.36.1.1\begin{equation} \label{chow-equation-chern-classes} \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*c_{r - i} = 0. \end{equation}With notation as above we set $c_ i(\mathcal{E}) \cap [X] = c_ i$ as an element of $\mathop{\mathrm{CH}}\nolimits _{n - i}(X)$. We call these the

*Chern classes of $\mathcal{E}$ on $X$*.The

*total Chern class of $\mathcal{E}$ on $X$*is the combination\[ c({\mathcal E}) \cap [X] = c_0({\mathcal E}) \cap [X] + c_1({\mathcal E}) \cap [X] + \ldots + c_ r({\mathcal E}) \cap [X] \]which is an element of $\mathop{\mathrm{CH}}\nolimits _*(X) = \bigoplus _{k \in \mathbf{Z}} \mathop{\mathrm{CH}}\nolimits _ k(X)$.

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