Definition 42.38.1. 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 sheaf of rank $r$ on $X$. We define, for every integer $k$ and any $0 \leq j \leq r$, an operation
\[ c_ j(\mathcal{E}) \cap - : Z_ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k - j}(X) \]
called intersection with the $j$th Chern class of $\mathcal{E}$.
Given an integral closed subscheme $i : W \to X$ of $\delta $-dimension $k$ we define
\[ c_ j(\mathcal{E}) \cap [W] = i_*(c_ j({i^*\mathcal{E}}) \cap [W]) \in \mathop{\mathrm{CH}}\nolimits _{k - j}(X) \]where $c_ j({i^*\mathcal{E}}) \cap [W]$ is as defined in Definition 42.37.1.
For a general $k$-cycle $\alpha = \sum n_ i [W_ i]$ we set
\[ c_ j(\mathcal{E}) \cap \alpha = \sum n_ i c_ j(\mathcal{E}) \cap [W_ i] \]
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