Definition 42.50.3. With $(S, \delta )$, $X$, $E \in D(\mathcal{O}_ X)$, and $i : Z \to X$ as in Situation 42.50.1.

1. If the restriction $E|_{X \setminus Z}$ is zero, then for all $p \geq 0$ we define

$P_ p(Z \to X, E) \in A^ p(Z \to X)$

by the construction in Lemma 42.50.2 and we define the localized Chern character by the formula

$ch(Z \to X, E) = \sum \nolimits _{p = 0, 1, 2, \ldots } \frac{P_ p(Z \to X, E)}{p!} \quad \text{in}\quad A^*(Z \to X) \otimes \mathbf{Q}$
2. If the restriction $E|_{X \setminus Z}$ is isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, then we define the localized $p$th Chern class $c_ p(Z \to X, E)$ by the construction in Lemma 42.50.2.

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