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.

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\[ ch(Z \to X, E) = \sum \nolimits _{p = 0, 1, 2, \ldots } \frac{P_ p(Z \to X, E)}{p!} \quad \text{in}\quad \prod \nolimits _{p \geq 0} A^ p(Z \to X) \otimes \mathbf{Q} \]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.

## Comments (0)