The Stacks project

Definition 42.46.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object.

  1. We say the Chern classes of $E$ are defined1 if there exists an envelope $f : Y \to X$ such that $Lf^*E$ is isomorphic in $D(\mathcal{O}_ Y)$ to a locally bounded complex of finite locally free $\mathcal{O}_ Y$-modules.

  2. If the Chern classes of $E$ are defined, then we define

    \[ c(E) \in \prod \nolimits _{p \geq 0} A^ p(X),\quad ch(E) \in \prod \nolimits _{p \geq 0} A^ p(X) \otimes \mathbf{Q},\quad P_ p(E) \in A^ p(X) \]

    by an application of Lemma 42.46.2.

[1] See Lemma 42.46.4 for some criteria.

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