The Stacks project

Lemma 42.46.4. 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. If one of the following conditions hold, then the Chern classes of $E$ are defined:

  1. 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. $E$ can be represented by a bounded complex of finite locally free $\mathcal{O}_ X$-modules,

  3. the irreducible components of $X$ are quasi-compact,

  4. $X$ is quasi-compact,

  5. there exists a morphism $X \to X'$ of schemes locally of finite type over $S$ such that $E$ is the pullback of a perfect object $E'$ on $X'$ whose chern classes are defined, or

  6. add more here.

Proof. Condition (1) is just Definition 42.46.3 part (1). Condition (2) implies (1).

As in (3) assume the irreducible components $X_ i$ of $X$ are quasi-compact. We view $X_ i$ as a reduced integral closed subscheme over $X$. The morphism $\coprod X_ i \to X$ is an envelope. For each $i$ there exists an envelope $X'_ i \to X_ i$ such that $X'_ i$ has an ample family of invertible modules, see More on Morphisms, Proposition 37.77.3. Observe that $f : Y = \coprod X'_ i \to X$ is an envelope; small detail omitted. By Derived Categories of Schemes, Lemma 36.36.7 each $X'_ i$ has the resolution property. Thus the perfect object $L(f|_{X'_ i})^*E$ of $D(\mathcal{O}_{X'_ i})$ can be represented by a bounded complex of finite locally free $\mathcal{O}_{X'_ i}$-modules, see Derived Categories of Schemes, Lemma 36.37.2. This proves (3) implies (1).

Part (4) implies (3).

Let $g : X \to X'$ and $E'$ be as in part (5). Then there exists an envelope $f' : Y' \to X'$ such that $L(f')^*E'$ is represented by a locally bounded complex $(\mathcal{E}')^\bullet $ of $\mathcal{O}_{Y'}$-modules. Then the base change $f : Y \to X$ is an envelope by Lemma 42.22.3. Moreover, the pulllback $\mathcal{E}^\bullet = g^*(\mathcal{E}')^\bullet $ represents $Lf^*E$ and we see that the chern classes of $E$ are defined. $\square$


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