The Stacks project

Remark 42.56.10. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme and let $D_{Z, perf}(\mathcal{O}_ X)$ be as in Remark 42.56.7. If $X$ is quasi-compact (or more generally the irreducible components of $X$ are quasi-compact), then the localized Chern classes define a canonical map

\[ c(Z \to X, -) : K_0(D_{Z, perf}(\mathcal{O}_ X)) \longrightarrow A^0(X) \times \prod \nolimits _{i \geq 1} A^ i(Z \to X) \]

by sending a generator $[E]$ on the left hand side to

\[ c(Z \to X, E) = 1 + c_1(Z \to X, E) + c_2(Z \to X, E) + \ldots \]

and extending multiplicatively (with product on the right hand side as in Remark 42.34.7). The quasi-compactness condition on $X$ guarantees that the localized chern classes are defined (Situation 42.50.1 and Definition 42.50.3) and that these localized chern classes convert distinguished triangles into the corresponding products in the bivariant chow rings (Lemma 42.52.4).

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