The Stacks project

Remark 42.56.11. 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 the irreducible components of $X$ are quasi-compact, then the localized Chern character defines a canonical additive and multiplicative map

by sending a generator $[E]$ on the left hand side to $ch(Z \to X, E)$ and extending additively. Namely, the condition on the irreducible components of $X$ guarantees that the localized chern character is defined (Situation 42.50.1 and Definition 42.50.3) and that these localized chern characters convert distinguished triangles into the corresponding sums in the bivariant chow rings (Lemma 42.52.5). The multiplication on $K_0(D_{Z, perf}(X))$ is defined using derived tensor product (Derived Categories of Schemes, Remark 36.38.9) hence $ch(Z \to X, \alpha \beta ) = ch(Z \to X, \alpha ) ch(Z \to X, \beta )$ by Lemma 42.52.6. If $X$ is quasi-compact, then the map $ch(Z \to X, -)$ has image contained in $A^*(Z \to X) \otimes \mathbf{Q}$; we omit the details.