Remark 42.56.12. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ and assume $X$ is quasi-compact (or more generally the irreducible components of $X$ are quasi-compact). With $Z = X$ and notation as in Remarks 42.56.10 and 42.56.11 we have $D_{Z, perf}(\mathcal{O}_ X) = D_{perf}(\mathcal{O}_ X)$ and we see that

$K_0(D_{Z, perf}(\mathcal{O}_ X)) = K_0(D_{perf}(\mathcal{O}_ X)) = K_0(X)$

see Derived Categories of Schemes, Definition 36.38.2. Hence we get

$c : K_0(X) \to \prod A^ i(X) \quad \text{and}\quad ch : K_0(X) \to \prod A^ i(X)$

as a special case of Remarks 42.56.10 and 42.56.11. Of course, instead we could have just directly used Definition 42.46.3 and Lemmas 42.46.7 and 42.46.11 to construct these maps (as this immediately seen to produce the same classes). Recall that there is a canonical map $K_0(\textit{Vect}(X)) \to K_0(X)$ which sends a finite locally free module to itself viewed as a perfect complex (placed in degree $0$), see Derived Categories of Schemes, Section 36.38. Then the diagram

$\xymatrix{ K_0((\textit{Vect}(X)) \ar[rd]_ c \ar[rr] & & K_0(D_{perf}(\mathcal{O}_ X)) = K_0(X) \ar[ld]^ c \\ & \prod A^ i(X) }$

commutes where the south-east arrow is the one constructed in Remark 42.56.4. Similarly, the diagram

$\xymatrix{ K_0((\textit{Vect}(X)) \ar[rd]_{ch} \ar[rr] & & K_0(D_{perf}(\mathcal{O}_ X)) = K_0(X) \ar[ld]^{ch} \\ & \prod A^ i(X) }$

commutes where the south-east arrow is the one constructed in Remark 42.56.5.

Comment #7544 by on

The target of $ch$ should be tensored with $\mathbf{Q}$ in several spots here.

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