Relations between rational algebraic K-theory and Chow groups

A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic $$ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*}(X) \otimes \mathbb{Q}$$ where $$X$$ is a smooth quasi-projective variety over a characteristic zero field $$k$$. This is generalized to higher K-groups and higher chow groups.

Is there any reference for the proof over general field / generalization to algebraic stacks over a field (like $$BG$$ where $$G$$ is a finite group) / good regular noetherian schemes (like integral model of curves)?

Maybe one need to develop variants of both sides e.g Weibel’s homotopy invariant algebraic K-theory to made such result holds in general.

Moreover, can we change $$\mathbb Q$$ to $$\mathbb Z[1/N]$$ where $$N$$ only depends on invariants e.g dimension of $$X$$.

• Fulton's intersection theory Example 15.2.16 contains the proof over an arbitrary field. Maybe (this paper) by Edidin and Graham is what you'd have in mind for BG. I don't know if anything exists (but I suppose it might) for bases that aren't a field. – Eoin Feb 6 at 6:05
• Oops, forgot to include the link (arxiv.org/abs/math/9905081). – Eoin Feb 6 at 7:20
• @Eoin Thanks, I will check the paper. – sawdada Feb 6 at 8:04
• It seems you did not do your research properly. Grothendieck's original SGA 6 deals with the case of an arbitrary regular scheme (or an arbitrary noetherian scheme with a dimension function if one wants full generality).Not speaking French is not an excuse since this is very well documented in the Stack-Project stacks.math.columbia.edu/tag/0FEW – Denis-Charles Cisinski Feb 6 at 10:08
• As for a version with integral coefficients, the key word is: motivic Atiyah-Hirzebruch spectral sequence. There is no isomorphism (actually no interesting map), but a rather a filtration of K-theory leading to a spectral sequence whose second page may be computed in terms of Bloch's higher Chow groups. – Denis-Charles Cisinski Feb 6 at 10:11