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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 $X$.

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    $\begingroup$ 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. $\endgroup$ – Eoin Feb 6 at 6:05
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    $\begingroup$ Oops, forgot to include the link (arxiv.org/abs/math/9905081). $\endgroup$ – Eoin Feb 6 at 7:20
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    $\begingroup$ @Eoin Thanks, I will check the paper. $\endgroup$ – sawdada Feb 6 at 8:04
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    $\begingroup$ 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 $\endgroup$ – Denis-Charles Cisinski Feb 6 at 10:08
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    $\begingroup$ 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. $\endgroup$ – Denis-Charles Cisinski Feb 6 at 10:11

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