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 $X$.