Recently I learned from the Stacks project that for every abelian category ${\mathcal A}$, there is a natural isomorphism $K\_0({\mathcal A})\cong K\_0(D^{b}(\mathcal A))$.
When we set $\mathcal A$ to be the abelian category $M(X)$ of coherent sheaves on a Noetherian scheme $X$,we get that $G\_0(X)\cong K\_0(D^{b}(M(X)))$.
Is it true that $K\_i({\mathcal A})\cong K\_i(D^{b}({\mathcal A}))$ for every abelian category ${\mathcal A}$ and $i\geq 0$?
I saw in Amnon Neeman’s 1997 paper K-theory for Triangulated Categories I(A):Homological Functors that my question was answered in the affirmative on page 341,as Theorem 7.1.
But I am new to the K-theory of triangulated categories.
So I would like to confirm that this is true.
Another question is, assuming $K\_i({\mathcal A})\cong K\_i(D^{b}({\mathcal A}))$ for every abelian category ${\mathcal A}$ and $i\geq 0$, does this isomorphism provide a practical method of computing the higher $G$-theory of a Noetherian scheme?



Can higher G-theory of Noetherian schemes be computed by derived categories?
Yeah this is true, as Neeman explains this follows from his more general theorem of the heart for the bounded derived category with the standard t-structure.


You can see a modern treatment ( with a shorter proof in the language of $\infty$-categories ) in Barwick's paper


*Barwick, Clark*, [**On exact (\infty)-categories and the theorem of the heart**](https://doi.org/10.1112/S0010437X15007447), Compos. Math. 151, No. 11, 2160-2186 (2015). [ZBL1333.19003](https://zbmath.org/?q=an:1333.19003).


There, in the first appendix, Barwick discusses applications of this theorem to calculating the G-theory of noetherian schemes, namely one of the claims is that the G-theory of such scheme coincides with the K-theory of an abelian category of Cohen–Macaulay complexes.


