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 Ktheory 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 Ktheory 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?
1 Answer
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 tstructure.
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, Compos. Math. 151, No. 11, 21602186 (2015). ZBL1333.19003.
There, in the first appendix, Barwick discusses applications of this theorem to calculating the Gtheory of noetherian schemes, namely one of the claims is that the Gtheory of such scheme coincides with the Ktheory of an abelian category of Cohen–Macaulay complexes.

$\begingroup$ Is there a way to compute the Ktheory of an abelian category of CohenMacaulay complexes for a normal variety? I am trying to compute the higher $G$theory of a simplicial toric variety. Thanks. $\endgroup$– BorisMay 9 at 12:46