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?


1 Answer 1


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, Compos. Math. 151, No. 11, 2160-2186 (2015). ZBL1333.19003.

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.

  • $\begingroup$ Is there a way to compute the K-theory of an abelian category of Cohen-Macaulay complexes for a normal variety? I am trying to compute the higher $G$-theory of a simplicial toric variety. Thanks. $\endgroup$
    – Boris
    May 9 at 12:46

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