This question is related to another question asked here. Let's assume we have an exact category $C$ that consists of specific vector bundles on a variety. Furthermore assume $C$ is idempotent complete but it is not closed under taking kernels of surjections. Now looking at bounded chain complexes $Ch^b(C)$ we have two choices of acyclic complexes. Let's assume $q_1$ refer when we choose those acyclic complexes that are acyclic as a complex of coherent sheaves and $q_2$ refer when we choose acyclic complexes that are acyclic and differentials have kernels and cokernels in $C$. Clearly there are more acyclic complexes in $q_1$ than $q_2$. By WaldhausenGillet, $K$theory of $q_2$ is $K(C)$. Is there any way to compare $K$theory of $q_1$ to $q_2$? (I know in Thomason's paper he mentions you can embed this in another abelian category that satisfies the condition of being closed under taking kernels, in that case it is gonna give $K(C)$, but of course in the process we are changing the acyclic complexes and the $K$theory space)
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1$\begingroup$ These should be related through a localization (long) exact sequence, I think. $\endgroup$ – მამუკა ჯიბლაძე Jan 20 at 7:52
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