ABSTRACT
Let P ⊂ H be full exact subcategories of an Abelian category A, both closed under extensions and inheriting their exact structure from A. Suppose that
(i) Every object M of H has a finite P-resolution. (ii) P is closed under kernels in H, that is, if L → M → N is an exact
sequence in H with M,N ∈ P , then L is also in P . Then KnP ∼= KnH for all n ≥ 0. (See [165] for the proof of this result.)
Remarks and Examples 6.1.1 (i) Let R be a regular Noetherian ring. Then by taking H = M(R), P = P(R) in (6.1)A, we have Kn(R) ∼= Gn(R) for all n ≥ 0.
