$+=Q$ Theorem, Part 2
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Given an exact category $\mathcal A$, there are two natural notions of K-groups on $\mathcal A$, one is given by the K-group of the Quillen exact structure, and the other is given by the K-group of the symmetric monoidal structure on its subcategory $S = i\mathcal A$, the isomorphism category of $\mathcal A$. It was observed by Quillen and later proven by Grayson using the $S^{-1}S$-construction in his 1976 paper that the two notions agree under mild assumptions, and in particular the K-groups of a ring $R$ agree with the K-groups of $\mathcal P(R)$, the category of finitely-generated projective $R$-modules. [Notes]