My question goes as follows: given a ring $R$ (with $1\neq 0$). Define $\mathbf{Perf}_{R}$ the the category of Prefect complexes over $R$. I want to prove that the Waldhausen $K$-theory of the category of perfect complexes is the same as the $K$-theory of $\mathbf{Proj}_{R}$ the category of finitely projective $R$-modules, i.e. $$ K(\mathbf{Perf}_{R})\sim K(\mathbf{Proj}_{R})$$ The proof that I am looking for is the one using the notion of $n$-spherical objects developed by Wladhausen in his famous article "Algebraic K-theory of spaces " (section 1.7 if I remember correctly).
Lets me try to adapt the Waldhausen theorem in the algebraic context. Let say we have a ring $R$. And Let $\mathbf{C}$ be an essentially small full subcategory of the category chain complexes over $R$. Such that $\mathbf{C}$ is a Waldausen category where weak equivalences are quasi-isomorphisms and cofibrations are those coming from the (projective) model structure on the category of chain complexes over $R$.
Definition $B$ is an $n$-spherical object in $\mathbf{C}$, if the homology $H_\ast (B)$ is concentrated in degree $n$. Lets denote by $\mathbf{C}^{n}$ the category of $n$-spherical objects in $\mathbf{C}$. If I an not wrong $\mathbf{C}^{n}$ is a waldhausen category where weak equivalences are quasi-iso and cofibrations are ordinary cofibrations such that the cofiber is also an object in $\mathbf{C}^{n}$.
Now the Wladhausen theorem says that $hocolim_{n}K(\mathbf{C}^{n})\sim K(\mathbf{C})$.
Since I'm not sure If my understanding of the theorem is corrected, I wanted to ask experts for some clarification. In case my interpretation of the theorem is completely wrong, I would be happy to get some help. Thank you!
