# Spherical objects and K-theory

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!

• Fair enough! So in the general version $K(\mathbf{C}^n)\sim K(\mathbf{C}_{strict}^n)$ where $\mathbf{C}_{strict}^n$ is the category of strict $n$-spherical objects in $\mathbf{C}$? – TTip Apr 3 at 21:43