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!


1 Answer 1


I believe the theorem you are trying to prove is due to Brinkmann who was an early student of Waldhausen.

What you are suggesting as the proof is only part of the story.

In addition to section 1.7, you will also need to apply section 1.8, which is about split cofibrations.

There is a difference between strictly spherical objects (i.e., complexes of projectives which are concentrated in a single degree) and spherical objects in your sense (in the sense of homology). You need to show that the K-theory of these coincide.

  • $\begingroup$ 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 $\mathbf{C}$? $\endgroup$
    – Let
    Apr 3, 2020 at 21:43
  • $\begingroup$ Under suitable assumptions yes. The assumption is that every cofibration sequence must split, I believe. $\endgroup$
    – John Klein
    Apr 5, 2020 at 1:08

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