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Consider the abelian category $\mathsf{finAb}$ of finite abelian groups. It is easy to prove that there is an isomorphism $\mathrm{ord} : K_0(\mathsf{finAb}) \to \mathbb{Q}^+$. Can you give a reference in the literature for that? More generally:

Question. Is there any more concrete description of the Waldhausen's K-theory space $\Omega |w S_{\bullet} \mathsf{finAb}|$? What is known about its homotopy groups, that is the $K$-theory groups $K_n(\mathsf{finAb})$ for $n>0$?

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up vote 11 down vote accepted

Everything is known. In fact as spectra we have canonically $K(\mathsf{finAb}) = \vee_p K(\mathbb{F}_p)$, and the spectra $K(\mathbb{F}_p)$ are identified in the work of Quillen (see e.g. In particular on $\pi_0$ we find $K_0(\mathsf{finAb}) = \oplus_p \mathbb{Z}$, agreeing with your claim, and on $\pi_n$ for $n>0$ we find that $K_n(\mathsf{finAb})$ is $0$ for $n$ even and is $\oplus_p \mathbb{Z}/(p^k-1)$ (non-canonically) for $n = 2k-1$.

To justify the claimed equality $K(\mathsf{finAb}) = \vee_p K(\mathbb{F}_p)$, note first that $\mathsf{finAb}$ is the filtered colimit over increasing finite sets of primes $P$ of the variant $\mathsf{finAb}_P$ where only products of $p$-groups for $p \in P$ are allowed; since K-theory commutes with filtered colimits, it then suffices to show that each $K(\mathsf{finAb}_P) = \prod_{p\in P} K(\mathbb{F}_p)$ and that for $P \subseteq P'$ this identification intertwines the inclusion $K(\mathsf{finAb}_P) \to K(\mathsf{finAb}_{P'})$ with the evident map $\prod_{p\in P} K(\mathbb{F}_p) \to \prod_{p\in P'} K(\mathbb{F}_p)$ which is zero outside of $P$.

But $\mathsf{finAb}_P$ is just the product over $p \in P$ of the categories $\mathsf{finAb}_p$, whose K-theory identifies with that of vector spaces over $\mathbb{F}_p$ by Quillen's devissage theorem. And K-theory commutes with finite products, so that's that.

Here I guess I was actually arguing using Quillen's Q-construction instead of Waldhausen's $S_{\bullet}$-construction. Otherwise I'm not sure how to justify the last step, the devissage. Actually I'm sure all of the above is in Quillen's paper on the Q-construction.

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Perhaps it should also be stated that when $w = i =$ isomorphisms, the Quillen $K$-theory space of an exact category $C$ has the homotopy type of Waldhausen's $\Omega |i S_\bullet C|$. – John Klein Apr 29 '12 at 23:00
@Dustin: Thanks! I've added the LaTeX. So the following question seems to remain: Is there some kind of devissage for Waldhausen's K-theory? – Martin Brandenburg Apr 30 '12 at 6:34
Martin: Yeah. As John points out, of course in some sense there is, since the S-dot construction identifies with the Q-construction in the case at hand. But one might ask what goes on when C is not an exact category. I would ask Clark Barwick, since I know he's been thinking about this sort of thing. – Dustin Clausen Apr 30 '12 at 19:14
A proof of Devissage that only uses the S-construction can be found in Ross Staffeldt's paper "On fundamental theorems of algebraic K-theory". Anyway, Waldhausen gives a natural zig-zag of weak equivalences between the S and the Q-constructions. – K.J. Moi Oct 22 '13 at 13:49

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