K-Theory space of finite abelian groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:13:55Z http://mathoverflow.net/feeds/question/95519 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95519/k-theory-space-of-finite-abelian-groups K-Theory space of finite abelian groups Martin Brandenburg 2012-04-29T20:34:26Z 2012-04-30T06:31:14Z <p>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:</p> <p><strong>Question.</strong> 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 $n>0$?</p> http://mathoverflow.net/questions/95519/k-theory-space-of-finite-abelian-groups/95520#95520 Answer by Dustin Clausen for K-Theory space of finite abelian groups Dustin Clausen 2012-04-29T21:09:23Z 2012-04-30T06:31:14Z <p>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. <a href="http://www.math.uiuc.edu/K-theory/1006/" rel="nofollow">http://www.math.uiuc.edu/K-theory/1006/</a>). 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 $n = 2k-1$.</p> <p>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 <code>$K(\mathsf{finAb}_P) = \prod_{p\in P} K(\mathbb{F}_p)$</code> and that for $P \subseteq P'$ this identification intertwines the inclusion <code>$K(\mathsf{finAb}_P) \to K(\mathsf{finAb}_{P'})$</code> with the evident map <code>$\prod_{p\in P} K(\mathbb{F}_p) \to \prod_{p\in P'} K(\mathbb{F}_p)$</code> which is zero outside of $P$.</p> <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.</p> <p>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.</p>