By a theorem of Keune (which is more of definition rather then theorem per se) K-groups of ring $R$ are Puppe's derived functors of pronilpotent completion of $GL(R)$. They are derived in the most straightforvard sense just as in case of abelian category: in absence of projective resolutions we have hunky dory simplicial ones. For any group $X$ viewed as degenerate simplicial group, taking Kan loops $G$ of classifying space $BX$ we functorially obtain its simplicial resolution $G B X$ with counit augmentation to $\pi_0(GBX) = X$ and higher homotopy groups vanishing. With this knowledge, we can upgrade any unital endofunctor $F: Grp \to Grp$ to its derived version $\mathcal L_{\bullet}F := \pi_{\bullet}(F(GBX))$. Keune's theorem follws directly from the facts that 1) +-construction is a homological equivalence and 2) all H-spaces, and $BGL(R)^+$ in particular, are $\mathbb Z$-local in Sullivan sense; apply Bousfield-Kan construction of pro-$p$-completion via $p$-nilpotent completion (in our case $p = 0$) of Kan loops and we're done.

Take a group $G$, and complete its rational group algebra $\mathbb QG$ in augmentation ideal. Define Malcev completion $\widehat G$ as group of grouplike elements in this complete Hopf algebra. It is, indeed, a functor and it is unital. Let's take a closer look at $\mathcal L_{\bullet} (\widehat{GL(R)})$.

So, what is this "rational homotopy K-theory"? It's probably wide known and (probably) related to Milnor K-groups.

Related question: can we say something about unstable version of these, i. e. $\mathcal L_{\bullet} (\widehat{GL(n, R)})$? It looks more computable, and for $GL(2, \mathbb Z)$ we have particularly nice resolvent of Gruenberg type obtained from combinatorial presentation. (Sidenote musings: shouldn't it resemble rationalisation of ordinary K-theory — at least in low degrees — for infinite fields?)

  • $\begingroup$ What is a reference for this theorem of Keune? $\endgroup$ – Jesse Silliman Dec 4 '16 at 22:50
  • $\begingroup$ @JesseSilliman afair it's in 341 issue of Lecture Notes in Mathematics. Google says it's named "Derived functors and algebraic K-theory". $\endgroup$ – Denis T. Dec 4 '16 at 23:04
  • $\begingroup$ Usually people mean a different thing when they talk about the "rational K-theory" of a ring (unless I misunderstood the construction you are talking about). $\endgroup$ – Denis Nardin Dec 4 '16 at 23:15
  • $\begingroup$ @DenisT. I would have guessed that these groups were exactly the groups $K_i(R) \otimes \mathbb{Q}$ for $i > 0$. Why do you only expect this for infinite fields? $\endgroup$ – Jesse Silliman Dec 4 '16 at 23:28
  • $\begingroup$ @JesseSilliman There's no homological stability of linear groups for arbitrary rings. DenisNardin Quotation marks are deliberate. Maybe I'm missing something obvious and it's really just $K \otimes \mathbb{Q}$. $\endgroup$ – Denis T. Dec 4 '16 at 23:33

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