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Let $R$ be the ring of integers in an algebraic number field. There are beautiful descriptions of $K_0(R)$ and $K_1(R)$. Namely, $\tilde{K}_0(R)$ is the class group of $R$ and $K_1(R)$ is the group of units of $R$. Question : Is there a nice description of $K_2(R)$ (or at least some reasonable conjectures)? I couldn't find much about this in Milnor's or Rosenberg's books on algebraic K-theory, so I expect that the answer is pretty complicated. Is it maybe at least known in some special cases (say, for $R$ the integers in a quadratic extension of $\mathbb{Q}$)?

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Thanks for all the great answers! – Andy Putman Jun 24 '10 at 15:57
No one seems to have mentioned explicitly the Quillen-Lichtenbaum conjecture, which gives a full answer. You should look it up, especially as it is now probably a theorem after the work of Voedvodsky and Rost. – Olivier Jun 30 '10 at 11:40
up vote 10 down vote accepted

It's a theorem of Garland that $K_2(R)$ is finite. Perhaps the best way to get a handle on it is to use Quillen's localization sequence $$0\rightarrow K_2(R)\rightarrow K_2(F)\stackrel{T}{\rightarrow} \oplus_v k(v)^*\rightarrow 0,$$ where $F$ is the fraction field and the $k(v)$ are the residue fields. The map $T$ is the sum of the tame symbols, which is surjective by a theorem of Matsumoto. The injectivity on the left follows from the vanishing of $K_2$ for finite fields.

This isn't much of an answer, but considering $K_2(R)$ as a subgroup of $K_2(F)$ seems a reasonable way to start some concrete considerations. For a detailed discussion of an algorithm that proceeds essentially along these lines ('Tate's method), see the paper

Belabas, Karim; Gangl, Herbert Generators and relations for $K_2( O_F)$. $K$-Theory 31 (2004), no. 3, 195--231.

Added, 8 July:

I'm sure most people know this, but I forgot to mention (for newcomers) the fact that $$K_2(F) = F^\times\otimes_{\mathbf Z} F^\times/\langle a\otimes(1-a)\mid a\not=0,1\rangle,$$ which I suppose motivates the original question, and makes it worthwhile to view $K_2(R)$ as a subgroup.

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Also, Tate has an explicit description of $K_2(F)$, where $F$ is a global field, in terms of Galois cohomology: $K_2(F)\cong\bigoplus_{\ell}H^2(G_F,\mathbf{Z}_{\ell})_{\mathrm{tor}}$, where $\ell$ runs over all primes distinct from the characteristic of $F$. See his paper "Relations between K_2 and Galois Cohomology", Inv Math 1976. – JBorger Jun 24 '10 at 7:30
Oops. That should be $\mathbf{Z}_{\ell}(2)$ instead of $\mathbf{Z}_{\ell}$. – JBorger Jun 24 '10 at 9:26
James: I agree that expressing $K_2$ in terms of Galois cohomology should be something 'explicit' in principle, but our current working knowledge of Galois cohomology seems primitive enough to make the utility of it entirely unclear. Recall that finiteness of $H^2(G_S, \mathbb{Z}_l(2)$, for example, uses the finiteness of $K_2$, even though one might have expected the application to run in the other direction. – Minhyong Kim Jun 24 '10 at 9:57
Good point. I agree. Nevertheless I thought Tate's theorem might be good to see for those not familiar with the subject. – JBorger Jun 24 '10 at 10:35
Garland's theorem as originally stated appears in: A Finiteness Theorem for K2 of a Number Field, Ann. of Math. 94, No. 3 (1971), 534-548 (online via JSTOR). From the review on MathSciNet you can also find some but not all earlier/later related papers, including one by Tate in Invent. Math. and a Bourbaki talk by Bass. As other answers and comments here indicate, the subject hasn't reached a definitive state yet. – Jim Humphreys Jul 8 '10 at 20:22

$K_1$ being the units, and Dirichlet's theorem on the unit group, generalize to odd K-groups via regulator maps and related conjectures and theorems (starting from work of Borel, Bloch, Beilinson, Zagier, and many others). For the even K-groups I don't know of any such description.

Charles Weibel's web page has a K-theory textbook with a chapter on $K_1$ and $K_2$, and a paper on the early history of K-theory. Both of those discuss $K_2$ of rings of integers in a number field, but I didn't see any characterization analogous to the one with regulators for $K_1$ or $K_3$.

I'm certainly not an expert on this subject but I have the feeling that if a snazzy description of $K_2({O_F})$ were proven or conjectured, it would be widely known.

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I think Kolster's survey is a good introduction to questions related to arithmetic interpretations of higher K-groups.

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I've never heard of $K_2(R)$ having a description as a more easily/elementarily described object attached to $R$. Borel showed that $K_2(R)$ is torsion. The order of $K_2(R)$ appears in "Theorem" 31 of Soulé's notes on higher algebraic K-theory of rings of algebraic integers where $K_2(R)$ (I think that "Theorem" 31 is called the Quillen-Lichtenbaum conjecture).

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For the conjectural order, see the Birch-Tate conjecture .

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