Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?

Question

With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?

Here, by "trivial examples" I mean for instance examples like $R = \mathbb{F}_q[t]$, or other examples that can easily be deduced from Quillen's computation of $K_i(\mathbb{F}_q)$, the zero ring, etc.

I believe there are no others but I'm not an expert (just curious) and I am having trouble deducing this from the literature.

For instance the $K$-groups of an algebraically closed field are divisible but (correct me if I'm wrong) the uniquely divisible part has not been determined.

Answer 1

For commutative rings, all the K-theory computations I am aware of are discussed in Chapter VI of Weibel's $K$-book (actually with the exception of finite fields).

Function rings: I would say that for smooth affine curves over finite fields, all the $K$-groups have been computed "essentially": Theorems VI.6.4 and VI.6.7 of the $K$-book completely describe the $K$-theory of smooth projective curves over finite fields. If $\overline{C}$ is a smooth projective curve over $\mathbb{F}_q$ which has an $\mathbb{F}_q$-rational point $P$, then you can split off one summand $K_\bullet(\mathbb{F}_q)$ to get a complete description of $K_\bullet(\mathbb{F}_q[\overline{C}\setminus\{P\}])$. For more general smooth affine curves, I believe one can work out the localization sequence and compute $K$-theory of those as well.

I do not think that there are many complete results for higher-dimensional varieties over finite fields, the basic obstruction would be Bass's conjecture or Parshin's conjecture (i.e., finite generation of K-theory of finitely generated commutative $\mathbb{F}_q$-algebras is expected but not yet proved, so all torsion computations leave a uniquely divisible indeterminacy).

Number rings: Sections VI.8 and VI.9 of the $K$-book have fairly complete calculations for $K$-theory of rings of $S$-integers in number fields (if you accept computations in terms of étale cohomology). These computations all follow from the solution of the Bloch-Kato conjecture which implied the Quillen-Lichtenbaum conjecture which enabled $K$-theory computations in terms of étale cohomology (roughly). The complete calculation of the $K_\bullet(\mathbb{Z})$ still depends on the Kummer-Vandiver conjecture in degrees $0$ mod $4$.

Further results: Everything else that I know is for special coefficients. With torsion coefficients, we know $K$-theory of algebraically closed fields, as you mentioned (Chapter VI.1). With torsion coefficients prime to the residue characteristic the K-theory of local fields and their dvrs can be computed using rigidity methods. For torsion coefficients at the residue characteristic, the $K$-theory of local fields is computed using trace methods (Chapter VI.7). Check out the $K$-book for more information. With a grain of salt, I would say that all other known computations of $K$-theory are deduced from the above via projective bundle formula, localization sequence or some such method.

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Non-commutative rings: I do not know so much about those. The Farrell-Jones conjecture has been proved for a couple of classes of groups (hyperbolic groups, CAT(0)-groups), see papers of Bartels, Lück, Reich. This is a major step toward computing $K$-theory of group rings of such groups. However, even with the Farrell-Jones conjecture, it still remains to compute the source of the assembly map, i.e., the equivariant homology. I do not know if the solution to the Farrell-Jones conjecture has been used in completely computing $K$-theory of group rings, you'd have to ask an expert in the field.

Of course, in the non-commutative setting there are also the flasque rings whose $K$-theory is trivial but I guess these qualify as "trivial examples".

Answer 2

In the ten years since this question was asked, there has been a lot of progress in algebraic $K$-theory. For example, Achim Krause, Ben Antieau, and Thomas Nikolaus came up with an algorithm to compute $K(\mathbb{Z}/p^n)$ and have carried it out for a large number of $(p,n)$, as shown in the tables at the end of that paper. Here's a video of a talk from a few days ago about this. Their techniques also apply to $O_K/\pi^n$, where $\pi$ is the uniformizer and $K$ is a finite extension of $\mathbb{Q}_p$.