Algebraic K-theory of a ring

Question

I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is: What is the list of rings such that all their algebraic $K$-theory groups are known? I have read that the $K$-theory of finite fields is known. Is there other (non trivial) examples?

Answer 1

The comments point out that this question is very similar to another question, but an attempt to close as a duplicate failed. So, rather than leave this question on the unanswered queue, I will try to answer.

I have read that the $K$-theory of finite fields is known.

Yes, this calculation is due to Dan Quillen, and you can read the answer here. From this, you can also deduce the algebraic $K$-theory of the rings $\mathbb{F}_q[t]$.

Is there other (non trivial) examples?

As I wrote on the other thread, recent work allows us to compute $K(\mathbb{Z}/p^n)$ for a large number of $(p,n)$, and the $K$-theory of $O_K/\pi^n$, where $\pi$ is the uniformizer and $K$ is a finite extension of $\mathbb{Q}_p$.

What is the list of rings such that all their algebraic -theory groups are known?

A fairly complete list can be found in Weibel's $K$-book, and is summarized at the other thread. It includes:

• Function rings like $\mathbb{F}_q[\overline{C} - \{P\}]$ where $\overline{C}$ is a smooth projective curve and $P$ is a rational point.
• Number rings (but the answers you get are in terms of etale cohomology). Note that we still do not know $K(\mathbb{Z})$, but we know that it contains a lot of arithmetic information.
• Algebraically closed fields in some cases, e.g., with torsion coefficients.
• Local fields and their DVRs in some cases (using trace methods)
• A few results for group rings of certain non-abelian groups like hyperbolic groups or CAT(0) groups.

Nowadays, researchers seem to be making big strides in algebraic $K$-theory calculations, thanks to improvements in trace methods, improvements in our understanding of topological cyclic homology, Galois descent, the redshift conjecture, and generally thanks to advances in the technical machinery that proofs in homotopy theory often boil down to. Readers can expect more results to follow in the years to come, so this answer will likely become out of date soon enough.