$K$-theory of formal power series.

Question

I was wondering whether there is a calculation of algebraic $K$-groups of the formal power series $\mathbb{F}_p[[x]]$?

Answer 1

Here are some thoughts which became a bit too long for a comment.

Since $\mathbb{F}_p[[X]]$ is local, we have $K_0(\mathbb{F}_p[[X]]) \simeq \mathbb Z$ and $K_1(\mathbb{F}_p[[X]]) \simeq (\mathbb{F}_p[[X]])_{\mathsf ab}^\times = \mathbb{F}_p[[X]]^\times$. This is proven for arbitrary, non necessarily commutative local rings in in Rosenberg's book.

Example 4.4.3 of Weibel's K-book, Chapter III cites a result of Drinfeld which says that $K_{-1}(A) = 0$ for all henselian local rings, which $\mathbb{F}_p[[X]]$ is an example of.

I don't know about the higher $K$-groups, they are famously hard to compute and usually involve some homotopical algebra, which I am not that familiar with.

One thing to note is that $\mathbb{F}_p[[X]]$ is regular noetherian, hence its $K$-theory coincides with Weibel's homotopy $K$-theory,

$$ K_\bullet(\mathbb{F}_p[[X]]) = KH_\bullet(\mathbb{F}_p[[X]]). $$

This is Corollary 12.3.1 of the $K$-book, Chapter IV. The advantage of the latter theory is that it is matricially stable, polynomially homotopy invariant, excisive, and commutes with filtering colimits.

Edit: for example, since $\mathbb{F}_p[[X]]$ sits in an exact sequence of non unital rings

$$ (X) \to \mathbb{F}_p[[X]] \to \mathbb{F}_p[[X]]/(X) \simeq \mathbb{F}_p, $$

we have a long exact sequence

$$ \ldots \to KH_\bullet((X)) \to KH_\bullet\mathbb{F}_p[[X]] \to KH_\bullet(\mathbb{F}_p) \to KH_{\bullet-1}((X)) \to \ldots. $$

A result due to Quillen characterizes the positive $K$-theory of finite fields (which by a previous argument, coincides with $KH$): for each $n \geq 1$ we have

$$ K_n(\mathbb F_p) = \begin{cases}\mathbb F_p^\times &\text{$n$ odd}\\ 0&\text{otherwise}\end{cases} $$

On the other hand, the negative $KH$-theory of a commutative artinian ring vanishes (e.g. by Example 12.5.1, Chapter IV of the $K$-book), so $K_n(\mathbb F_p) = 0$ for negative $n$.

I don't know how much information this gives; for what it's worth we see that for a vast amount of indices the $K$-theory of $\mathbb F_p[[X]]$ is the same as the $KH$-theory of its maximal ideal $(X)$ viewed as a rng.

It would be interesting to know any results on the polynomial homotopy type of these algebras.

Edit': upon further inspection of the aformentioned Chapter IV, Example 12.5.2 shows that $KH_n(R) = 0$ for $n < 0$ and $R$ commutative, noetherian and $1$-dimensional.

Hence $K_n(\mathbb F_p[[X]]) = 0$ for negative $n$, and by excision, the same holds for the negative groups of $KH_\bullet((X))$.

Edit'': to sum up, so far we have $$ K_n(\mathbb F_p[[X]]) = K_n(\mathbb{F}_p) = \delta_{n,0}\mathbb Z $$

for all $n \leq 0$, $K_1(\mathbb F_p[[X]]) = \mathbb F_p[[X]]^\times$ and exact sequences

$$ 0 \to KH_{2n-1}((X)) \to K_{2n-1}(\mathbb F_p[[X]]) \to \mathbb F_p^\times \to K_{2n-2}((X)) \to KH_{2n-2}(\mathbb F_p[[X]]) \to 0 $$

for each $n > 1$.