# $K$-theory of formal power series.

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

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 $$n > 1$$.