5
$\begingroup$

Are $K^{MW}_*(\mathbb{F_q})$ and $K^{MW}_n(\mathbb{F_q})$ already known? Where can I read about it?

Improve this question
$\endgroup$

2 Answers 2

Reset to default
8
$\begingroup$

I wanted to leave this as a comment, but I don't have enough reputation points yet.

There is a short exact sequence for every $n$ $$ 0 \to I^{n - 1}(k) \to K_n^{MW}(k) \to K_n^M(k) \to 0 $$ This is proven independently of Morel's article by Gille, Scully, Zhong in "Milnor-Witt $K$-groups of local rings.", see Def.3.7, Thm.3.8, Thm.5.4.

The Milnor $K$-theory of a finite field appears in Milnor's original article "Algebraic $K$-theory of quadratic forms" as Example 1.5.

For the Witt groups and powers of augmentation ideals, see page 36, 37 and Theorem 3.5 in Lam's book "Introduction to quadratic forms over fields".

The culmination of these is:

$$ K^{MW}_{\geq 0}(\mathbf{F}_q) = (\mathbb{Z} \oplus \mathbb{Z} / 2) \oplus (\mathbb{F}_q^*) \oplus 0 \oplus 0 \dots$$ and for $n < 0$ we have $$K^{MW}_n(\mathbf{F}_q) = \mathbb{Z} / 4$$ if $q$ is 3 mod 4, and $$K^{MW}_n(\mathbf{F}_q) = \mathbb{Z} / 2[\epsilon] / \epsilon^2$$ if $q$ is 1 mod 4.

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ We can even be more explicit. Let $\omega$ be a generator of $\mathbf{F}_q^*$. Then $\omega$ as an element of $K_1^{MW}$ is (obviously) $[\omega]$, the copy of $\mathbf{Z} / 2$ in $K_0^{MW}$ is additively generated by $\eta [\omega]$, if $n > 0$ and $q$ is 3 mod 4 then $K_{-n}^{MW}$ is additively generated by $\eta^n$, and if $q$ is 1 mod 4 then the $\epsilon$ in $K_{-n}^{MW}$ corresponds to $\eta^{n + 1}[\omega]$. $\endgroup$
    – Shane Kelly
    Commented May 5, 2016 at 20:02
7
$\begingroup$

The Milnor-Witt K-theory for finite fields can be put together using $$ K^{MW}_n\cong K^M_n\times_{I^n/I^{n+1}}I^n $$ using the known results for Milnor K-theory and the Witt ring. A treatment of the Witt ring dealing with all characteristics can be found in Elman-Karpenko-Merkurjev "The algebraic and geometric theory of quadratic forms".

Addendum: the formula for Milnor-Witt K-theory can be found in Morel's paper "Sur les puissances de l'idéal fondamental de l'anneau de Witt".

Improve this answer
$\endgroup$
1
  • $\begingroup$ Is there an article which proves this formula? I have not found it in the book you've mentioned. $\endgroup$
    – T. Viktor
    Commented Jan 24, 2016 at 15:49

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .