Milnor K-Groups

Milnor K-theory provides a purely algebraic, symbol-based approach to K-theory of fields. Defined via generators and relations, the Milnor K-groups $K_n^M(F)$ capture essential arithmetic information and serve as the domain for higher residue symbols and the Bloch--Kato conjecture.

Definition

Definition4.1Milnor K-theory of a field

For a field $F$ , the Milnor K-ring is the graded ring

$K_*^M(F) = T^*(F^{\times}) / \langle a \otimes (1-a) : a \in F \setminus \{0, 1\} \rangle$

where $T^*(F^{\times}) = \bigoplus_{n \geq 0} (F^{\times})^{\otimes n}$ is the tensor algebra of the abelian group $F^{\times}$ over $\mathbb{Z}$ .

The $n$ -th Milnor K-group is the degree- $n$ component:

$K_n^M(F) = \underbrace{F^{\times} \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} F^{\times}}_{n} \Big/ \langle \cdots \otimes a \otimes \cdots \otimes (1-a) \otimes \cdots \rangle$

where the relation holds whenever $a$ and $1-a$ appear as consecutive tensor factors.

The image of $a_1 \otimes \cdots \otimes a_n$ in $K_n^M(F)$ is denoted $\{a_1, \ldots, a_n\}$ and called a symbol.

RemarkLow-degree groups

In low degrees:

$K_0^M(F) = \mathbb{Z}$ (the empty tensor product).
$K_1^M(F) = F^{\times}$ (no relation in degree 1).
$K_2^M(F) = F^{\times} \otimes F^{\times} / \langle a \otimes (1-a) \rangle$ , which agrees with Quillen's $K_2(F)$ by Matsumoto's theorem.

For $n \geq 3$ , $K_n^M(F)$ does not agree with Quillen's $K_n(F)$ in general. There is a natural map $K_n^M(F) \to K_n(F)$ (defined by iterated products), but it is neither injective nor surjective for $n \geq 3$ .

Basic properties

Definition4.2Symbol relations

The Steinberg relation $\{a, 1-a\} = 0$ in $K_2^M(F)$ extends to multilinear Steinberg relations in $K_n^M$ . From the definition, one derives:

Anticommutativity: $\{a_1, \ldots, a_i, a_{i+1}, \ldots, a_n\} = -\{a_1, \ldots, a_{i+1}, a_i, \ldots, a_n\}$
Vanishing of diagonal: $\{\ldots, a, \ldots, a, \ldots\} = 0$ when any two entries coincide. More precisely, $\{a, a\} = \{a, -1\}$ (which may be nonzero).
Sum relation: $\{a, -a\} = 0$ for all $a \in F^{\times}$ .
Multilinear Steinberg: $\{a_1, \ldots, a_n\} = 0$ whenever $a_i + a_j = 1$ for some $i < j$ (not necessarily consecutive).

The graded ring $K_*^M(F)$ is graded-commutative: $\alpha \cdot \beta = (-1)^{|\alpha||\beta|} \beta \cdot \alpha$ .

ExampleMilnor K-theory of finite fields

For a finite field $\mathbb{F}_q$ :

$K_n^M(\mathbb{F}_q) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{F}_q^{\times} \cong \mathbb{Z}/(q-1) & n = 1 \\ 0 & n \geq 2 \end{cases}$

The vanishing for $n \geq 2$ follows from the fact that the norm map $N: K_n^M(\mathbb{F}_{q^m}) \to K_n^M(\mathbb{F}_q)$ is multiplication by $m$ and $K_n^M(\mathbb{F}_q)$ is $(q-1)$ -torsion, combined with the fact that $\mathbb{F}_q^{\times}$ is cyclic (so $\{a, b\} = \{g^i, g^j\} = ij\{g, g\} = ij\{g, -1\}$ for a generator $g$ , and this is zero since $-1 = g^{(q-1)/2}$ and $(q-1)\{g, g\} = 0$ ).

Computations

ExampleMilnor K-theory of ℝ

For the real numbers:

$K_n^M(\mathbb{R}) \cong \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{R}^{\times} & n = 1 \\ \mathbb{Z}/2\mathbb{Z} & n \geq 2 \end{cases}$

For $n \geq 2$ , the generator is $\{-1, -1, \ldots, -1\}$ ( $n$ copies). To see $K_2^M(\mathbb{R}) = \mathbb{Z}/2$ : any symbol $\{a, b\}$ with $a, b > 0$ vanishes (since positive reals are squares and $\{a^2, b\} = 2\{a, b\}$ , but also $\{a, b\} = \{a, b\}$ is a "continuous" function of $a$ and $b$ , which must be constant on connected components).

For $n \geq 2$ : $K_n^M(\mathbb{R}) = \{0, \{-1, \ldots, -1\}\}$ since $\{-1, -1, \ldots, -1\}^2 = 0$ by anticommutativity.

ExampleMilnor K-theory of local fields

For a non-archimedean local field $F$ with residue field $k$ and uniformizer $\pi$ :

$K_n^M(F) / p \cong K_n^M(k) / p \oplus K_{n-1}^M(k) / p$

for primes $p \neq \operatorname{char}(k)$ . The first factor comes from symbols in units $\{u_1, \ldots, u_n\}$ and the second from symbols $\{\pi, u_1, \ldots, u_{n-1}\}$ .

For $F = \mathbb{Q}_p$ and $n = 2$ : $K_2^M(\mathbb{Q}_p) / p \cong \mathbb{F}_p^{\times} / (\mathbb{F}_p^{\times})^p \oplus \mathbb{F}_p^{\times} / (\mathbb{F}_p^{\times})^p$ , which detects ramification in $K_2$ .

Norms and transfers

Definition4.3Norm map in Milnor K-theory

For a finite field extension $L/F$ of degree $d$ , there is a norm map

$N_{L/F}: K_n^M(L) \to K_n^M(F)$

characterized (for $n = 1$ ) by the field norm $N_{L/F}: L^{\times} \to F^{\times}$ and extended to higher degrees by the bass--tate construction:

For a simple extension $L = F(\alpha)$ with minimal polynomial $f(t) = \prod (t - \alpha_i)$ :

$N_{L/F}(\{a, \alpha\}) = \{a, (-1)^d N_{L/F}(\alpha)\} = \{a, f(0) \cdot (-1)^d\}$

for $a \in F^{\times}$ . The norm satisfies the projection formula: $N_{L/F}(\{a\} \cdot \beta) = \{a\} \cdot N_{L/F}(\beta)$ for $a \in F^{\times}$ and $\beta \in K_{n-1}^M(L)$ .

The composition $K_n^M(F) \xrightarrow{\text{res}} K_n^M(L) \xrightarrow{N} K_n^M(F)$ is multiplication by $[L:F]$ .

RemarkMilnor conjecture on quadratic forms

Milnor conjectured a deep connection between $K_*^M(F)/2$ and the graded Witt ring $\operatorname{gr}^* I(F) = \bigoplus I^n(F)/I^{n+1}(F)$ , where $I(F)$ is the fundamental ideal in the Witt ring $W(F)$ . Specifically:

$K_n^M(F) / 2 \cong I^n(F) / I^{n+1}(F) \cong H^n(F, \mathbb{Z}/2)$

where the second isomorphism is the Galois cohomology of $F$ . The map $K_n^M(F)/2 \to H^n_{\text{et}}(F, \mu_2^{\otimes n})$ sending $\{a_1, \ldots, a_n\} \mapsto (a_1) \cup \cdots \cup (a_n)$ is the norm residue homomorphism. This was proved by Voevodsky (Fields Medal, 2002).