K₂ and the Steinberg Group

The group $K_2(R)$ , introduced by Milnor, measures the non-obvious relations among elementary matrices. It is defined via the Steinberg group and connects algebraic K-theory to central extensions, Brauer groups, and reciprocity laws in number theory.

The Steinberg group

Definition1.8Steinberg group

The Steinberg group $\operatorname{St}_n(R)$ is the group with generators $x_{ij}(\lambda)$ for $1 \leq i \neq j \leq n$ and $\lambda \in R$ , subject to the Steinberg relations:

$x_{ij}(\lambda) \cdot x_{ij}(\mu) = x_{ij}(\lambda + \mu)$
$[x_{ij}(\lambda), x_{jk}(\mu)] = x_{ik}(\lambda\mu)$ for $i, j, k$ distinct
$[x_{ij}(\lambda), x_{kl}(\mu)] = 1$ for $j \neq k$ and $i \neq l$

The stable Steinberg group is $\operatorname{St}(R) = \varinjlim_n \operatorname{St}_n(R)$ .

There is a natural surjection $\phi: \operatorname{St}(R) \twoheadrightarrow E(R)$ sending $x_{ij}(\lambda) \mapsto e_{ij}(\lambda)$ .

Definition1.9K₂ of a ring

The second algebraic K-group is

$K_2(R) = \ker\bigl(\phi: \operatorname{St}(R) \to E(R)\bigr).$

This is the center of $\operatorname{St}(R)$ , and the sequence

$1 \to K_2(R) \to \operatorname{St}(R) \to E(R) \to 1$

is the universal central extension of the perfect group $E(R)$ . Thus $K_2(R) = H_2(E(R); \mathbb{Z}) = H_2(GL(R); \mathbb{Z})$ .

Symbols and computations

Definition1.10Steinberg symbol

For a commutative ring $R$ and units $a, b \in R^{\times}$ , the Steinberg symbol $\{a, b\} \in K_2(R)$ is defined as

$\{a, b\} = h_{12}(a) \, h_{12}(b) \, h_{12}(ab)^{-1}$

where $h_{ij}(u) = w_{ij}(u) \, w_{ij}(-1)$ and $w_{ij}(u) = x_{ij}(u) \, x_{ji}(-u^{-1}) \, x_{ij}(u)$ for $u \in R^{\times}$ .

The Steinberg symbol is bimultiplicative and antisymmetric:

$\{ab, c\} = \{a, c\}\{b, c\}$ and $\{a, bc\} = \{a, b\}\{a, c\}$
$\{a, b\} = \{b, a\}^{-1}$
$\{a, 1-a\} = 1$ for $a, 1-a \in R^{\times}$ (the Steinberg identity)

ExampleK₂ of a field (Matsumoto's theorem)

For a field $F$ , Matsumoto's theorem states that $K_2(F)$ is the abelian group generated by symbols $\{a, b\}$ for $a, b \in F^{\times}$ , subject to:

Bimultiplicativity: $\{ab, c\} = \{a, c\}\{b, c\}$
Steinberg relation: $\{a, 1-a\} = 1$ for $a \neq 0, 1$

Thus $K_2(F) = (F^{\times} \otimes_{\mathbb{Z}} F^{\times}) / \langle a \otimes (1-a) : a \neq 0, 1 \rangle$ .

Computations:

$K_2(\mathbb{F}_q) = 0$ for all finite fields $\mathbb{F}_q$ (every symbol is trivial)
$K_2(\mathbb{R})$ contains $\{-1, -1\}$ of order 2 (related to the real Brauer group)
$K_2(\mathbb{Q})$ is large: $K_2(\mathbb{Q}) \cong \{+1, -1\} \oplus \bigoplus_p \mathbb{F}_p^{\times}$

The Hilbert symbol and reciprocity

ExampleConnection to the Hilbert symbol

For a local field $F$ (e.g., $\mathbb{Q}_p$ or $\mathbb{R}$ ), the tame symbol provides a homomorphism

$\partial: K_2(F) \to k^{\times}$

where $k$ is the residue field. For $F = \mathbb{Q}_p$ with uniformizer $p$ :

$\partial(\{p^a u, p^b v\}) = (-1)^{ab} u^b v^{-a} \mod p$

for $u, v \in \mathbb{Z}_p^{\times}$ .

The Hilbert symbol $(a, b)_v \in \{\pm 1\}$ for $a, b \in \mathbb{Q}_v^{\times}$ is defined by $(a, b)_v = 1$ iff $ax^2 + by^2 = z^2$ has a nontrivial solution in $\mathbb{Q}_v$ . It factors through $K_2(\mathbb{Q}_v)$ via

$K_2(\mathbb{Q}_v) \to \mu_2, \quad \{a, b\} \mapsto (a, b)_v.$

Hilbert reciprocity: $\prod_v (a, b)_v = 1$ for all $a, b \in \mathbb{Q}^{\times}$ , where $v$ ranges over all places of $\mathbb{Q}$ . This is equivalent to quadratic reciprocity and is a shadow of the reciprocity map in class field theory.

K₂ of number fields

RemarkTate's computation and Garland's theorem

For a number field $F$ with ring of integers $\mathcal{O}_F$ :

Garland's theorem: $K_2(\mathcal{O}_F)$ is finite.
Tate's computation: $K_2(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$ , generated by $\{-1, -1\}$ .
More generally, $K_2(\mathcal{O}_F)$ is related to the Brauer group and class field theory.

The Quillen--Lichtenbaum conjecture (now largely proved via motivic cohomology) relates $K_2(\mathcal{O}_F)$ to special values of Dedekind zeta functions. For a totally real field:

$|K_2(\mathcal{O}_F)| \sim_{\mathbb{Q}^{\times}} \zeta_F(-1)$

(up to powers of 2), connecting algebraic K-theory to analytic number theory.

ExampleK₂(ℤ) = ℤ/2ℤ

The group $K_2(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$ is generated by $\{-1, -1\}$ . To see this is nontrivial, we use the Hilbert symbol at the real place: $(-1, -1)_{\infty} = -1$ since $-x^2 - y^2 = z^2$ has no nontrivial real solution.

The computation uses the exact sequence from the localization sequence:

$K_2(\mathbb{Z}) \to K_2(\mathbb{Q}) \xrightarrow{\oplus \partial_p} \bigoplus_p \mathbb{F}_p^{\times} \to K_1(\mathbb{Z}) \to K_1(\mathbb{Q})$

combined with careful analysis of tame symbols. Bass and Tate showed $K_2(\mathbb{Z}) = \{1, \{-1,-1\}\}$ .