K2 of a ring

Remarks

The group $K_2(R)$ is the center of $St(R)$; in particular it is abelian. There is an exact sequence $$1 \to K_2(R) \to St(R) \to GL(R) \to K_1(R) \to 1.$$ Since $\phi$ is a universal central extension, $K_2(R) \cong H_2(E(R), \Z)$. It agrees with Quillen's $\pi_2$ of the plus construction $BGL(R)^+$.

Example: $K_2(\Z) \cong \Z/2\Z$, generated by $(x_{12}(1) \, x_{21}(-1) \, x_{12}(1))^4$; this element maps to the identity matrix but is not trivial in $St(\Z)$.

Example: for a finite field $\F_q$, $K_2(\F_q) = 0$.

Non-example: an element such as $x_{12}(1)$ of $St(\Z)$ is not in $K_2(\Z)$, since $\phi(x_{12}(1)) = e_{12}(1)$ is not the identity matrix.

For a commutative ring $R$ and units $u, v$, there is a Steinberg symbol $\set{u, v} \in K_2(R)$, and for a field these symbols generate $K_2$.