K-theory with finite coefficients

Remarks

The cofibre sequence $K(R) \xrightarrow{\ell} K(R) \to K(R)/\ell$ gives the universal coefficient sequence $$0 \to K_n(R)/\ell \to K_n(R; \Z/\ell) \to K_{n-1}(R)[\ell] \to 0,$$ where $A[\ell]$ is the subgroup of elements of $A$ killed by $\ell$, part of the torsion subgroup. For $n \ge 2$, one can also write $K_n(R; \Z/\ell) = [P^n(\ell), BGL(R)^+]$, pointed homotopy classes of maps from the mod $\ell$ Moore space $P^n(\ell) = S^{n-1} \cup_\ell e^n$ into the plus construction.

Example: for a field $F$, $K_0(F) = \Z$ is torsion-free, so $K_1(F; \Z/\ell) \cong K_1(F)/\ell = F^\times/(F^\times)^\ell$. For $F = \Q$ and $\ell = 2$ this is infinite, generated by the classes of $-1$ and of the primes.

Non-example: $K_n(R; \Z/\ell)$ is in general not $K_n(R) \otimes \Z/\ell$. For $R = \C$, the group $K_2(\C)$ is divisible, since $\set{a, b} = \ell \set{c, b}$ when $a = c^\ell$, so $K_2(\C)/\ell = 0$. But $K_1(\C)[\ell]$ is the group $\mu_\ell$ of $\ell$-th roots of unity, so $K_2(\C; \Z/\ell) \cong \mu_\ell \cong \Z/\ell$.