Negative K-theory

Remarks

The definition is motivated by degree one. There, the same cokernel, built from $K_1$ instead of $K_{-n}$, is isomorphic to $K_0(R)$, by the Bass–Heller–Swan decomposition of $K_1(R[t, t^{-1}])$.

Example: let $F$ be a field. The rings $F[t]$ and $F[t, t^{-1}]$ are principal ideal domains, so both have $K_0 \cong \Z$, generated by the class of the free module of rank $1$. The inclusion sends one generator to the other, so the map is onto, and $K_{-1}(F) = 0$. More generally, $K_{-n}(R) = 0$ for all $n \ge 1$ when $R$ is a regular Noetherian ring.

Example with a nonzero group: let $k$ be a field and let $R = k[x, y]/(y^2 - x^3 - x^2)$, the coordinate ring of a nodal cubic. The normalization $k[s]$, with $x = s^2 - 1$ and $y = s(s^2 - 1)$, gives a Milnor square, and its Mayer–Vietoris sequence shows $K_{-1}(R) \cong \Z$.

Non-example: the cokernel of the map from $K_{-n}(R[t])$ alone is not the right group. It also contains the image of $K_{-n}(R[t^{-1}])$, which need not lie in the image of $K_{-n}(R[t])$; both polynomial subrings must be used.