Goodwillie's theorem on relative K-theory

Remarks

The isomorphism is given by a relative Chern character, a refinement of the Dennis trace map. The proof works with simplicial rings and reduces to the rational homology of general linear groups. Example: let $A = \Q[\varepsilon]/(\varepsilon^2)$ and $I = (\varepsilon)$. Then $K_1(A, I) = 1 + I$, and $1 + a\varepsilon \mapsto a$ identifies it with $\Q$, while $HC_0(A, I) = I \cong \Q$, as the theorem predicts. Nilpotence is needed: for $A = \Q[t]$ and $I = (t)$, homotopy invariance gives $K_n(\Q[t], (t)) = 0$, but $HC_0(\Q[t], (t)) = t\Q[t]$ is not zero.

Going deeper

For a student

The theorem is the rational form of a general principle: relative K-theory for nilpotent extensions is controlled by trace invariants. Its integral form is the Dundas–Goodwillie–McCarthy theorem, where cyclic homology is replaced by topological cyclic homology. For $\Q$-algebras the relative K-groups of a nilpotent ideal are already rational vector spaces, so there the theorem computes them exactly.