Date of this Version
Invent math (2010) 181: 421–448
The K-theory of a polynomial ring R[t ] contains the K-theory of R as a summand. For R commutative and containing Q, we describe K∗(R[t ])/K∗(R) in terms of Hochschild homology and the cohomology of Kähler differentials for the cdh topology.
We use this to address Bass’ question, whether Kn(R) = Kn(R[t ]) implies Kn(R) = Kn(R[t1, t2]). The answer to this question is affirmative when R is essentially of finite type over the complex numbers, but negative in general.