Bass' NK groups and cdh-fibrant Hochschild homology

Abstract

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.