## Mathematics, Department of

#### Date of this Version

2010

#### Citation

Invent math (2010) 181: 421–448

#### 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 *K _{n}(R) *=

*K*[

_{n}(R*t*]

*)*implies

*K*=

_{n}(R)*K*[

_{n}(R*t*

_{1}

*, t*

_{2}]

*)*. The answer to this question is affirmative when

*R*is essentially of finite type over the complex numbers, but negative in general.

## Comments

© The Author(s) 2010.