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 $K_{\ast }(R[t])/K_{\ast }(R)$ in terms of Hochschild homology and the cohomology of Kähler differentials for the $cdh$ topology. We use this to address Bass' question, on whether $K_n(R)=K_n(R[t])$ implies $K_n(R)=K_n(R[t_1,t_2])$. The answer is positive over fields of infinite transcendence degree; the companion paper arXiv:1004.3829 provides a counterexample over a number field.

The article was split into two parts on referee's suggestion in 4/2010. This is the first part; the second can be found at arXiv:1004.3829