Bass' $NK$ groups and $cdh$-fibrant Hochschild homology

TL;DR

This paper investigates the relationship between algebraic K-theory of polynomial rings and Hochschild homology, providing new descriptions in the context of the cdh topology and addressing Bass' question about K-theory invariance.

Contribution

It offers a new description of K-theory differences using Hochschild homology and cdh cohomology, and resolves Bass' question in certain cases.

Findings

K-theory of R[t] relates to Hochschild homology and Kähler differentials.

Over fields of infinite transcendence degree, K_n(R)=K_n(R[t]) implies K_n(R)=K_n(R[t_1,t_2]).

Counterexamples exist over number fields.

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 K\"ahler 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.