### (I) Bass' NK groups and cdh-fibrant Hochschild homology; and (II) A negative answer to a question of Bass, by Guillermo Cortiñas, Christian Haesemeyer, Mark Walker, and Chuck Weibel

[Apr 28, 2010: In preparation for publication, the original preprint of Feb 13, 2008, has
been subdivided into two papers, both of which are presented here.
The first part (I) will appear in *Inventiones Math.*]

In 1972, Bass asked if K_n(R)=K_n(R[x]) implies K_n(R)=K_n(R[x,y]).

We give a counterexample which is the coordinate ring of an isolated surface
singularity over the rationals: K_0(R)=K_0(R[x]) yet K_0(R) ≠ K_0(R[x,y]).
This is contained in paper (II) posted below.
(In paper (I) we show that the answer to Bass' question is affirmative when R is
essentially of finite type over the complex numbers.)

The counterexample relies upon paper (I), in which we give explicit formulas for
NK_{n}(R)=K_{n}(R[x])/K_{n}(R) in terms of
Hochschild homology and the cohomology of Kähler differentials
for the cdh topology, where R is a commutative ring containing the rationals.
These formulas imply that:

K_{n}(R)=K_{n}(R[x,y]) holds if and only if both
K_{n}(R)=K_{n}(R[x]) *and*
K_{n-1}(R)=K_{n-1}(R[x]).

Guillermo Cortiñas <gcorti@dm.uba.ar>

Christian Haesemeyer <chh@math.uic.edu>

Mark Walker <mwalker5@math.unl.edu>

Chuck Weibel <weibel@math.rutgers.edu>