(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 NKn(R)=Kn(R[x])/Kn(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:

Kn(R)=Kn(R[x,y]) holds if and only if both Kn(R)=Kn(R[x]) and Kn-1(R)=Kn-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>