BASS’ NK GROUPS AND cdh-FIBRANT HOCHSCHILD HOMOLOGY

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 Kn(R) = Kn(R[t]) implies Kn(R) = Kn(R[t1, t2]). The answer to this question is affirmative when R is essentially of finite type over the complex numbers, but negative in general. In 1972, H. Bass posed the following question (see [2], question (VI)n): Does Kn(R) = Kn(R[t]) imply that Kn(R) = Kn(R[t1, t2])? One can rephrase the question in terms of Bass’ groups NKn, introduced in [1]: Does NKn(R) = 0 imply that N Kn(R) = 0? More generally, for any functor F from rings to an abelian category, Bass defines NF (R) as the kernel of the map F (R[t]) → F (R) induced by evaluation at t = 0, and NF = N(NF ). Bass’ question was inspired by Traverso’s theorem [24], from which it follows that N Pic(R) = 0 implies NPic(R) = 0. In this paper, we give a new interpretation of the groups NKn(R) in terms of Hochschild homology and the cohomology of Kähler differentials for the cdh topology, for commutative Q-algebras. This allows us to give a counterexample to Bass’ question in the companion paper [6] (see Theorem 0.2 below). To state our main structural theorem, recall from [28] that each NKn(R) has the structure of a module over the ring of big Witt vectors W (R). It is convenient to use the countably infinite-dimensional Q-vector spaces tQ[t] and Ω Q[t]. If M is any R-module, then M ⊗ tQ[t] and M ⊗Ω Q[t] are naturally W (R)-modules by [10]. Theorem 0.1. Let R be a commutative ring containing Q. Then there is a W (R)module isomorphism NKn(R) ∼= (NKn(R)⊗ tQ[t]) ⊕ ( NKn−1(R)⊗ Ω 1 Q[t] ) . Thus Kn(R) = Kn(R[t1, t2]) iff NKn(R) = NKn−1(R) = 0 iff N Kn(R) = 0. In addition, the following are equivalent for all p > 0: a) Kn(R) = Kn(R[t1, ..., tp]) b) NKn(R) = 0 and Kn−1(R) = Kn−1(R[t1, ..., tp−1]). c) NKq(R) = 0 for all q such that n− p < q ≤ n. Date: March 11, 2010. Cortiñas’ research was partially supported by Conicet and partially supported by grants PICT 2006-00836, UBACyT X051, PIP 112-200801-00900, and MTM2007-64704. Haesemeyer’s research was partially supported by NSF grant DMS-0652860. Walker’s research was partially supported by NSF grant DMS-0601666. Weibel’s research was supported by NSA grant MSPF-04G-184 and the Oswald Veblen Fund.