K-REGULARITY, cdh-FIBRANT HOCHSCHILD HOMOLOGY, AND A CONJECTURE OF VORST

It is a well-known fact that algebraicK-theory is homotopy invariant as a functor on regular schemes; if X is a regular scheme, then the natural map Kn(X) → Kn(X×A) is an isomorphism for all n ∈ Z. This is false in general for nonregular schemes and rings. To express this failure, Bass introduced the terminology that, for any contravariant functor P defined on schemes, a scheme X is called P-regular if the pullback maps P(X) → P(X × A) are isomorphisms for all r ≥ 0. If X = Spec(R), we also say that R is P-regular. Thus regular schemes are Kn-regular for every n. In contrast, it was observed as long ago as in [2] that a nonreduced affine scheme can never be K1-regular. In particular, if A is an Artinian ring (that is, a 0-dimensional Noetherian ring), then A is regular (that is, reduced) if and only if A is K1-regular. In [17], Vorst conjectured that for an affine scheme X, of finite type over a field F and of dimension d, regularity and Kd+1-regularity are equivalent; Vorst proved this conjecture for d = 1 (by proving that K2-regularity implies normality). In this paper, we prove Vorst’s conjecture in all dimensions provided the characteristic of the ground field F is zero. In fact we prove a stronger statement. We say that X is regular in codimension < n if Sing(X) has codimension ≥ n in X. Note that for all n ∈ Z, if a ring R is Kn-regular, then it is Kn−1-regular. This is proved in [17] for n ≥ 1 and in [6, 4.4] for n ≤ 0. Let FK denote the presheaf of spectra such that FK(X) is the homotopy fiber of the natural mapK(X) → KH(X), whereK(X) is the algebraicK-theory spectrum of X and KH(X) is the homotopy K-theory of X defined in [19]. We write FK(R) for FK(Spec(R)). Theorem 0.1. Let R be a commutative ring which is essentially of finite type over a field F of characteristic 0. Then the following hold. (a) If FK(R) is n-connected, then R is regular in codimension < n. (b) If R is Kn-regular, then R is regular in codimension < n. (c) (Vorst’s conjecture) If R is K1+dim(R)-regular, then R is regular. It was observed in [19, Proposition 1.5] that if X is Kn-regular, then Ki(X) → KHi(X) is an isomorphism for i ≤ n and a surjection for i = n+1, so that FK(X)