$K$-regularity, $cdh$-fibrant Hochschild homology, and a conjecture of Vorst

Abstract

It is a well-known fact that algebraic if-theory is homotopy invariant as a functor on regular schemes; if X is a regular scheme, then the natural map Kn(X) ? Kn(X x ?1) is an isomorphism for all n eZ. This is false in general for nonregular schemes and rings. To express this failure, Bass introduced the terminology that, for any contravari ant functor V defined on schemes, a scheme X is called V-regular if the pullback maps V(X) ?> V(X x Ar) are isomorphisms for all r > 0. If X = Spec(i?), we also say that R is P-regular. Thus regular schemes are ifn-regular for every n. In contrast, it was observed as long ago as in [2] that a nonreduced affine scheme can never be K\-regular. In particular, if A is an Artinian ring (that is, a O-dimensional Noetherian ring), then A is regular (that is, reduced) if and only if A is ifi-regular. In [17], Vorst conjectured that for an affine scheme X, of finite type over a field F and of dimension d, regularity and K?+i-regularity are equivalent; Vorst proved this conjecture for d = 1 (by proving that ^-regularity implies normality). In this paper, we prove Vorst's conjecture in all dimensions provided the char acteristic of the ground field F is zero. In fact we prove a stronger statement. We say that X is regular in codimension n in X. Note that for all n Z, if a ring R is ifn-regular, then it is Kn-i-regular. This is proved in [17] for n > 1 and in [6, 4.4] for n KH(X), where K(X) is the algebraic if-theory spectrum of X and KH(X) is the homotopy if-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 Tk(R) is n-connected, then R is regular in codimension KHi(X) is an isomorphism for i < n and a surjection for i = n +1, so that Fk(X)