We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.

Abstract. We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M∞-stable, homotopy-invariant, excisive Ktheory of algebras over a fixed unital ground ring H, (A, B) → kk * (A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel's homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then kk * (H, A) = KH * (A). We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk.

Abstract. We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities.The goal of this paper is to prove Haesemeyer's Theorem [18, 3.12] for toric schemes in any characteristic. It is proven below as Corollary 14.4.Theorem 0.1. Assume k is a commutative regular noetherian ring containing an infinite field and let G be a presheaf of spectra defined on the category of schemes of finite type over k. If G satisfies the Mayer-Vietoris property for Zariski covers, finite abstract blow-up squares, and blow-ups along regularly embedded closed subschemes, then G satisfies the Mayer-Vietoris property for all abstract blow-up squares of toric k-schemes obtained from subdividing a fan.The application we have in mind is to understand the relationship between the algebraic K-theory K * (X) = π * K(X) and topological cyclic homology T C * (X) = {π * T C ν (X, p)} of a toric scheme over a regular ring of characteristic p (and in particular of toric varieties over a field of characteristic p). Thus we consider the presheaf of homotopy fibers {F ν (X)} of the map of pro-spectra from K(X) to {T C ν (X, p)}. Work of Geisser-Hesselholt [11, Thm. B], [12] shows that this homotopy fiber (regarded as a pro-presheaf of spectra) satisfies the hypotheses of Theorem 0.1 and hence a slight modification of the proof of our theorem implies that it satisfies the Mayer-Vietoris property for all abstract blow-up squares of toric schemes. We will give a rigorous proof of this in Corollary 14.8 below.One major tool in our proof will be a theorem of Bierstone-Milman [1] which says that the singularities of a toric variety (or scheme) can be resolved by a sequence of blow-ups X C → X along a center C that is a smooth, equivariant closed subscheme of X along which X is normally flat. If one only had to consider toric schemes, this would allow one to use Haesemeyer's original argument to prove Theorem 0.1,

Let $f:A \to B$ be a ring homomorphism of not necessarily unital rings and $I\triangleleft A$ an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups $K_*(A:I) \to K_*(B:f(I))$ to be an isomorphism; it is measured by the birelative groups $K_*(A,B:I)$. We show that these are rationally isomorphic to the corresponding birelative groups for cyclic homology up to a dimension shift. In the particular case when A and B are $\Q$-algebras we obtain an integral isomorphism.Comment: Final version to appear in Inventione