TORIC VARIETIES, MONOID SCHEMES AND cdh DESCENT

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 TC∗(X) = {π∗TC (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(X)} of the map of pro-spectra from K(X) to {TC(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 XC → 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, Date: June 6, 2012. 2000 Mathematics Subject Classification. 19D55, 14M25, 19D25, 14L32.