We show that if X is a toric scheme over a regular ring containing a field then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic 0. The affi
ne case of our result was conjectured by Gubeladze.
We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topologic
al cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps.
The goal of this paper is to present proofs of two results of Markus Rost: the Chain Lemma and the Norm Principle. These are the final steps needed to complete the publishable verification of the Bloch-Kato conjecture, that the norm residue maps are
isomorphisms between Milnor K-theory $K_n^M(k)/p$ and etale cohomology $H^n(k,mu_p^n)$ for every prime p, every n and every field k containing 1/p. Our proofs of these two results are based on Rosts 1998 preprints, his web site and Rosts lectures at the Institute for Advanced Study in 1999-2000 and 2005.
