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Register a new userNorm Varieties and the Chain Lemma (after Markus Rost)
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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.
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We introduce a new perspective on the $K$-theory of exact categories via the notion of a CGW-category. CGW-categories are a generalization of exact categories that admit a Qullen $Q$-construction, but which also include examples such as finite sets and varieties. By analyzing Quillens proofs of devissage and localization we define ACGW-categories, an analogous generalization of abelian categories for which we prove theorems analogous to devissage and localization. In particular, although the category of varieties is not quite ACGW, the category of reduced schemes of finite type is; applying devissage and localization allows us to calculate a filtration on the $K$-theory of schemes of finite type. As an application of this theory we construct a comparison map showing that the two authors definitions of the Grothendieck spectrum of varieties are equivalent.
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 topological 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.
We compute Witt groups of maximal isotropic Grassmannians, aka. spinor varieties. They are examples of type D homogenuous varieties. Our method relies on the Blow-up setup of Balmer-Calm`es, and we investigate the connecting homomorphism via the projective bundle formula of Walter-Nenashev, the projection formula of Calm`es-Hornbostel and the excess intersection formula of Fasel. The computation in the Type D case can be presented by so called even shifted young diagrams.
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 affine case of our result was conjectured by Gubeladze.
Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$ over a field $k$ of characteristic~0. We calculate the $K$-theory of $R$ in terms of the geometry of the projective embedding of $X$. In particular, if $X$ is a curve then we calculate $K_0(R)$ and $K_1(R)$, and prove that $K_{-1}(R)=oplus H^1(C,cO(n))$. The formula for $K_0(R)$ involves the Zariski cohomology of twisted Kahler differentials on the variety.
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