Rational Witt vectors and associated sheaves

TL;DR

This paper explores the sheafification of rational Witt vectors and related maps in various topologies, revealing new geometric insights and connections to algebraic K-theory.

Contribution

It introduces a sheaf-theoretic framework for rational Witt vectors, proves that certain associated schemes are ind-schemes, and links these to algebraic K-theory and correspondences.

Findings

W_{rat}(K) equals global sections of a sheaf for fields and Dedekind rings

W_{rat}(A) for normal domains relates to universal finite correspondences

Provides a geometric interpretation of Almkvist's theorem on cyclic K-theory

Abstract

We study the sheafification of $W_{\mathrm{rat}} (\mathcal{O})$ and of the maps $\underline{\mathbb{Z}} \mathcal{O} \to W_{\mathrm{rat}} (\mathcal{O})$ and $W_{\mathrm{rat}} (\mathcal{O}) \to W_J (\mathcal{O})$ in various Grothendieck topologies, both subcanonical and non-subcanonical. Here, for a commutative ring $A$, $\underline{\mathbb{Z}} A$ is the reduced monoid algebra on $(A , \cdot)$ and $W_{\mathrm{rat}} (A)$ is the subring of rational functions in the big Witt ring $W (A)$. Moreover, $W_J$ is the ind-scheme representing $W_{\mathrm{rat}}$ on Fatou rings which was introduced by Hazewinkel and which we prove to be an ind-ring scheme. It turns out, for example that for any field $K$, we have $W_{\mathrm{rat}} (K) = \Gamma (\mathrm{spec}\, K , (\underline{\mathbb{Z}}\mathcal{O})^{\sharp})$ where $\sharp$ denotes the associated sheaf in the finite flat topology. More generally, this is true for Dedekind rings. By comparing our results with work of Suslin and Voevodsky we found an isomorphism of $W_{\mathrm{rat}} (A)$ for normal domains $A$ with a ring of universally integral finite relative correspondences. This gives a new geometric interpretation of Almkvist's theorem on cyclic $K$-theory for such rings and suggests a number of interesting questions.