Rational Witt vectors and associated sheaves

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.