Rational Witt vectors and associated sheaves
Christopher Deninger

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 and of the maps and in various Grothendieck topologies, both subcanonical and non-subcanonical. Here, for a commutative ring , is the reduced monoid algebra on and is the subring of rational functions in the big Witt ring . Moreover, is the ind-scheme representing 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 , we have where denotes the associated sheaf in the finite flat topology. More generally,…
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