Local compactness as the K(1)-local dual of finite generation

TL;DR

This paper demonstrates that in the K(1)-local homotopy category, the K-theory of finitely generated modules over certain rings is dual to the K-theory of locally compact modules, revealing a deep duality in algebraic K-theory.

Contribution

It establishes a new duality between K-theory of finitely generated and locally compact modules over localizations of number rings in the K(1)-local setting.

Findings

K-theory of finitely generated R-modules is dual to that of locally compact R-modules.

The duality extends to p-adic and finite fields.

The duality is realized in the K(1)-local homotopy category.

Abstract

Suppose R is any localization of the ring of integers of a number field. We show that the K-theory of finitely generated R-modules, and the K-theory of locally compact R-modules, are Anderson duals in the K(1)-local homotopy category. The same is true for p-adic and finite fields.