Algebraic K-theory of the infinite place

TL;DR

This paper explores the algebraic K-theory of generalized archimedean valuation rings within Durov's compactification, revealing connections to stable homotopy groups and highlighting pathological behavior of the residue field at infinity.

Contribution

It establishes a link between algebraic K-theory of archimedean valuation rings and stable homotopy groups, and analyzes the K-theoretic properties of the residue field at infinity.

Findings

K-theory of valuation rings relates to stable homotopy groups

Residue field at infinity exhibits pathological K-theoretic behavior

Provides new insights into Durov's compactification framework

Abstract

In this note, we show that the algebraic K-theory of generalized archimedean valuation rings occurring in Durov's compactification of the spectrum of a number ring is given by stable homotopy groups of certain classifying spaces. We also show that the "residue field at infinity" is badly behaved from a K-theoretic point of view.