(non)commutative f-un geometry

TL;DR

This paper explores a noncommutative extension of F-un geometry by utilizing dual coalgebras, clarifying Habiro-type rings' role, and applying it to the study of group representations and profinite completions.

Contribution

It introduces a noncommutative generalization of affine schemes over the field with one element using dual coalgebras, linking to group representations.

Findings

Clarifies the role of Habiro-type rings in the commutative case

Provides a framework for noncommutative F-un geometry

Connects the theory to representations of discrete groups

Abstract

Stressing the role of dual coalgebras, we modify the definition of affine schemes over the 'field with one element'. This clarifies the appearance of Habiro-type rings in the commutative case, and, allows a natural noncommutative generalization, the study of representations of discrete groups and their profinite completions being our main motivation.