A point-free approach to the Nakaoka spectrum of a Tambara functor

TL;DR

This paper introduces a point-free framework for the Nakaoka spectrum of a Tambara functor, establishing its topological properties and connecting algebraic and geometric perspectives.

Contribution

It constructs the frame of radical Tambara ideals and proves the Nakaoka spectrum is a spectral space, extending recent results.

Findings

The frame of radical Tambara ideals is spatial and coherent.

The Nakaoka spectrum is shown to be a spectral space.

Points of the frame correspond to Nakaoka primes.

Abstract

For $G$ a finite group and $T$ a $G$-Tambara functor, we construct the frame $\mathop{RadId}_G(T)$ of radical Tambara ideals and show that its points are the Nakaoka primes. We show that this frame is spatial and coherent, and deduce that the Nakaoka spectrum is a spectral space, recovering a recent result of Chan and Spitz.