Spectra of bi-incomplete Tambara functors

TL;DR

This paper defines the spectrum of prime ideals for bi-incomplete Tambara functors, generalizing previous notions, and provides computational tools for analyzing various examples.

Contribution

It introduces a unified spectrum concept for bi-incomplete Tambara functors, extending prior work by Lewis and Nakaoka, with practical computational methods.

Findings

Defined the spectrum of prime ideals for bi-incomplete Tambara functors.

Generalized Lewis and Nakaoka's notions of prime ideals.

Developed computational tools for examples of interest.

Abstract

Bi-incomplete Tambara functors are equivariant generalizations of commutative rings. The most common forms of bi-incomplete Tambara functors are coefficient systems of commutative rings, Green functors, and Tambara functors. In the 1980s, Lewis introduced prime ideals in Green functors, and in the 2010s, Nakaoka introduced prime ideals in Tambara functors. In this work, we define the spectrum of prime ideals for an arbitrary bi-incomplete Tambara functor, simultaneously generalizing Lewis and Nakaoka's notions. We then produce many computational tools which we apply to several examples of interest.