The spectrum of the Burnside Tambara functor

TL;DR

This work determines the Nakaoka spectrum of the Burnside Tambara functor $\underline{A}_G$ for any finite group $G$, showing that prime ideals are precisely $\mathfrak{p}_{H,p}$ with $H\le G$ and $p$ prime or $0$. It develops and leverages the ghost of the Burnside Tambara functor to control spectrum structure, proving that the ghost map $\chi$ is a Tambara map and using it to classify containment relations among primes. The main result is a complete description of $\mathrm{Spec}(\underline{A}_G)$, including explicit poset relations and Noetherian topology, with detailed examples for several non-abelian groups such as dihedral groups, $Q_8$, $A_4$, and $GL_3(\mathbb{F}_2)$. The findings provide a foundational algebraic framework for equivariant geometry over Tambara functors and connect to broader programs in equivariant tensor-triangular geometry and stable homotopy theory.

Abstract

We compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group. Our proof uses recent advances in the commutative algebra of Tambara functors, as well as a Tambara functor analogue of ghost coordinates which works over arbitrary finite groups and clarifies some previous computations. As examples, we explicitly compute the spectrum of the Burnside Tambara functor over all dihederal groups, the quaternion group $Q_8$, the alternating group $A_4$, and the general linear group $GL_3(F_2)$.

Theorems & Definitions (87)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Definition 2.1
• Definition 2.4
• Theorem 2.5: dress:1971
• Definition 2.6
• Remark 2.7
• Example 2.8
• Example 2.9