Inverting Integers in Tambara Functors

TL;DR

This paper proves that in Tambara functors, the invertibility of an element is independent of the subgroup, allowing for a well-defined localization and showing that norm functors commute with inverting elements.

Contribution

It establishes the invariance of element invertibility across subgroups in Tambara functors and demonstrates the compatibility of norm functors with localization.

Findings

Invertibility of elements is subgroup-independent in Tambara functors.

Localization at an element is well-defined across all subgroups.

Norm functors commute with inverting elements.

Abstract

Let $G$ be a finite group, and $k$ an integer. In this note, we show that for any $G$-Tambara functor $T$ and any subgroups $H_1, H_2 \leq G$, $k$ is a unit in $T(G/H_1)$ if and only if $k$ is a unit in $T(G/H_2)$. In other words, one may speak unambiguously of the localization $T[1/k]$. As a consequence, the norm functors $N_H^G$ commute with inverting $k$.