Inverting Integers in Tambara Functors
Ben Spitz
TL;DR
This work proves that inverting an integer $k$ in a $G$-Tambara functor $T$ yields a localization that is independent of the level at which the inversion is performed: $T[1/k_e]$ is canonically isomorphic to $T[1/k_G]$, and more generally $k$ is a unit at all levels iff it is a unit at any single level. The authors leverage the Burnside Tambara functor, the ghost map, and Dress’s prime-ideal description to establish the equivalence of unit conditions across $T(G/G)$, $T(G/H)$ for all $H\le G$, and $T(G/e)$, with a contradiction argument when assuming $k$ fails to be a unit in the relevant localization. The results unify and simplify localization arguments in equivariant algebra, align with related work on $G$-spectra localization and Tambara-localizations, and imply that norm constructions $N_H^G$ preserve localization by inverting $k$. Overall, the paper clarifies the intrinsic level-independence of integer localization in Tambara functors and provides groundwork for streamlined computations in equivariant algebraic contexts.
Abstract
Let $G$ be a finite group. A $G$-Tambara functor $T$ consists of collection of commutative rings $T(G/H)$ (one for each subgroup $H$ of $G$), together with certain structure maps, satisfying certain axioms. In this note, we show that, for any integer $k$, 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 $T(G/H_2)$.