Radicals and Nilpotents in Equivariant Algebra
David Chan, Ben Spitz
TL;DR
This work extends the commutative-algebraic intuition of nilpotence to $G$-Tambara functors by establishing that the nilradical is computed levelwise and equals the intersection of all prime Tambara ideals. It also introduces and analyzes kilpotents, elements inverted to zero, via a levelwise criterion that distinguishes them from nilpotents. The authors prove that the Nakaoka spectrum $\mathrm{Spec}(T)$ is a spectral space, developing a concrete prime-ideal theory for Tambara functors and linking radical constructions to levelwise radicals. A detailed Kilradical theory is developed, providing explicit criteria for kilpotents and illustrating the distinction between nilpotence and kilpotence in key examples like the Burnside Tambara functor and $\mathrm{RO}(G)$-graded stable stems. The results have implications for localization and equivariant phenomena in representation theory and equivariant stable homotopy theory.
Abstract
Associated to each Tambara functor $T$ is its Nakaoka spectrum $\mathrm{Spec}(T)$, analogous to the Zariski spectrum of a commutative ring. We establish that this topological space is spectral. This result follows from an analysis of the notion of nilpotence in Tamabra functors. We prove that the nilradical of a Tambara functor $T$ (the intersection of all of its prime ideals) is computed levelwise, i.e. consists precisely of the nilpotent elements in $T$. In contrast to ordinary commutative algebra, the nilpotents of $T$ are not the same as the elements $x$ such that $T[1/x] = 0$; we therefore also give a classification of these elements. As a corollary, we observe that the set of these elements in $π_\star^s$ (the equivariant stable stems, viewed as an $\mathrm{RO}(G)$-graded Tambara functor) forms an ideal.