Radicals and Nilpotents in Equivariant Algebra

TL;DR

This paper studies the structure of Tambara functors, proving their spectra are spectral spaces, characterizing nilpotent elements, and contrasting their nilpotents with elements inverted to zero, with implications for equivariant stable stems.

Contribution

It establishes the spectrality of Nakaoka spectra for Tambara functors and characterizes nilpotent elements levelwise, providing new insights into equivariant algebraic structures.

Findings

The Nakaoka spectrum of a Tambara functor is spectral.

Nilradical of a Tambara functor is computed levelwise.

Nilpotents differ from elements inverted to zero in Tambara functors.

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 $\pi_\star^s$ (the equivariant stable stems, viewed as an $\mathrm{RO}(G)$-graded Tambara functor) forms an ideal.