Radicals and Nilpotents in Equivariant Algebra

Associated to each Tambara functor $T$ is its Nakaoka spectrum $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(T)$, 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 $T[1\mathrm{/}x]=0$; 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 $\mathrm{R}\mathrm{O}(G)$-graded Tambara functor) forms an ideal.