Radicals and nilpotents in equivariant algebra

Associated to each Tambara functor T is its Nakaoka spectrum \,Spec\,(T), 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 Tambara 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 $$T[1/x] = 0;$$ 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 _ ˢ π ⋆ s (the equivariant stable stems, viewed as an RO(G) RO ( G ) -graded Tambara functor) forms an ideal.