Genuine‐commutative structure on rational equivariant K$K$‐theory for finite abelian groups

Abstract

In this paper, the authors build on their previous work to show that periodic rational G $G$ -equivariant topological K $K$ -theory has a unique genuine-commutative ring structure for G $G$ a finite abelian group. This means that every genuine-commutative ring spectrum whose homotopy groups are those of K U Q , G $KU_{\mathbb {Q},G}$ is weakly equivalent, as a genuine-commutative ring spectrum, to K U Q , G $KU_{\mathbb {Q},G}$ . In contrast, the connective rational equivariant K $K$ -theory spectrum does not have this type of uniqueness of genuine-commutative ring structure.