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

TL;DR

This paper proves that for finite abelian groups, the rational equivariant topological K-theory spectrum has a unique genuine-commutative ring structure, unlike the connective version, highlighting a special property of the periodic case.

Contribution

It establishes the uniqueness of the genuine-commutative ring structure on periodic rational G-equivariant K-theory for finite abelian groups.

Findings

Periodic rational G-equivariant K-theory has a unique genuine-commutative ring structure.

Connective rational equivariant K-theory does not have this uniqueness.

The result builds on previous work to clarify the structure of equivariant K-theory spectra.

Abstract

In this paper, we build on the work from our previous paper (arXiv:2002.01556) to show that periodic rational $G$-equivariant topological $K$-theory has a unique genuine-commutative ring structure for $G$ a finite abelian group. This means that every genuine-commutative ring spectrum whose homotopy groups are those of $KU_{\mathbb{Q},G}$ is weakly equivalent, as a genuine-commutative ring spectrum, to $KU_{\mathbb{Q},G}$. In contrast, the connective rational equivariant $K$-theory spectrum does not have this type of uniqueness of genuine-commutative ring structure.