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

Abstract

In this paper, we build on the work from [10] 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 Q,G is weakly equivalent, as a genuine-commutative ring spectrum, to KU Q,G . In contrast, the connective rational equivariant K-theory spectrum does not have this type of uniqueness of genuine-commutative ring structure.

Total power operations in spectral sequences

“…This is a refinement of the G-equivariant spectra KU G , in the sense that there are symmetric monoidal functors U G : Glob → Sp G and KU G U G KU. For our purposes, we may take this as the definition of the G-spectra KU G , to be assured that the G-E ∞ structure on KU G is compatible with the ultracommutative ring structure of KU; it is not obvious whether the G-E ∞ structure on KU G is unique, see for instance [BHIKM21].…”