Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum

Abstract

We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, we obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category.

36 pages. v2: Slightly simpler (but equivalent) reformulation of main theorem. Revisions in response to my PhD committee's report and a referee report. v3: Final version, to appear in "Advances in Mathematics"

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• G-symmetric monoidal categories of modules over equivariant commutative ring spectra
• An introduction to algebraic models for rational G-spectra
• Idempotent characters and equivariantly multiplicative splittings of K-theory
• Genuine-commutative ring structure on rational equivariant $K$-theory for finite abelian groups