On the $RO(Q)$-graded coefficients of Eilenberg-MacLane spectra

TL;DR

This paper computes the $RO(Q)$-graded coefficients of Eilenberg-MacLane $Q$-spectra using the Tate diagram, revealing their structure as modules and, in some cases, their multiplicative properties.

Contribution

It provides explicit calculations of $RO(Q)$-graded coefficients for Eilenberg-MacLane spectra and describes their module and multiplicative structures.

Findings

Computed $RO(Q)$-graded coefficients using the Tate diagram.

Described the structure as modules over the Burnside Mackey functor.

Inferred multiplicative structures for Green functor cases.

Abstract

Let $Q$ denote the cyclic group of order two. Using the Tate diagram we compute the $RO(Q)$-graded coefficients of Eilenberg-MacLane $Q$-spectra and describe their structure as a module over the coefficients of the Eilenberg-MacLane spectrum of the Burnside Mackey functor. If the underlying Mackey functor is a Green functor, we also infer the multiplicative structure on the $RO(Q)$-graded coefficients.