Equivariant and nonequivariant module spectra

Abstract

Let G be a compact Lie group, let R G , be a commutative algebra over the sphere G-spectrum S G , and let R be its underlying nonequivariant algebra over the sphere spectrum S. When R G is split as an algebra, as holds, for example, for R G = MU G . we show how to “extend scalars” to construct a split R G -modale M G = R G Λ R M from an R-module M. We also show how to compute the coefficients M ∗ G in terms of the coefficients R ∗ G, R ∗, and M ∗ . This allows the wholesale construction of highly structured equivariant module spectra from highly structured nonequivariant module spectra. In particular, it applies to construct MU G -modules from MU-modules and therefore gives conceptual constructions of equivariant Brown-Peterson and Morava K-theory spectra.