In “The localization of spectra with respect to homology”, Bousfield describes localizations with respect to Moore Spectra. Given a spectrum $E$, and a group $M$, he describes the spectrum with coefficients $EM$. An example of this can be found in Adams's blue book, where he defines the $K$-theory spectrum with coefficients in the group $\mathbb{Z}\left[\frac{1}{k}\right]$. In the $G$-equivariant case, where the underlying group is a finite/compact Lie group, is it possible to define $G$-equivariant spectrum with coefficients in an abelian group?
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4$\begingroup$ Yes, the category $\mathcal{S}_G$ of $G$-spectra is enriched over the category $\mathcal{S}$ of spectra, so for $E\in\mathcal{S}_G$ and $M\in\text{Ab}$ you can define the Moore spectrum $SM\in\mathcal{S}$ and then $EM=E\wedge SM\in\mathcal{S}_G$. $\endgroup$– Neil StricklandCommented Sep 26, 2022 at 12:19
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