Units of equivariant ring spectra

Abstract

It is well known that very special $\Gamma$–spaces and grouplike $E_{\infty }$–spaces both model connective spectra. Both these models have equivariant analogues in the case when the group acting is finite. Shimakawa defined the category of equivariant $\Gamma$–spaces and showed that special equivariant $\Gamma$–spaces determine positive equivariant spectra. Costenoble and Waner [Trans. Amer. Math. Soc. 326 (1991) 485-505] showed that grouplike equivariant $E_{\infty }$–spaces determine connective equivariant spectra.

We show that with suitable model category structures the category of equivariant $\Gamma$–spaces is Quillen equivalent to the category of equivariant $E_{\infty }$–spaces. We define the units of equivariant ring spectra in terms of equivariant $\Gamma$–spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.