Additive power operations in equivariant cohomology

Abstract

Let $G$ be a finite group and $E$ be an $H_\infty$-ring $G$-spectrum. For any $G$-space $X$ and positive integer $m$, we give an explicit description of the smallest Mackey ideal $\underline{J}$ in $\underline{E}^0(X\times B\Sigma_m)$ for which the reduced $m$th power operation $\underline{E}^0(X) \to \underline{E}^0(X \times B\Sigma_m )/\underline{J}$ is a map of Green functors. We obtain this result as a special case of a general theorem that we establish in the context of $G\times\Sigma_m$-Green functors. This theorem also specializes to characterize the appropriate ideal $\underline{J}$ when $E$ is a $G_\infty$-ring in global spectra. We give example computations for the sphere spectrum, complex $K$-theory, and Morava $E$-theory.