The Tate spectrum of the higher real $K$-theories at height $n=p-1$

Abstract

Let $E_n$ be Morava $E$-theory and let $G \subset G_n$ be a finite subgroup of $G_n$, the extended Morava stabilizer group. Let $E_{n}^{tG}$ be the Tate spectrum, defined as the cofiber of the norm map $N:(E_n)_{hG} \to E_n^{hG}$. We use the Tate spectral sequence to calculate $\pi_*E_{p-1}^{tG}$ for $G$ a maximal finite $p$-subgroup, and $p$ an odd prime. We show that $E_{p-1}^{tG} \simeq \ast$, so that the norm map gives equivalence between homotopy fixed point and homotopy orbit spectra. Our methods also give a calculation of $\pi_*E_{p-1}^{hG}$, which is a folklore calculation of Hopkins and Miller.