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 ${\pi }_{\ast }{E}_{p-1}^{tG}$ 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 ${\pi }_{\ast }{E}_{p-1}^{hG}$, which is a folklore calculation of Hopkins and Miller.

16 pages; comments welcome. Updated to fix some typos

References in corpus (1)

• Finite subgroups of extended Morava stabilizer groups