Tate Cohomology Lowers Chromatic Bousfield Classes

Let G be a finite group. We use the results of [5] to show that the Tate homology of E(n) local spectra with respect to G produces E(n− 1) local spectra. We also show that the Bousfield class of the Tate homology of LnX (for X finite) is the same as that of Ln−1X. To be precise, recall that Tate homology is a functor from G-spectra to G-spectra. To produce a functor PG from spectra to spectra, we look at a spectrum as a naive G-spectrum on which G acts trivially, apply Tate homology, and take G-fixed points. This composite is the functor we shall actually study, and we’ll prove that 〈PG(LnX)〉 = 〈Ln−1X〉 when X is finite. When G = Σp, the symmetric group on p letters, this is related to a conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald’s functor RP−∞(−)).