Tate cohomology lowers chromatic Bousfield classes

Abstract

Let G G be a finite group. We use recent results of J. P. C. Greenlees and H. Sadofsky to show that the Tate homology of E ( n ) E(n) local spectra with respect to G G produces E ( n − 1 ) E(n-1) local spectra. We also show that the Bousfield class of the Tate homology of L n X L_{n}X (for X X finite) is the same as that of L n − 1 X L_{n-1}X . To be precise, recall that Tate homology is a functor from G G -spectra to G G -spectra. To produce a functor P G P_{G} from spectra to spectra, we look at a spectrum as a naive G G -spectrum on which G G acts trivially, apply Tate homology, and take G G -fixed points. This composite is the functor we shall actually study, and we’ll prove that ⟨ P G ( L n X ) ⟩ = ⟨ L n − 1 X ⟩ \langle P_{G}(L_{n}X) \rangle = \langle L_{n-1}X \rangle when X X is finite. When G = Σ p G = \Sigma _{p} , the symmetric group on p p letters, this is related to a conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald’s functor R P − ∞ ( − ) {\mathbf R}P_{-\infty }(-) ).