Tate cohomology lowers chromatic Bousfield classes

Abstract:

Let $G$ be a finite group. We use recent results of J. P. C. Greenlees and H. Sadofsky to show that the Tate homology of $E(n)$ local spectra with respect to $G$ produces $E(n-1)$ local spectra. We also show that the Bousfield class of the Tate homology of $L_{n}X$ (for $X$ finite) is the same as that of $L_{n-1}X$. To be precise, recall that Tate homology is a functor from $G$-spectra to $G$-spectra. To produce a functor $P_{G}$ 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 $\langle P_{G}(L_{n}X) \rangle = \langle L_{n-1}X \rangle$ when $X$ is finite. When $G = \Sigma _{p}$, the symmetric group on $p$ letters, this is related to a conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald’s functor ${\mathbf R}P_{-\infty }(-)$).