Tate Cohomology Lowers Chromatic Bousfield Classes
By Mark Hovey and Hal Sadofsky
Let $G$ be a finite group. We use the results of \cite{greenlees-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 $RP_{-\infty}(-)).$