97d:55010 55N22

Greenlees, J. P. C. ^{(}4-SHEF^{)}; Sadofsky, Hal ^{(}1-JHOP^{)} The Tate spectrum of v_{n}-periodic complex oriented theories. Math. Z. 222 (1996), no. 3, 391–405. The Tate spectrum was introduced by the first author in joint work with Peter May. It is a functor which takes G-equivariant spectra to G-equivariant spectra. It is quite surprising to the reviewer that applying the Tate spectrum to a spectrum X thought of as a Gspectrum with trivial action and then taking G-fixed points to recover a non-equivariant spectrum t_{G}X gives interesting results, but this paper demonstrates that it does. Actually, Greenlees and May already knew that when G is a cyclic group of order p, t_{G}X is the inverse limit of the tower used by Mahowald to define the root invariant.

The main theorem of this paper is that t_{G}K(n) is trivial. This theorem follows from the fact that K(n)_{∗}BG is finite-dimensional for all finite G. This result has several interesting consequences. The main corollary of this paper is that if E is complex-oriented and v_{n}- periodic, then t_{G}E is v_{n−1}-periodic. This is evidence for the general slogan that t_{G} reduces chromatic periodicity.

This paper has opened the way to a great deal of further work. For example, the second author has studied the formal group associated to t_{G}E in joint work with M. Ando and J. Morava. The Holy Grail in this area would be an understanding of t_{G}L_{n}S^{0}, which is conjectured to be L_{n−1}S^{0} wedged with many other terms. This is related to the chromatic splitting conjecture of Hopkins. Mark Hovey (1-MIT)

\begin{matrix}c \\ ◯ \end{matrix} Copyright American Mathematical Society 1997