Completions of $\mathbb{Z}/(p)$–Tate cohomology of periodic spectra

Abstract

We construct splittings of some completions of the $\mathbb{Z}∕\left(p\right)$–Tate cohomology of $E\left(n\right)$ and some related spectra. In particular, we split (a completion of) $tE\left(n\right)$ as a (completion of) a wedge of $E\left(n-1\right)$s as a spectrum, where $t$ is shorthand for the fixed points of the $Z∕\left(p\right)$–Tate cohomology spectrum (ie the Mahowald inverse limit ${invlim}_k\left(\left(P_{-k}\wedge \Sigma E\left(n\right)\right)\right)$). We also give a multiplicative splitting of $tE\left(n\right)$ after a suitable base extension.