Completions of Z/(p)-Tate cohomology of periodic spectra

We construct splittings of some completions of the Z/(p)-Tate cohomology of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n-1)'s as a spectrum, where t is shorthand for the fixed points of the Z/(p)-Tate cohomology spectrum (ie Mahowald's inverse limit of P_{-k} smash SE(n)). We also give a multiplicative splitting of tE(n) after a suitable base extension.