Tate cohomology of theories with one-dimensional coefficient ring (Q1377995)

In a previous paper [Math. Z. 222, No. 3, 391-405 (1996; Zbl 0849.55005)], for a finite group \(G\) and a prime \(p\), the authors proved that if \(E\) is mod \(p\) Morava \(K\)-theory (whose coefficient ring \(K(n)^*= \mathbb{F}_p[v_n,v_n^{-1}]\) is a graded field), then the associated \(G\)-equivariant Tate theory is trivial and the representation spectrum is equivariantly contractible, \(t(K(n))\simeq *\). In the present paper, the authors calculate the \(E\)-Tate cohomology \(t(E)^*_G\) and the \(E\)-homology \(E_*(BG_+)\) as functors of the augmented commutative ring \(E^*(BG_+)\) when \(E^*(\cdot)\) is a complex oriented, \(v_n\)-periodic cohomology theory with one-dimensional graded coefficient ring \(E^*\). In calculating \(E_*(BG_+)\) from \(E^*(BG_+)\), the authors make use of Krull dimension, and the Tate theory.