Tate cohomology of Hopf algebras

We discuss how to define Tate cohomology for a Hopf algebra \(\Gamma\) over a commutative ring \(k\) from two perspectives: via complete resolutions and via the Tate complex. In particular, we consider the case \(\Gamma = \pi_*(\mathbb{S}G)\), where \(\mathbb{S}G\) denotes the spherical group ring of a compact Lie group, which is a Hopf algebra over \(\pi_*(\mathbb{S})\) under suitable flatness conditions. We look at some computations for the case \(G=\mathbb{T}=U(1)\), which gives us the second page of a homotopically based Tate spectral sequence for the circle group.