A Multiplicative Tate Spectral Sequence for Compact Lie Group Actions

Given a compact Lie group G and a commutative orthogonal ring spectrum R such that R[G]_* = _*(R ? G₊) is finitely generated and projective over _*(R), we construct a multiplicative G-Tate spectral sequence for each R-module X in orthogonal G-spectra, with E²-page given by the Hopf algebra Tate cohomology of R[G]_* with coefficients in _*(X). Under mild hypotheses, such as X being bounded below and the derived page RE vanishing, this spectral sequence converges strongly to the homotopy _*(XtG) of the G-Tate construction X^tG = [EG ? F(EG+, X]^G. See more