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 XtG = [EG ? F(EG+, X]G.