A Multiplicative Tate Spectral Sequence for Compact Lie Group Actions

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 π*(X).

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Given a compact Lie group G and a commutative orthogonal ring spectrum R such that R[G]* = π*(R ? G+) is finitely generated and projective over π*(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 π*(X). Under mild hypotheses, such as X being bounded below and the derived page RE∞ vanishing, this spectral sequence converges strongly to the homotopy π*(XtG) of the G-Tate construction XtG = [EG ? F(EG+, X]G.