A Multiplicative Tate Spectral Sequence for Compact Lie Group Actions

Abstract

Given a compact Lie group G G and a commutative orthogonal ring spectrum R R such that R [ G ] ∗ = π ∗ ( R ∧ G + ) R[G]_* = \pi _*(R \wedge G_+) is finitely generated and projective over π ∗ ( R ) \pi _*(R) , we construct a multiplicative G G -Tate spectral sequence for each R R -module X X in orthogonal G G -spectra, with E 2 E^2 -page given by the Hopf algebra Tate cohomology of R [ G ] ∗ R[G]_* with coefficients in π ∗ ( X ) \pi _*(X) . Under mild hypotheses, such as X X being bounded below and the derived page R E ∞ RE^\infty vanishing, this spectral sequence converges strongly to the homotopy π ∗ ( X t G ) \pi _*(X^{tG}) of the G G -Tate construction X t G = [ E G ~ ∧ F ( E G + , X ) ] G X^{tG} = [\widetilde {EG}\wedge F(EG_+, X)]^G .