A multiplicative Tate spectral sequence for compact Lie group actions

TL;DR

This paper develops a multiplicative Tate spectral sequence for compact Lie group actions on orthogonal ring spectra, providing a tool to compute the homotopy groups of the Tate construction with convergence under mild conditions.

Contribution

It introduces a new multiplicative spectral sequence for G-Tate constructions in orthogonal spectra, linking Tate cohomology with homotopy groups of the Tate spectrum.

Findings

Spectral sequence converges strongly under mild hypotheses.

E^2-page given by Hopf algebra Tate cohomology.

Applicable to a broad class of G-spectra with finitely generated projective coefficients.

Abstract

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