A multiplicative Tate spectral sequence for compact Lie group actions

Abstract

Given a compact Lie group $G$ and a commutative orthogonal ring spectrum $R$ such that $R[G{]}_{\ast }={\pi }_{\ast }(R\wedge G_{+})$ is finitely generated and projective over ${\pi }_{\ast }(R)$, 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 $R[G{]}_{\ast }$ with coefficients in ${\pi }_{\ast }(X)$. Under mild hypotheses, such as $X$ being bounded below and the derived page $R{E}^{\infty }$ vanishing, this spectral sequence converges strongly to the homotopy ${\pi }_{\ast }(X^{tG})$ of the $G$-Tate construction $X^{tG}=[\widetilde{EG}\wedge F(E{G}_{+},X){]}^G$.

134 pages

References in corpus (2)

• Global homotopy theory
• Topological Hochschild homology and the Hasse-Weil zeta function

Cited by in corpus (1)

• Algebraic K-theory of elliptic cohomology