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 $R[G]_* = π_*(R \wedge G_+)$ 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 $E^2$-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 $RE^\infty$ vanishing, this spectral sequence converges strongly to the homotopy $π_*(X^{tG})$ of the $G$-Tate construction $X^{tG} = [\widetilde{EG} \wedge F(EG_+, X)]^G$.

Figures (1)

• Figure 1: The bicomplex $(U_{*,*}, \partial^h, \partial^v)$ for $\Gamma = k[s]/(s^2=\eta s)$

Theorems & Definitions (235)

• Theorem 1.1
• Theorem 1.2: Pareigis
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof