A lift from group cohomology to spectra for trivial profinite actions

Abstract

Let $G$ be a profinite group, $X$ a discrete $G$-spectrum with trivial action, and $X^{hG}$ the continuous homotopy fixed points. For any $N \trianglelefteq_o G$ ("$o$" for open), $X = X^N$ is a $G/N$-spectrum with trivial action. We construct a zigzag $\text{colim}\,_N \,X^{hG/N} \buildrel\Phi\over\longrightarrow \text{colim}\,_N \,(X^{hN})^{hG/N} \buildrel\Psi\over\longleftarrow X^{hG}$, where $\Psi$ is a weak equivalence. When $\Phi$ is a weak equivalence, this zigzag gives an interesting model for $X^{hG}$ (for example, its Spanier-Whitehead dual is $\text{holim}\,_N \,F(X^{hG/N}, S^0)$). We prove that this happens in the following cases: (1) $|G| < \infty$; (2) $X$ is bounded above; (3) there exists $\{U\}$ cofinal in $\{N\}$, such that for each $U$, $H^s_c(U, \pi_\ast(X)) = 0$, for $s > 0$. Given (3), for each $U$, there is a weak equivalence $X \buildrel\simeq\over\longrightarrow X^{hU}$ and $X^{hG} \simeq X^{hG/U}$. For case (3), we give a series of corollaries and examples. As one instance of a family of examples, if $p$ is a prime, $K(n_p,p)$ the $n_p$th Morava $K$-theory $K(n_p)$ at $p$ for some $n_p \geq 1$, and $\mathbb{Z}_p$ the $p$-adic integers, then for each $m \geq 2$, (3) is satisfied when $G \leqslant \prod_{p \leq m} \mathbb{Z}_p$ is closed, $X = \bigvee_{p > m} (H\mathbb{Q} \vee K(n_p,p))$, and $\{U\} := \{N_G \mid N_G \trianglelefteq_o G\}$.