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 $N{⊴}_{o}G$ ("$o$" for open), $X=X^N$ is a $G/N$-spectrum with trivial action. We construct a zigzag $\text{colim}\,_N \,X^{hG/N} \buildrelΦ\over\longrightarrow \text{colim}\,_N \,(X^{hN})^{hG/N} \buildrelΨ\over\longleftarrow X^{hG}$, 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 $U$, $H_c^s(U,{\pi }_{\ast }(X))=0$, 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 $n_p\ge 1$, and ${\mathbb{Z}}_p$ the $p$-adic integers, then for each $m\ge 2$, (3) is satisfied when $G⩽{\prod }_{p\le m}{\mathbb{Z}}_p$ is closed, $X={\bigvee }_{p>m}(H\mathbb{Q}\vee K(n_p,p))$, and $\{U\}:=\{N_G\mid N_G{⊴}_{o}G\}$.

arXiv comment18 pages; submitted for publication