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Register a new userA lift from group cohomology to spectra for trivial profinite actions
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Daniel G. Davis
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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 $text{colim},_N ,X^{hG/N} buildrelPhioverlongrightarrow text{colim},_N ,(X^{hN})^{hG/N} buildrelPsioverlongleftarrow X^{hG}$, 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 $X buildrelsimeqoverlongrightarrow X^{hU}$ 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, $X = bigvee_{p > m} (Hmathbb{Q} vee K(n_p,p))$, and ${U} := {N_G mid N_G trianglelefteq_o G}$.
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