2019/08/05 by Davis, Daniel G.
#55P42 #55P91 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.01898
Let G be a profinite group, X a discrete G-spectrum with trivial action, and XhG the continuous homotopy fixed points. For any N \trianglelefteqo G ("o" for open), X = XN is a G/N-spectrum with trivial action. We construct a zigzag colim N XhG/N \buildrelΦ\over\longrightarrow colim N (XhN)hG/N \buildrelΨ\over\longleftarrow XhG, where Ψ is a weak equivalence. When Φ is a weak equivalence, this zigzag gives an interesting model for XhG (for example, its Spanier-Whitehead dual is holim N F(XhG/N, S0)). We prove that this happens in the following cases: (1) |G| < ∞; (2) X is bounded above; (3) there exists \U\ cofinal in \N\, such that for each U, Hsc(U, π_∗(X)) = 0, for s > 0. Given (3), for each U, there is a weak equivalence X \buildrel≃\over\longrightarrow XhU and XhG ≃ XhG/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(np,p) the npth Morava K-theory K(np) at p for some np ≥ 1, and ℤp the p-adic integers, then for each m ≥ 2, (3) is satisfied when G \leqslant ∏p ≤ m ℤp is closed, X = \bigveep > m (Hℚ \vee K(np,p)), and \U\ := \NG | NG \trianglelefteqo G\.