Several homotopy fixed point spectral sequences in telescopically localized algebraic K-theory

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Let n ≥ 1, p a prime, and T(n) any representative of the Bousfield class of the telescope v−1n F(n) of a finite type n complex. Also, let En be the Lubin-Tate spectrum, K(En) its algebraic K-theory spectrum, and Gn the extended Morava stabilizer group, a profinite group. Motivated by an Ausoni-Rognes conjecture, we show that there are two spectral sequences<br> IEs,t2 ⇒ πt−s((LT(n+1)K(En))hGn) ⇐ IIEs,t2<br> with common abutment π∗(−) of the continuous homotopy fixed points of LT(n+1)K(En), where IEs,t2 is continuous cohomology with coefficients in a certain tower of discrete Gn-modules. If the tower satisfies the Mittag-Leffler condition, then there are isomorphisms with continuous cochain cohomology groups:<br> IE∗,∗2 ≅ H∗cts(Gn, π∗(LT(n+1)K(En))) ≅ IIE∗,∗2.<br> We isolate two hypotheses, the first of which is true when (n, p) = (1, 2), that imply (LT(n+1)K(En))hGn ≃ LT(n+1)K(LK(n)S0). Also, we show that there is a spectral sequence<br> Hscts(Gn, πt(K(En) ⊗ T(n + 1))) ⇒ πt−s((K(En) ⊗ T(n + 1))hGn).

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