Let $n \geq 1$, $p$ a prime, and $T(n)$ any representative of the Bousfield class of the telescope $v_n^{ -1}F(n)$ of a finite type $n$ complex. Also, let $E_n$ be the Lubin-Tate spectrum, $K(E_n)$ its algebraic $K$-theory spectrum, and $G_n$ the extended Morava stabilizer group, a profinite group. Motivated by an Ausoni-Rognes conjecture, we show that there are two spectral sequences \[{ ^{ I}}\mspace{ -3mu}E_2^{ s,t} \Longrightarrow \pi_{ t-s}((L_{ T(n+1)}K(E_n))^{ hG_n}) \Longleftarrow { ^{ II}}\mspace{ -2mu}E_2^{ s,t}\] with common abutment $\pi_\ast(-)$ of the continuous homotopy fixed points of $L_{ T(n+1)}K(E_n)$, where ${ ^{ I}}\mspace{ -3mu}E_2^{ s,t}$ is continuous cohomology with coefficients in a certain tower of discrete $G_n$-modules. If the tower satisfies the Mittag-Leffler condition, then there are continuous cochain cohomology groups \[{ ^{ I}}\mspace{ -3mu}E_2^{ \ast,\ast} \cong H^\ast_\mathrm{ cts}(G_n, \pi_\ast(L_{ T(n+1)}K(E_n))) \cong { ^{ II}}\mspace{ -2mu}E_2^{ \ast,\ast}.\] We isolate two hypotheses, the first of which is true when $(n,p) = (1,2)$, that imply $(L_{ T(n+1)}K(E_n))^{ hG_n} \simeq L_{ T(n+1)}K(L_{ K(n)}S^0)$. Also, we show that there is a spectral sequence \[H^s_\mathrm{ cts}(G_n, \pi_t(K(E_n) \otimes T(n+1))) \Longrightarrow \pi_{ t-s}((K(E_n) \otimes T(n+1))^{ hG_n}).\]
Categories: math.AT math.KT