A construction of some objects in many base cases of an Ausoni-Rognes conjecture

Abstract

Let $p$ be a prime, $n \geq 1$, $K(n)$ the $n$th Morava $K$-theory spectrum, $\mathbb{G}_n$ the extended Morava stabilizer group, and $K(A)$ the algebraic $K$-theory spectrum of a commutative $S$-algebra $A$. For a type $n+1$ complex $V_n$, Ausoni and Rognes conjectured that (a) the unit map $i_n: L_{K(n)}(S^0) \to E_n$ from the $K(n)$-local sphere to the Lubin-Tate spectrum induces a map \[K(L_{K(n)}(S^0)) \wedge v_{n+1}^{-1}V_n \to (K(E_n))^{h\mathbb{G}_n} \wedge v_{n+1}^{-1}V_n\] that is a weak equivalence, where (b) since $\mathbb{G}_n$ is profinite, $(K(E_n))^{h\mathbb{G}_n}$ denotes a continuous homotopy fixed point spectrum, and (c) $\pi_\ast(-)$ of the target of the above map is the abutment of a homotopy fixed point spectral sequence. For $n = 1$, $p \geq 5$, and $V_1 = V(1)$, we give a way to realize the above map and (c), by proving that $i_1$ induces a map \[K(L_{K(1)}(S^0)) \wedge v_{2}^{-1}V_1 \to (K(E_1) \wedge v_{2}^{-1}V_1)^{h\mathbb{G}_1},\] where the target of this map is a continuous homotopy fixed point spectrum, with an associated homotopy fixed point spectral sequence. Also, we prove that there is an equivalence \[(K(E_1) \wedge v_{2}^{-1}V_1)^{h\mathbb{G}_1} \simeq (K(E_1))^{\widetilde{h}\mathbb{G}_1} \wedge v_2^{-1}V_1,\] where $(K(E_1))^{\widetilde{h}\mathbb{G}_1}$ is the homotopy fixed points with $\mathbb{G}_1$ regarded as a discrete group.