For primes $pgeq 5 $, $K(KU_p)$ -- the algebraic $K$-theory spectrum of $(KU)^{wedge}_p$, Morava $K$-theory $K(1)$, and Smith-Toda complex $V(1)$, Ausoni and Rognes conjectured (alongside related conjectures) that $L_{K(1)}S^0 mspace{-1.5mu}xrightarrow{mspace{-2mu}text{unit} , i}~mspace{-7mu}(KU)^{wedge}_p$ induces a map $K(L_{K(1)}S^0) wedge v_2^{-1}V(1) to K(KU_p)^{hmathbb{Z}^times_p} wedge v_2^{-1}V(1)$ that is an equivalence. Since the definition of this map is not well understood, we consider $K(L_{K(1)}S^0) wedge v_2^{-1}V(1) to (K(KU_p) wedge v_2^{-1}V(1))^{hmathbb{Z}^times_p}$, which is induced by $i$ and also should be an equivalence. We show that for any closed $G < mathbb{Z}^times_p$, $pi_ast((K(KU_p) wedge v_2^{-1}V(1))^{hG})$ is a direct sum of two pieces given by (co)invariants and a coinduced module, for $K(KU_p)_ast(V(1))[v_2^{-1}]$. When $G = mathbb{Z}^times_p$, the direct sum is, conjecturally, $K(L_{K(1)}S^0)_ast(V(1))[v_2^{-1}]$ and, by using $K(L_p)_ast(V(1))[v_2^{-1}]$, where $L_p = ((KU)^{wedge}_p)^{hmathbb{Z}/((p-1)mathbb{Z})}$, the summands simplify. The Ausoni-Rognes conjecture suggests that in [(-)^{hmathbb{Z}^times_p} wedge v_2^{-1}V(1) simeq (K(KU_p) wedge v_2^{-1}V(1))^{hmathbb{Z}^times_p},] $K(KU_p)$ fills in the blank; we show that for any $G$, the blank can be filled by $(K(KU_p))^mathrm{dis}_mathcal{O}$, a discrete $mathbb{Z}^times_p$-spectrum built out of $K(KU_p)$.
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