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A computational reduction for many base cases in profinite telescopic algebraic $K$-theory (2101.11205v2)

Published 27 Jan 2021 in math.AT and math.KT

Abstract: For primes $p\geq 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 $L_{K(1)}S0 \mspace{-1.5mu}\xrightarrow{\mspace{-2mu}\text{unit} \, i}~\mspace{-7mu}(KU){\wedge}_p$ induces a map $K(L_{K(1)}S0) \wedge v_2{-1}V(1) \to K(KU_p){h\mathbb{Z}\times_p} \wedge v_2{-1}V(1)$ that is an equivalence. Since the definition of this map is not well understood, we consider $K(L_{K(1)}S0) \wedge v_2{-1}V(1) \to (K(KU_p) \wedge v_2{-1}V(1)){h\mathbb{Z}\times_p}$, 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, $K(L{K(1)}S0)_\ast(V(1))[v_2{-1}]$ and, by using $K(L_p)\ast(V(1))[v_2{-1}]$, where $L_p = ((KU){\wedge}_p){h\mathbb{Z}/((p-1)\mathbb{Z})}$, the summands simplify. The Ausoni-Rognes conjecture suggests that in [(-){h\mathbb{Z}\times_p} \wedge v_2{-1}V(1) \simeq (K(KU_p) \wedge v_2{-1}V(1)){h\mathbb{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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