A computational reduction for many base cases in profinite telescopic algebraic $K$-theory

Abstract

For primes $p\ge 5$, $K(K{U}_p)$ -- the algebraic $K$-theory spectrum of $(KU{)}_p^{\wedge }$, 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 $K(L_{K(1)}{S}^0)\wedge v_2^{-1}V(1)\to K(K{U}_p{)}^{h{\mathbb{Z}}_p^{\times }}\wedge v_2^{-1}V(1)$ that is an equivalence. Since the definition of this map is not well understood, we consider $K(L_{K(1)}{S}^0)\wedge v_2^{-1}V(1)\to (K(K{U}_p)\wedge v_2^{-1}V(1){)}^{h{\mathbb{Z}}_p^{\times }}$, which is induced by $i$ and also should be an equivalence. We show that for any closed $G<{\mathbb{Z}}_p^{\times }$, ${\pi }_{\ast }((K(K{U}_p)\wedge v_2^{-1}V(1){)}^{hG})$ is a direct sum of two pieces given by (co)invariants and a coinduced module, for $K(K{U}_p{)}_{\ast }(V(1))[v_2^{-1}]$. When $G={\mathbb{Z}}_p^{\times }$, 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 $L_p=((KU{)}_p^{\wedge }{)}^{h\mathbb{Z}/((p-1)\mathbb{Z})}$, the summands simplify. The Ausoni-Rognes conjecture suggests that in \[(-)^{h\mathbb{Z}^\times_p} \wedge v_2^{-1}V(1) \simeq (K(KU_p) \wedge v_2^{-1}V(1))^{h\mathbb{Z}^\times_p},\] $K(K{U}_p)$ fills in the blank; we show that for any $G$, the blank can be filled by $(K(K{U}_p){)}_{\mathcal{O}}^{\mathrm{dis}}$, a discrete ${\mathbb{Z}}_p^{\times }$-spectrum built out of $K(K{U}_p)$.

20 pages; resubmitted for publication; Section 1.1 is new; updated several references

References in corpus (4)

• Towards solid abelian groups: A formal proof of Nöbeling's theorem
• Chromatic fracture cubes
• Cubical and cosimplicial descent
• A construction of some objects in many base cases of an Ausoni-Rognes conjecture