2020/05/08 by Daniel G. Davis, Davis, Daniel G.
Mathematics · Physics and Astronomy · #19D55 #55N15 #55P42 #55T99 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.2005.04190
openalex publication_date 2020/05/08 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Let p be a prime, n \≥ 1, K(n) the nth Morava K-theory spectrum,\n mathbbGn the extended Morava stabilizer group, and K(A) the algebraic\nK-theory spectrum of a commutative S-algebra A. For a type n+1 complex\nVn, Ausoni and Rognes conjectured that (a) the unit map in: LK(n)(S0)\n\→ En from the K(n)-local sphere to the Lubin-Tate spectrum induces a map\n\K(LK(n)(S0))
wedge vn+1-1Vn
to (K(En))^h
mathbbGn
wedge\nvn+1-1Vn that is a weak equivalence, where (b) since mathbbGn is\nprofinite, (K(En))^h mathbbGn denotes a continuous homotopy fixed point\nspectrum, and (c) \π_\∗(-) of the target of the above map is the abutment\nof a homotopy fixed point spectral sequence. For n = 1, p \≥ 5, and V1\n= V(1), we give a way to realize the above map and (c), by proving that i1\ninduces a map \K(LK(1)(S0))
wedge v2-1V1
to (K(E1)
wedge\nv2-1V1)^h
mathbbG1, where the target of this map is a continuous\nhomotopy fixed point spectrum, with an associated homotopy fixed point spectral\nsequence. Also, we prove that there is an equivalence \(K(E1)
wedge\nv2-1V1)^h
mathbbG1
simeq (K(E1))^
widetildeh
mathbbG1\n
wedge v2-1V1, where (K(E1))^ widetildeh mathbbG1 is the\nhomotopy fixed points with mathbbG1 regarded as a discrete group.\n