2024/02/01 by Tobias Barthel, Tomer M. Schlank, Barthel, Tobias +5 · 1 voice
Mathematics · #14G22 #55Q45 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Holomorphic and Operator Theory #Number Theory (math.NT) #math.AG #math.AT #math.NT
paper · pdf · doi:10.48550/arxiv.2402.00960
openalex publication_date 2024/02/01 · arxiv published 2024/02/01 · arxiv updated 2025/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We compute the rational homotopy groups of the K(n)-local sphere for all heights n and all primes p, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion of the Witt vectors into the Lubin-Tate ring induces a split injection on continuous stabilizer cohomology with torsion cokernel of bounded exponent, thereby proving Hopkins' chromatic splitting conjecture and the vanishing conjecture of Beaudry-Goerss-Henn rationally. The key ingredients are the equivalence between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze-Weinstein, integral p-adic Hodge theory, and an integral refinement of a theorem of Tate on the Galois cohomology of non-archimedean fields.