On the rationalization of the K(n)-local sphere

Barthel, Tobias; Schlank, Tomer M.; Stapleton, Nathaniel; Weinstein, Jared

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On the rationalization of the K(n)-local sphere

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Barthel, Tobias
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.48550/arXiv.2402.00960
(Preprint)

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2402.00960.pdf
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Barthel, T., Schlank, T. M., Stapleton, N., & Weinstein, J. (submitted). On the rationalization of the K(n)-local sphere.

Cite as: https://hdl.handle.net/21.11116/0000-000F-E55C-8

Abstract

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.