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

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arxiv 2402.00960 v2 pith:VCGRIAVY submitted 2024-02-01 math.AT math.AGmath.NT

Tobias Barthel , Tomer M. Schlank , Nathaniel Stapleton , Jared Weinstein This is my paper · ORCID
classification math.ATmath.AGmath.NT
keywords cohomologyconjectureintegrallocallubin-tatespheretoweradic
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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.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A fully faithful p-adic Riemann-Hilbert functor for coadmissible D-cap-modules

    math.AG 2025-06 unverdicted novelty 8.0 of 10

    A p-adic Riemann-Hilbert correspondence is established: after base change to the overconvergent almost de Rham period sheaf, the solution functor becomes fully faithful on coadmissible D-modules.

  2. Higher Semiadditive Character Theory

    math.AT 2026-07 accept novelty 7.0 of 10

    Every ∞-commutative monoid has a universal (n−t)-fold semiadditive character that blue-shifts height, recovers the transchromatic character on Morava E-theory, and computes L_Q(S^A_{K(n)}) via GL_{n−t}(Z_p)-fixed points.

  3. Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory

    math.AT 2025-08 conditional novelty 7.0 of 10

    Derived level structures in spectral algebraic geometry are representable, yielding Jacquet-Langlands spectra and a proposed Jacquet-Langlands dual of Morava E-theory.

  4. Modeling Group Actions on Stacks (Especially the Lubin-Tate Action)

    math.AG 2025-06 conditional novelty 3.0 of 10

    A survey codifying geometric modeling and two-tower methods for group actions on formal moduli stacks, with the Lubin-Tate action as the guiding example.

  5. Periodic phenomena in stable motivic homotopy theory

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    A survey of periodic phenomena in stable motivic homotopy theory, organizing known motivic Adams spectral sequence computations and open problems; no new theorem is proven.

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