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.
