We calculate the homotopy type of and at the prime 2, where is localization with respect to Morava –theory and localization with respect to –local –theory. In we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology , where is the Morava stabilizer group and is the ring of functions on the height Lubin–Tate space. We show that the inclusion of the constants induces an isomorphism on group cohomology, a radical simplification.
Agnès Beaudry. Paul G Goerss. Hans-Werner Henn. "Chromatic splitting for the –local sphere at ." Geom. Topol. 26 (1) 377 - 476, 2022. https://doi.org/10.2140/gt.2022.26.377