The chromatic splitting conjecture concerns the overlap 
between height-
localization and lower chromatic heights. Its weak sphere
formulation asks whether the canonical unit 
has a homotopy retraction. A strong
formulation specifies a decomposition into shifted local spheres. An ordinary finite-assembly question instead asks for membership in the
thick subcategory generated by 
through 
in the category
of 
-local
modules over 
. These requirements retain different information about
maps and attaching maps.
Chromatic Splitting Conjecture
See also
Bousfield Localization, Cofiber, Morava K-Theory, Sphere Spectrum, Thick SubcategoryExplore with Wolfram|Alpha
References
Hovey, M. and Strickland, N. P. Morava K-Theories and Localisation. Providence, RI: Amer. Math. Soc., 1999. https://doi.org/10.1090/memo/0666.Cite this as:
Weisstein, Eric W. "Chromatic Splitting Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChromaticSplittingConjecture.html