Chromatic homotopy theory

In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and tmf.
Chromatic convergence theorem.
In algebraic topology, the chromatic convergence theorem states the homotopy limit of the chromatic tower (defined below) of a finite "p"-local spectrum formula_1 is formula_1 itself. The theorem was proved by Hopkins and Ravenel.
Statement.
Let formula_3 denotes the Bousfield localization with respect to the Morava E-theory and let formula_1 be a finite, formula_5-local spectrum. Then there is a tower associated to the localizations 
called the chromatic tower, such that its homotopy limit is homotopic to the original spectrum formula_1.
The stages in the tower above are often simplifications of the original spectrum. For example, formula_8 is the rational localization and formula_9 is the localization with respect to "p"-local "K"-theory.
Stable homotopy groups.
In particular, if the formula_5-local spectrum formula_1 is the spectrum of the stable formula_5-local sphere spectrum formula_13, then the homotopy limit of this sequence is the original formula_5-local sphere spectrum. This is a key observation for studying stable homotopy groups of spheres using chromatic homotopy theory.