In algebraic topology, the
chromatic convergence theorem states the
homotopy limit of the chromatic tower (defined below) of a finite
p-local spectrum X is X itself. The theorem was proved by Hopkins and Ravenel.
Statement Let L_{E(n)} denotes the
Bousfield localization with respect to the
Morava E-theory and let X be a finite, p-local spectrum. Then there is a tower associated to the localizations :\cdots \rightarrow L_{E(2)} X \rightarrow L_{E(1)} X \rightarrow L_{E(0)} X called the
chromatic tower, such that its homotopy limit is homotopic to the original spectrum X. The stages in the tower above are often simplifications of the original spectrum. For example, L_{E(0)} X is the rational localization and L_{E(1)} X is the localization with respect to
p-local K-theory.
Stable homotopy groups In particular, if the p-local spectrum X is the stable p-local
sphere spectrum \mathbb{S}_{(p)}, then the homotopy limit of this sequence is the original p-local sphere spectrum. This is a key observation for studying stable homotopy groups of spheres using chromatic homotopy theory. == See also ==