Chromatic homotopy theory is a branch of algebraic topology that studies stable homotopy groups of spheres and related phenomena through the lens of chromatic filtration. It originated from attempts to better understand the relationship between stable homotopy theory and complex-oriented cohomology theories, particularly in the context of the stable homotopy category.
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Chromatic homotopy theory organizes stable homotopy theory by a prime number and a hierarchy of heights . Its main probes include Morava K-theory, whose standard coefficient ring for isThe index is the height of the associated formal group law. Rational homology plays the height-zero role. These different probes detect different parts of a spectrum, rather than replacing all stable homotopy information by one ordinary homology group.[ref]
Morava E-Theory and Morava K-Theory by Jacob Lurie. Lecture 22, Definition 5 and coefficient calculation, page 3
A basic chromatic question asks which heights detect a given finite spectrum: its type is the least with , when such an exists. Rational information already detects the sphere spectrum at height zero, whereas purely torsion phenomena require other probes. The chromatic viewpoint makes periodic patterns and localization of spectra into organizing tools for difficult stable computations.