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Chromatic homotopy theory organizes stable homotopy theory by a prime number $p$ and a hierarchy of heights $n$. Its main probes include Morava K-theory, whose standard coefficient ring for $n\ge 1$ 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]

A basic chromatic question asks which heights detect a given finite spectrum: its type is the least $n$ with $K(n{)}_{\ast }(X)\ne 0$, when such an $n$ 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.

Jacob Lurie is a mathematician whose work develops higher category theory and its applications to algebraic geometry and homotopy theory. His 2009 paper sketches a proof of the cobordism hypothesis, a classification of extended topological quantum field theories.^[ref]

The organizing idea is that the value on a point, with the appropriate duality properties, can determine the field theory on higher-dimensional manifolds and their cobordisms. Cutting and gluing become composition laws in a higher category. This explains why passing from whole manifolds to pieces of every dimension changes the algebraic framework. His chromatic lecture notes give another bridge between algebra and topology, constructing Morava K-theory from structured topological spectra.^[ref]

At a prime number $p$ and height $n\ge 1$, Morava K-theory is a periodic generalized homology theory represented by a topological spectrum $K(n)$ with coefficient graded ringHere multiplication by $v_n$ is invertible and implements periodicity. Every nonzero homogeneous coefficient is invertible, so this is a graded field, although it is not a field as an ungraded ring.^[ref]

For $p=2,n=1$, the period is two: the coefficient group is ${\mathbb{F}}_2$ in each even degree and zero in odd degrees. Distinct heights supply different tests for topological spectra; vanishing of $K(n{)}_{\ast }(X)$ means that this particular probe misses $X$, not that $X$ is necessarily trivial. This is the organizing idea of chromatic homotopy theory.