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A generalized homology theory obeys the homotopy, exactness, suspension and wedge axioms, without imposing the ordinary dimension axiom. A represented homology theory is obtained by taking stable homotopy groups after a smash product of spectra.

Given a topological spectrum $E$, its localization of spectra sends $X$ to an $E$-local topological spectrum $L_{E}X$ through a map $X\to L_{E}X$ inducing an isomorphism on $E$-homology. Here $E$-local means that every map from an $E$-acyclic topological spectrum to $L_{E}X$ is nullhomotopic; $A$ is $E$-acyclic when $E\wedge A$ is trivial. This is Bousfield localization.^[ref]

The map is universal for maps from $X$ to $E$-local objects. For rational homology, localization rationalizes homotopy groups: ${\pi }_j(L_{H\mathbb{Q}}X)={\pi }_j(X)\otimes \mathbb{Q}$. Thus the sphere spectrum loses its finite torsion information and retains $\mathbb{Q}$ in degree zero. Localization deliberately preserves the information seen by a chosen generalized homology theory while discarding information invisible to it.

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.

A topological spectrum $E$ represents the reduced generalized homology theory ${\tilde{E}}_k(X)={\pi }_k(E\wedge {\Sigma }^{\infty }X)$ on based spaces, where the smash product of spectra is derived. It is covariant in $X$.