Morava K-theory

In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number p, it consists of theories K for each nonnegative integer n, each a spectrum (homotopy theory)|ring spectrum] in the sense of homotopy theory. published the first account of the theories.

Overview

The theory K agrees with singular homology with rational coefficients, whereas K is a summand of mod-p complex K-theory. The theory K has coefficient ring
where v_n has degree 2. In particular, Morava K-theory is periodic with this period, in much the same way that complex K-theory has period 2.
These theories have several remarkable properties.

They have Künneth isomorphisms for arbitrary pairs of spaces: that is, for X and Y CW complexes, we have
They are "fields" in the category of ring spectra. In other words every module spectrum over K is free, i.e. a wedge of suspensions of K.
They are complex oriented, and the formal group they define has height n.
Every finite p-local spectrum X has the property that K_∗ = 0 if and only if n is less than a certain number N, called the type of the spectrum X. By a theorem of Devinatz-Hopkins-Smith, every thick subcategory of the category of finite p-local spectra is the subcategory of type-n spectra for some n.