Morava K-theory explained

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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.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 (which is suppressed in the notation), it consists of theories K(n) for each nonnegative integer n, each a ring spectrum in the sense of homotopy theory. published the first account of the theories.

Details

The theory K(0) agrees with singular homology with rational coefficients, whereas K(1) is a summand of mod-p complex K-theory. The theory K(n) has coefficient ring

F_p[''v''_''n'',''v''_''n''^−1]

where v_n has degree 2(pⁿ - 1). 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

K(n) _{*(X} x Y) \cong K(n) _{*(X)} ⊗ _{K(n) _{*}} K(n) _{*(Y).}

• They are "fields" in the category of ring spectra. In other words every module spectrum over K(n) is free, i.e. a wedge of suspensions of K(n).
• They are complex oriented (at least after being periodified by taking the wedge sum of (pⁿ - 1) shifted copies), and the formal group they define has height n.
• Every finite p-local spectrum X has the property that K(n)_∗(X) = 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.

See also

• Chromatic homotopy theory
• Morava E-theory

References

• • Hovey-Strickland, "Morava K-theory and localisation"