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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.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 (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. (Johnson Wilson) 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

[math]\displaystyle{ K(n)_*(X \times Y) \cong K(n)_*(X) \otimes_{K(n)_*} K(n)_*(Y). }[/math]

• 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

• Johnson, David Copeland; Wilson, W. Stephen (1975), "BP operations and Morava's extraordinary K-theories.", Math. Z. 144 (1): 55−75, doi:10.1007/BF01214408 
• Hovey-Strickland, "Morava K-theory and localisation"
• Ravenel, Douglas C. (1992), Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, 128, Princeton University Press 
• Würgler, Urs (1991), "Morava K-theories: a survey", Algebraic topology Poznan 1989, Lecture Notes in Math., 1474, Berlin: Springer, pp. 111–138, doi:10.1007/BFb0084741, ISBN 978-3-540-54098-4