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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.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 (1975) published the first account of the theories.

Overview

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.

Properties

• They have Künneth isomorphisms for arbitrary pairs of spaces: that is, for X and Y CW complexes, we have

$K(n)_{*}(X\times Y)\cong K(n)_{*}(X)\otimes _{K(n)_{*}}K(n)_{*}(Y).$

• For every prime number $p$ and every natural number $n\in \mathbb {N}$, Morava K-theory $K(n)$ is a "field" in the category of ring spectra, meaning the graded ring $\pi _{0}K(n)$ is even a graded field, meaning every nonzero homogeneous element in it is invertible. Therefore every module spectrum over $K(n)$ is free, meaning it's 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.

K(n)-local spheres

For the sphere spectrum $\mathbb {S}$ (with $\mathbb {S} _{n}=S^{n}$), its Bousfield localizations $L_{K(n)}\mathbb {S}$ are also called $K(n)$-local spheres.

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&minus, 75, doi:10.1007/BF01214408, MR 0377856
• Hovey-Strickland, "Morava K-theory and localisation"
• Lurie, Jacob (2010-04-27). "Morava E-Theory and Morava K-Theory (Lecture 22)" (PDF). people.math.harvard.edu. Harvard University. Retrieved 2026-08-21.
• Lurie, Jacob (2010-04-27). "The Bousfield Classes of E(n) and K(n) (Lecture 23)" (PDF). people.math.harvard.edu. Harvard University. Retrieved 2026-08-21.
• Lurie, Jacob (2010-04-27). "Uniqueness of Morava K-Theory (Lecture 24)" (PDF). people.math.harvard.edu. Harvard University. Retrieved 2026-08-21.
• Ravenel, Douglas C. (1992), Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, vol. 128, Princeton University Press, MR 1192553
• Würgler, Urs (1991), "Morava K-theories: a survey", Algebraic topology Poznan 1989, Lecture Notes in Math., vol. 1474, Berlin: Springer, pp. 111–138, doi:10.1007/BFb0084741, ISBN 978-3-540-54098-4, MR 1133896

External links

• Morava K-theory on the nLab