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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.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

## 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

Fp[vn,vn−1]

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