nLab
Morava K-theory

Definition

For each prime integer $p$ there exists a sequence of generalized homology theories (equivalently spectra) $\{K(n)\}$ indexed by the non-negative integers, with the following properties:

1. $K(0{)}_{*}(X)=H_{*}(X;\mathbb{Q})$ and ${\overline{K(0)}}_{*}(X)=0$ when ${\overline{H}}_{*}(X)$ is all torsion.
2. $K(1{)}_{*}(X)$ is one of $p-1$ isomorphic summands of mod-$p$ complex topological K-theory.
3. $K(0{)}_{*}(\mathrm{pt}.)=\mathbb{Q}$ and for $n\ne 0$, $K(n{)}_{*}(\mathrm{pt}.)=\mathbb{Z}/(p)[v_n,v_n^{-1}]$ where $\mid v_n\mid =2p^n-2$. This ring is a graded field in the sense that every graded module over it is free. $K(n{)}_{*}(X)$ is a module over $K(n{)}_{*}(\mathrm{pt}.)$.
4. There is a Künneth isomorphism: $K(n{)}_{*}(X\times Y)\cong K(n{)}_{*}(X){\otimes }_{K(n{)}_{*}(\mathrm{pt}.)}K(n{)}_{*}(Y).$
5. Let $X$ be a p-local finite CW-complex. If ${\overline{K(n)}}_{*}(X)$ vanishes then so does ${\overline{K(n-1)}}_{*}(X)$.
6. If $X$ as above is not contractible then ${\overline{K(n)}}_{*}(X)=K(n{)}_{*}(\mathrm{pt}.)\otimes {\overline{H}}_{*}(X;\mathbb{Z}/(p))$.

Connection to Bousfield lattice

It is known that in the Bousfield lattice of the stable homotopy category, the Bousfield classes of the Morava K-theories are minimal. It is conjectured by Mark Hovey and John Palmieri that the Boolean algebra contained in the Bousfield lattice is atomic and generated by the Morava K-theories and the spectra $A(n)$ which measure the failure of the telescope conjecture.

Related concepts

• K-theory, topological K-theory, algebraic K-theory

• Morava E-theory

References

• D. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Mathematics Studies 128, Princeton University Press (1992).

• Morava E-theory and Morava K-theory Lecture notes (pdf)