# Morava K-theory

## Definition

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

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

## 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 $A\left(n\right)$ which measure the failure of the telescope conjecture.

## 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)

Revised on April 10, 2013 15:20:24 by Urs Schreiber (82.169.65.155)