Morava K-theory

Remarks

$K(n)$ is a ring spectrum and is complex oriented. Its formal group law can be chosen with $p$-series $[p](x) = v_n x^{p^n}$, so it has height $n$. Every graded module over $K(n)_*$ is free, since $K(n)_*$ is a graded field. This gives, for spaces $X$ and $Y$, a Künneth isomorphism $K(n)_*(X \times Y) \cong K(n)_*(X) \otimes_{K(n)_*} K(n)_*(Y)$. The theory $K(n)$ is $2(p^n - 1)$-periodic. By convention $K(0)$ is rational cohomology and $K(\infty)$ is mod $p$ cohomology.

Example: for $p = 2$ and $n = 1$, mod $2$ complex K-theory $KU/2$ has coefficients $\F_2[u, u^{-1}]$ with $\abs{u} = 2 = 2(2^1 - 1)$. Its formal group law is the multiplicative one, whose $2$-series is $u x^2$ modulo $2$. So $KU/2$ is a model of $K(1)$ at $2$.

Non-example: for an odd prime $p$, the spectrum $KU/p$ is not $K(1)$. Its coefficients $\F_p[u, u^{-1}]$ have a unit in degree $2$, while $v_1$ has degree $2(p - 1) > 2$. In fact $KU/p$ splits as a sum of $p - 1$ suspensions of $K(1)$.