I am learning about Morava E-theory, but I am confused about how to use Landweber exact functor theorem to construct it.
To be precise, suppose $k$ is a perfect field with characteristic $p > 0$ and $R = W(k)[[v_1, \cdots, v_{n-1}]]$ is the Lubin-Tate ring. This ring is Landweber exact, i.e., the sequence $p,v_1,v_2,\cdots,v_{n-1}$ is regular. Let $\tilde{R} = R[\beta^{\pm 1}]$ with $\deg \beta = 2$. Then by Landweber exact functor theorem, we get a generalized homology theory $(E_n)_*$ such that $(E_n)_*(X) = MU_*(X) \otimes_{MU_*} \tilde{R}_*$. However, when we talk about Morava E-theory, we actually mean a cohomology theory. My question is that how can we turn theis homology theory into a cohomology theory? More generally, can we use Landweber exact functor theorem to define a cohomology theory?
PS. I learnt from nlab that over finite CW-complexes by Spanier-Whitehead duality there is a form of Brown representability theorem for covariant functors. Thus, there is a spectrum representing the above homology theory for finite CW-complexes. However, when $X$ is not finite, the homology theory represented by the spectrum may not be the same with the one constructed by Landweber exact functor theorem?
