I'm wrapping up a summer project that involved a computation in Morava $E$-theory. As background knowledge I had to look into how the Johnson-Wilson theories $E(n)$ and Morava $K$-theories were constructed. This was manageable since I'd been part-way down that road already and there's lots of support in, for example, the form of Hopkins' course notes. Then, in early May I spent some time digging around for a construction of Morava $E$-theory, which led me to some conclusions:

- The words "Morava $E$-theory" don't determine what object you're talking about; there's a whole bunch of slightly different Morava $E$-theories.
- People frequently conflate Morava $E$-theory with (completed) Johnson-Wilson theory. One source even claimed (without citation) that one was a finite free module over the other, and that I therefore shouldn't worry about the difference.

In the end, I didn't need to really know much about $E_n$ beyond a couple formal properties to work out the broad strokes of my computation, so I let the whole thing slide and pretended there existed a spectrum that did what I'd hoped.

However, I'm now getting to a point where understanding what I'm actually doing would be valuable. In decreasing order of importance, can someone provide a reference that...

- ... constructs a family of Morava $E$-theories. Any family would be a start! I am particularly interested in one with a coefficient ring of the form $\mathbb{Z}_p[\![v_1, \ldots, v_{n-1}]\!][v_n^{\pm 1}]$.
- ... illustrates that $\mathrm{spf}\,(E_n)^* \mathbb{C}P^\infty$ is the universal deformation of $\mathrm{spf}\,K(n)^* \mathbb{C}P^\infty$ to formal groups over formal spectra of complete, local rings with residue field $\mathbb{F}_p[v_n^{\pm 1}]$. The remark above about the comparison between Johnson-Wilson theory and Morava $E$-theory made me particularly uncomfortable in this respect; it's not clear to me that the formal group associated to Johnson-Wilson theory should be thought of as the universal deformation of the Honda formal group. Clearing that up would be nice too.
- ... also shows that the reduction of the universal deformation to the "mod $p$" case exists as a spectrum, and the reduction map exists as a map of spectra. That is, there is a complex-oriented, structured ring spectrum $E_n/p$ with coefficient ring $\mathbb{F}_p[\![v_1, \ldots, v_{n-1}]\!][v_n^{\pm 1}]$ whose associated formal group is the universal deformation of the Honda formal group to complete, local rings of characteristic $p$.
- ... also shows that the reduction of the universal deformation modulo the $n$
^{th}power of its maximal ideal exists as a spectrum, and the reduction map exists as a map of spectra. - ... demonstrates this fact about $E_n$ being a finite free module over $E(n)$, at least for an appropriate interpretation of the symbol "$E_n$".

It may not be the case that points 3 and 4 are even true, but I'm hopeful. Still, surely this is all catalogued somewhere!