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## Construction of Morava E-theory

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!

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excellent questions. i have not really been able to find constructions either. – Sean Tilson Aug 22 2010 at 3:45
also, which set of notes are you referring to? COCTALOS? – Sean Tilson Aug 22 2010 at 4:06
Yes, COCTALOS is what I meant; among other things, those notes cover the Landweber exact functor theorem, which is what you need to build Johnson-Wilson theory (but only as spectra, not as $E_\infty$-rings, which I do actually need). I'm not sure what Tyler was referring to, since they don't talk about Hopkins-Miller theory, and I'm unaware of any course notes by Hopkins that do -- but that note by Rezk I link to in my comment is pretty nice. – Eric Peterson Aug 22 2010 at 4:47
Just in case others stumble across this like I did, Coctalos is a course Hopkins taught at MIT in 1999. It's short for "COMPLEX ORIENTED COHOMOLOGY THEORIES AND THE LANGUAGE OF STACKS" and can be found at math.rochester.edu/people/faculty/doug/… – David White Feb 26 at 21:15

So far as what Morava E-theory should be: Morava E-theory always implicitly comes with a choice of a perfect residue field of positive characteristic and a formal group law of finite height over this field. Sometimes people take a very specific formal group law, but there is no reason to be restrictive. The underlying homotopy type of spectrum may not change in a very interesting way, but the multiplicative structure does (just as many rings that are different in interesting ways may have the same underlying abelian groups). This actually becomes a more serious issue with things like $BP\langle n\rangle$ and the Johnson-Wilson theories because there are multiple possible inequivalent orientations that look basically the same when you write down the rings of homotopy groups.

• Constructing a family of Morava E-theories. (The coefficient ring you list is implicitly the coefficient ring of a completed Johnson-Wilson theory, and that's only if I interpret your power series brackets as "invert vn and then complete" due to the grading issue.) Johnson-Wilson theories and completed Johnson-Wilson theories, with the names you've given to the generators, all satisfy Landweber's criterion and so you can produce them the easy way as spectra via the Landweber exact functor theorem. Morava E-theories are slightly more difficult; they're still Landweber exact, but to prove that you need to know that the universal deformation ring of a formal group law of finite height has a particular structure relative to the coordinates of its p-series. However, if you want to construct any of these as genuine commutative ring spectra then you've got to use the Goerss-Hopkins-Miller theorem on the Morava E-theories, use it with its functoriality properties for the completed Johnson-Wilson theories, and you're out of luck for the uncompleted Johnson-Wilson theories except in a handful of very specific cases (some of which may require you to be slightly flexible about what "Johnson-Wilson" means). For references for the Hopkins-Miller theorem in the associative case, there are of course Charles Rezk's Notes on the Hopkins-Miller theorem, although you have a far better source on hand. The Goerss-Hopkins paper Moduli problems for structured ring spectra covers the obstruction theory for making these objects commutative.
• So far as illustration goes, the method I was proposing in the previous paragraph started with the universal deformation ring and produced the E-theory as its manifestation, meaning that by the time you get here you've already finished this step. Note that you have to be a little careful about the grading issue. The coefficient ring of a Morava E-theory in its full, graded glory is the universal deformation ring of a formal group law equipped with a choice of generator of the relative cotangent space (a trivial torsor for the multiplicative group over the universal deformation of the formal group law).
• The mod-p reduction exists as a spectrum, but not as a commutative ring spectrum. Any commutative ring spectrum with $p=0$ in its homotopy groups is a module over the Eilenberg-Mac Lane spectrum ${\rm H}\mathbb{F}_p$ (this is because this Eilenberg-Mac Lane spectrum is actually the free algebra over the little 2-disks operad with $p=0$!) and these residue fields don't qualify. You can show that these residue objects exist using obstruction theory; there are a number of references but let me specifically plug Vigleik Angeltveit's Topological Hochschild homology and cohomology of A-infinity ring spectra.
• Modulo the n'th power of the maximal ideal is a little trickier and I would have to go check to see if the obstruction theory necessarily worked out. If you're willing to instead kill the n'th powers of the generators of the maximal ideal then the obstruction theory from the previous point still works, because you are killing a regular sequence of generators. Again, this only produces associative objects rather than commutative ones.
• The fact that $E_n$ is a finite free module over the completed $E(n)$ is a formal consequence of the computation of the homotopy groups of both involved. The left-hand object has homotopy groups $$\mathbb{Z}_{p^n}[\![u_1,\ldots,u_{n-1}]\!][u^{\pm 1}]$$ with $|u_i| = 0, |u| = 2$ and the right-hand object has homotopy groups as the subring $$\mathbb{Z}[v_1,\cdots,v_n,v_n^{-1}]^\wedge$$ where $v_i = u^{p^i - 1} u_i$ for $1 \leq i < n$, $v_n = u^{p^n - 1}$, and the completion is taken with respect to the intersection of the maximal ideal with this subring. Once you have figured out what this entails, you find that the homotopy groups are a finite free module and as a consequence the spectrum itself is a finite free module.
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Thanks! There's a lot here, I'll start digesting. I've thumbed through a lot of Rezk's written materials before, and he avoids saying much about Hopkins-Miller in those notes, but he does have some "Notes on the Hopkins-Miller theorem" at citeseerx.ist.psu.edu/viewdoc/… which you might have been thinking of, which mention $A_\infty$-rings on the first page, and which I've never read. -- And, now that I've read the first two pages, this is very much the sort of thing I was hoping for. I'll have to tell Rezk thank you too. :) – Eric Peterson Aug 22 2010 at 3:07
Can I interpret part of your answer as saying the following: whenever we have a fgl over R that satisfies the hypothesis of the Landweber Exact Functor theorem then we can use that and Brown representability to have the specified homotopy groups and fgl? This is sort of a black box approach and doesnt answer more delicate questions, but is this in the right direction? – Sean Tilson Aug 22 2010 at 4:24
@Lennart: At p=2 this essentially follows from Mahowald's construction of HZ/2 as a Thom spectrum over $\Omega^2 S^3$ (see "Ring spectra which are Thom complexes); at odd primes it is a p-local Thom spectrum There's a recent paper on the arXiv by Blumberg-Cohen-Schlichtkrull (arxiv.org/abs/0811.0553) that discusses these results on pp3-4. The work in these results amounts to understanding how the lowest-excess Dyer-Lashof operations act on the dual Steenrod algebra. – Tyler Lawson Aug 22 2010 at 18:33
If anyone's still wondering: turns out that Baker has worked out the existence of $A_\infty$ structures on the intermediate spectra $E(n) / I_n^r$ in maths.gla.ac.uk/~ajb/dvi-ps/ainfty.pdf , and Wuethrich has checked that the system is one of $MU$-algebras in addition to $S$-algebras in arxiv.org/pdf/math/0607110v4.pdf . – Eric Peterson Mar 12 2012 at 17:24
@Akhil: you can also look at Mark Steinberger's section of the H_\infty ring spectra volume (it is available online at Peter May's website). That is more in line with the last sentence of Tyler's response to Lennart. – Sean Tilson Sep 11 at 18:30