1. Tyler Lawson
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5. Morava K- and E-theory

This is part of an algebraic topology problem list, maintained by Mark Hovey.

General references for this section are the memoir of the AMS by Hovey-Strickland, and the Picard group paper by Hopkins-Mahowald-Sadofsky.

1. Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree. This would follow from the chromatic splitting conjecture, I think. (I believe there is an argument for this in Devinatz' last paper on the generating hypothesis). I believe we know almost nothing about these groups, though we probably know they are pro-p-groups. I am not even sure about that right now. An unfortunate point here is that this is false if * is allowed to range over the Picard group instead of just the integers (Hovey-Strickland, based on Shimomura's calculation of L_2 S^0 at p>3).
   
   
2. Show that the Picard group is finitely generated over the p-adics. I don't think this is known even for the algebraic Picard group, which is obtained purely through group cohomology. So I think there might be some room for some purely algebraic understanding of profinite group cohomology here.
   
   
3. Elucidate the connection between the Morava stabilizer groups and the K(n)-local category. The first such problem, which is certainly not very hard and is no doubt known to Hopkins and some others, is to determine the structure of E*E and L_K(n)(E_*E), where E is Morava E-theory. You can take Morava E-theory to be the K(n)-localization of E(n), or you can take it to be the one related to the Lubin-Tate moduli space. The answer is supposed to be: for E^*E, you should get the twisted completed group ring E[[S]] of the stabilizer group--recall the stabilizer group acts on E, and that is where the twisting comes in. The completion affects both E and S and is an inverse limit over open subgroups of finite index. The answer for L_K(n)(E_*E), by which I mean pi_*(L_K(n) (E smash E)), should be C(S,E), continuous functions from S to E. Part of this problem is choosing exactly which E you want to work with and defining all these terms precisely. Actually, it would be better to do this for every reasonable choice of Morava E-theory E. Neil Strickland and I could probably have done this in our memoir on the K(n)-local category, but we thought the paper was long enough. But nobody else has done it, so maybe not. Even though this is folklore, I think it would be of value to get it all straight.
   
   
4. As a rule, I am not happy about the arbitrary nature of some of the constructions in the K(n)-local category. Consider the spectral sequence, for example, which relates the continuous cohomology of S with coefficients in pi_*(L_K(n) (E smash X)) to pi_* L_K(n) X, for example. The usual construction of this is to take the Adams-Novikov spectral sequence for E smash X smash a type n, and then take an inverse limit. I have never liked this, and it certainly only works for dualizable X. Is there some more natural construction? I have the same objection to the more general Devinatz-Hopkins construction of fixed point spectra designed to yield a spectral sequence starting with continuous cohomology rather than discrete cohomology.
   
   
5. Find the shadow of the thick subcategory theorem in the K(n)-local category. There is only one thick subcategory of small spectra in the K(n)-local category, corresponding to the type n finites. But the finite spectra of smaller type are still there--they are dualizable, but not small. My own conjecture, with Strickland, is that the ideals of dualizable spectra give you the expected filtration--an ideal is a thick subcategory closed under smashing with any dualizable spectrum.
   
   
6. We now know that Morava E-theory admits an action of the stabillizer group S. This is the famous Hopkins-Miller result, which one day I hope will see the light of day. Their proof relies on calculating the A_infinity automorphism group of E and showing that it is a homotopy discrete group whose pi_0 is isomorphic as an abstract group to the stabilizer group. As a method, this leaves a lot to be desired. The biggest problem here is: give a construction of Morava E-theory that makes it obvious that there is an action of the stabilizer group on it. Presumably this will require going from Lubin-Tate moduli spaces to spectra using some sort of infinite loop space technology. Note that the action should be continuous in an appropriate sense.
   
   
7. Presumably one should be able to form a category of E-S module spectra; spectra with an action of the ring spectrum E and a compatible action of the group S. Is this possible--and is it worthwhile? This one is going to take you into formal issues that one can spend a lot of time and verbiage on.
   
   
8. Understand the relationship between the K(n)-local category and some sort of (algebraic) derived category of E_*-S-modules. Jens Franke has claimed there is an equivalence there, but I don't really understand what he does. (This is likely a deficiency in me) In particular, I don't even know what the right algebraic category is. E_*E is a Hopf algebroid, so maybe we should consider comodules over it. But it is really pi_* L_K(n)(E smash E) that we should be considering, and I don't know what sort of algebraic gadget that is.
   
   
9. One of the corollaries of the Hopkins-Miller theorem, together with the Devinatz-Hopkins fixed point business, is that the famous class zeta in continuous H^1 of the stabilizer group survives to give a class in homotopy of degree -1. This is trivial at large primes, where the spectral sequence collapses, but at small primes it was not known. The chromatic splitting conjecture asserts that there should be other such classes, in degree -3, -5,...,-2n+1, so presumably in continuous H^3, H^5,...,H^2n-1 of S with trivial coefficients. Find these classes, and see if Hopkins-Miller allows you to determine that they survive to homotopy classes at all primes.
   
   
10. Bousfield has give a description of the E(1)-local category in terms of algebraic data related to K-theory. Franke claims to have generalized all this, but once again, I do not understand what he does. Bousfield describes the isomorphism classes, but not the maps. In the K(1)-local category, it seems to me, things might be a bit simpler. Is there some more natural description of Bousfield's work in the K(1)-local category? It is always dangerous to try to improve on Bousfield, and certainly no one has succeeded in the past 10 plus years.

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