July 17, 2005

Dear Mike,

This note consists of some questions about your work on the problem of realizing E_{n} as a G_{n}-spectrum with a continuous structured action. The note also contains a discussion of my work on this problem.

1. Trying to understand your approach to this problem

Mark Behrens sent me a sketch of a proof, based on your argument, that the pro-object {L_{n}M_{I}}_{I} is an H_{∞} object in the category of pro-spectra, where I = (p^{i_{0}}, v_{1}^{i_{1}}, ..., v_{n−1}^{i_{n−1}}) ⊂ BP_{∗}, M_{I} is the corresponding generalized Moore spectrum, and {I} is a cofinal collection of ideals such that L_{K(n)}S^{0} = holim_{I} L_{n}M_{I}. My understanding is that your actual argument was that E_{n} is a continuous G_{n}-E_{∞} ring spectrum, in some sense, by using pro-spectra.

I have been thinking about how to realize E_{n} as a continuous G_{n}-A_{∞} ring spectrum since 2000. Thus, I am very interested in learning more about what your ideas are regarding this problem. Also, because this problem is important to me, I am eager to learn what progress you have made on this problem.

Based on the above result that Mark told me, one might hope to argue in the following way. (Thus, if a step has not been rigorously verified, then it seems plausible.) The speculative argument begins with some definitions.

Definition 1.1. Let G be a profinite group. If S is a G-set, then S is a discrete G-set if the action map G × S → S is continuous, where S is given the discrete topology. If X is a (naive) G-symmetric spectrum of simplicial sets, such that, for each k ≥ 0, X_{k} is a simplicial discrete G-set, then X is a discrete G-spectrum.

We need the following categories.

Definition 1.2. Let Spt_{G} denote the category of discrete G-spectra. Let Spt_{G}^{a} be the category of discrete G-spectra that are symmetric ring spectra (that is, A_{∞} ring spectra in symmetric spectra) such that the G-action is through maps of symmetric ring spectra. Also, let (pro−Spt_{G})_{C_{∞}} be the category of objects in pro−Spt_{G}, the category of pro-objects in Spt_{G}, that are C_{∞} objects, where C is H or E.

Let {U_{i}}_{i≥0} be a descending chain of open normal subgroups of G_{n}, such that ⋂_{i} U_{i} = {e}. Let E_{∞} be the category of commutative S-algebras. By [7], the G_{n}/U_{i}-action on E_{n}^{hU_{i}} is given by a functor

the groupoid {G_{n}/U_{i}} → E_{∞}, G_{n}/U_{i} ↦ E_{n}^{hU_{i}}.

Composing this functor with the functor {(−) ∧ M_{I}} yields a functor

the groupoid {G_{n}/U_{i}} → (pro−Spt_{G_{n}/U_{i}})_{H_{∞}}, G_{n}/U_{i} ↦ {E_{n}^{hU_{i}} ∧ M_{I}}.

By the projection G_{n} → G_{n}/U_{i}, for each i,

{E_{n}^{hU_{i}} ∧ M_{I}} ∈ (pro−Spt_{G_{n}})_{H_{∞}}.

Without loss of generality, one can assume that certain cofibrancy conditions are satisfied so that

{colim_{i}(E_{n}^{hU_{i}} ∧ M_{I})} ∈ (pro−Spt_{G_{n}})_{H_{∞}},

where the colimit is formed in Spt_{G_{n}} (which is formed in symmetric spectra).

By [7], E_{n} ∧M_{I} ≃ colim_{i}(E_{n}^{hU_{i}} ∧M_{I}). Since holim_{I}(E_{n} ∧M_{I}) is the G_{n}-spectrum E_{n}, holim_{I} colim_{i}(E_{n}^{hU_{i}} ∧ M_{I}) ≃ E_{n} is a weak equivalence that respects the G_{n}- actions. Thus, one can say that E_{n} is a continuous G_{n}-H_{∞} spectrum, where this terminology indicates precisely that

{colim_{i}(E_{n}^{hU_{i}} ∧ M_{I})} ∈ (pro−Spt_{G_{n}})_{H_{∞}}.

Thus, the diagram {colim_{i}(E_{n}^{hU_{i}} ∧ M_{I})} has enough structure so that

holim_{I} colim_{i}(E_{n}^{hU_{i}} ∧ M_{I})

is an H_{∞} spectrum such that the G_{n}-action is through H_{∞} maps of spectra. (If there’s not enough structure, for this to be true, then perhaps one can show that

holim_{I} colim_{i}(E_{n}^{hU_{i}} ∧ M_{I})

is an A_{∞} spectrum with a G_{n}-action through A_{∞} maps.) Recall from [4] that E_{n} ≃ holim_{I} colim_{i}(E_{n}^{hU_{i}} ∧ M_{I}) makes E_{n} a continuous G_{n}-spectrum. Similarly, the fact that holim_{I} colim_{i}(E_{n}^{hU_{i}} ∧ M_{I}) is H_{∞} makes E_{n} a continuous G_{n}-H_{∞} spectrum (or hopefully, at least a continuous G_{n}-A_{∞} spectrum).

The above argument is what I’ve guessed might be your strategy for producing a continuous structured action. In addition to what Mark told me about your argument, I’ve seen Rognes’s comment that “Hopkins has suggested that a weaker form of structured commutativity, in terms of pro-spectra, may instead be available” [9, pg. 25].

Given the above, I’m wondering: is the above argument the kind of argument that you have in mind? Is your actual argument that it should all go through in the E_{∞} setting, and not just in the H_{∞} setting; that is, are you able to show that

{colim_{i}(E_{n}^{hU_{i}} ∧ M_{I})} ∈ (pro−Spt_{G_{n}})_{E_{∞}}?

I really would like to learn about your ideas for this problem, because this kind of realization problem is difficult and I don’t want to pursue a strategy that is erroneous or inefficient - in §3, I explain where I am at in my own approach to this problem, which was, for the most part, done while I was doing my Ph.D. with Paul.

2. Useful observations regarding this problem

The following definition is useful.

Definition 2.1. Let

F_{n} = colim_{i} E_{n}^{hU_{i}}.

After trying various realization strategies, I decided that the best route was to try to realize (E_{n})_{∗}/I by a discrete G_{n}-symmetric ring spectrum E_{n}/I, that is, by a spectrum E_{n}/I ∈ Spt_{G_{n}}^{a}.

Before describing my work on this problem in §3, below I make a series of remarks that help to frame the problem.

(2.2) By taking cofibrant replacements and working with cofibrations, as needed, the discrete G_{n}-spectrum F_{n} is a commutative symmetric ring spectrum such that the discrete G_{n}-action is by maps of commutative symmetric ring spectra. (Jeff Smith explained to me that there is a model category structure on commutative symmetric ring spectra that makes this work.) Thus, the interesting spectrum F_{n} is a discrete G_{n}-commutative symmetric ring spectrum; the continuous action on F_{n} is E_{∞}.

(2.3) By [6], the diagram {M_{I}} can be assumed to be a diagram of homotopy ring spectra. Then, by [8, Lemma 2.2], the pro-object {F_{n} ∧ M_{I})} is a diagram of discrete G_{n}-spectra that are homotopy ring spectra.

Therefore, presumably after running through the above arguments (including those in Remark (2.2)) with a little care, one obtains that the discrete G_{n}-action on F_{n}∧M_{I} is through maps of homotopy ring spectra, so that E_{n}∧M_{I} ≃ F_{n}∧M_{I} is a discrete G_{n}-homotopy ring spectrum. However, though this would give a continuous weakly structured action, we really only care about structure that occurs on the point-set level, so that this is not really interesting.

(2.4) It has been known for a long time that E_{n} ∧M_{I} almost definitely is an A_{∞} ring spectrum, since its close relative E(n)/I_{n}^{k} is A_{∞}, by [2]. Now, by [1], it is a theorem that E_{n} ∧ M_{I} is an A_{∞} ring spectrum. However, this is far from knowing that G_{n} acts discretely on F_{n} ∧ M_{I} by maps of A_{∞} ring spectra.

(2.5) I am excited about the idea that there is a spectrum E_{n}/I ∈ Spt_{G_{n}}^{a}, because this result should have a surprising consequence (first observed by Charles Rezk). Suppose that E_{n}/I ∈ Spt_{G_{n}}. Then the forgetful functor

U: Spt_{G_{n}}^{a} → symmetric ring spectra

should be well-behaved enough, so that the construction

holim_{Δ} Map_{c}(G_{n}^{•+1}, (E_{n}/I)_{f,G_{n}})^{G_{n}} ≃ (E_{n}/I)^{hG_{n}}

can be done entirely in Spt_{G_{n}}^{a}, so that (E_{n}/I)^{hG_{n}} ≃ E_{n}^{hG_{n}} ∧ M_{I} is an A_{∞} ring spectrum. This implies that E_{n}^{hG_{n}} ∧ M_{I} ≃ L_{K(n)}M_{I} is A_{∞}.

Paul, Charles, and Jeff have told me that it would be quite interesting if L_{K(n)}M_{I} turns out to be A_{∞}. More generally, (E_{n}/I)^{hG} ≃ E_{n}^{hG} ∧ M_{I}, for any closed G in G_{n}, would be A_{∞}.

(2.6) The implications of the conjecture that E_{n}/I is a discrete G_{n}-symmetric ring spectrum for Rognes’s Galois extensions and associative Galois extensions are explored in my manuscript [5].

(2.7) Jim and Haynes told me that it is very doubtful that E_{n} ∧ M_{I} has the homotopy type of an E_{∞} ring spectrum. Charles sketched for me an argument that seems to imply that, for all n, p, and I, E_{n} ∧ M_{I} fails to have the homotopy type of an E_{∞} ring spectrum. (Charles said this argument was yours; Jim said the argument has an important antecedent in work of Mark Steinberger.)

There is no reason to hope that E_{n}/I is a discrete G_{n}-commutative symmetric ring spectrum. Thus, the only way to show that holim_{I}(E_{n}/I) makes E_{n} a continuous G_{n}-E_{∞} ring spectrum is to show that, thanks to enough structure being present in each E_{n}/I, the homotopy limit holim_{I}(E_{n}/I) yields a G_{n}-action through E_{∞} maps. Over the years, I have tried to set up various obstruction theoretic machines, in the spirit of Hopkins-Miller and Goerss-Hopkins, and I believe that the machinery required for the type of realization problem described in the preceding sentence would be extremely technically formidable.

The fact that {E_{n}/I} cannot be E_{∞} has the effect of making our problem more well-defined, and, together with Remark (2.4), it indicates that trying to show that E_{n}/I ∈ Spt_{G_{n}}^{a} is a plausible first step in producing a continuous structured action.

(2.8) Jeff has done work leading him to believe that, roughly speaking, something like the following theorem is true: “Let C be a symmetric monoidal model category of R-modules, where R is a monoid (in a symmetric monoidal model category D). If [Σ^{n}R, R]_{Ho(C)} = 0, whenever n is odd, and if α ⊂ π_{∗}(R) is a regular sequence, then R/α is also a monoid in C.”

Jeff and I have talked about how this theorem could be useful for the problem of constructing a continuous structured action. (An easy corollary of this theorem is that E_{n}/I is A_{∞}, which, as noted earlier, is already known.) For example, let C be the category _{F_{n}}Spt_{G_{n}}^{a} of discrete G_{n}-symmetric ring spectra that are twisted F_{n}-modules (that is, the module structure map F_{n} ∧ F_{n} → F_{n} is G_{n}-equivariant, where the source has the diagonal action). However, π_{∗}(F_{n}^{hG_{n}}) is not known, so that the hypotheses of the theorem cannot be verified to conclude that F_{n}/I ≃ E_{n}/I ∈ Spt_{G_{n}}^{a}. Thus, this theorem is not known to be helpful. (Also, it not known that I is a regular sequence in π(F_{n}).)

(2.9) The previous remark points out that it might be useful to know what π_{∗}(F_{n}) is. Paul computed this in the case n = 1 and p = 2 and obtained that

π_{∗}(F_{1}) = \begin{cases}ℤ/p^{∞} & ∗ = odd, ≠ −1, \\ ℤ_{p} & ∗ = 0, \\ ℚ_{p} & ∗ = −1, \\ 0 & otherwise. \end{cases}

This computation led Paul to suspect that F_{n} and the spectrum E_{n}/I_{n}^{∞} (which plays a role in Neil Strickland’s proof of Gross-Hopkins duality [10, pp. 1029-1031]) are closely related to each other. (Jeff also thinks that F_{n} and E_{n}/I_{n}^{∞} seem to be close to each other.) Note that F_{n} and E_{n}/I_{n}^{∞} cannot be identical to each other, because the identify M_{n}E_{n} = Σ^{−n}E/I_{n}^{∞} (by [10, pg. 1030]) shows that E/I_{n}^{∞} is K(n)-local, whereas F_{n} is not K(n)-local (see [4]).

In this context, it is natural to compute π_{∗}(F_{n}^{hG_{n}}) using its descent spectral sequence. Also, I have wondered if F_{n}^{hG_{n}} is closely related to L_{n}S^{0}. I think that a close relationship between F_{n}^{hG_{n}} and L_{n}S^{0} would be the discrete analogue of the equivalence that comes from this relationship’s I_{n}-adic completion

L_{K(n)}(L_{n}S^{0}) = L_{K(n)}S^{0} = E_{n}^{hG_{n}} = L_{K(n)}(F_{n}^{hG_{n}}),

where the last identity is shown in [4]. Once, Paul quickly did a descent spectral sequence computation and concluded that, for n = 1 and p > 2, F_{n}^{hG_{n}} and L_{n}S^{0} are probably not the same.

(2.10) In trying to realize E_{n}/I ∈ Spt_{G_{n}}^{a}, most of the time when one wants to (a) prove something about the G_{n}-action on E_{n} ∧ M_{I}, one can just as well obtain the desired result by instead (b) proving the analogous thing about the G_{n}/U_{i}-action on E_{n}^{hU_{i}} ∧ M_{I}, for each i.

An advantage of (a) is that G_{n}/U_{i} is a finite discrete group, so that its topology is simpler. For example, in (b) one simply asks for a finite group to act by A_{∞}- maps, but in (a), the A_{∞}-action must also respect the discrete topology on all the (E_{n}/I)_{k} and the profinite topology of G_{n}. Also, conceptually, it is easier to work with the category of modules over π_{∗}(E_{n}^{hU_{i}})[G_{n}/U_{i}] than with (E_{n})_{∗}[[G_{n}]], because with the former twisted group ring, one can ignore the topology. This makes the homological algebra involved in (b) simpler than the homological algebra for (a). When working with the homological algebra for (a), I often was forced to work with an Ext, whose topological algebra was so complicated that I never succeeded in getting it to behave as needed.

However, a disadvantage of (b) is that, in general, π_{∗}(E_{n}^{hU_{i}} ∧M_{I}) is not explicitly known, whereas, in (a), π_{∗}(E_{n} ∧ M_{I}) is complete known. In trying (b), I almost always find myself in a situation where I cannot proceed beyond the first step, due to the lack of computational knowledge about π_{∗}(E_{n}^{hU_{i}} ∧ M_{I}). This disadvantage of (b) has always meant that I have to proceed with (a), and abandon (b).

3. My approach to this problem

Now I briefly describe my work in trying to realize E_{n}/I as a discrete G_{n}- symmetric ring spectrum. For simplicity, let’s view the Hopkins-Miller and Goerss/Hopkins machinery for constructing A_{∞} and E_{∞}-actions of G_{n} on E_{n} as consisting of a universal coefficient spectral sequence and an obstruction theory for realizing F over an Adams spectrum E. For example, in the Hopkins-Miller theorem, E = F = E_{n}.

I have made two major efforts towards obtaining E_{n}/I ∈ Spt_{G_{n}}^{a}. In both efforts, F = E_{n}/I. In the first attempt, I let E = E_{n} and tried for a long time to build a universal coefficient sequence for Spt_{G_{n}} of the form

Ext_{(E_{n})_{∗}[[G_{n}]]}^{∗,∗}(Map_{c}(G_{n}, (E_{n})_{∗}/I, (E_{n})_{∗}) ⇒ π_{∗}Map_{G_{n}}(E_{n}/I, E_{n}/I).

But I never succeeded in constructing this spectral sequence because I could not get the requisite homological algebra of topological modules to work out.

Then I tried a different strategy: build the obstruction theory for E = F = E_{n}/I. To begin with a (partly) developed a version of André-Quillen cohomology for twisted discrete associative R-G-algebras, where R is a twisted discrete commutative G-ring. The Hopkins/Miller and Goerss/Hopkins machinery and [3] imply that the obstructions to the existence of E_{n}/I ∈ Spt_{G_{n}}^{a} are in

R^{t+2}Der_{(E_{n})_{∗}/I}^{G_{n}}((E_{n}/I)_{∗}(E_{n}/I), E_{n,∗+t}/I), t ≥ 0.

I tried to compute these obstructions for the case n = 1 and p > 2, but I never succeeded in this.

Since finishing my Ph.D. two years ago, I have not tried to take the two efforts above any further. Regarding the second effort, I concluded that (E_{n}/I)_{∗}(E_{n}/I) is probably too nasty to work with successfully. Regarding the first effort, at one point, I thought that I had succeeded in getting the requisite homological algebra to work out, but I did not have time to work through it carefully. The constraints of time during my Ph.D. forced me to stop the first effort completely and work on the second one.

In the past two years, I have primarily taken a step back and tried to reflect on what is the best strategy for tackling this problem. I have decided that the best way to proceed is to go back to effort one, where E = E_{n} and F = E_{n}/I, and see if I was correct about finally getting the homological algebra to work out. This way seems best because it seems that it would be closest to the Hopkins/Miller and Goerss/Hopkins approaches. Unfortunately, the level of difficulty of this approach seems such that it would be unwise to take a major chunk of time and pursue it, at this stage of my career. Instead, I believe that I should instead focus on my other projects, which are easier and which allow me to have publications.

Sometime after making this (disappointing) decision, Mark Behrens told me about your alternative strategy of producing a continuous structured action. As far as I can tell your strategy does not help with the problem of realizing E_{n}/I ∈ Spt_{G_{n}}^{a}, but it does seem to be a doable approach for obtaining an interesting result about E_{n}.

Because I love this problem and your strategy offers a way of making progress in the immediate future, I am very interested in learning about it.

Sometimes I wonder if the following approach to E_{n}/I ∈ Spt_{G_{n}}^{a} offers a better way than that given by E = E_{n}, F = E_{n}/I. Since E_{n} ∧ M_{I} is an L_{K(n)}S^{0}-module and L_{K(n)}S^{0} is a trivial discrete G_{n}-spectrum, let E = L_{K(n)}S^{0} and F = E_{n}/I. Ignoring the fact that I don’t know if L_{K(n)}S^{0} is an Adams spectrum (it needs to be one), the obstructions to the existence of E_{n}/I ∈ Spt_{G_{n}}^{a} are in

R^{t+2}Der_{π_{∗}(L_{K(n)}S^{0})}^{G_{n}}((E_{n})_{∗}/I, (E_{n})_{∗+t}/I), t ≥ 0.

If we can let E = L_{K(n)}S^{0} in the Hopkins/Miller and Goerss/Hopkins machinery, and if one is in the range where π_{∗}(L_{K(n)}S^{0}) is computationally accessible, then I wonder if these obstruction groups might be computable; because the action on π_{∗}(L_{K(n)}S^{0}) is trivial, its profinite topology should not present as many problems when one tries to work out the homological algebra. I wonder if you have any thoughts about whether or not this approach might be a good one.

Note that E_{n}^{hU_{i+j}} is an E_{n}^{hU_{i}}-module, for any j ≥ 0. Combining Remark (2.10) with the above strategy, leads one to note that another strategy, which depends on whether or not E_{n}^{hU_{i}} is an Adams spectrum to just get off the ground, is to let E = E_{n}^{hU_{i}} and F = E_{n}^{hU_{i+j}}/I, where j ≥ 0. In this case, the obstructions are

R^{t+2}Der_{π_{∗}(E_{n}^{hU_{i}})}^{U_{i}/U_{i+j}}(π_{∗}(E_{n}^{hU_{i}} ∧ E_{n}^{hU_{i+j}} ∧ M_{I}), π_{∗+t}(E_{n}^{hU_{i+j}}/I)),

where U_{i}/U_{i+j} is a finite group.

Another approach, which might computationally be the best, but which technically would be the most difficult to construct, is to apply the “look at the associated Morava module” technique. If X → Y is a map between K(n)-local spectra, often it is easiest to prove that this is a weak equivalence by showing that the associated map of Morava modules L_{K(n)}(E_{n} ∧ X) → L_{K(n)}(E_{n} ∧ Y) is a weak equivalence. Thus, one might redo the Hopkins/Miller machinery for realizing F by working over E, so that the machinery realizes F by looking at L_{K(n)}(E_{n} ∧ F), while working over L_{K(n)}(E_{n} ∧ E_{n}). Then the obstruction groups might look like

R^{t+2}Der_{Map_{c}(G_{n},(E_{n})_{∗})}^{G_{n}}(Map_{c}(G_{n}^{2}, (E_{n})_{∗}/I), Map_{c}(G_{n}, (E_{n})_{∗+t}/I) and

R^{t+2}Der∏_{G_{n}/U_{i}}(E_{n})_{∗}^{U_{i}/U_{i+j}}(Map_{c}(G_{n}/U_{i} × G_{n}/U_{i+j}, (E_{n})_{∗}/I), ∏_{G_{n}/U_{i+j}}(E_{n})_{∗+t}/I),

depending on which strategy is chosen.

Sincerely, Daniel

References

[1] Vigleik Angeltveit. A_{∞} obstruction theory and the strict associativity of E/I. Preprint available at the Hopf Topology Archive, 2005.

[2] Andrew Baker. A_{∞} structures on some spectra related to Morava K-theories. Quart. J. Math. Oxford Ser. (2), 42(168):403–419, 1991.

[3] D. Blanc, W. G. Dwyer, and P. G. Goerss. The realization space of a Π-algebra: a moduli problem in algebraic topology. Topology, 43(4):857–892, 2004.

[4] Daniel G. Davis. Homotopy fixed points for L_{K(n)}(E_{n} ∧ X) using the continuous action. To appear in the Journal of Pure and Applied Algebra, accepted July 13, 2005.

[5] Daniel G. Davis. Rognes’s theory of Galois extensions and the continuous action of G_{n} on E_{n}. Inactive manuscript, 14 pp., available at www.math.purdue.edu/^{∼}dgdavis, May 14, 2004.

[6] Ethan S. Devinatz. Small ring spectra. J. Pure Appl. Algebra, 81(1):11–16, 1992.

[7] Ethan S. Devinatz and Michael J. Hopkins. Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology, 43(1):1–47, 2004.

[8] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and algebras in stable homotopy theory. American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole.

[9] John Rognes. Galois extensions of structured ring spectra. Preprint submitted, February 8, 2005.

[10] N. P. Strickland. Gross-Hopkins duality. Topology, 39(5):1021–1033, 2000.