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This Example is meant to reinterpret the Adams spectral sequence against 1.4.10, but makes no mention of it.

The discussion of the role of the *-product in the unstable context is unclear. Does it actually appear in the cosimplicial object? Why do we actually pass to *-indecomposables? Clear things up around 4.1.2-3.

"There is a functional f_c given by" is redundant phrasing. Maybe just "The functional f_c given by"?

Maybe describe how the left- and right-units act? This factorization into a tensor product is otherwise too mysterious to be useful.

(1): isomorphic as schemes, but not as group schemes

There is a subsection called “Variations on These Results” where B-L and Lurie would both fit in.

The argument in 5.6 is (nearly?) complete for E-theories of height ≤ 2, but this is not enough to conclude anything about elliptic cohomology theories more generally. Matt has suggested an approach along the lines of fiber-by-fiber flatness criteria. Investigate this and close this argument.

We define a map i and name its induced map on KU-cohomology i^* in a diagram, but then we write that second map as a bare i in the centered equation block below.

The definitions which connect the formally defined object (\moduli{fg])^\wedge_\Gamma to formal group laws are a bit off. I want to capture two main features:

1. The restriction of M_fg to Gamma should be equivalent to Spec k // Aut Gamma. Correspondingly, the completion of M_fg at Gamma will also be a stack, not a scheme.
2. The use of star-isomorphisms indeed makes the Lubin-Tate theorem produce a scheme, but that presents the Lubin-Tate stack only after quotienting by the inherited Aut Gamma action.

You can stack the two product factors.

The sheaf (O_top)^_p is being constructed on (M_ell)^_p, not M_ell itself.

Mentioning tmf itself here would be very out of sequence, but something of some sort ought to be said.

We toss out a fancy definition of the Thom spectrum associated to an infinite loop map in A.4 without a lot of justification. Someone asked for clarification about this here: https://mathoverflow.net/questions/337277/equivalent-definitions-of-thom-spectra , and Dylan gave a nice overview answer that could probably be further condensed into an inline explanation.

PJ, a 2018 Behrens student, wrote his thesis on reproducing the A-H-S result using the Hopf ring methods of R-W-Y:

https://curate.nd.edu/show/hd76rx9419z .

Since we spend all this effort introducing Hopf ring methods in Ch 4, it would be good to digest these results and include a summary of them. Such a summary would perhaps belong at the very end of 5.5.

Footnote 5.12 warns that Ga is not a p-divisible group, but doesn’t explain why that might be an issue: it’s because Ga^(^2) doesn’t make sense in isolation. Clarify this.

Piotr (indirectly) pointed out that the origin of a deformation's commutativity is not clear in our discussion.

In 2.4.18, we remark that the Steenrod squares also appear via power operations. We can actually justify this: we've calculated HF2^* BC_2 to be a domain, so that ordinary power operations include into the Tate target, and the stable operations with which it's tagged must be the ones we've already named as Steenrod operations.

Remark 2.6.6 claims that there is a simpler version of Quillen's argument for MO. As far as I'm aware, this version has not appeared in print, and we give only the barest justification that it indeed ought to be simpler. It might be worth working this out in full and including a lengthier description of these methods here.

In B.2’s subsection Why Formal Groups?, we name Priddy’s and Beardsley’s theorems as evidence for a conjecture. We should also name the Bockstein family on HF2.

It does, if A is cocommutative or the comodules are cobimodules. Specify.

Even if the proof is by induction, it’s inappropriate for it to be in the statement.

Both the co- and contravariant statements of the Dieudonné correspondence are missing the uniform, reduced, and 1-dimensional hypotheses.

Sometimes Delta means the monadic unit for the composite adjunction introduced at the start of the section, and sometimes it means the "usual" diagonal on a space (the comultiplication for the Cartesian comonad).

It's probably hopeless to actually change the symbols so that they aren't both Delta. Instead, I propose adding a footnote that warns the user about potential ambiguities.

It isn’t a digression.

There is a result which describes the Hopf ring F_* E_* where F is complex-orientable and E is Landweber flat. We cite this result in a footnote, and we probably come close to proving it in 4.3, but we fall short. If possible, it would make a nice extension to 4.3.

Paul vanKoughnett asked me this.

In Figure 3.2, the rightmost element is labeled as needing to be divided by three. I think this may already be tracked in alpha_3/2.

We assert that the Lubin-Tate cover is related to the Lubin-Tate stack by a quotient, but we don't give a reference or explicit justification.

One of the big consequences of Thom's calculation of the unoriented bordism ring was an analysis of when a homology class is the image of the fundamental class of a manifold (cf. Théorème II.27). Include some material on this.

Is the argument for the Hopf algebra extension in 1.4.20 actually right? Is there a way to make it shorter?

The argument in 3.6.20 includes use of the cup product, which is enough to conclude that the attaching map is unstably nontrivial. Make it clear that we’re studying the stable class.

There’s no constraint on k. Does k need to be such that E_0 BU[2k, ∞) is free?

Charles says that Neil worked out analogues of his subgroup and isogeny theorems for ordinary cohomology, and that these (mostly) appeared in a course he taught in Boston some time ago. Charles gives a rough sketch of some of the ingredients in that MathOverflow conversation, and Neil has some private notes on the Steenrod algebra that cover a strongly overlapping set of facts.

I think it would be good to incorporate these facts (and, perhaps, proofs) into the document, probably into A.2.

Make the region after 4.4.7 clearer: the action does restrict to the familiar ell-series when ell is an integer, but p-adic convergence lets you say more than this, so that at least Z_p acts. In fact, I think Wp(k) is supposed to act, but in order to show this you have to construct a lot of other series that are not obvious.

Probably drop the earlier one.

Different people make different claims about the realizability of tmf as an object in spectral algebraic geometry. Find someone who has sorted this out conclusively and update this remark to match.

Dexter pointed out that there is some slight of hand around the application of the "Künneth isomorphism" in order to deduce the Segal condition for various contexts appearing in the book. Namely, the typical Künneth map is

E ⊗ X ⊗ E ⊗ Y → E ⊗ E ⊗ X ⊗ Y → E ⊗ X ⊗ Y

but the relevant map in the context setting is

X ⊗ E ⊗ E ⊗ Y → X ⊗ E ⊗ Y ,

which is slightly different and can cause a headache if not expecting it. Make note of this in a footnote somewhere.

The Ravenel-Wilson calculation has geometric implications in the form of the Conner-Floyd conjecture. I should try to summarize this and include it either at the end of 4.6 or in a new 4.7.

In the definition of the action of psi^p preceding A.3.13, there’s a missing hat over C^(p).

We introduce the (U)FH in order to get a Segal condition on certain (unstable) descent objects. In the unstable setting, we advocate passing to the *-indecomposables of cooperations and claim that this can be made to give a category scheme in the unproven (and uncited) Lemma 4.1.16.

Is it really true that passing *-indecomposables through a cosimplicial object satisfying the Segal condition produces a cosimplicial object also satisfying a Segal condition? Or does this just happen to hold in these sufficiently free settings, like when working with HF2?

Revisit Example 1.4.20, the homology of kO, as part of / directly following Example 3.1.18, which describes descent along the unit of HF2.

Is this actually used? Does it deserve a name?

The constant group scheme example that closes the Definition is not formal Lie, so doesn’t deserve a hat.

We discuss the unstable Adams spectral sequence in 4.1. It'd be nice to include a pair of Adams charts for comparison with the stable ones in Chapter 1.

Mention somewhere that the A_infty assumption is overkill; cf. constructions of Miller and Devinatz.

This is his preferred citation: http://scholar.google.com/citations?user=BWmMhLMAAAAJ&hl=en .

Piotr Pstragowski recently found an error in 3.2: the beta-family in H^2(Ga; Ga) starts at index j = 1 rather than j = 0. This has a cascade of effects: it shortens H^2(G; Ga) in 3.4 to be only (d-1)-dimensional, which means that our account of its deformation theory does not correctly capture deformations away from p = 0.

In conversation with Piotr, I linked to a discussion by Mike Hopkins of the Kodaira-Spencer map for Lubin-Tate theory. For one, it could be enlightening to include a discussion of this map. For another, the difference between Ga (as in 3.2) and Ga (x) Lie G (as in the K-S map, hence as in 3.4) might explain the twist in Lazarev’s equivalence of chain complexes.

If this turns out to be so, there’s also a sentence to delete in B.2.