A lift from group cohomology to spectra for trivial profinite actions

Daniel G. Davis

A lift from group cohomology to spectra for trivial profinite actions

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that, for a profinite group G and a discrete G-spectrum X with trivial action, the continuous homotopy fixed points X^{hG} are weakly equivalent to the colimit colim_N X^{hG/N} over open normal subgroups whenever G is…

desk verdict Genuinely new colimit model for continuous homotopy fixed points under trivial profinite actions; the main theorems look sound, with one compressed proof step worth asking the author to expand. read the letter →

arxiv 1908.01898 v1 pith:UHI57S2J submitted 2019-08-05 math.AT

Daniel G. Davis This is my paper · ORCID

classification math.AT MSC 55P9120J0655T99

keywords profinitegroupscontinuoushomotopyfixedpointsdiscreteG-spectratrivialgroupactioncohomologypointspectralsequenceMoravaK-theorydescent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks when homotopy fixed points of a profinite group acting trivially on a spectrum can be assembled from homotopy fixed points of finite quotients. It constructs a zigzag connecting $\operatorname{colim}_N X^{hG/N}$ to $X^{hG}$, and proves that the comparison map $\Phi$ is a weak equivalence in three cases: $G$ finite, $X$ bounded above, or a cohomological vanishing condition on a cofinal family of open normal subgroups. In the vanishing case, each such subgroup $U$ satisfies $X \simeq X^{hU}$ and $X^{hG} \simeq X^{hG/U}$. This matters because it lifts the classical algebraic isomorphism $H^*_c(G,\pi_t(X)) \cong \operatorname{colim}_N H^*(G/N,\pi_t(X))$ to the level of spectra, making infinite-group homotopy fixed points computable from finite-group data.

What carries the argument

The central object is the zigzag $$\operatorname{colim}_N $X^{{hG/N}}$ \xrightarrow{\Phi} \operatorname{colim}_N ($X^{{hN}}$)^{hG/N} \xleftarrow{\Psi} $X^{{hG}}$,$$ where $\Psi$ is a weak equivalence and $\Phi$ is the map whose failure or success is at issue. The engine of the proof is the continuous homotopy fixed point spectral sequence, a cohomology-to-homotopy bookkeeping device whose second page is $H^s_c(U,\pi_t(X))$. Under the vanishing hypothesis this spectral sequence collapses at the $E_2$-page, giving $\pi_t(X^{hU}) \cong \pi_t(X)$ and forcing $\Phi$ to be a weak equivalence; for bounded-above spectra, a colimit-homotopy-limit interchange and the equivalence $X^{hG} \simeq \operatorname{holim}_\Delta \operatorname{Map}_c(G^\bullet, X^f)$ play the same role.

What would settle it

A single pair $(G,X)$ satisfying either the bounded-above hypothesis or the cohomological vanishing hypothesis of Theorem 1.3, for which the homotopy groups of $X^{hG}$ and $\operatorname{colim}_N X^{hG/N}$ differ in some degree, would refute the paper's central claim. Concretely, compute $\pi_t$ of both sides in the first degree where they might differ for a candidate such as $X=K(n)$ and $G=\mathbb{Z}_\ell$ with $\ell \neq p$.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.3, is that if a cofinal collection $\{U\}$ of open normal subgroups of a profinite group $G$ satisfies $H^s_c(U,\pi_t(X))=0$ for all $s>0$ and all $t$, then the comparison map $\Phi$ is a weak equivalence. Consequently $X^{hG} \simeq \operatorname{colim}_N X^{hG/N}$, $X^{hG/U} \simeq X^{hG}$, and $X \simeq X^{hU}$ for each such $U$. A separate theorem, Theorem 1.2, proves the same equivalence for every bounded-above spectrum $X$ with trivial $G$-action, with no cohomological hypothesis. In the zigzag, the map $\Psi$ is always a weak equivalence, so the entire content is whether $\Phi$ is one.

Load-bearing premise

The whole argument leans on a prior technical result guaranteeing that a certain cohomology-to-homotopy calculation converges for profinite groups; if that convergence fails for these spectra, the main theorems lose their proof.

Editorial extensions

If this is right

Whenever the cohomological vanishing condition holds, continuous homotopy fixed points for an infinite profinite group are computed as a filtered colimit of homotopy fixed points for finite quotients, and each finite-quotient fixed point spectrum is itself equivalent to $X^{hU}$.
For spectra whose homotopy groups are torsion-free divisible, every profinite group $G$ satisfies $X \simeq X^{hG}$; in particular the rational Eilenberg-Mac Lane spectrum is homotopy fixed under every profinite group.
For spectra whose homotopy groups split as a torsion-free divisible group plus a $J$-torsion group, where the primes in $J$ do not divide the order of $G$, the conclusion holds for every closed subgroup $H$ and every open normal subgroup $N_H$.
Whenever $\Phi$ is a weak equivalence, the Spanier-Whitehead dual satisfies $D(X^{hG}) \simeq \operatorname{holim}_N F(X^{hG/N},S^0)$, and $X^{hG} \simeq \operatorname{colim}_N F(B(G/N)_+,X^f)$, tying infinite-group fixed points to classifying spaces of finite quotients.
In a chromatic example, Morava $K$-theory $K(n)$ at a prime $p$ satisfies $K(n) \simeq K(n)^{h\mathbb{Z}_\ell}$ for a different prime $\ell$, and $K(n)^{h(\mathbb{Z}_\ell \times \mathbb{Z}/p^r)} \simeq K(n)^{h\mathbb{Z}/p^r}$, showing that the extra procyclic factor can be invisible to homotopy fixed points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

This suggests a broader descent criterion: any condition that forces the continuous homotopy fixed point spectral sequence for open normal subgroups to collapse to its $H^0$ term will make $\Phi$ a weak equivalence; the paper's vanishing hypotheses are one sufficient family, and other vanishing inputs could be substituted.
The chromatic equivalence $K(n)^{h\mathbb{Z}_\ell} \simeq K(n)$ for $\ell \neq p$ indicates that $p$-local spectra can be insensitive to procyclic groups at other primes, which may refine expectations about which closed subgroups of the Morava stabilizer group are visible through homotopy fixed points.
The presentation $X^{hG} \simeq \operatorname{colim}_N F(B(G/N)_+,X^f)$ offers a concrete route to computing profinite-group homotopy fixed points by finite classifying spaces, potentially connecting to algorithmic profinite group homology computations.
A testable boundary case: for a spectrum such as a $K(n)$-local sphere and a profinite group $G$ where the hypotheses fail, the difference between $\pi_*(X^{hG})$ and $\operatorname{colim}_N \pi_*(X^{hG/N})$ should be detectable in the first nonzero degree of the appropriate spectral sequence, exhibiting exactly where Theorem 1.3 stops.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this paper proves a genuinely new colimit model for continuous homotopy fixed points under trivial profinite actions, and the main theorems look sound.

What is actually new is the zigzag and the conditions under which the map Phi is a weak equivalence. The algebraic fact that continuous cohomology of a profinite group is a colimit of cohomology of finite quotients is standard. Lifting that to spectra, X^{hG} ≃ colim_N X^{hG/N}, is new. The bounded-above case (Theorem 1.2) and the vanishing-condition case (Theorem 1.3) are substantial, and the corollaries give concrete chromatic examples, including K(n)^{hZ_l} ≃ K(n) when l and p are distinct primes. That example is a good sanity check and will be useful to people computing homotopy fixed points.

The paper does several things well. The construction of Psi is explicit, and the proof that it is a weak equivalence is careful. The paper is honest about where the profinite HFPSS comes from; the main external dependencies are the author's own prior work with Behrens and Davis, and the citations are located precisely. The colim/holim interchange argument in Theorem 4.1 is neat.

Soft spots, in proportion: the proof of Theorem 1.3 rests on imported HFPSS convergence. That is not a flaw, but a reader not steeped in Behrens-Davis will have to work to check that the convergence is strong enough for the collapse argument to give the edge isomorphism. The most compressed step is the deduction in Section 3 that lambda_{U'} is a weak equivalence from the isomorphism in (3.3). The chain connecting (XfG)^{U'}, X^{hU'}, and the homotopy limit is terse; I had to fill in a diagram, and I believe it goes through. The identifications in Section 2 are also informal at points, though they look correct. None of this rises to a serious flaw. The citation pattern is self-heavy, but for foundational machinery that is reasonable.

This paper is for homotopy theorists working with profinite group actions, especially in chromatic homotopy theory and descent. It deserves a serious referee. If I were the editor I would send it out, with the main request being to expand the compact step in Section 3 and to state explicitly which convergence result from Davis [5] is being imported.

Referee Report

0 major / 5 minor

Summary. The paper studies a profinite group G and a discrete G-spectrum X with trivial G-action. It constructs a zigzag colim_N X^{hG/N} -> colim_N (X^{hN})^{hG/N} <- X^{hG}, where the right-hand map Ψ is proved to be a weak equivalence, and it asks when the left-hand map Φ is a weak equivalence. The main results are Theorem 1.2 (Φ is a weak equivalence if X is bounded above) and Theorem 1.3 (if there is a cofinal collection {U} of open normal subgroups of G with H^s_c(U,π_t(X))=0 for all s>0 and all t, then Φ is a weak equivalence, X^{hG} ≃ colim_N X^{hG/N}, X^{hG/U} ≃ X^{hG}, and X ≃ X^{hU}). Section 2 constructs Φ and Ψ and proves the finite-group case; Section 3 proves Theorem 1.3 by collapsing a continuous homotopy fixed point spectral sequence; Section 4 proves Theorem 1.2 via a colimit/homotopy-limit interchange. Corollaries and examples apply the results to rational spectra, torsion-free divisible homotopy groups, and Morava K-theory spectra, including equivalences of the form K(n)^{hZ_ℓ} ≃ K(n) for distinct primes p and ℓ.

Significance. If the results hold, the paper provides a useful criterion under which continuous homotopy fixed points for a profinite group can be computed from finite quotients, thereby lifting group-cohomology vanishing from algebra to spectra. The paper is transparent about its main external input: the existence and conditional convergence of the continuous HFPSS for profinite groups, imported from [2], [4], and [5]. The internal arguments in Sections 3 and 4 are coherent, and the paper identifies exactly which prior results it needs. A particular strength is that the main theorems are accompanied by concrete, checkable examples, including explicit Morava K-theory computations.

minor comments (5)

[Section 3, before Eq. (3.1)] The sentence 'Since {U''} is a subset of {U}, H^s_c(U'',π_t(X))=0 ... and therefore, the conditionally convergent HFPSS (3.1) exists' conflates two different roles: the existence and conditional convergence of (3.1) are supplied by the cited results [5, page 911], [2, proof of Theorem 3.2.1], and [4, proof of Theorem 7.4], while the vanishing hypothesis is used to collapse the spectral sequence at E2. Rewording this sentence would make the logical structure of the proof clearer.
[Section 2, after Definition 2.1] The identifications of the source and target of Φ with colim_N X^{hG/N} and colim_N (X^{hN})^{hG/N} are up to natural weak equivalences rather than isomorphisms; since the text explicitly promises to explain these identifications, it would be helpful to state this point explicitly in Definition 2.1 or the paragraph immediately following it.
[Section 4, proof of Theorem 1.2] The appeal to [16, Proposition 3.4] for the map s1 would be easier to verify if the statement of that proposition and its hypotheses were quoted explicitly, since s1 is a filtered-colimit/homotopy-limit interchange for a diagram of cosimplicial spectra.
[Example 1.9] The indexing of J′ as a disjoint union, and the sentence explaining that n_p for p∈P′∩J are distinct elements in {n_q | q∈J′}, is confusing; a cleaner indexing, such as parameterizing the summands by pairs (p,0) and (p,1), would improve readability.
[Example 1.10, final sentence] The phrase 'G does not belong to {U}' is ambiguous, since the members of {U} are open normal subgroups of G; the intended statement is that no cofinal collection satisfying the vanishing hypothesis can contain the full subgroup G, and this should be spelled out.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper's results are proved from prior fixed-point spectral sequence machinery and internal diagram chases, not from the conclusions themselves.

full rationale

The derivation chain is not circular. The map Ψ is proved to be a weak equivalence in Section 2 by showing that each canonical map ι_U from a limit to a homotopy limit is a weak equivalence, using the cited equivalence holim_Δ Map((G/U)^•,(XfG)^U) ≃ X^{hG} from Behrens–Davis [2]; this is prior model-categorical input, not the paper's target statement. Theorems 1.2 and 1.3 do not assume Φ. Theorem 1.3 assumes only the vanishing of continuous cohomology H^s_c(U,π_t(X)) for s>0 on a cofinal family of open normal subgroups, and then invokes the conditionally convergent homotopy fixed point spectral sequence (3.1), imported from Davis [5] and Behrens–Davis [2], to obtain collapse and the edge isomorphism. The conclusion X ≃ X^{hU} follows from identifying that edge map with the unit map, and the equivalence of Φ follows because the map of E2-terms induced by λ_{U'} is an isomorphism. Theorem 1.2 proceeds by colim/holim interchange together with the bounded-above equivalence X^{hG} ≃ holim_Δ Map_c(G^•,XfG), again cited from [5]. None of these cited results is the paper's conclusion: they are general convergence and model-structure theorems whose stated assumptions do not include Φ being a weak equivalence. The central spectral-sequence input comes from the author's own prior work, but that is ordinary reliance on previously proved machinery rather than a reduction of the target statement to itself. No parameter is fitted and later called a prediction, and no object is defined in terms of the claimed equivalence. Therefore there is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities or free parameters. It relies on established model category machinery and spectral sequence results, including two results from the author's own prior work, cited transparently. The central claim is not assumed in these inputs.

assumptions (4)

domain assumption The model structure on discrete G-spectra (ΣSp^G) and the right Quillen functor (-)^G exist as in Behrens-Davis [2, Section 3.1].
Used throughout to define X^{hG} = (X_fG)^G and to know that homotopy fixed points are a right derived functor. This is prior work by the author and Behrens.
domain assumption The continuous homotopy fixed point spectral sequence for profinite groups exists and converges conditionally, as established in Davis [5] and Behrens-Davis [2].
Used in Section 3 (equation (3.1)) and Section 4 to compute π_*(X^{hG}) and to prove Theorem 1.2. This is from the author's own prior paper, but it is not the target result.
standard math Mitchell's Proposition 3.4 [16] about filtered colimits commuting with homotopy limits for bounded-above spectra.
Used in Theorem 1.2 to show the colim/holim interchange map s1 is a weak equivalence.
standard math Wilson's Lemma 10.2.1 [22] on vanishing of continuous cohomology of profinite groups with torsion coefficients when p does not divide the order.
Used in Corollary 1.8 and Example 1.10 to verify the cohomological vanishing hypotheses of Theorem 1.3 for J-torsion coefficient groups.

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Pith. "Pith review of A lift from group cohomology to spectra for trivial profinite actions." pith.science (2026). https://pith.science/paper/UHI57S2J

@misc{pith2026190801898,
  author       = {Pith},
  title        = {Pith review of: A lift from group cohomology to spectra for trivial profinite actions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UHI57S2J}},
  note         = {Machine review of arXiv:1908.01898}
}

abstract

Let $G$ be a profinite group, $X$ a discrete $G$-spectrum with trivial action, and $X^{hG}$ the continuous homotopy fixed points. For any $N \trianglelefteq_o G$ ("$o$" for open), $X = X^N$ is a $G/N$-spectrum with trivial action. We construct a zigzag $\text{colim}\,_N \,X^{hG/N} \buildrel\Phi\over\longrightarrow \text{colim}\,_N \,(X^{hN})^{hG/N} \buildrel\Psi\over\longleftarrow X^{hG}$, where $\Psi$ is a weak equivalence. When $\Phi$ is a weak equivalence, this zigzag gives an interesting model for $X^{hG}$ (for example, its Spanier-Whitehead dual is $\text{holim}\,_N \,F(X^{hG/N}, S^0)$). We prove that this happens in the following cases: (1) $|G| < \infty$; (2) $X$ is bounded above; (3) there exists $\{U\}$ cofinal in $\{N\}$, such that for each $U$, $H^s_c(U, \pi_\ast(X)) = 0$, for $s > 0$. Given (3), for each $U$, there is a weak equivalence $X \buildrel\simeq\over\longrightarrow X^{hU}$ and $X^{hG} \simeq X^{hG/U}$. For case (3), we give a series of corollaries and examples. As one instance of a family of examples, if $p$ is a prime, $K(n_p,p)$ the $n_p$th Morava $K$-theory $K(n_p)$ at $p$ for some $n_p \geq 1$, and $\mathbb{Z}_p$ the $p$-adic integers, then for each $m \geq 2$, (3) is satisfied when $G \leqslant \prod_{p \leq m} \mathbb{Z}_p$ is closed, $X = \bigvee_{p > m} (H\mathbb{Q} \vee K(n_p,p))$, and $\{U\} := \{N_G \mid N_G \trianglelefteq_o G\}$.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

Daniel G. Davis. Homotopy ﬁxed points for proﬁnite groups emulate homotopy ﬁxed points for discrete groups. New York J. Math. , 19:909–924, 2013

work page 2013

[2] [2]

Mark Behrens and Daniel G. Davis. The homotopy ﬁxed point s pectra of proﬁnite Galois extensions. Trans. Amer. Math. Soc. , 362(9):4983–5042, 2010

work page 2010

[3] [4]

Daniel G. Davis. Homotopy ﬁxed points for LK(n)(En∧ X) using the continuous action. J. Pure Appl. Algebra , 206(3):322–354, 2006

work page 2006

[4] [1]

Chromatic structures in stable homotopy theory

Tobias Barthel and Agn` es Beaudry. Chromatic structures in stable homotopy theory. Available at arXiv:1901.09004v2 [math.AT], 2019

work page Pith review arXiv 1901

[5] [3]

Rational h omotopy of the (homotopy) ﬁxed point sets of circle actions

Urtzi Buijs, Yves F´ elix, and Aniceto Murillo. Rational h omotopy of the (homotopy) ﬁxed point sets of circle actions. Adv. Math. , 222(1):151–171, 2009

work page 2009

[6] [6]

Devinatz and Michael J

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

work page 2004

[7] [7]

W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar. Duality in a lgebra and topology. Adv. Math., 200(2):357–402, 2006

work page 2006

[8] [8]

W. G. Dwyer and C. W. Wilkerson. Homotopy ﬁxed-point metho ds for Lie groups and ﬁnite loop spaces. Ann. of Math. (2) , 139(2):395–442, 1994

work page 1994