{"id":"c830ac46-99e3-49c9-97ba-de56e42e01f9","arxiv_id":"1908.01898","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For profinite groups G acting trivially on a spectrum X, the paper gives conditions under which X^{hG} is weakly equivalent to the colimit over finite quotients of X^{hG/N}.","lead":"The paper proves that for certain infinite profinite groups acting trivially on a spectrum, the continuous homotopy fixed points can be computed from the fixed points of finite quotient groups. This gives homotopy theorists a new tool for calculations in chromatic homotopy theory, such as for Morava K-theory spectra.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's verdict is ACCEPT with moderate confidence, and the reader's weakest_assumption is exactly the same point I identified: the continuous homotopy fixed point spectral sequence imported from [5] and [2]. Stress-testing the proof structure confirms this is the only load-bearing external dependency: Theorem 1.3(a)-(c), Corollaries 1.4 and 1.8, Examples 1.9-1.10, and Theorem 1.2 via [5, condition (iii)] all flow through the existence and convergence of this HFPSS. The internal logic is coherent: if E2^{s>0}=0 and the spectral sequence converges to the abutment in the required sense, then the edge map is an isomorphism, λ_{U'} is a weak equivalence, and the desired weak equivalences follow. I found no unsupported step beyond the imported theorem, and the citations are precise enough to be checked. Therefore I do not recommend changing the ACCEPT verdict; the concern is a dependency to verify rather than a demonstrated defect.","tokens_in":16861,"tokens_out":44332,"duration_ms":513744,"concrete_test":"Inspect the statement of [5, page 911, condition (ii)/(1.2)] and verify that for each U' in Theorem 1.3 the cofinal family {U∩U'} satisfies its hypotheses, including any countability or finiteness conditions, and that the theorem gives enough convergence for the collapse E2^{s>0}=0 to imply π_t(X^{hU'})≅H^0_c(U',π_t(X)) with the edge map (3.3) an isomorphism. If the cited result provides only conditional convergence, check separately that the lim^1 obstruction vanishes when E2^{s>0}=0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. Theorems 1.2 and 1.3 rest on the continuous homotopy fixed point spectral sequence imported from Davis [5] (page 911, conditions (ii)/(iii)) and Behrens-Davis [2, proof of Theorem 3.2.1]. In Section 3, the collapse of (3.1) at E2 is used to identify π_t(X^{hU'}) with H^0_c(U',π_t(X)) and to obtain the edge isomorphism (3.3), which then forces λ_{U'} and hence X≃X^{hU'}. I found no internal inconsistency in this argument: if the cited HFPSS exists with enough convergence that E2^{s>0}=0 implies the edge map is an isomorphism, the steps in Sections 3 and 4 are sound. The only soft spot is that this central machinery is imported rather than re-derived, and the cited references are to the author's own prior work. Because the dependency is explicitly located and appears to supply exactly the needed condition, I do not treat it as a discovered flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17044,"tokens_out":23990,"duration_ms":248991,"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.","major_comments":[],"minor_comments":[{"comment":"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":"Section 3, before Eq. (3.1)"},{"comment":"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":"Section 2, after Definition 2.1"},{"comment":"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.","section":"Section 4, proof of Theorem 1.2"},{"comment":"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.","section":"Example 1.9"},{"comment":"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.","section":"Example 1.10, final sentence"}],"recommendation":"minor_revision","confidential_remarks":"The paper relies heavily on the author's own prior work ([2], [4], [5]) for the profinite model structure and the continuous HFPSS. This reliance is explicit and seems appropriate, but it is the main place where correctness depends on conditions imported by page number; the editor may wish to ensure the cited conditions indeed cover the class of discrete G-spectra used here. The examples and corollaries are valuable and the internal proofs are coherent; I found no reason to doubt the central claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nSoft 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.\n\nThis 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.","headline":"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.","tokens_in":17592,"tokens_out":7229,"would_cite":true,"duration_ms":71751,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P91","20J06","55T99"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["profinite groups","continuous homotopy fixed points","discrete G-spectra","trivial group action","continuous group cohomology","homotopy fixed point spectral sequence","Morava K-theory","descent"],"falsifier":"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 $\\ell \\neq p$.","tokens_in":16668,"feed_emoji":"🔄","tokens_out":6844,"duration_ms":66483,"temperature":0.7,"pith_summary":"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.","feed_headline":"Profinite homotopy fixed points reduce to finite quotients","feed_subtitle":"When cohomology vanishes or the spectrum is bounded above, the infinite-group homotopy fixed points become a colimit of finite-quotient…","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the model category of discrete $G$-spectra, the continuous homotopy fixed point construction, the $\\operatorname{Map}_c$ cosimplicial machinery, and the homotopy spectral sequence used throughout.","marker":"[2]"},{"why":"Gives the existence and conditional convergence of the continuous homotopy fixed point spectral sequence for profinite groups, on which the proof of Theorem 1.3 rests.","marker":"[5]"},{"why":"Provides the proof of the spectral sequence result cited as Theorem 7.4, used to identify the collapsing spectral sequence (3.1) and its consequences.","marker":"[4]"},{"why":"Gives the finite-group homotopy fixed point spectral sequence and the equivalence $X^{hK} \\simeq F(BK_+,X^f)$ used for finite quotients and examples.","marker":"[8]"},{"why":"Establishes the stable model category of symmetric spectra of simplicial sets in which all spectra in the paper live.","marker":"[13]"},{"why":"Supplies Proposition 3.4, used to show that the colimit-homotopy-limit interchange map is a weak equivalence under the bounded-above hypothesis.","marker":"[16]"},{"why":"Provides standard profinite group cohomology facts, including the colimit description over finite quotients and vanishing for torsion-free divisible modules.","marker":"[19]"},{"why":"Provides the injectivity of restriction for torsion coefficients, used in Corollary 1.8 and Example 1.10 to establish the cohomological vanishing hypotheses.","marker":"[22]"}],"fun_headline_variants":["Profinite homotopy fixed points: finite quotients suffice","Bounded-above spectra reduce profinite fixed points to finite quotients","Cohomology vanishing: profinite fixed points are finite quotients","Trivial profinite actions: homotopy fixed points from finite quotients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Profinite homotopy fixed points: finite quotients suffice","Bounded-above spectra reduce profinite fixed points to finite quotients","Cohomology vanishing: profinite fixed points are finite quotients","Trivial profinite actions: homotopy fixed points from finite quotients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001553,"raw_usage":{"total_tokens":6339,"prompt_tokens":1212,"completion_tokens":5127,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":828,"completion_tokens_details":{"reasoning_tokens":5049}},"tokens_in":828,"tokens_out":5127,"duration_ms":32045,"temperature":1.0,"reasoning_tokens":5049,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:00:16.230264+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $\\ell \\neq p$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the model category of discrete $G$-spectra, the continuous homotopy fixed point construction, the $\\operatorname{Map}_c$ cosimplicial machinery, and the homotopy spectral sequence used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the existence and conditional convergence of the continuous homotopy fixed point spectral sequence for profinite groups, on which the proof of Theorem 1.3 rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the proof of the spectral sequence result cited as Theorem 7.4, used to identify the collapsing spectral sequence (3.1) and its consequences."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the finite-group homotopy fixed point spectral sequence and the equivalence $X^{hK} \\simeq F(BK_+,X^f)$ used for finite quotients and examples."},{"cited_title":"Symmetric spe ctra","cited_arxiv_id":null,"evidence_quote":"Establishes the stable model category of symmetric spectra of simplicial sets in which all spectra in the paper live."},{"cited_title":"Mitchell","cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 3.4, used to show that the colimit-homotopy-limit interchange map is a weak equivalence under the bounded-above hypothesis."},{"cited_title":"Profinite groups","cited_arxiv_id":null,"evidence_quote":"Provides standard profinite group cohomology facts, including the colimit description over finite quotients and vanishing for torsion-free divisible modules."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the injectivity of restriction for torsion coefficients, used in Corollary 1.8 and Example 1.10 to establish the cohomological vanishing hypotheses."}],"review_version":1}