{"id":"2a34e349-c7c3-4222-bb50-8795f780a59c","arxiv_id":"2608.02390","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For oriented P-divisible groups over noetherian E-infinity rings, family-completion of tempered cohomology modules is equivalent to algebraic completion at the corresponding ideal, generalizing Atiyah–Segal and AHJM.","lead":"This paper proves that for any tempered cohomology theory, completing equivariant modules at a family of subgroups agrees with algebraic completion at an intersection of augmentation ideals, extending the Adams–Haeberly–Jackowski–May and Atiyah–Segal theorems and giving a family completion theorem for equivariant topological modular forms. A generalist might read it because it reveals a deep structural uniformity across equivariant cohomology theories, from K-theory to TMF.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.7's support classification is load-bearing for Theorem 3.20, but its point-counting proof has a sketched lift/zig-zag step that is not fully justified.","rationale":"The reader's verdict identifies the same broad region — the support theory of Proposition 3.7 and its external inputs — as the weakest point. I agree that the proof of the converse of Theorem 3.20 is only as secure as Proposition 3.7. However, the reader frames the risk as failure of external inputs such as [GLP24, prop. 15.14], [BDL26, prop. 5.4.4], and [Lur18a, prop. 2.3.9]. My concern is more specific and internal: even assuming those inputs, the proof of Proposition 3.7 contains a sketched zig-zag/lift step that is never verified. This is load-bearing because the induction in Theorem 3.20 needs the exact equality of preimages W = Z^G(F|_H), which in turn needs the support of every point to be a single conjugacy class.\n\nI am not claiming the theorem is false; the argument may well be repairable, and the surrounding recollement machinery looks coherent. But the central claim should not be fully accepted until this step is either proved in detail or checked in a concrete nonabelian example. Hence I recommend CONDITIONAL: accept the paper's framework and results conditional on a complete justification of the point-counting step in Proposition 3.7.","tokens_in":40411,"tokens_out":27543,"duration_ms":259319,"concrete_test":"Run Proposition 3.7 in the smallest nonabelian case: G = S_3, base R = KU, and G = μ_{P^∞}. Compute |G{BS_3}| as colim_{G/H∈Orb_ab(G)} |G{BH}| and explicitly compute the images of |G{BC_2}| (the three conjugate C_2's) and |G{BC_3}|. The support theorem predicts these images are disjoint: no point is supported on both a C_2 and the C_3. Pick an explicit point in the intersection if one exists; if such a point exists, Proposition 3.7 is false and Theorem 3.20 fails. If no intersection exists, repeat the point-counting calculation for the zig-zag connecting a C_2 and the C_3 through the orbit category and verify that the claimed prime-ideal transport cannot be performed — this would expose the missing compatibility condition.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The converse of Theorem 3.20 reduces the comparison of the ideals J = res(I_G(F)) and I_G(F|_H) to Proposition 3.7's conclusion that every point of |G{BG}| has support a single conjugacy class of abelian subgroups. If that conclusion fails, the equality of preimages W = Z^G(F|_H) in the induction breaks, and the proof of the converse collapses.\n\nThe vulnerable point is inside the proof of Proposition 3.7. After reducing to a complete local base and passing to the faithfully flat splitting algebra S, the proof claims\n\n|Inj(\\hat V, G)_S|_n = ∐_{α: \\hatΛ↠V} |Spec S|_n.\n\nThis is justified by showing |Spec S(G^∘_S)^V|_n ≃ {n} using [Lur18a, prop. 2.3.9]. But between the choice of a prime ideal p_V ⊆ k(x_V)⊗_{k(m)}k(n) and the conclusion that y_V and y_W lie in the same component, the argument says only that a zig-zag of subconjugations connects x_V and x_W and that one may 'transport p_V' along this zig-zag. This transport requires compatible lifts of the chosen points through the colimit presentation G{BG} ≃ colim_{G/H∈Orb_ab(G)} G{BH}; the compatibility is asserted, not proved. The whole weight of the support theorem rests on this step, because it is what forces the components attached to non-conjugate V and W to be identified.\n\nThis is not merely an external-input issue: even if [GLP24], [BDL26], and [Lur18a] are all correct, Proposition 3.7 as written contains an unverified internal gluing step. A concrete failure would make the support set contain non-conjugate abelian subgroups, and then Theorem 3.20's induction cannot go through.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a family completion theorem for tempered cohomology. For a finite group G, a family F of subgroups, and an oriented P-divisible group G over a noetherian E-infinity ring R, it compares the geometric family-completion recollement R(gE_F) with the algebraic ideal-completion recollement associated to I_G(F), and shows they are equivalent (Theorem 3.20). This recovers the Adams-Haeberly-Jackowski-May theorem for equivariant K-theory and the Atiyah-Segal theorem as special cases. The proof uses a new support theory for points of the tempered character stack G{BG}, and a local-to-global descent argument for locally noetherian geometric stacks, yielding a version for genuine equivariant topological modular forms (Theorem 3.29). The paper is long and contains substantial foundational material on recollements and spectral algebraic geometry in appendices.","tokens_in":1418,"tokens_out":1709,"duration_ms":260330,"significance":"If the main theorem is correct, it is a significant advance: it gives a uniform family completion statement for a large class of equivariant cohomology theories, with the classical AHM theorem as a special case and with new applications to TMF. The support theory for tempered character stacks, modeled on Segal's analysis of representation rings, is a genuinely new and promising tool. The paper is carefully structured, explicitly citing the external inputs [GLP24], [BDL26], and [Lur18a]; no machine-checked proofs are provided, but the proof is presented in enough detail that a referee can identify the exact places needing scrutiny. The appendices, especially Appendix B, are largely self-contained and useful in their own right. However, two load-bearing points in the proof of the main theorem need repair before the paper can be accepted.","major_comments":[{"comment":"The final step of Proposition 3.7 rests on an unproved compatibility of chosen lifts. After fixing a prime p_V and a zig-zag of subconjugations connecting x_V and x_W, the text says: 'We then choose lifts and restrict to transport p_V along the zig-zag' and asserts that this ensures y_V and y_W map to the same point. But the proof does not show that these lifts are compatible with the colimit presentation G{BG} ≃ colim over Orb_ab(G) G{BH}, nor with the faithfully flat base change to S. This compatibility is exactly what forces the components attached to non-conjugate V and W to be identified. Since Proposition 3.7 feeds into Propositions 3.13 and 3.15, the induction in Theorem 3.20 inherits this gap. The step should be written out in full, or replaced by an explicit argument using the universal property of the colimit.","section":"Section 3.1 (proof of Proposition 3.7)"},{"comment":"The assertion 'we have gE_P = gE_P tensor gE_AB' is false. For a proper nonabelian subgroup H of G, (gE_P)^H = S^0, while (gE_P tensor gE_AB)^H = (gE_P)^H tensor (gE_AB)^H = S^0 tensor * = *. The subsequent conclusion that AB-completeness of every R(G)^G-module gives X = 0 is therefore not justified as written. The intended argument can likely be repaired, for instance by showing directly that every R(G)^G-module is P-complete using the AB-nilpotence of R(G)^G in the sense of [MNN17], but the present proof is not valid at this point.","section":"Section 3.2 (Theorem 3.20, nonabelian case)"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical errors: 'isomoprhic', 'constrctions', 'noethrian', 'P-divisable', 'deontet', 'catgeory', 'seperation', 'continiuous', 'surejctions', 'singelton', 'unqiely', and others. These should be corrected before final publication.","section":"Throughout"},{"comment":"The abstract uses the same letter G for the P-divisible group and for the finite group; in the body the former is a different symbol. Harmonize the notation for clarity.","section":"Abstract"},{"comment":"The sentence 'If M is an I_P-nilpotent R(G)^A-module, then M^A is an I_P-complete R(G)^A-module' is terse. It would help the reader to see the identification of M^A with the A-fixed points and why the relevant completeness is checked there.","section":"Section 3.2, proof of Lemma 3.19"},{"comment":"Minor wording: 'with each R_i noethrian' should be 'noetherian.' Also, the notion of LNG-stack is central, so it may be worth stating explicitly that the transition maps being flat implies the canonical maps Spec R_i to M are flat.","section":"Section 3.3, Definition 3.22"},{"comment":"The appendix is long and some statements are not used later (for example, Example A.20). This is not a defect, but a short indication of which parts are needed for the main results would improve readability.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The reader's accept verdict is understandable, and the overall strategy of the paper is plausible. But the two issues listed above are load-bearing: the support-classification step in Proposition 3.7 is not fully justified, and the nonabelian step in Theorem 3.20 contains a false equality and an invalid inference. Both appear fixable within the paper's framework, so I do not recommend rejection. The author should also double-check the application of [GLP24, prop. 15.14] and [BDL26, prop. 5.4.4], since the induction relies on them essentially. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Tokic proves a real generalization of the classical family completion theorems. Theorem 3.20 identifies geometric F-completion with algebraic I_G(F)-completion for tempered cohomology of finite groups, recovering AHJM for mu over KU and Atiyah-Segal as the trivial-family case. Proposition 3.21 (bounded height detection) and Theorem 3.29 (LNG stacks, including genuine equivariant TMF) are genuinely new. The support theory for the tempered character stack is the right tool, and the paper is honestly written: limitations about filtrations and the qualitative nature of the 0-affine case are stated plainly, and the appendices do real work.\n\nWhat I'd want a referee to focus on is Proposition 3.7. The stress-test note is on target: after reducing to the faithful flat splitting algebra S, the proof shows |Spec S(G^circ)^V|_n is a point and then asserts that a chosen prime p_V can be transported along a zig-zag of subconjugations to p_W, with compatible lifts through the colimit presentation. That compatibility is not proved. It is load-bearing: if the components attached to non-conjugate V and W were not identified, the support of a point could contain non-conjugate abelian subgroups, and the induction in Theorem 3.20 would collapse. I don't think the step is wrong—the intended argument via colimit finality and careful base change is plausible, and the rest of the paper is coherent—but it is the point where I'd want the author to write out the diagram chase before publication.\n\nThe other dependence is external: the abelian case uses [GLP24, prop. 15.14] and the nonabelian step uses [BDL26, prop. 5.4.4]. Those are heavy, not machine-checked, but they are cited as lemmas rather than assumed conclusions, and the paper's own contribution is not circular. The reader's moderate confidence is about right.\n\nBottom line: this is a serious paper for people in equivariant stable homotopy and chromatic homotopy theory. It deserves a proper peer review, and I'd expect acceptance after the Proposition 3.7 gap is closed. I'd cite it once it's out.","headline":"A substantial family completion theorem for tempered cohomology; the main theorem is new and credible, but Proposition 3.7's support classification has a compressed gluing step that should be expanded before I'd call the proof complete.","tokens_in":41369,"tokens_out":2846,"would_cite":true,"duration_ms":27344,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N91","55P91"],"pacs":[],"model":"deepseek-v4-flash","headline":"Tempered cohomology satisfies a family completion theorem: geometric completion at a family of subgroups agrees with algebraic completion at an ideal.","keywords":["tempered cohomology","family completion theorem","equivariant K-theory completion","character stacks","support theory","recollements","equivariant topological modular forms","P-divisible groups"],"falsifier":"Produce a finite group G and a family F for which some I_G(F)-complete R(G)^G-module is not F-complete; equivalently, find a module whose algebraic completion at the ideal vanishes but whose geometric family completion does not. The theorem claims this is impossible, and checking a small nonabelian example, such as G of order 8 with F the trivial family, would settle it.","tokens_in":40263,"feed_emoji":"🧮","tokens_out":5748,"duration_ms":72633,"temperature":0.7,"pith_summary":"The paper establishes a family completion theorem for tempered cohomology, a framework that encompasses equivariant K-theory, equivariant elliptic cohomology, and genuine equivariant topological modular forms. The result says that two different ways of completing modules over the equivariant coefficient ring coincide: completing with respect to a family F of subgroups (the geometric side) agrees with completing at the ideal I_G(F) defined by restrictions to those subgroups (the algebraic side). This recovers the classical completion theorem for equivariant K-theory as a special case and extends the statement to general locally noetherian geometric base stacks. The proof works by studying the tempered character stack G{BG} and proving that its points have support in a single conjugacy class of abelian subgroups.","feed_headline":"Geometric and algebraic completion agree for tempered cohomology","feed_subtitle":"One theorem covers K-theory, elliptic cohomology, and genuine equivariant TMF.","key_machinery":"The tempered character stack G{BG}, built by left Kan extension from the affinization of tempered cohomology on abelian subgroups, is the central object. A support theory for its points is developed: proposition 3.7 shows that the support of any point is a single conjugacy class of abelian subgroups, and proposition 3.15 identifies the preimage of V(I_G(F)) under the affinization map with the vanishing locus of points supported inside F. The proof of the main theorem uses this to compare restrictions of ideals under change of groups, avoiding a direct comparison of the two ideals, and then assembles the result via recollement basechange.","core_discovery":"The central claim, Theorem 3.20, is an equivalence of recollements: for a finite group G and a family F, the geometric family-completion recollement associated to the idempotent algebra gEF, after tensoring up to modules over R(G)^G, is equivalent to the algebraic ideal-completion recollement associated to I_G(F), the intersection of the kernels of the restriction maps from π0R(G)^G to π0R(G)^H for H in F. In concrete terms, an R(G)^G-module is F-complete exactly when it is I_G(F)-complete. Theorem 3.29 extends this from affine bases to locally noetherian geometric stacks, replacing the ideal by an open substack of the tempered character stack, which yields a family completion theorem for ge","pith_inferences":["Since the theorem is stated as an equivalence of completed objects rather than of filtrations, a natural extension would be to refine the argument to compare the filtrations on the two completions, strengthening the classical pro-group version.","The support stratification by conjugacy classes of abelian subgroups suggests a full 'prime spectrum' description for tempered cohomology of nonabelian groups, going beyond the abelian case treated so far.","Because the base-stack version replaces ideals by open substacks, a concrete computational form for TMF may require a 0-affine descent argument to identify those open substacks with ideal completions.","The statement assumes families are closed under subgroups and conjugation; testing whether the theorem survives for arbitrary families would clarify how essential that hypothesis is."],"forward_implications":["The classical completion theorem for equivariant K-theory, and its extension to arbitrary families of subgroups, follow as special cases of one statement.","For genuine equivariant topological modular forms over the moduli stack of elliptic curves, the family completion theorem holds, giving a parallel to the K-theory story.","Bounded-height tempered cohomology is right Kan extended from abelian groups generated by at most n elements; for equivariant TMF, this recovers detection from groups generated by at most two elements.","The support-theoretic description of character stacks gives a new structural handle on the prime-like points of tempered cohomology, in the spirit of the classical description of the complex representation ring."],"fun_headline_variants":["Geometric and algebraic completion coincide for tempered cohomology","Family completion theorem: geometry meets algebra","Tempered cohomology: one theorem for K-theory and TMF","Atiyah-Segal generalized: family completion via geometry","Completion in tempered cohomology: ideal equals geometric"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The hard converse direction depends on imported structural facts about tempered cohomology modules—that certain nilpotent modules are generated by modules induced from proper subgroups, and that connected-étale splitting holds after faithful flat base change—so if those facts fail in this setting, the induction collapses.","fun_headline_variants_meta":{"raw":{"variants":["Geometric and algebraic completion coincide for tempered cohomology","Family completion theorem: geometry meets algebra","Tempered cohomology: one theorem for K-theory and TMF","Atiyah-Segal generalized: family completion via geometry","Completion in tempered cohomology: ideal equals geometric"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000944,"raw_usage":{"total_tokens":3933,"prompt_tokens":873,"completion_tokens":3060,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":2980}},"tokens_in":617,"tokens_out":3060,"duration_ms":19238,"temperature":1.0,"reasoning_tokens":2980,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:09:29.683700+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce a finite group G and a family F for which some I_G(F)-complete R(G)^G-module is not F-complete; equivalently, find a module whose algebraic completion at the ideal vanishes but whose geometric family completion does not. The theorem claims this is impossible, and checking a small nonabelian example, such as G of order 8 with F the trivial family, would settle it.","supporting_citations":[],"review_version":1}