{"id":"dddc2786-75d1-434b-8361-90db0cd7067b","arxiv_id":"2501.15584","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For three rank-2 toral groups of mixed type, rational G-spectra over full subgroups are Quillen equivalent to an explicit abelian category A(G|full).","lead":"Mathematicians constructed a small algebraic description for the rational homotopy of spaces with a two-dimensional torus symmetry plus a reflection, including the normalizer of the maximal torus in U(2). It is a step toward a complete algebraic model for U(2) and SU(3) equivariant spectra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 8.7's final step—cellular triviality of X''' from 'torsion free and injective'—is unproved and the stated implication does not obviously hold for the displayed dual cells; Proposition 8.1 and hence Theorem 7.1 rest on it.","rationale":"Proposition 8.1 is the bridge from the algebraic module category over π^G_*((S^0)_⌟)[W] to DG-A(G|full). It is the only step that quantifies over arbitrary modules; the preceding sections identify the correct ring spectra, the pullback cube, generators, formality, and the homology functor, but they all take place in module categories that are known to be Quillen equivalent by Shipley-type results. The final equivalence in the Section 3 string is entirely the content of Proposition 8.1, and Lemma 8.7 is its core. The manuscript's proof of Lemma 8.7 is a sketched adaptation of the torus argument; the last paragraph replaces a computation for [Dσ_G,·] with the assertion that all terms are torsion-free and injective. That assertion is doing the work of showing the residual object is cellularly trivial, but no argument is given for why a free entry of a dual cell cannot map into a torsion-free injective object. This is especially delicate because the module categories are full of infinite products of localizations, where injectivity and vanishing of Hom from dual cells are not formal consequences of torsion-freeness. The concrete test isolates this exact computation. If the computation confirms the sketch, the theorem is supported; if not, the central equivalence has a genuine gap. The reader's CONDITIONAL verdict remains appropriate, so no adjustment is needed.","tokens_in":20542,"tokens_out":12321,"duration_ms":116131,"concrete_test":"Take the algebraic category alone. Choose X = f1(M) with M a nontrivial torsion Q[c]-module supported at a single finite subgroup F; apply the three killing steps of Lemma 8.7 literally using the rings of Section 2, and compute Hom_{DG-A}(Dσ_G, X''') and Hom(Dσ_H, X''') for the three types of dual cells, including all suspensions. If any computed Hom is nonzero, Proposition 8.1 fails and Theorem 7.1 is unsupported; if all vanish, write out the injectivity and torsion-freeness proof for the localized product terms and identify exactly which dual-cell modules (free versus torsion) are being used.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 7.1 depends on the five equivalences in Section 3; the last one, 'Equivalence 5', is Proposition 8.1. The proof of Proposition 8.1 is Lemma 8.7. After killing G-, Z- and ~T-evaluations, the residual object X''' is alleged to be cellularly trivial because '[σ_F,X''']^G = 0 ... since φ_1 X''' is torsion free and injective', and similarly for 1-dimensional H. This is the only step that shows an arbitrary object of the module category is cellularly equivalent to an object of the standard model, so without it the cellular skeleton identification is exactly what is missing. The implication is not automatic. In the one-dimensional case the cell Dσ_1 is the torsion module Q ≅ Q[c]/(c), and Hom(Q[c]/(c), M) = 0 follows from M being c-torsion-free. But for H containing Z, the cell Dσ_H of Definition 6.3 has a free E^{-1}OF ⊗_{OF/Z} Q_H entry at VZ. A free module maps nontrivially into many torsion-free injective localizations; injectivity of the target does not by itself make Hom from a free source vanish. The one-line appeal to 'torsion free and injective' therefore does not prove [σ_H,X'''] = 0 for the cells that matter. One also needs the injectivity of the localized product terms (e.g., E^{-1}I^{-1}OF) in the category of qce diagrams, which is asserted but not shown. The manuscript says the argument is 'a 2-dimensional version of a standard argument'; the rank-1 warm-up (Example 8.6) does a genuine homotopy-pullback computation for [Dσ_G,·], but Lemma 8.7 replaces that computation by a blanket claim. Since all terms of the diagram are infinite products of localizations, the missing verification is substantial.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs an explicit algebraic model for the category of rational G-spectra over full subgroups for three rank-2 toral groups of mixed type: G = O(2) × T, G = Pin(2) × T, and G = N_{U(2)}(T^2). The model A(G|full) is defined in Section 2 as a category of quasicoherent, extended diagrams of modules over an inflation diagram of rings built from products of Q and Q[c] indexed by conjugacy classes of subgroups. The main result, Theorem 7.1, asserts Quillen equivalences G-spectra|full ≃ DG−A(G|full). The proof follows the Greenlees–Shipley strategy: the sphere is expressed as a homotopy pullback of isotropically simple ring spectra (Section 5), the resulting diagram is shown to be formal (Section 6), the homology functor is constructed (Section 7), and a Cellular Skeleton Theorem (Proposition 8.1) identifies the cellularization of the module category with the standard model. Several steps are only sketched, with references to the torus case and to the author's prior and in-preparation work.","tokens_in":20981,"tokens_out":8899,"duration_ms":75162,"significance":"The paper addresses a class of groups that is genuinely new in this program: the mixed case Λ_Q = Q ⊕ Q~ has not been treated before, and the normalizer N_{U(2)}(T^2) is the most complicated block in the proposed model for rational U(2)- and SU(3)-spectra. If Theorem 7.1 is correct, the model is explicit and calculable, and the paper would be an important step toward the general abelian model conjecture. The strategy is coherent and the topology-to-algebra link is supported by the established methods of [14], [15], and [16]. The paper is not self-contained in a few key places: the formality lemma and the cellular skeleton argument are delegated to sketched adaptations, and the latter contains an unmet mathematical obligation. These points are load-bearing, so the central claim is not yet fully established as written.","major_comments":[{"comment":"The assertion that the residual object X''' is cellularly trivial is the key step proving Proposition 8.1, but the argument given is incomplete. For a 1-dimensional subgroup H containing Z, the dual cell Dσ_H of Definition 6.3 has a nonzero free entry E^{-1}OF ⊗_{OF/Z} Q_H at VZ; the statement that '[σ_H, X''']^G = 0 ... because all terms are I-torsion free and injective' does not follow, since Hom from a free module to an injective module need not vanish. The proof must supply the actual computation of Hom(Dσ_H, X''') in the diagram category, including a justification that the localized product terms are injective objects there, and similarly for the cells containing ~T. Until this is provided, the Cellular Skeleton Theorem and hence Theorem 7.1 are not established.","section":"Section 8, Lemma 8.7"},{"comment":"The formality of the diagram of ring spectra is a load-bearing part of Equivalence 4, but the proof is one sentence: 'The argument is the same as for the torus.' The current diagram contains R(V~T) = e_G D S(∞V(Z))_+, which is not a torus-type ring, and the finite W-action requires a Maschke argument in the diagram category. The reduction to [15, 10.1–10.2] should be written out, or at least the specific inductive steps and the shape of the diagram should be identified, so that the reader can verify that the torus argument applies verbatim.","section":"Section 6.C, Lemma 6.4"},{"comment":"The proof of the sphere pullback is a case check that concludes, for H containing ~T, that the square is 'the well known pullback square for the dihedral part of O(2).' This is not a proof: the square involves R(V~T) and the product S∞V(Z) ∧ R(V1) × e_G R(V1), and the identification with the dihedral pullback should be verified explicitly, particularly because the three groups G have different component group actions. The reader needs to see the argument that the relevant H-fixed point spectra give the stated coefficient rings and that the square is indeed a pullback for all H.","section":"Section 5.A, Proposition 5.1"}],"minor_comments":[{"comment":"The diagrams in Definition 6.3 would be much easier to read if the vertex labels (VG, VZ, V~T, V1) were printed in the diagrams or listed in a table immediately after the definition; as typeset, the location of each entry is ambiguous.","section":"Definition 6.3"},{"comment":"In Lemma 4.7, the notation DE⟨D⟩ is used before it is defined; define it as the product over dihedral subgroups, or replace it with the explicit product.","section":"Section 4.A"},{"comment":"The proof of Lemma 2.1 says the results are 'observed directly from the calculations in Sections 9 and 11 of [6]' but gives no specific pointers; adding precise lemma numbers from [6] would help the reader verify the height-0 and height-1 claims.","section":"Section 2.A"},{"comment":"There are several typos, e.g., 'mutliplicatively' in Section 2.G, 'muliplicatively' in the abstract, and 'Analagously' in Section 4.A; these should be corrected.","section":"Throughout"},{"comment":"In the statement of Proposition 6.2, the notation R(V_H(∞,n)) is used, but the corresponding ring was denoted R(VZ) in Section 4; the relationship between these should be stated explicitly.","section":"Proposition 6.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is part of a series and relies on several in-preparation works by the same author ([8]–[12]). The central theorem of this paper does not appear to depend on those in-preparation papers, but the framing does; the editor may wish to consider whether the paper should be judged as a standalone article. My main concern is the unproved last step of Lemma 8.7; if the author can supply the missing computation, the result is likely correct. The paper would benefit from a longer, fully detailed Section 8."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something real: it writes down an explicit algebraic model A(G|full) for the three mixed-type rank-2 toral groups and states Theorem 7.1, the Quillen equivalence. The Type 1/Type 2 subgroup pattern, the handling of the non-normal ~T component, and the choice of the inflation diagram are new and look sensible. The paper is also honest about what comes from prior work and what is in preparation; the strategy follows the Greenlees-Shipley torus template, but the adaptation is not a trivial copy.\n\nThe soft spot is exactly where the stress-test note lands: Lemma 8.7. The proof that an arbitrary module is cellularly equivalent to an object of the standard model after killing G, Z, and ~T evaluations is the load-bearing step for Proposition 8.1 and therefore for Theorem 7.1. The final paragraph claims cellular triviality of X''' because all terms are 'torsion free and injective.' That is not enough. For the dual cell attached to a one-dimensional H containing Z, the entry at VZ is a free E^{-1}OF-module (tensor Q_H). A free module has nontrivial maps into many torsion-free injective localizations; injectivity of the target does not make Hom from a free source vanish. The worked one-dimensional warm-up (Example 8.6) is genuinely a homotopy-pullback computation, but the analogous computation for the Z-containing cells is replaced by a blanket assertion. This needs a real fix, not a small edit.\n\nTwo smaller items: Lemma 6.4 (formality) and Proposition 5.1 (sphere pullback) are asserted rather than derived, with citations to the toral case. Those are probably fine in context, but a referee will want the details or precise pointers. And the heavy reliance on in-preparation papers [8]–[12] limits the standalone checkability of the paper, though that is structural to the program.\n\nWho should read this: specialists in equivariant homotopy and the algebraic models program. The central theorem is plausible and the explicitly constructed model is useful, but the proof as written is not complete. I would send it to peer review, with a referee specifically asked to verify or repair Lemma 8.7. If that gap closes, this is a solid contribution to the program.","headline":"The explicit model for the three mixed-type groups is a genuine advance, but the key Cellular Skeleton step, Lemma 8.7, has a real gap in the stated proof and needs referee attention before the theorem is trusted.","tokens_in":21506,"tokens_out":2871,"would_cite":false,"duration_ms":30375,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P91","55P42","55N91"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the three rank-2 toral groups of mixed type, rational G-spectra on full subgroups reduce to an explicit small abelian category.","keywords":["rational equivariant spectra","toral groups","algebraic models","Quillen equivalence","cellular skeleton","full subgroups","injective model structure","SU(3) blocks"],"falsifier":"Take the module $X$ described in Remark 8.10 (constant $\\mathbb{Q}$ on the finite subgroups $H^s(1,n)$, zero elsewhere) and compute its cellularization with respect to the dual basic cells of Definition 6.3. If the result is not isomorphic in the derived category to an object of the standard model, Proposition 8.1 and the theorem fail. Equivalently, any non-contractible rational $G$-spectrum over full subgroups whose homology $\\pi^A_*$ is the zero object of $A(G|\\mathrm{full})$ would disprove the Quillen equivalence.","tokens_in":20311,"feed_emoji":"📐","tokens_out":14404,"duration_ms":118999,"temperature":0.7,"pith_summary":"For the three rank-2 toral groups of mixed type - the split and non-split extensions with identity component a 2-torus and component group of order 2, represented by $O(2)\\times T$, $Pin(2)\\times T$, and the normalizer $N_{U(2)}(T^2)$ - this paper proves that rational $G$-spectra with full isotropy are Quillen equivalent to the derived category of a small, explicitly written abelian category $A(G|\\mathrm{full})$. The point of the model is calculability: rational cohomology theories on these groups become diagrams of modules over products of polynomial rings, with the topology encoded in a finite sheaf condition. The normalizer case is the most complicated block in the analysis of rational $U(2)$- and $SU(3)$-spectra, so this removes the main obstruction there. The proof decomposes the sphere as a pullback of four isotropically simple formal ring spectra and then shows, via the Cellular Skeleton Theorem, that the resulting algebraic module category collapses to $A(G|\\mathrm{full})$.","feed_headline":"Rank-2 toral spectra reduce to one small algebraic model","feed_subtitle":"Full rational spectra for O(2)×T, Pin(2)×T and the U(2) normalizer become a small diagram category.","key_machinery":"The load-bearing object is the standard model $A(G|\\mathrm{full})$: a square diagram of modules over the ring $\\tilde{\\mathcal{O}}_F$, required to be quasicoherent, meaning $N(V_Z)=E^{-1}N(V_1)$, $N(V_{\\tilde T})=I^{-1}N(V_1)$, and $N(V_G)=E^{-1}I^{-1}N(V_1)$, and extended, meaning each entry is obtained by tensor product from a module over a smaller ring. This is the algebraic shadow of the decomposition of the sphere as a homotopy pullback of the four ring spectra $S^{\\infty V(Z)}\\wedge e_G S^0$, $S^{\\infty V(Z)}\\wedge DE\\langle D\\rangle$, $e_G D S^{\\infty V(Z)}_+$, and $DE_{F+}$, with homotopy rings $\\mathbb{Q}$, $\\prod \\mathbb{Q}$, $\\prod \\mathbb{Q}[c]$, and $\\prod \\mathbb{Q}[c]$ respectively. The Cellular Skeleton Theorem (Proposition 8.1) is the step that lets the topological decomposition be replaced by the abelian category; it is proved using right adjoints $f_G$, $f_Z$, $f_{\\tilde T}$, $f_1$ to evaluation at the four strata of the subgroup space, together with the fact that every object has an injective resolution of length at most 2.","core_discovery":"The central claim, stated as Theorem 7.1, is a Quillen equivalence $G\\text{-spectra}|\\mathrm{full}\\simeq DG\\text{-}A(G|\\mathrm{full})$ for each of the three groups. The category $A(G|\\mathrm{full})$ is the standard model of Section 2: quasicoherent and extended diagrams of modules over a ring diagram with vertices $E^{-1}I^{-1}\\mathcal{O}_F$, $E^{-1}\\mathcal{O}_F$, $I^{-1}\\mathcal{O}_F$, and $\\mathcal{O}_F$, where $\\mathcal{O}_F$ is a product of copies of $\\mathbb{Q}[c]$ indexed by the finite full subgroups, and $E^{-1}$ and $I^{-1}$ are algebraic localizations matching geometric ones. The proof has two halves: first, the sphere is shown to be a homotopy pullback of four commutative ring spectra that are isotropically simple and intrinsically formal, so modules over the sphere become modules over a punctured cube of rings; second, the Cellular Skeleton Theorem identifies the cellularization of that module category with the derived category of the standard model. The model has injective dimension 2, so homology isomorphisms are the weak equivalences and injective resolutions provide the computational formalism.","pith_inferences":["An extension the paper leaves implicit: the same four-vertex inflation square should govern any module category built from these ring spectra, so explicit Borel cohomology and free-loop-space computations for these groups could be read off directly from $A(G|\\mathrm{full})$.","A testable extension is to use the generator checks of Lemma 7.3 as a template: compute $\\pi^A_*$ on a specific spectrum, such as the localization of the sphere at a finite subgroup, and verify the resulting diagram is quasicoherent and extended.","Because the injective dimension is 2, the model predicts a two-stage Postnikov-style decomposition for every rational cohomology theory on these groups; making that decomposition explicit would connect the model to classical Borel cohomology and change-of-rings computations."],"forward_implications":["For each of the three groups, any rational $G$-spectrum with full isotropy is represented up to weak equivalence by a small algebraic diagram, so rational equivariant cohomology can be computed inside $A(G|\\mathrm{full})$.","Because $A(G|\\mathrm{full})$ has injective dimension 2, the derived category admits the injective model structure with homology isomorphisms as weak equivalences, and every object has an explicit three-term injective resolution.","The $N_{U(2)}(T^2)$ case provides the full-subgroup block of rational $U(2)$-spectra and the most complicated toral block of rational $SU(3)$-spectra, so the models for those groups inherit a complete description of this block.","The same strategy - sphere as a pullback of isotropically simple formal rings, then a cellular skeleton argument - applies uniformly to all three cases, so the three models share one shape even where their subgroup lattices differ."],"supporting_citations":[{"why":"determines the conjugacy classes of subgroups and the spaces $V^G_{\\mathrm{full}}$ that index the model.","marker":"[6]"},{"why":"supplies the torus case method of sphere-as-pullback, formality, and the cellular skeleton argument that this paper adapts.","marker":"[15]"},{"why":"gives the equivalence between modules over a homotopy pullback ring and diagrams of modules, used at Equivalence 1.","marker":"[14]"},{"why":"provides the theorem that $H\\mathbb{Z}$-algebra spectra are differential graded algebras, used at Equivalence 3.","marker":"[16]"},{"why":"gives the fixed-point adjunction for equivariant module spectra, used at Equivalence 2.","marker":"[13]"},{"why":"identifies the Balmer spectrum of finite rational $G$-spectra, fixing the sheaf-theoretic shape of the model.","marker":"[5]"},{"why":"states the conjecture that rational $G$-spectra are Quillen equivalent to DG objects of an abelian category, which the theorem proves in these cases.","marker":"[3]"}],"fun_headline_variants":["One algebraic model tames all three rank-2 toral spectra","Rational G-spectra for U(2) normalizer: one small diagram","Mixed toral spectra fold into one algebraic model","Three groups, one model: rational toral spectra collapse"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the claim in Section 8 that every algebraic module is cellularly equivalent to an object of the small standard model; the proof of that claim is sketched rather than written out, and it depends on a leftover object being trivial because all its pieces are torsion free and injective.","fun_headline_variants_meta":{"raw":{"variants":["One algebraic model tames all three rank-2 toral spectra","Rational G-spectra for U(2) normalizer: one small diagram","Mixed toral spectra fold into one algebraic model","Three groups, one model: rational toral spectra collapse"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001139,"raw_usage":{"total_tokens":4704,"prompt_tokens":894,"completion_tokens":3810,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":3739}},"tokens_in":510,"tokens_out":3810,"duration_ms":28232,"temperature":1.0,"reasoning_tokens":3739,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:08:58.635625+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the module $X$ described in Remark 8.10 (constant $\\mathbb{Q}$ on the finite subgroups $H^s(1,n)$, zero elsewhere) and compute its cellularization with respect to the dual basic cells of Definition 6.3. If the result is not isomorphic in the derived category to an object of the standard model, Proposition 8.1 and the theorem fail. Equivalently, any non-contractible rational $G$-spectrum over full subgroups whose homology $\\pi^A_*$ is the zero object of $A(G|\\mathrm{full})$ would disprove the Quillen equivalence.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the equivalence between modules over a homotopy pullback ring and diagrams of modules, used at Equivalence 1."},{"cited_title":"HZ-algebra spectra are differential graded alg ebras","cited_arxiv_id":null,"evidence_quote":"provides the theorem that $H\\mathbb{Z}$-algebra spectra are differential graded algebras, used at Equivalence 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the fixed-point adjunction for equivariant module spectra, used at Equivalence 2."}],"review_version":1}