Rational G-spectra for rank 2 toral groups of mixed type

J.P.C.Greenlees

REVIEW 3 major objections 5 minor 2 cited by

Rational G-spectra for rank 2 toral groups of mixed type

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For the three rank-2 toral groups of mixed type, rational G-spectra on full subgroups reduce to an explicit small abelian category.

desk verdict 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. read the letter →

arxiv 2501.15584 v1 pith:53SLCXSY submitted 2025-01-26 math.AT

J.P.C.Greenlees This is my paper · ORCID

classification math.AT MSC 55P9155P4255N91

keywords rationalequivariantspectratoralgroupsalgebraicmodelsQuillenequivalencecellularskeletonfullsubgroupsinjectivemodelstructureSU(3)blocks

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

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 $A(G|\mathrm{full})$.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

Two 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section 8, Lemma 8.7] 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 6.C, Lemma 6.4] 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 5.A, Proposition 5.1] 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.

minor comments (5)

[Definition 6.3] 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 4.A] 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 2.A] 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.
[Throughout] 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.
[Proposition 6.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the algebraic model is constructed explicitly and independently before the spectral comparison, and the load-bearing citations are independent published results rather than the target theorem.

full rationale

The standard model A(G|full) is explicitly defined in Section 2 as a category of qce diagrams of modules over the inflation diagram OF/G -> OF/Z, OF/~T -> OF, before any Quillen equivalence with G-spectra is asserted; it is not fitted to the spectra it later models. Theorem 7.1 is proved through a chain of five equivalences that cite independent results: [14, 4.1] for modules over homotopy pullback diagrams, [13] for fixed-point adjunctions, [16] for HZ-algebra spectra, and [15] for the torus strategy. These are parameter-free theorems whose assumptions do not contain the target rank-2 mixed-type claim. The self-citations, including [6] for the subgroup classification and [15] for the standard argument, are load-bearing but they supply independent mathematical inputs rather than the conclusion of the present paper. The genuine weakness is in Section 8: Lemma 8.7's final step, asserting that X''' is cellularly trivial from "torsion free and injective" terms, is not fully justified, and Proposition 8.1 is under-written; however, this is an omitted proof or correctness gap, not a circular reduction. No equation or definition in the paper is equivalent to the theorem by construction, and no fitted quantity is renamed as a prediction. Therefore the paper does not exhibit circularity, though it does contain significant proof gaps.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical free parameters are fitted to data; the model is constructed from subgroup data and ring spectra. The algebraic objects introduced are definitions within an existing framework, not empirical entities. The main burden is carried by prior results, especially self-cited preprints and in-preparation papers, rather than by new ungrounded entities.

assumptions (6)

domain assumption The classification and parametrization of subgroups H-lambda(m,n) from [6, Sections 9 and 11] determines V_full and the Thomason stratification used in Section 2.
Self-cited preprint by the same author; all subsequent definitions of the standard model depend on this data, and no full derivation is repeated.
standard math The Balmer spectrum of finite rational G-spectra is XG = Sub(G)/G with the h-topology and Zariski topology, as in [5] and [1, 11.5].
Used in Section 1.B to write A(G) as sheaves over XG and to identify Thomason height with dimension in Lemma 2.1.
standard math Modules over a homotopy pullback ring are equivalent to cellularized diagrams of modules over the constituent rings, [14, 4.1].
Equivalence 1 in Section 3; the crux is that the sphere spectrum is a homotopy pullback of the ring spectra R(VK).
ad hoc to paper The Greenlees-Shipley formality and cellular skeleton arguments for the torus [15] extend to the mixed cases in Lemma 6.4 and Section 8.
The paper says the argument is 'the same as for the torus' and treats finite group actions as harmless by Maschke's Theorem, but the full verification is not given.
standard math Known pullback squares for the circle group and for the dihedral part of O(2) hold equivariantly in the localized contexts needed in Proposition 5.1.
Used to check the sphere decomposition for proper subgroups containing Z or ~T.
ad hoc to paper The in-preparation papers [8] through [12] provide the general abelian-model framework and the assembled U(2) and SU(3) models that frame this result.
Section 1.C says this paper is the third of five and that the general framework [10] and assembly [8], [9] are not yet public, so the broader significance is currently unverifiable.

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Pith. "Pith review of Rational G-spectra for rank 2 toral groups of mixed type." pith.science (2026). https://pith.science/paper/53SLCXSY

@misc{pith2026250115584,
  author       = {Pith},
  title        = {Pith review of: Rational G-spectra for rank 2 toral groups of mixed type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/53SLCXSY}},
  note         = {Machine review of arXiv:2501.15584}
}

read the original abstract

We give an explicit and calculable algebraic model for the block of rational G-spectra on full subgroups when G has identity component a 2-torus T, and component group of order 2 acting non-trivially on H_1(T). The example of particular interest is the normalizer of the maximal torus in U(2), which constitutes one of the most complicated blocks in the analysis of SU(3). This builds on the determination of subgroups up to conjugacy in arXiv 2501.06914

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Rational SU(3)-equivariant cohomology theories
math.AT 2025-02 conditional novelty 6.0 of 10

Rational SU(3)-equivariant spectra are Quillen equivalent to differential graded objects of an explicitly constructed abelian category A(SU(3)).
An algebraic model for rational U(2)-spectra
math.AT 2025-02 conditional novelty 6.0 of 10

The category of rational U(2)-spectra is shown to decompose into seven blocks, each with an explicit algebraic model, yielding a calculable algebraic model for the whole category.

Reference graph

Works this paper leans on

16 extracted references · 4 canonical work pages · cited by 2 Pith papers

[8]

J. P. C. Greenlees. An algebraic model for rational U (2)-spectra. In preparation, 15pp , 2025

work page 2025
[12]

J. P. C. Greenlees. Rational G-spectra for components with ﬁnite Weyl groups. In preparation, 4pp , 2025

2025
[14]

J. P. C. Greenlees and B. Shipley. Homotopy theory of modules o ver diagrams of rings. Proc. Amer. Math. Soc. Ser. B , 1:89–104, 2014

work page 2014
[15]

J. P. C. Greenlees and B. Shipley. An algebraic model for rationa l torus-equivariant spectra. J. Topol., 11(3):666–719, 2018

2018
[16]

HZ-algebra spectra are diﬀerential graded alg ebras

Brooke Shipley. HZ-algebra spectra are diﬀerential graded alg ebras. American Journal of Mathematics , 129(2):351–379, 2007. Mathematics Institute, Zeeman Building, Coventry CV4, 7AL , UK Email address : john.greenlees@warwick.ac.uk 24

work page 2007
[1]

Balchin, T

S. Balchin, T. Barthel, and J. P. C. Greenlees. Prismatic decompo sitions and rational G-spectra. Preprint, 59pp, arXiv 2311.18808 , 2023

arXiv 2023
[2]

Greenlees, and Magdalena K¸ edziorek

David Barnes, J.P.C. Greenlees, and Magdalena K¸ edziorek. An alg ebraic model for rational toral G- spectra. Algebr. Geom. Topol., 19(7):3541–3599, 2019

2019
[3]

J. P. C. Greenlees. Triangulated categories of rational equivar iant cohomology theories. Oberwolfach Reports, pages 480–488, 2006. (cit. on p. 2) , 2006

2006

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[4]

J. P. C. Greenlees. Rational equivariant cohomology theories wit h toral support. Algebr. Geom. Topol., 16(4):1953–2019, 2016

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J. P. C. Greenlees. The Balmer spectrum of rational equivariant cohomology theories. J. Pure Appl. Algebra, 223(7):2845–2871, 2019

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J. P. C. Greenlees. Spaces of subgroups of toral groups. Preprint, 38pp, arXiv:2501.06914 , 2025

2025
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J. P. C. Greenlees. Algebraic models for one-dimensional catego ries of rational G-spectra. Preprint, 28pp, arXiv:2501.11200, 2025

2025 arXiv
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J. P. C. Greenlees. An algebraic model for rational SU (3)-spectra. In preparation, 11pp , 2025

2025
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J. P. C. Greenlees. An abelian model for rational G-spectra for a compact Lie group G. In preparation, 27pp
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J. P. C. Greenlees. An algebraic model for rational G-spectra for ﬁnite central extensions of a torus. In preparation, 18pp
[13]

J. P. C. Greenlees and B. Shipley. Fixed point adjunctions for eq uivariant module spectra. Algebr. Geom. Topol., 14(3):1779–1799, 2014

2014

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [8]

J. P. C. Greenlees. An algebraic model for rational U (2)-spectra. In preparation, 15pp , 2025

work page 2025

[2] [12]

J. P. C. Greenlees. Rational G-spectra for components with ﬁnite Weyl groups. In preparation, 4pp , 2025

2025

[3] [14]

J. P. C. Greenlees and B. Shipley. Homotopy theory of modules o ver diagrams of rings. Proc. Amer. Math. Soc. Ser. B , 1:89–104, 2014

work page 2014

[4] [15]

J. P. C. Greenlees and B. Shipley. An algebraic model for rationa l torus-equivariant spectra. J. Topol., 11(3):666–719, 2018

2018

[5] [16]

HZ-algebra spectra are diﬀerential graded alg ebras

Brooke Shipley. HZ-algebra spectra are diﬀerential graded alg ebras. American Journal of Mathematics , 129(2):351–379, 2007. Mathematics Institute, Zeeman Building, Coventry CV4, 7AL , UK Email address : john.greenlees@warwick.ac.uk 24

work page 2007

[6] [1]

Balchin, T

S. Balchin, T. Barthel, and J. P. C. Greenlees. Prismatic decompo sitions and rational G-spectra. Preprint, 59pp, arXiv 2311.18808 , 2023

arXiv 2023

[7] [2]

Greenlees, and Magdalena K¸ edziorek

David Barnes, J.P.C. Greenlees, and Magdalena K¸ edziorek. An alg ebraic model for rational toral G- spectra. Algebr. Geom. Topol., 19(7):3541–3599, 2019

2019

[8] [3]

J. P. C. Greenlees. Triangulated categories of rational equivar iant cohomology theories. Oberwolfach Reports, pages 480–488, 2006. (cit. on p. 2) , 2006

2006