The Telescope Conjecture for Global Representations and FI-modules

Peng Xu

REVIEW 3 major objections 4 minor 26 references

The Telescope Conjecture for Global Representations and FI-modules

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper classifies the localizing ideals of derived categories of global representations over several infinite families of finite groups and uses the classification to prove the telescope conjecture for these categories and for derived F

desk verdict The classifications are new and the telescope proofs hold up — the reader's main objection dissolves once naturality in A(E_p) is taken seriously; this deserves a serious referee. read the letter →

arxiv 2607.15586 v1 pith:5YWNN6P5 submitted 2026-07-17 math.CT math.RT

Peng Xu This is my paper · ORCID

classification math.CTmath.RT MSC 18G8020C99

keywords telescopeconjectureglobalrepresentationslocalizingidealsFI-modulesVI-modulestensortriangulargeometryhomologicalsupportderivedcategories

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 aims to settle the telescope conjecture for derived categories of global representations over three infinite families of finite groups — cyclic groups of prime order plus the trivial group, cyclic p-groups, and elementary abelian p-groups — and for the derived category of FI-modules. A sympathetic reading: it claims that in each of these categories every smashing tensor ideal is generated by compact objects, and that the full lattice of localizing tensor ideals is classified by homological support into subsets (or open subsets) of an explicit spectrum. This matters because these categories are not rigidly compactly generated, so the standard stratification machinery that proves the telescope conjecture in rigid settings does not apply; the paper offers a different mechanism based on characteristic objects, torsion theory, and a generic quotient. If correct, the result also covers derived VI-modules via Pontryagin duality and provides the first large tensor-triangular classification in a non-rigid global representation setting.

What carries the argument

The load-bearing machinery is the homological support hsupp(X) = {G | X(G) ≠ 0}, together with characteristic objects χ_{G,V} concentrated at a single group. For essentially finite families a localizing lemma shows any object is generated by the χ_{G,k} for G in its support. For the infinite families, the argument separates torsion from non-torsion localizing ideals: torsion ideals are generated by χ_{i,k}, while non-torsion ideals contain a compact generator e_Δ or M_Δ and are generated by it plus the χ_{i,k} in the support. A key technical point is the identification, for noetherian families C_p, E_p, and FI, of the parameter space as the tensor abelian spectrum Spc(A^c) rather than the Ba

What would settle it

Compute Hom_{D(E_p)}(χ_{i,k}, e_n) for i≥n in the derived category of global representations of elementary abelian p-groups. The evaluation e_n((Z/p)^i) is nonzero, giving a nonzero map χ_{i,k} → e_n in degree zero, contradicting the identity used in the proof of Corollary 5.5; the same computation with FI-modules (replacing e_n by M_n and χ_{i,k} by the FI-characteristic object) tests Corollary 6.14.

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Extended reading notes

Core claim

Homological support — the set of groups G with X(G) ≠ 0 — is shown to completely classify localizing tensor ideals. For cyclic prime-order groups plus the trivial group, every localizing ideal is generated by characteristic objects χ_{G,k} for G in a subset of P^*, recovering arbitrary subsets of the Balmer spectrum. For cyclic p-groups, only open subsets occur. For elementary abelian p-groups and FI-modules, the classifying space is the tensor abelian spectrum of the noetherian heart, and every non-torsion localizing ideal is generated by one compact generator together with the χ_{i,k} in its support. The telescope conjecture, that every smashing ideal is compactly generated, follows in eac

Load-bearing premise

The proof that torsion localizing ideals in D(E_p) and D(FI) are not smashing depends on the assertion that Hom_{D(E_p)}(χ_{i,k}, Σ^s e_n)=0 for all i,n,s; but this appears false for i≥n because e_n((Z/p)^i) ≠ 0, so the argument for Corollary 5.5 (and its FI analogue in Corollary 6.14, which invokes 'the same argument') has no support as written.

Editorial extensions

If this is right

If Theorem 1.4 is correct, the telescope conjecture holds for the derived categories of global representations over C_p^r, C_p, E_p, and FI-modules, and via Pontryagin duality for VI-modules.
The bijections in Theorems 3.11, 4.15, 5.3, and 6.13 mean the entire lattice of localizing tensor ideals is explicitly parametrized by subsets or open subsets of a known spectrum, not just the smashing ones.
Because the classification is support-theoretic, it provides a substitute for Balmer–Favi stratification in these non-rigid categories, where the compact objects are not rigid.
The Balmer spectrum of D(E_p)^c is not N^*, so the big category's geometry is governed by the tensor abelian spectrum; this changes how one predicts smashing ideals from compact data.
Every smashing ideal in these categories is generated by compact objects, matching the predicted form of the telescope conjecture.

Reading between the lines

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

If correct, the support-theoretic form of the classification suggests that for non-rigid categories the relevant geometry is the tensor abelian spectrum of the noetherian heart; this may be the right invariant for other global families and representation-stability categories.
The same torsion/non-torsion dichotomy might extend to any widely closed family U with a locally noetherian heart and a semisimple generic quotient, which would make the telescope conjecture a consequence of two ingredients: compact generation of non-torsion ideals and an orthogonality statement for characteristic objects.
A direct computation of Hom_{D(E_p)}(χ_{i,k}, e_n) for i≥n would test the non-smashing torsion step; since e_n((Z/p)^i) is nonzero, this is a concrete and inexpensive check.

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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

Bottom line: this paper is a genuine advance, and I don't agree with the reader's rejection. The one objection that looked load-bearing — Hom(χ_i, Σ^s e_n)=0 failing when i≥n — does not land. A map χ_i→e_n in the functor category is an element x∈e_n(E_i) satisfying α_*x=0 for every surjection α:E_j→E_i with j>i. The induced map α_*:e_n(E_i)→e_n(E_j) sends a basis surjection f to fα, and that is injective on the basis, so x must be zero. Since e_n is injective and projective, higher Exts vanish too. The same argument goes through for FI with M_n. So Corollaries 5.5 and 6.14 are not undermined by the reported counterexample.

What is genuinely new: the localizing ideal classifications for D(C_pr), D(C_p), D(E_p), and D_FI, and the resulting telescope conjecture for non-rigid categories. Previous work mostly settled thick ideals in the compact part or prime spectra; the step to all localizing ideals is substantial. The torsion-generic quotient is a real extension, and the homological-support bijections are likely to be useful.

The soft spots are real but smaller than the reader thinks. Lemma 6.12 and parts of Lemma 6.6 are left to the reader, and Corollary 6.14 leans on 'the same argument' from Corollary 5.5. That is sloppy for a paper with this much machinery; a referee should ask for the arguments in full. The dependence on [26] is heavy and worth making explicit, but it is not circular: [26] supplies tensor-abelian geometry, not the localizing-ideal classifications. The paper also has places where 'by the same proof' conceals bookkeeping, but I did not find a mathematical gap after the naturality check.

Who it is for: people in tensor triangular geometry and representation stability. It is a serious paper with significant results, and it deserves a serious referee rather than a desk rejection. I would send it out, with a request that the omitted lemmas be proved and the reliance on [26] be stated cleanly.

Referee Report

3 major / 4 minor

Summary. The paper studies the derived categories D(U) of global representations over a characteristic-zero field k for several infinite families U, as well as the derived category of FI-modules. It claims a complete classification of localizing tensor ideals by homological support (Theorems 3.11, 4.15, 5.3, 6.13) and deduces the telescope conjecture for D(C_pr), D(C_p), D(E_p), the derived category of VI-modules, and the derived category of FI-modules (Theorem 1.4 and Corollaries 5.6, 6.14). The strategy is to separate each localizing ideal into a torsion part generated by characteristic objects and a non-torsion part controlled by a cofinite support and a single compact object such as e_\Delta or M_\Delta.

Significance. If the results were correct, this would be a significant advance: the telescope conjecture is usually proved in rigidly-compactly generated settings, while the categories considered here are generally not rigid. A complete classification of localizing ideals is also stronger than the telescope conjecture itself. The sections on C_pr, C_p, and E_p are developed in detail and appear to contain substantial work. However, the FI-section contains a false generation lemma (Lemma 6.6(4) and its consequence Lemma 6.9), and because the proof of the FI classification and telescope conjecture relies essentially on that lemma, the advertised Theorem 1.4 for D_FI is not established as written.

major comments (3)

[§6, Lemma 6.6(4) and Lemma 6.9] Lemma 6.6(4) states D(FI_{\ge n}) = Loc<M_n>. This is false. Let \chi_2 be the FI-module with \chi_2(2)=k (trivial S_2-action) and \chi_2(m)=0 for m\ne 2. In D(FI_{\ge 2}), the object \chi_2 is nonzero and has homological support {2}. The class of objects with cofinite-or-empty homological support is a localizing subcategory; it contains M_2 because hsupp(M_2)=\{m\ge 2\} is cofinite in {m\ge 2}. Hence every object of Loc<M_2> has cofinite-or-empty support. But {2} is not cofinite, so \chi_2\notin Loc<M_2>. Thus D(FI_{\ge 2})\ne Loc<M_2>. The same example disproves Lemma 6.9: in D_FI, min hsupp(\chi_2)=2, yet \chi_2\notin Loc<M_2>. Lemma 6.9 is used essentially in Proposition 6.11 and again in the proof of Theorem 6.13, so the classification of localizing ideals and the telescope conjecture for D_FI are unsupported.
[§6, Theorem 6.13 and Corollary 6.14] The proof of Theorem 6.13 relies on the false assertion that (j_m)_!(j_m)^* M_{\Delta} \in Loc<M_m>; this is justified only by Lemma 6.6(4). Since that lemma is false, the containment is not established. Likewise, Proposition 6.11, which is used to show M_{\Delta}\in Loc<m_L> in Corollary 6.14, depends on Lemma 6.9. Consequently the FI-module part of the main theorem, including the telescope conjecture for D_FI, does not follow from the arguments given. A replacement argument for the FI case would be needed.
[§6, Lemma 6.12] Lemma 6.12 is stated without proof: 'We omit the proof of the following lemma, which is the FI-analogue of Lemma 5.2.' This lemma is load-bearing in Theorem 6.13, where it is used to assert that a non-torsion localizing ideal has cofinite support. If it is standard, a precise reference should be supplied; otherwise a proof is required. The current text leaves an essential step unsupported.

minor comments (4)

[§5, Corollary 5.5] There is a typo in the sentence 'Hence L^\perp=D(C_p)': it should be D(E_p).
[§3, Proposition 3.2] The displayed diagram in the proof is garbled in the text; the triangle and the map g_n are hard to read. Please reformat.
[§2, Conventions] The statement 'All categories considered in this paper are assumed to be small' is in tension with constructions such as Fun(U^op, Mod_k), which is not small. This is probably a harmless convention, but it should be stated precisely.
[§6, Lemma 6.6] Parts (1)-(3) and (5) of Lemma 6.6 are standard, but part (4) is false as discussed. The 'proof omitted' note is not acceptable for a false statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the derivation rests on independent compact-object and tensor-abelian classifications, and no prediction reduces to a fitted input.

full rationale

I walked the claimed derivation chain. Theorems 3.11, 4.15, 5.3, and 6.13 are proved by support-theoretic arguments that reduce localizing ideals to homological support and then identify the support lattices using prior compact-object results ([5], [6], [11]) and the tensor-abelian spectrum from [26]. At no point is the target classification (localizing ideals / telescope conjecture) assumed as an input, nor is a parameter fitted to data and then renamed a prediction. The author's self-citation [26] is load-bearing for the E_p, VI, and FI parts, but it is an independent prior classification of prime Serre ideals / tensor-abelian spectra, not a restatement of the telescope conjecture or of the localizing-ideal classification; it is therefore real evidence under the stated rule and does not by itself constitute circularity. The manuscript flags two omitted/standard arguments: Lemma 4.12 (Gabriel correspondence, 'We leave the details to the reader') and Lemma 6.12 ('We omit the proof... FI-analogue of Lemma 5.2'); these are proof gaps, not circular steps. Similarly, Corollary 5.5's asserted identity Hom_{D(E_p)}(χ_{i,k},Σ^s e_n)=0 and Corollary 6.14's 'same argument' are correctness risks if the identity fails, but the theorem is not equivalent to that identity by construction. No self-definitional, fitted-input, ansatz-smuggling, or renaming pattern is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted numerical parameters appear. The paper's input consists of structural hypotheses on U and prior classification theorems, including the author's [26]. The central classification is not derived by fitting constants.

assumptions (7)

domain assumption k is a field of characteristic zero
Used throughout; guarantees group algebras k[Out(G)] are semisimple and Schur's lemma applies (Section 1 Conventions, Lemma 3.1).
domain assumption U is widely closed and the tensor unit in D(U) is compact (Hypothesis 2.4)
All main theorems assume this hypothesis; it ensures compact generators e_G (Proposition 2.5).
standard math Gabriel correspondence: localizing ideals of a locally noetherian Grothendieck category correspond to Serre ideals of its noetherian objects
Invoked in Lemma 4.12 and used in Theorems 4.15, 5.3, and 6.13; proof left to the reader.
domain assumption Derived Gabriel localization gives a tensor equivalence D(A)/T_D ≅ D(A/T_A)
Used without citation in the proofs of Theorem 4.15 and Theorem 6.13 to reduce to the torsion-free quotient.
domain assumption Classification of prime Serre ideals and tensor abelian spectrum of A(C_p)^c and A(E_p)^c from Xu [26]
The author's own to-appear paper supplies Spc(A^c) ≅ N^* and the prime Serre ideals; used as a black box in Sections 4-6.
domain assumption Uniform torsion bounds for finitely generated torsion FI/global modules ([13, Theorem B], [20, Theorem B])
Used to show non-zero perfect complexes have non-torsion cohomology (Theorem 6.4 and Lemma 5.2).
domain assumption Semisimplicity of A(C_p)/T_A and triviality of its localizing ideals
Asserted in the proof of Theorem 4.15; if false, the generic-point argument fails.

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Cite this review

Pith. "Pith review of The Telescope Conjecture for Global Representations and FI-modules." pith.science (2026). https://pith.science/paper/5YWNN6P5

@misc{pith2026260715586,
  author       = {Pith},
  title        = {Pith review of: The Telescope Conjecture for Global Representations and FI-modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YWNN6P5}},
  note         = {Machine review of arXiv:2607.15586}
}

read the original abstract

In this paper, we classify the localizing ideals of the derived category D(U) of global representations over a field k of characteristic zero, for various infinite families U of finite groups. These families include elementary abelian p-groups, cyclic p-groups, and cyclic groups of prime order together with the trivial group. We deduce that the telescope conjecture holds for these D(U). In particular, via Pontryagin duality, our results for elementary abelian p-groups also establish the telescope conjecture and the corresponding classification for derived VI-modules. We also prove that the telescope conjecture holds for the derived category of FI-modules.

Discussion (0). Continue with ORCID to comment.

Reference graph

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2025