Residual regularity in tensor triangular geometry

Emmy Van Rooy

Residual regularity in tensor triangular geometry

T0 review · 0 major / 1 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read Residual regularity descends and ascends via finite separable extensions, classifying all finite groups whose derived category of permutation modules satisfies the property.

desk verdict Van Rooy defines residual regularity, proves its stability under finite separable extensions, and classifies the groups with residually regular permutation module categories. read the letter →

arxiv 2605.27244 v1 pith:F5DQBU4Q submitted 2026-05-26 math.CT math.AGmath.RT

Emmy Van Rooy This is my paper · ORCID

classification math.CTmath.AGmath.RT

keywords residualregularitytensortriangulatedcategoriesfiniteseparableextensionspermutationmodulesderivedgroupsclassification

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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 defines residual regularity as a new notion of regularity for tensor triangulated categories. It proves that this property descends and ascends along finite separable extensions. It applies the result to classify all finite groups for which the derived category of permutation modules is residually regular. A sympathetic reader would care because the stability gives a practical way to move the property between categories and the classification identifies exactly when it holds in this representation-theoretic setting.

What carries the argument

Residual regularity, the newly defined property of tensor triangulated categories that is shown to transfer along finite separable extensions and to classify the relevant groups.

What would settle it

A finite separable extension where residual regularity fails to descend or ascend, or a finite group outside the classified list whose derived category of permutation modules is residually regular.

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Extended reading notes

Core claim

We introduce residual regularity as a new notion of regularity for tensor triangulated categories. We show that residual regularity descends and ascends via finite separable extensions and we classify all finite groups whose derived category of permutation modules is residually regular.

Load-bearing premise

The definition of residual regularity must pick out a meaningful and non-vacuous property on the tensor triangulated categories under study.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main takeaway is that this paper defines a new notion of residual regularity in tensor triangulated categories, establishes its stability under finite separable extensions, and classifies the finite groups for which the derived category of permutation modules satisfies the property.

What is new is the definition and the two theorems that follow from it. The descent and ascent results are the sort of technical statements that allow the notion to be applied more broadly, and the classification provides specific content by identifying which groups work.

The paper does well in setting up a property that behaves nicely with respect to extensions. This is a common technique in the field, and applying it here gives the definition some structure right away. The classification is also a positive step because it turns the abstract definition into something that can be checked for particular examples.

On the soft spots, the main one is the motivation for the new notion. Without more context on why residual regularity is needed beyond existing concepts like regularity in tt-geometry, it is not clear how much this reorganizes the area. The classification will be key to showing it is non-vacuous, but if it turns out to coincide with groups having semisimple or regular representation categories in a known way, the contribution is more limited. The abstract does not indicate any computational or external verification, so the strength rests entirely on the internal proofs.

This paper is aimed at researchers in tensor triangular geometry and related areas of algebraic K-theory or representation theory. A reader who follows papers on properties of triangulated categories with tensor structure would get the most out of it. It is worth sending to peer review because the results are stated clearly enough to be evaluated by experts in the subfield, even if the ultimate impact stays within that community.

Referee Report

0 major / 1 minor

Summary. The manuscript introduces a new notion of residual regularity for tensor triangulated categories. It proves that this property descends and ascends along finite separable extensions and classifies all finite groups whose derived category of permutation modules satisfies residual regularity.

Significance. If residual regularity is a meaningful and non-vacuous property, the stability results under finite separable extensions and the classification for permutation-module categories would constitute a useful contribution to tensor triangular geometry, particularly for understanding regularity phenomena in derived categories of group representations. The classification result would serve as evidence that the definition is not vacuous.

minor comments (1)

The abstract does not indicate whether the definition of residual regularity is accompanied by concrete examples or computations that would allow readers to verify non-vacuousness independently of the classification theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. The recommendation is marked uncertain, which appears to hinge on whether residual regularity is a meaningful notion. The classification of all finite groups whose derived permutation module categories are residually regular provides concrete evidence that the property is non-vacuous and distinguishes interesting examples, supporting the utility of the descent/ascent results under finite separable extensions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper introduces a new definition of residual regularity and derives its stability under finite separable extensions together with a classification of finite groups for which the derived category of permutation modules satisfies the property. These steps are internal to the definition and its consequences; no load-bearing claim reduces to a fitted parameter, self-citation chain, or renaming of prior results. The classification demonstrates non-vacuousness rather than presupposing it, rendering the derivation self-contained.

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

Abstract-only review supplies no explicit free parameters, background axioms, or invented entities; the new notion itself is a definition rather than a postulated object with independent evidence.

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Cite this review

Pith. "Pith review of Residual regularity in tensor triangular geometry." pith.science (2026). https://pith.science/paper/F5DQBU4Q

@misc{pith2026260527244,
  author       = {Pith},
  title        = {Pith review of: Residual regularity in tensor triangular geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F5DQBU4Q}},
  note         = {Machine review of arXiv:2605.27244}
}

read the original abstract

We investigate a new notion of regularity for tensor triangulated categories, called residual regularity. We show that residual regularity descends and ascends via finite separable extensions and we classify all finite groups whose derived category of permutation modules is residually regular.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 4 canonical work pages

[1]

The spectrum of prime ideals in tensor triangulated categories

[Bal05] Paul Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math.588 (2005), pp. 149–168. [Bal07] Paul Balmer. Supports and filtrations in algebraic geometry and modular representation theory.Amer. J. Math.129.5 (2007), pp. 1227–1250. [Bal10] Paul Balmer. Tensor triangular geometry.Proceedings of the International ...

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[2]

A guide to tensor-triangular classification.Handbook of homo- topy theory

[Bal20a] Paul Balmer. A guide to tensor-triangular classification.Handbook of homo- topy theory. CRC Press. CRC Press, Boca Raton, FL, 2020, pp. 145–162. [Bal20b] Paul Balmer. Homological support of big objects in tensor-triangulated cat- egories.J. ´Ec. polytech. Math.7 (2020), pp. 1069–1088. [Bal20c] Paul Balmer. Nilpotence theorems via homological resi...

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Permutation modules, Mackey functors, and Artin motives.Representations of algebras and related structures

[BG23] Paul Balmer and Martin Gallauer. Permutation modules, Mackey functors, and Artin motives.Representations of algebras and related structures. EMS Ser. Congr. Rep. EMS Press, Berlin, 2023, pp. 37–75. [BG25a] Paul Balmer and Martin Gallauer. The geometry of permutation modules. Invent. Math.241.3 (2025), pp. 841–928. 30 REFERENCES [BG25b] Paul Balmer ...

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[4]

Geometric Points in Tensor Triangular Geometry

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work page arXiv
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arXiv:1402.6038 [math.KT]. [BIKP24] Dave Benson, Srikanth Iyengar, Henning Krause and Julia Pevtsova. Locally dualisable modular representations and local regularity

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[Bur11] William Burnside

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[Nee26] Amnon Neeman. Triangulated categories with a single compact generator, and two Brown representability theorems.Invent. Math.244.2 (2026), pp. 531–

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Quasi-Galois theory in triangulated categories

[Orl16] Dmitri Orlov. Smooth and proper noncommutative schemes and gluing of DG categories.Adv. Math.302 (2016), pp. 59–105. REFERENCES 31 [Pau15] Bregje Pauwels. “Quasi-Galois theory in triangulated categories”. PhD thesis. University of California, Los Angeles,

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[Pau17] Bregje Pauwels. Quasi-Galois theory in symmetric monoidal categories.Al- gebra Number Theory11.8 (2017), pp. 1891–1920. [Rou08] Rapha¨ el Rouquier. Dimensions of triangulated categories.J. K-Theory1.2 (2008), pp. 193–256. [San22] Beren Sanders. A characterization of fi...

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Emmy V an Rooy, UCLA Mathematics Department, Los Angeles, CA 90095-1555, USA Email address:emmyvr@ucla.edu URL:http://www.math.ucla.edu/~emmyvr

arXiv:2605.08868 [math.CT]. Emmy V an Rooy, UCLA Mathematics Department, Los Angeles, CA 90095-1555, USA Email address:emmyvr@ucla.edu URL:http://www.math.ucla.edu/~emmyvr

arXiv

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

The spectrum of prime ideals in tensor triangulated categories

[Bal05] Paul Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math.588 (2005), pp. 149–168. [Bal07] Paul Balmer. Supports and filtrations in algebraic geometry and modular representation theory.Amer. J. Math.129.5 (2007), pp. 1227–1250. [Bal10] Paul Balmer. Tensor triangular geometry.Proceedings of the International ...

2005

[2] [2]

A guide to tensor-triangular classification.Handbook of homo- topy theory

[Bal20a] Paul Balmer. A guide to tensor-triangular classification.Handbook of homo- topy theory. CRC Press. CRC Press, Boca Raton, FL, 2020, pp. 145–162. [Bal20b] Paul Balmer. Homological support of big objects in tensor-triangulated cat- egories.J. ´Ec. polytech. Math.7 (2020), pp. 1069–1088. [Bal20c] Paul Balmer. Nilpotence theorems via homological resi...

2020

[3] [3]

Permutation modules, Mackey functors, and Artin motives.Representations of algebras and related structures

[BG23] Paul Balmer and Martin Gallauer. Permutation modules, Mackey functors, and Artin motives.Representations of algebras and related structures. EMS Ser. Congr. Rep. EMS Press, Berlin, 2023, pp. 37–75. [BG25a] Paul Balmer and Martin Gallauer. The geometry of permutation modules. Invent. Math.241.3 (2025), pp. 841–928. 30 REFERENCES [BG25b] Paul Balmer ...

2023

[4] [4]

Geometric Points in Tensor Triangular Geometry

arXiv:2603.25664 [math.AT]. [BL14] Clark Barwick and Tyler Lawson. Regularity of structured ring spectra and localization in K-theory

work page arXiv

[5] [5]

Regularity of structured ring spectra and localization in K-theory

arXiv:1402.6038 [math.KT]. [BIKP24] Dave Benson, Srikanth Iyengar, Henning Krause and Julia Pevtsova. Locally dualisable modular representations and local regularity

work page Pith review arXiv

[6] [6]

[Bur11] William Burnside

arXiv:2401.00130 [math.RA]. [Bur11] William Burnside. Theory of groups of finite order (2nd edition). Cambridge University Press, Cambridge,

work page arXiv

[7] [7]

A family of infinite degree tt-rings.Bull

[G´ om24] Juan Omar G´ omez. A family of infinite degree tt-rings.Bull. Lond. Math. Soc.56.2 (2024), pp. 518–522. [GS20] J. P. C. Greenlees and Greg Stevenson. Morita theory and singularity cat- egories.Adv. Math.365 (2020), pp. 107055,

2024

[8] [8]

18417 [math.RT]

arXiv:2512 . 18417 [math.RT]. [Kra00] Henning Krause. Smashing subcategories and the telescope conjecture—an algebraic approach.Invent. Math.139.1 (2000), pp. 99–133. [Kra20] Henning Krause. Completing perfect complexes.Math. Z.296.3-4 (2020). With appendices by Tobias Barthel and Bernhard Keller, pp. 1387–1427. [Nee96] Amnon Neeman. The Grothendieck dual...

2000