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arxiv: 2605.19983 · v1 · pith:VXFHNQBCnew · submitted 2026-05-19 · 🧮 math.CT · math.AG· math.AT· math.RT

Tensor triangular geometry -- Notes for an Oberwolfach Seminar

Henning Krause This is my paper

Pith reviewed 2026-05-20 03:29 UTC · model grok-4.3

classification 🧮 math.CT math.AGmath.ATmath.RT
keywords tensor triangular geometrytriangulated categoriessupporttensor idealsmodular representation theoryfinite groupsthick idealslocalising ideals
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The pith

Notions of support classify thick and localising tensor ideals in categories from modular representation theory of finite groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The notes develop support functions for objects in tensor triangulated categories. These functions are then used to classify the thick tensor ideals and the localising tensor ideals that arise when the categories come from the modular representations of finite groups. A reader would care because the classification turns algebraic ideal structure into geometric data on a spectrum, giving a uniform way to understand invariants in representation theory.

Core claim

By equipping triangulated categories with suitable support maps, the thick tensor ideals correspond to certain subsets of the spectrum and the localising tensor ideals are likewise determined by their supports, yielding a complete classification for the stable module categories that appear in the modular representation theory of finite groups.

What carries the argument

Support for an object in a tensor triangulated category, which assigns a closed subset of the spectrum that detects membership in tensor ideals.

If this is right

  • Thick tensor ideals in these categories are in bijection with closed subsets defined by supports.
  • Localising tensor ideals are classified by the same support data.
  • The classification applies directly to stable module categories over finite group algebras.
  • Support detects vanishing of objects and ideal membership uniformly across examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same support machinery could be tested on categories from algebraic geometry or homotopy theory to see if analogous classifications hold.
  • Explicit computations for small groups such as cyclic or elementary abelian p-groups would provide concrete checks of the support-to-ideal correspondence.
  • Links to existing classifications in derived categories of schemes might emerge once the support spectra are compared.

Load-bearing premise

The triangulated categories arising from modular representations of finite groups possess well-defined and sufficiently rich support functions that permit a complete classification of their tensor ideals.

What would settle it

A concrete thick tensor ideal in the stable module category of a finite group whose support does not match any closed subset predicted by the classification would disprove the claim.

read the original abstract

These are the notes from lectures I gave at the Oberwolfach Seminar "Tensor Triangular Geometry and Interactions" which was held in October 2025. The aim of these notes is twofold: We develop notions of support for triangulated categories, and we apply them to classify thick and localising tensor ideals of categories that arise in modular representation theory of finite groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. These are lecture notes from an Oberwolfach Seminar on Tensor Triangular Geometry and Interactions. The notes develop notions of support for triangulated categories from standard axioms and apply them to classify thick and localising tensor ideals in categories arising in the modular representation theory of finite groups.

Significance. As an expository summary of established results, the notes offer a structured resource that connects abstract tensor triangular geometry with concrete classification problems in representation theory. The grounding in prior literature and focus on support-based classifications of tensor ideals provide clear pedagogical value for the field.

minor comments (2)
  1. [Introduction] The introduction could more explicitly distinguish which classification theorems are direct citations of prior work versus any re-presentations or simplifications introduced for the seminar audience.
  2. [Support notions] In the sections developing support notions, a short table or diagram comparing the Balmer spectrum construction to related support theories (e.g., from algebraic geometry) would improve clarity for readers crossing fields.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the lecture notes and for recommending minor revision. We appreciate the recognition of the notes' value as an expository resource connecting tensor triangular geometry with classification results in modular representation theory.

Circularity Check

0 steps flagged

Expository notes develop standard support notions from axioms with no circular reduction

full rationale

The paper consists of lecture notes that introduce support for triangulated categories directly from standard axioms and apply the resulting classifications to external examples drawn from modular representation theory of finite groups. All derivations rest on established category-theoretic constructions (such as Balmer spectra and related support theories) rather than any self-definitional loop, fitted input renamed as prediction, or load-bearing self-citation chain. The weakest assumption—that the relevant categories admit sufficiently rich support—is justified by independent prior literature, rendering the derivation chain self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The notes rest on standard axioms of triangulated categories and tensor structures in representation theory; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Triangulated categories admit notions of support that can be developed and used for classification.
    Invoked as the starting point for the development described in the abstract.

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Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar, and C. Miller,Homology of perfect complexes, Adv. in Math.223(2010), 1731–1781

    work page 2010

  2. [2]

    Balmer,The spectrum of prime ideals in tensor triangulated categories, J

    P. Balmer,The spectrum of prime ideals in tensor triangulated categories, J. Reine & Angew. Math.588(2005), 149–168

    work page 2005

  3. [3]

    ,Spectra, spectra, spectra—tensor triangular spectra versus Zariski spectra of endo- morphism rings, Algebr. Geom. Topol.10(2010), no. 3, 1521–1563

    work page 2010

  4. [4]

    Balmer and G

    P. Balmer and G. Favi,Generalized tensor idempotents and the telescope conjecture, Proc. London Math. Soc. (3)102(2011), no. 6, 1161–1185

    work page 2011

  5. [5]

    D. J. Benson, J. F. Carlson, and J. Rickard,Thick subcategories of the stable module category, Fundamenta Mathematicae153(1997), 59–80

    work page 1997

  6. [6]

    D. J. Benson, S. B. Iyengar, and H. Krause,Local cohomology and support for triangulated categories, Ann. Scient. ´Ec. Norm. Sup. (4)41(2008), 575–621

    work page 2008

  7. [7]

    Topology4(2011), 641–666

    ,Stratifying triangulated categories, J. Topology4(2011), 641–666

    work page 2011

  8. [8]

    of Math.174(2011), 1643–1684

    ,Stratifying modular representations of finite groups, Ann. of Math.174(2011), 1643–1684

    work page 2011

  9. [9]

    Reine & Angew

    ,Colocalising subcategories and cosupport, J. Reine & Angew. Math.673(2012), 161–207

    work page 2012

  10. [10]

    ,A local–global principle for small triangulated categories, Math. Proc. Camb. Phil. Soc.158(2015), no. 3, 451–476

    work page 2015

  11. [11]

    J. F. Carlson,Cohomology and induction from elementary abelian subgroups, Quarterly J. Math. (Oxford)51(2000), 169–181

    work page 2000

  12. [12]

    J. F. Carlson and S. B. Iyengar,Thick subcategories of the bounded derived category of a finite group, Trans. Amer. Math. Soc.367(2015), no. 4, 2703–2717

    work page 2015

  13. [13]

    ,Hopf algebra structures and tensor products for group algebras, New York Journal of Mathematics23(2017), 351–364

    work page 2017

  14. [14]

    Evens,Cohomology of groups, Oxford University Press, 1991

    L. Evens,Cohomology of groups, Oxford University Press, 1991

    work page 1991

  15. [15]

    Hochster,Prime ideal structure in commutative rings, Trans

    M. Hochster,Prime ideal structure in commutative rings, Trans. Amer. Math. Soc.142 (1969), 43–60

    work page 1969

  16. [16]

    M. J. Hopkins,Global methods in homotopy theory, Homotopy Theory, Durham 1985 (E. Rees and J. D. S. Jones, eds.), London Math. Soc. Lecture Note Series, vol. 117, Cambridge University Press, 1987

    work page 1985

  17. [17]

    Hovey, J

    M. Hovey, J. H. Palmieri, and N. P. Strickland,Axiomatic stable homotopy theory, vol. 128, Mem. Amer. Math. Soc., no. 610, American Math. Society, 1997

    work page 1997

  18. [18]

    Krause,Homological theory of representations, Cambridge Studies in Advanced Mathe- matics, Cambridge University Press, 2022

    H. Krause,Homological theory of representations, Cambridge Studies in Advanced Mathe- matics, Cambridge University Press, 2022

    work page 2022

  19. [19]

    ,Central support for triangulated categories, Int. Math. Res. Not. (2023), no. 22, 19773–19800

    work page 2023

  20. [20]

    Neeman,The chromatic tower forD(R), Topology31(1992), 519–532

    A. Neeman,The chromatic tower forD(R), Topology31(1992), 519–532

    work page 1992

  21. [21]

    D. G. Quillen,The spectrum of an equivariant cohomology ring, I, Ann. of Math.94(1971), 549–572

    work page 1971

  22. [22]

    of Math.94(1971), 573– 602

    ,The spectrum of an equivariant cohomology ring, II, Ann. of Math.94(1971), 573– 602

    work page 1971

  23. [23]

    Serre,Sur la dimension cohomologique des groupes profinis, Topology3(1965), 413–420

    J.-P. Serre,Sur la dimension cohomologique des groupes profinis, Topology3(1965), 413–420

    work page 1965

  24. [24]

    Stevenson,Some notes on tensor triangular geometry, arXiv:2602.08480

    G. Stevenson,Some notes on tensor triangular geometry, arXiv:2602.08480

    work page arXiv

  25. [25]

    R. W. Thomason,The classification of triangulated subcategories, Compositio Math.105 (1997), 1–27

    work page 1997

  26. [26]

    tom Dieck,Algebraic topology, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2008

    T. tom Dieck,Algebraic topology, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2008. F akult¨at f ¨ur Mathematik, Universit ¨at Bielefeld, D-33501 Bielefeld, Germany Email address:hkrause@math.uni-bielefeld.de

    work page 2008