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arxiv: 2606.27063 · v1 · pith:WJ7OTJAGnew · submitted 2026-06-25 · 🧮 math.RT

Brick infinite algebras admit infinitely many non-τ-rigid bricks

Kaveh Mousavand , Charles Paquette This is my paper

Pith reviewed 2026-06-26 02:30 UTC · model grok-4.3

classification 🧮 math.RT
keywords bricksτ-rigid modulesbrick-finite algebrasrepresentation theory of algebrasbrick-Brauer-Thrall conjectures
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The pith

Any brick-infinite algebra admits infinitely many bricks that are not τ-rigid.

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

The paper proves that for a finite-dimensional algebra over an algebraically closed field, the condition that all but finitely many bricks are τ-rigid forces the algebra to have only finitely many bricks altogether. The argument proceeds by establishing the contrapositive statement that brick-infinite algebras must contain infinitely many non-τ-rigid bricks. This settles, without any tameness hypothesis, a question previously resolved only for E-tame algebras and supplies a weaker form of an open conjecture that replaces τ-rigidity by rigidity.

Core claim

If all but finitely many bricks of the algebra A are τ-rigid, then A is brick-finite. Equivalently, any brick-infinite algebra admits infinitely many bricks which are not τ-rigid.

What carries the argument

The direct implication, proved in full generality, that an algebra with only finitely many non-τ-rigid bricks must be brick-finite.

If this is right

  • Any algebra possessing infinitely many bricks must possess infinitely many that are not τ-rigid.
  • The result confirms a weaker form of the conjecture that almost all bricks being rigid implies the algebra is brick-finite.
  • The statement holds for arbitrary finite-dimensional algebras and removes the E-tameness restriction used in earlier work.

Where Pith is reading between the lines

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

  • τ-rigidity functions as a coarse but effective filter separating algebras with finite brick sets from those with infinite ones.
  • Analogous statements might be examined for other module properties such as rigidity or finite projective dimension.

Load-bearing premise

A is a finite-dimensional algebra over an algebraically closed field.

What would settle it

An explicit brick-infinite algebra in which all but finitely many bricks are τ-rigid would falsify the claim.

read the original abstract

Let $A$ be a finite dimensional algebra over an algebraically closed field. Motivated by some foundational interactions between bricks and $\tau$-rigid modules, we prove, in full generality, that if all but finitely many bricks of the algebra $A$ are $\tau$-rigid, then $A$ is brick-finite. Equivalently, any brick-infinite algebra admits infinitely many bricks which are not $\tau$-rigid. Because $\tau$-rigidity implies rigidity, our result verifies a weaker version of an open conjecture which states that if (almost) all bricks over $A$ are rigid, then $A$ should be brick-finite. In retrospect, this work strengthens some previous results and contributes to the recent studies of a series of challenging problems, all tied to the $2$nd brick-Brauer-Thrall conjecture. More specifically, without any tameness assumption, we settle a question that was previously known only for $E$-tame algebras.

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. The manuscript proves that for any finite-dimensional algebra A over an algebraically closed field, if all but finitely many bricks are τ-rigid then A is brick-finite. Equivalently, every brick-infinite algebra admits infinitely many non-τ-rigid bricks. The result is obtained in full generality without tameness hypotheses and is presented as verifying a weaker form of the open 2nd brick-Brauer-Thrall conjecture on rigid bricks; it settles a question previously known only for E-tame algebras.

Significance. If the derivation is correct, the result is significant: it removes the E-tameness restriction from a previously known case and supplies a general statement linking brick-infiniteness directly to the existence of infinitely many non-τ-rigid bricks. The work strengthens earlier results on bricks and τ-rigid modules and advances the study of the 2nd brick-Brauer-Thrall conjecture in representation theory of algebras.

minor comments (2)
  1. [Abstract] Abstract, final paragraph: the phrase 'settle a question that was previously known only for E-tame algebras' would benefit from an explicit citation to the E-tame result being extended.
  2. [Introduction] The notation for τ-rigidity and brick-finiteness is used throughout without a dedicated preliminary subsection; a short paragraph collecting the relevant definitions and standard facts would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. We are grateful for the recommendation to accept and for recognizing the result as a general strengthening of prior work on the 2nd brick-Brauer-Thrall conjecture without tameness assumptions.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states a direct theorem in full generality: for any finite-dimensional algebra A over an algebraically closed field, if all but finitely many bricks are τ-rigid then A is brick-finite (equivalently, brick-infinite algebras have infinitely many non-τ-rigid bricks). This is presented as an independent proof that weakens the 2nd brick-Brauer-Thrall conjecture without tameness assumptions and without reducing to fitted parameters, self-definitions, or load-bearing self-citations. No equations or steps in the provided abstract or description exhibit a reduction by construction to the inputs; prior results are strengthened rather than presupposed as the sole justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the result rests on standard background results in the representation theory of finite-dimensional algebras over algebraically closed fields; no free parameters, invented entities, or ad-hoc axioms are visible.

axioms (1)
  • standard math Finite-dimensional algebras over algebraically closed fields admit Auslander-Reiten theory and the notions of bricks and τ-rigidity as standard.
    Invoked implicitly as the setting for the theorem.

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discussion (0)

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Reference graph

Works this paper leans on

22 extracted references · 4 canonical work pages · 2 internal anchors

  1. [1]

    Adachi, O

    T. Adachi, O. Iyama, I. Reiten, -tilting theory, Compos. Math. 150 (2014), no. 3, 415--452

    2014

  2. [2]

    Bicompact torsion classes and conjectures on brick infinite algebras

    S. Asai, Bicompact torsion classes and conjectures on brick infinite algebras, arXiv:2604.04505

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    Asai, The wall-chamber structures of the real Grothendieck groups, Adv

    S. Asai, The wall-chamber structures of the real Grothendieck groups, Adv. Math. 381 (2021), Paper No. 107615

    2021

  4. [4]

    Assem, D

    I. Assem, D. Simson, A. Skowro\'nski, Elements of the representation theory of associative algebras, Vol. 1, Cambridge University Press, Cambridge, 2006

    2006

  5. [5]

    Auslander, S

    M. Auslander, S. O. Smalø, Almost split sequences in subcategories, J. Algebra 69 (1981) 426–454. Addendum; J. Algebra 71 (1981), 592–594

    1981

  6. [6]

    Demonet, Combinatorics of mutations in representation theory, Habilitation (2017), Available on Laurent Demonet’s website

    L. Demonet, Combinatorics of mutations in representation theory, Habilitation (2017), Available on Laurent Demonet’s website

    2017

  7. [7]

    Demonet, O

    L. Demonet, O. Iyama, G. Jasso, -tilting finite algebras, bricks, and g -vectors, Int. Math. Res. Not. (2019), no. 3, 852--892

    2019

  8. [8]

    Dr\"axler, Auslander-Reiten quivers of algebras whose indecomposable modules are bricks, Bull

    P. Dr\"axler, Auslander-Reiten quivers of algebras whose indecomposable modules are bricks, Bull. Lond. Math. Soc. 23 (1991), 141--145

    1991

  9. [9]

    King, Moduli of representations of finite dimensional algebras, Quart

    A. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530

    1994

  10. [10]

    Mizuno, Classifying -tilting modules over preprojective algebras of Dynkin type, Math

    Y. Mizuno, Classifying -tilting modules over preprojective algebras of Dynkin type, Math. Z. 277 (2014), no. 3-4, 665–690

    2014

  11. [11]

    Mousavand, -tilting finiteness of non-distributive algebras and their module varieties, Journal of Algebra 608 (2022), 673--690

    K. Mousavand, -tilting finiteness of non-distributive algebras and their module varieties, Journal of Algebra 608 (2022), 673--690

    2022

  12. [12]

    Mousavand, C

    K. Mousavand, C. Paquette, Geometric interactions between bricks and -rigidity, arXiv:2311.14863 (2023)

    work page arXiv 2023

  13. [13]

    Mousavand, C

    K. Mousavand, C. Paquette, Hom-orthogonal modules and brick-Brauer-Thrall conjectures, J. Algebra 686 (2026), 650--676

    2026

  14. [14]

    Mousavand, C

    K. Mousavand, C. Paquette, Minimal ( -)tilting infinite algebras, Nagoya Math. J. 249 (2023), 221--238

    2023

  15. [15]

    Mousavand, C

    K. Mousavand, C. Paquette, On the bricks (Schur representations) of finite dimensional algebras, arXiv:2508.11789 (2025)

    work page arXiv 2025

  16. [16]

    Marks, J

    F. Marks, J. S t ov\'i c ek, Torsion classes, wide subcategories and localisations, Bull. Lond. Math. Soc. 49 (2017), no. 3, 405--416

    2017

  17. [17]

    Pfeifer, Remarks on -tilted versions of the second Brauer-Thrall Conjecture, Bulletin of the London Mathematical Society, Vol

    C. Pfeifer, Remarks on -tilted versions of the second Brauer-Thrall Conjecture, Bulletin of the London Mathematical Society, Vol. 57, Issue 5 (2025), 1568-1583

    2025

  18. [18]

    C. M. Ringel, Bricks in hereditary length categories, Result. Math., 6(1983), 64--70

    1983

  19. [19]

    Infinite $\tau$-tilting theory

    K. Schlegel, Infinite -tilting theory, arXiv:2606.23297 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  20. [20]

    Simson, A

    D. Simson, A. Skowro\'nski, Elements of the representation theory of associative algebras, Vol. 3, Cambridge University Press, Cambridge, 2007

    2007

  21. [21]

    Schroll, H

    S. Schroll, H. Treffinger, Y. Valdivieso, On band modules and -tilting finiteness, Mathematische Zeitschrift 299 (2021), 4567--4575

    2021

  22. [22]

    Wang, -tilting finiteness of two-point algebras I, Math

    Q. Wang, -tilting finiteness of two-point algebras I, Math. J. of Okayama Univ. (2022), 117–141

    2022