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arxiv: 2604.04505 · v1 · submitted 2026-04-06 · 🧮 math.RT

Bicompact torsion classes and conjectures on brick infinite algebras

Sota Asai This is my paper

Pith reviewed 2026-05-10 20:09 UTC · model grok-4.3

classification 🧮 math.RT
keywords torsion classesbicompact torsion classesfunctorially finitehereditary algebrasbrick infinite algebrasDemonet conjectureEnomoto conjecture
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The pith

Bicompact torsion classes are conjectured to coincide exactly with functorially finite torsion classes over finite-dimensional algebras.

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

The paper defines a torsion class in mod A as bicompact when it is the smallest torsion class containing a single module M and the dual condition holds in the opposite category. It conjectures that these bicompact classes are precisely the functorially finite ones. The claim is established for hereditary algebras by using their homological properties and separately for semistable torsion classes. This yields the meta-result that the Demonet conjecture on brick infiniteness implies the Enomoto conjecture.

Core claim

Bicompact torsion classes, those generated by a single module both directly and dually, are conjectured to be exactly the functorially finite torsion classes. The equality holds when A is hereditary or when the torsion class is semistable, which in turn shows that Demonet’s conjecture implies Enomoto’s conjecture on algebras with infinitely many bricks.

What carries the argument

Bicompact torsion class: a torsion class that is the smallest one containing some module M and satisfies the dual generation condition.

If this is right

  • For every hereditary algebra, bicompact torsion classes are functorially finite.
  • Semistable torsion classes that are bicompact are functorially finite.
  • Demonet’s conjecture on brick-infinite algebras implies Enomoto’s conjecture.
  • Compactness of a torsion class implies functorial finiteness in the settings where the equality is proven.

Where Pith is reading between the lines

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

  • The conjecture, if true in full generality, would give a single-module test for functorial finiteness of torsion classes.
  • The link between the two brick-infinite conjectures may extend to other classes of algebras closed under derived equivalence.
  • Counterexamples, if they exist, are likely to appear first among non-hereditary algebras with complicated torsion lattices.

Load-bearing premise

The closure properties and Hom-vanishing conditions that make the proofs work for hereditary algebras and semistable torsion classes extend to arbitrary finite-dimensional algebras.

What would settle it

An explicit finite-dimensional algebra together with a torsion class that is bicompact but not functorially finite, or functorially finite but not bicompact.

read the original abstract

A torsion class $\mathcal{T}$ of the module category $\operatorname{\mathsf{mod}} A$ of a finite dimensional algebra $A$ over a field $K$ is said to be compact if there exists a module $M \in \operatorname{\mathsf{mod}} A$ such that $\mathcal{T}$ is the smallest torsion class containing $M$. If a torsion class satisfies this and the dual condition, then we call it a bicompact torsion class. We conjecture that bicompact torsion classes are precisely functorially finite torsion classes, and prove it for hereditary algebras and also for semistable torsion classes. This gives that Demonet Conjecture implies Enomoto Conjecture, both of which are important conjectures on brick infiniteness.

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

1 major / 2 minor

Summary. The manuscript defines a torsion class T in mod A (A finite-dimensional over a field) to be compact if it is the smallest torsion class containing some module M, and bicompact if it is both compact and dually compact. It conjectures that bicompact torsion classes coincide exactly with functorially finite torsion classes, proves the equivalence when A is hereditary (using the absence of relations and direct-sum decompositions) and when T is semistable (using Hom-vanishing between stable and unstable summands), and deduces that this identification would imply the Demonet conjecture entails the Enomoto conjecture on brick-infinite algebras.

Significance. If the conjecture holds in generality, it would relate two central open questions on brick infiniteness by identifying compactness properties of torsion classes with the existence of left and right approximations. The verified cases supply rigorous, self-contained arguments that exploit standard closure properties of hereditary categories and Hom-vanishing for semistable classes; these constitute concrete progress and may guide attempts to remove the restrictions.

major comments (1)
  1. [section deriving the implication between Demonet and Enomoto conjectures] In the section deriving the implication between the Demonet and Enomoto conjectures, the claim that the bicompact-functorially-finite equivalence yields Demonet implies Enomoto is not fully justified for arbitrary finite-dimensional algebras, because the equivalence is established only for hereditary algebras and semistable torsion classes; the manuscript should clarify whether the two conjectures are understood to apply only in those restricted settings or whether an additional reduction argument covers the general case.
minor comments (2)
  1. [Abstract] The abstract would benefit from a one-sentence reminder of the statements of the Demonet and Enomoto conjectures to make the implication immediately intelligible to readers outside the immediate subfield.
  2. [Introduction] Notation for the smallest torsion class generated by a module (e.g., the symbol used for the closure operation) should be introduced explicitly at the first occurrence rather than relying on context.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address the major comment below and have revised the text for clarity.

read point-by-point responses

  1. Referee: [section deriving the implication between Demonet and Enomoto conjectures] In the section deriving the implication between the Demonet and Enomoto conjectures, the claim that the bicompact-functorially-finite equivalence yields Demonet implies Enomoto is not fully justified for arbitrary finite-dimensional algebras, because the equivalence is established only for hereditary algebras and semistable torsion classes; the manuscript should clarify whether the two conjectures are understood to apply only in those restricted settings or whether an additional reduction argument covers the general case.

    Authors: We thank the referee for this observation. The equivalence between bicompact and functorially finite torsion classes is conjectural in general and is proven only for hereditary algebras and semistable torsion classes. The section derives the logical implication Demonet implies Enomoto under the assumption that the bicompact-functorially finite identification holds for arbitrary finite-dimensional algebras. The Demonet and Enomoto conjectures are general statements about brick-infinite algebras; our argument shows that the general conjecture would entail the desired implication between them. The special-case proofs provide supporting evidence for the conjecture but are not used to cover the general derivation. We have revised the manuscript to state this distinction explicitly and to clarify that no reduction argument to the special cases is claimed for the implication itself. revision: yes

Circularity Check

0 steps flagged

No circularity: conjecture with direct proofs in restricted cases using standard torsion class properties

full rationale

The paper states a conjecture that bicompact torsion classes equal functorially finite torsion classes in mod A, proves the equivalence only for hereditary algebras (via single-module generation and absence of relations) and semistable torsion classes (via Hom-vanishing), and deduces a conditional implication between Demonet and Enomoto conjectures. All steps are direct verifications relying on closure under extensions, quotients, and Hom properties that are standard and external to the paper. No equations reduce by construction, no parameters are fitted then renamed as predictions, and no load-bearing self-citations or imported uniqueness theorems appear. The general case is left open, so the derivation chain does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of torsion classes in mod A and on the known properties of hereditary algebras and semistable torsion classes; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math A torsion class is closed under extensions and quotients.
    Standard definition invoked throughout the abstract.
  • domain assumption Hereditary algebras and semistable torsion classes satisfy additional Hom-vanishing and closure properties used in the proofs.
    These are the restricted settings in which the equality is proved.

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