Second Brick Brauer-Thrall Conjecture

• The Second Brick Brauer-Thrall Conjecture is a set of statements in algebra predicting that an infinite number of non-isomorphic bricks (indecomposable modules with scalar endomorphisms) exist in representation-infinite settings.
• It extends the classical Brauer-Thrall Theorem by applying to contexts like totally reflexive, maximal Cohen–Macaulay, and derived module categories through explicit module constructions and homological methods.
• Recent research links brick-finiteness to τ-tilting theory and geometric orbit properties, offering concrete criteria and stability conditions across various classes of algebras.

The Second Brick Brauer–Thrall Conjecture refers to a cluster of interrelated statements in the representation theory of algebras and their module categories, all of which concern the abundance and distribution of indecomposable modules, specifically bricks (Schur representations), of fixed size. These conjectures generalize the classical Second Brauer–Thrall Theorem, extending its reach to settings such as totally reflexive modules, maximal Cohen–Macaulay modules, derived categories, and the theory of bricks and τ-tilting. This article presents precise formulations, methods, and major results associated with the Second Brick Brauer–Thrall phenomenon, tracing its influence through commutative and noncommutative algebra, as well as modern representation-theoretic geometry.

1. Classical Formulation, Extensions, and Definitions

The classical Second Brauer–Thrall Conjecture asserts: if a finite-dimensional algebra $A$ over an infinite field has infinite representation type, then there exists an infinite sequence of positive integers $n_1 < n_2 < \cdots$ such that for each $i$, there are infinitely many non-isomorphic indecomposable $A$-modules of dimension $n_i$.

A brick is an indecomposable module $X$ such that $\mathrm{End}_A(X) \cong k$, i.e., $X$ has only scalar endomorphisms; bricks correspond to Schur representations. The Second Brick Brauer–Thrall Conjecture (2nd bBT) posits: if there are infinitely many pairwise non-isomorphic bricks in $\mathrm{mod}\,A$, then for some positive integer $d$ there are infinitely many bricks of dimension $d$.

Major extensions of the conjecture appear in:

• Totally reflexive modules over non-Gorenstein local rings with $m^3=0$ (Christensen et al., 2010, Celikbas et al., 2012)
• Maximal Cohen–Macaulay (MCM) modules over Cohen–Macaulay local rings (Leuschke et al., 2012)
• Representation-finite and minimal representation-infinite algebras via ray-categories (Bongartz, 2013, Bongartz, 2016)
• Strongly unbounded type in abelian length categories and brick-finiteness criteria (Krause, 2013, Mousavand et al., 30 Jul 2024)
• Derived module categories and the dichotomy between derived discrete and strongly derived unbounded algebras (Zhang et al., 2013)
• Bricks and $\tau$-tilting theory, especially in special biserial and minimal $\tau$-tilting infinite algebras (Schroll et al., 2019, Mousavand et al., 2021, Pfeifer, 2023, Mousavand et al., 15 Aug 2025)
• Constructible subcategories with connections to the Ziegler spectrum (Schlegel et al., 29 Jul 2025)

2. Criteria, Equivalences, and Key Theorems

2.1 Structure and Equivalences

The existence of infinite families of bricks in a fixed dimension is frequently characterized via Hom-orthogonality, semibricks, and geometric properties of orbits in representation varieties. For a finite-dimensional algebra $A$ over an algebraically closed field and fixed $d > 0$, the following are equivalent (Mousavand et al., 30 Jul 2024):

• There are infinitely many non-isomorphic bricks of dimension $d$.
• There exists an infinite Hom-orthogonal subset (semibrick) in the set $\operatorname{brick}(A,d)$.

In the presence of a generalized standard component in the Auslander–Reiten quiver, brick-finiteness, representation-finiteness, and orbit properties are shown to be equivalent (Mousavand et al., 30 Jul 2024):

Property	Equivalent Condition	Reference
A is representation-finite	Brick-finite: $\forall d$, $\operatorname{brick}(A,d)$ finite	(Mousavand et al., 30 Jul 2024)
All $\operatorname{GL}(d)$-orbits dense	For each $d$, all orbits in every irreducible component dense	(Mousavand et al., 30 Jul 2024)
Finitely many $\theta$-stable bricks	For each $d$, only finitely many $\theta$-stable bricks for all $\theta$	(Mousavand et al., 30 Jul 2024)

In the context of tame and minimal brick-infinite algebras (Mousavand et al., 15 Aug 2025), the following are equivalent:

• $A$ admits infinitely many bricks of fixed $k$-dimension $d$.
• $A$ admits an infinite set of bricks of projective dimension at most one.
• There exists an infinite set of $\tau$-shifts of bricks contained in the injective-orthogonal subcategory $\mathcal{F}_I$.

2.2 Module Constructions and Parameter Families

For totally reflexive modules, explicit families $M_n(w,x,y,z)$ of indecomposables with periodic (period $\leq 2$) minimal free resolutions and constant Betti numbers are constructed for all $n > 0$ (Christensen et al., 2010). Under the presence of an exact zero divisor and infinite residue field, one obtains:

• For each $n$, infinitely many pairwise non-isomorphic modules, all minimally generated by $n$ elements.
• When the residue field is infinite, these families are parametrized by elements of the field or its units.

For artin algebras and finite-dimensional $A$ of infinite representation type, representation embedding theory (Bongartz, 2016) yields:

• Explicit functors $H(X) \cong M \otimes_{k[T]} X$ embedding the Kronecker or polynomial ring module category into $\mathrm{mod}\,A$, guaranteeing parameter families (often 1-dimensional) of indecomposables of fixed dimension.

For special biserial algebras, the existence of a band module $M(b,\lambda,1)$ that is a brick for some $\lambda \in k$ leads to an infinite family of such bricks parameterized by $k$ (Schroll et al., 2019).

3. Geometric and Homological Approaches

3.1 Orbit Structure and Stability

Bricks with open orbits in representation varieties correspond to the "discrete" locus; brick-infinite algebras are characterized by the failure of brick-discreteness (Mousavand et al., 15 Aug 2025). Minimal extending bricks have open orbits, and the existence of infinitely many bricks of unbounded dimensions forces unboundedness in the chain of minimal extending bricks, providing a bridge to τ-tilting theory (Mousavand et al., 2021).

Stability conditions, using weights $\theta \in K_0^{\mathrm{proj}}(A)$, provide geometric methods to construct infinite families of stable bricks. For an appropriate $\theta$, the condition

$$\theta([X]) = \dim_k \operatorname{Hom}_A(P_0, X) - \dim_k \operatorname{Hom}_A(P_1, X)$$

along with $\theta([X]) = 0$ and $\theta([Y]) < 0$ for every proper submodule $Y \subset X$, guarantees $X$ is $\theta$-stable (and thus a brick). In the presence of generalized standard AR components, this yields infinite families of θ-stable bricks of fixed dimension (Mousavand et al., 30 Jul 2024).

3.2 Ziegler Spectrum and Constructibility

For constructible subcategories of $\mathrm{Mod}\,A$, the Ziegler spectrum provides a topological tool to relate finiteness properties to geometric accumulation points. The key result is that, under suitable assumptions (e.g., $k$ perfect, Krull–Gabriel dimension defined for $A \otimes_k \bar{k}$), infinite type for a constructible subcategory implies strong unboundedness: infinitely many indecomposables for infinitely many dimensions (Schlegel et al., 29 Jul 2025). Without these conditions, this implication can fail.

4. Derived, Categorical, and τ-Tilting Extensions

4.1 Derived Categories

In the bounded derived category of a finite-dimensional algebra, the second Brauer–Thrall type theorem states that the category is either derived discrete (having only finitely many indecomposables for each cohomological range) or strongly derived unbounded (having infinitely many indecomposables for infinitely many cohomological ranges) (Zhang et al., 2013). Numerical invariants—cohomological length, width, and range—enable this dichotomy, making the derived setting structurally parallel to the brick-conjecture context.

4.2 τ-Tilting, Bricks, and Geometric Reformulations

Brick-finiteness is deeply linked to τ-tilting finiteness: for special biserial algebras, τ-tilting finiteness is equivalent to non-existence of any band module that is a brick (Schroll et al., 2019). Stable and τ-reduced versions of the second Brauer–Thrall Conjecture employ moduli spaces of stable modules and generically τ-reduced components, respectively, linking the conjecture to the geometric theory of $g$-vectors and τ-tilting fans (Pfeifer, 2023). For $E$-tame algebras, these geometric criteria can be reversed: τ-reduced families give rise to families of stable modules of positive dimension in their moduli space.

Minimal τ-tilting infinite algebras serve as testing grounds: showing the conjecture for these minimal objects suffices to verify the general case (Mousavand et al., 2021).

5. Impact, Generalizations, and Open Problems

5.1 Brick–Finiteness: Algebraic and Geometric Criteria

Brick-finite algebras admit only finitely many isomorphism classes of bricks in each fixed dimension, with a sharp upper bound on the size of semibricks depending on the number of simple modules (Mousavand et al., 30 Jul 2024). Geometrically, in brick-finite algebras, the orbits of non-isomorphic modules must always "interact": for all $X \not\cong Y$, $\operatorname{Hom}_A(X, Y) \neq 0$ in any irreducible component; otherwise, one would obtain infinitely many bricks. This generalizes a classical characterization of local algebras.

5.2 Counterexamples, Failure, and Role of Additional Hypotheses

Variants of the conjecture fail without further assumptions. In the constructible subcategory setting (Schlegel et al., 29 Jul 2025), infinite type need not imply strong unboundedness unless conditions on the field (perfect, infinite) and Krull–Gabriel dimension are met. This demonstrates that the conjecture's scope is nuanced and tied to deeper homological or algebraic properties. Similar subtleties arise in the derived setting and in the failure of the strong semibrick conjecture in pathological cases (Mousavand et al., 30 Jul 2024).

5.3 Tame and Wild Dichotomy

For tame algebras and minimal brick-infinite tame algebras, the 2nd bBT conjecture has been verified to hold: the presence of bricks of projective dimension at most one already brings forth the required infinite family of bricks in a fixed dimension, and connections to homological properties facilitate their detection (Mousavand et al., 15 Aug 2025). A plausible implication is that, for broader classes of "tame" or "brick-tame" algebras, the 2nd bBT conjecture may likewise hold true, reducing the focus to their more tractable structure.

6. Summary and Current Status

The Second Brick Brauer–Thrall Conjecture asserts a deep link between representation-infinite behavior and the existence of infinite families of bricks in fixed dimension, often governed by underlying algebraic, geometric, or homological properties. Explicit constructions, geometric stratifications, representation embeddings, and links to invariants such as τ-tilting fans, $g$-vectors, or Ziegler spectrum topology all serve as both methods of verification and structural indicators. Recent works have verified the conjecture in significant new classes—algebras with generalized standard components, tame minimal brick-infinite algebras, and in the context of constructible subcategories under additional hypotheses (Mousavand et al., 30 Jul 2024, Mousavand et al., 15 Aug 2025, Schlegel et al., 29 Jul 2025).

Current research focuses on further geometric reformulations (stability and τ-reduced families), explicit equivalences via Hom-orthogonality, and reduction to minimal classes (e.g., minimal τ-tilting infinite algebras), with ongoing work aiming to connect brick-infinite behavior to torsion theory, open orbits, and homological invariants in a unified framework. Several open questions remain regarding the universality of the conjecture in arbitrary algebras and its precise homological and geometric triggers, particularly in wild or non-tame contexts.