Brick-Finite Algebras

• Brick-finite algebras are finite-dimensional algebras that admit only finitely many isomorphism classes of bricks, where each brick has a division (scalar) endomorphism ring.
• They are characterized by their equivalence to τ-tilting finiteness and the finiteness of torsion classes and wide subcategories, simplifying module classification.
• Their study bridges algebraic, geometric, and combinatorial methods to provide actionable insights into representation theory and classification problems.

A brick-finite algebra is a finite-dimensional algebra over a field admitting only finitely many isomorphism classes of bricks, where a brick is a module whose endomorphism ring is a division algebra. The structure and characterization of brick-finite algebras interconnects deep aspects of representation theory, relative homological algebra, the theory of torsion classes, and invariant theory, with extensions to geometric and combinatorial frameworks. Brick-finite algebras provide a unifying language—generalizing the theory of semisimple algebras and classical representation-finite algebras—and are decisive in τ-tilting theory and related classification problems.

1. Definition and Fundamental Properties

Let $A$ be a finite-dimensional $k$-algebra over an algebraically closed field $k$. An $A$-module $M$ is a brick if $\End_A(M)$ is a division algebra; over $k$ this forces $\End_A(M)\cong k$, i.e., every endomorphism is scalar. The brick-finiteness property is then: $A\ \text{ is brick-finite} \iff |\brick(A)| < \infty,$ where $\brick(A)$ denotes the set of isomorphism classes of bricks in $k$0 (Mousavand et al., 15 Aug 2025).

Bricks generalize simples (since simples are always bricks by Schur’s Lemma) and play a key role in spectral and categorical decompositions. The brick-finite property is closely tied to global finiteness conditions: in particular, a finite-dimensional $k$1 is brick-finite if and only if it is τ-tilting finite and, in representation-finite cases, every indecomposable is a brick (Sentieri, 2020, Mousavand et al., 15 Aug 2025).

2. Equivalence to τ-Tilting Finiteness and Torsion Theory

The central result is the equivalence between brick-finiteness and τ-tilting finiteness, as formalized in the brick–τ-rigid correspondence (Sentieri, 2020, Mousavand, 2019, Nasr-Isfahani, 15 Nov 2025): $k$2 A finite-dimensional algebra is τ-tilting finite if it admits only finitely many basic support τ-tilting modules up to isomorphism. Equivalently, the lattice of torsion classes $k$3 in mod $k$4 is finite; this is also equivalent to having only finitely many isoclasses of bricks.

The torsion-theoretic perspective is reinforced via the bijections:

• The set of basic support τ-tilting modules bijects with the set of functorially finite torsion classes.
• The minimal labelings of covers in $k$5 correspond bijectively to bricks.
• Every functorially finite torsion class is generated by a finite semibrick; thus, brick-finiteness coincides with the property that every wide subcategory of $k$6 is functorially finite (Nasr-Isfahani, 15 Nov 2025).

The table below summarizes some key equivalent criteria for brick-finiteness (Nasr-Isfahani, 15 Nov 2025):

Criterion	Invariant/Condition	Paper Reference
Brick-finite	$k$7	(Mousavand et al., 15 Aug 2025, Nasr-Isfahani, 15 Nov 2025)
τ-tilting finite	Finitely many support τ-tilting modules	(Sentieri, 2020, Mousavand, 2019)
Finite torsion lattice	$k$8 finite/combinatorially finite	(Sentieri, 2020)
All semibricks finite	$k$9 finite	(Nasr-Isfahani, 15 Nov 2025)
Functorial finiteness	Every wide subcat is functorially finite	(Nasr-Isfahani, 15 Nov 2025)

3. Geometric and Hom-Orthogonality Characterizations

Algebraic and geometric characterizations root brick-finiteness in both module-theoretic and orbit-theoretic frameworks (Mousavand et al., 2024):

• Hom-orthogonality: A brick-finite algebra of rank $k$0 admits no Hom-orthogonal set of size $k$1 among modules. A maximal such set (semibrick) is the simple modules. A Hom-orthogonal set is a set of pairwise non-isomorphic modules $k$2 such that $k$3 for $k$4.
• Orbit geometry: In each irreducible component $k$5, $k$6 (i.e., all orbits are dense), and for all $k$7, $k$8. Infinite families of same-dimensional bricks manifest as existence of Hom-orthogonal orbits, i.e., infinite curves of orthogonal modules in the variety (Mousavand et al., 2024).
• Auslander–Reiten theory: The presence of a generalized standard (e.g., tubular or preprojective) component in the AR quiver forces brick-infinity except in representation-finite cases.

This dual algebraic-geometric perspective allows for the detection of brick-infinity and provides algorithmic and conceptual control over classification, especially in tame and biserial settings (Mousavand et al., 2022, Sengupta et al., 2024).

4. Semibricks, Wide Subcategories, and Lattice Theory

A semibrick is a set of pairwise Hom-orthogonal bricks; the collection $k$9 is pivotal in understanding wide subcategories and lattice-theoretic invariants (Asai, 2016, Nasr-Isfahani, 15 Nov 2025). Key facts:

• Brick-finite $A$0 every semibrick is finite, and $A$1 itself is finite.
• Under Ringel–Asai bijections, semibricks correspond to functorially finite wide subcategories, and via τ-tilting/silting theory to combinatorial structures such as $A$2-tilting cones and the $A$3-vector/c-vector fans.
• The lattice of torsion classes $A$4 is finite, semidistributive, and has trim (in the sense of extremal and left-modular) structure precisely when $A$5 is brick-directed (i.e., contains no cycle of nonzero morphisms among bricks) (Asai et al., 16 Jun 2025).

These structural results are extended by results affirming that in the brick-finite case, any chain of wide subcategories or any chain in the κ-order on torsion classes eventually stabilizes (Nasr-Isfahani, 15 Nov 2025).

5. Characterizations in Biserial and String Algebras; Generic Bricks

For special biserial and string algebras, brick-finiteness admits purely combinatorial/testable criteria:

• A biserial algebra is brick-finite iff it admits no band whose (one-parameter) family consists entirely of bricks, i.e., no generic brick (Mousavand et al., 2022).
• Minimal brick-infinite biserials are gentle algebras of generalized barbell or affine type (e.g., Kronecker); all other mild/special biserials are brick-finite (Mousavand, 2019).
• In string algebras with acyclic quivers, brick-infinity is characterized combinatorially via weakly perfectly clustering condition on associated words (zigzags and crowns), providing an explicit algorithm for detection (Sengupta et al., 2024).
• In tame settings, brick-infinity is equivalent to the existence of a generic brick $A$6 with $A$7; such $A$8 parametrizes a $A$9-family of bricks of constant dimension (Bautista et al., 2024).

6. Brauer–Thrall Phenomena and Role in Module Theory

Brick-finiteness encodes a "brick version" of Auslander’s theorem and the first Brauer–Thrall theorem: For any finite-dimensional $M$0, brick-infinity is equivalent to the existence of bricks of arbitrarily large length (Mousavand et al., 2021). The following equivalence holds: $M$1 This result demonstrates that brick-infinite algebras can never have bounded length of bricks—unboundedness of length is an exact obstruction.

The open conjectural "Second brick Brauer–Thrall" posits that if $M$2 is brick-infinite, then for some $M$3 there are infinitely many non-isomorphic bricks of dimension $M$4 (Mousavand et al., 15 Aug 2025).

7. Examples, Constructions, and Applications

• Dynkin algebras are always brick-finite: every indecomposable is simple and hence a brick; the number is precisely the number of simple modules.
• Kronecker algebra is the minimal brick-infinite example: it admits a 1-parameter family of simple regular bricks.
• Local algebras: $M$5 is brick-finite (unique simple).
• Wind-wheel algebras: brick-finite specific domestic special biserials with explicit enumeration of bricks (Asai et al., 16 Jun 2025).
• Glue constructions: arbitrary brick-finite algebras of given rank and representation type can be built by gluing along quivers and preserving brick-directedness (Asai et al., 16 Jun 2025).

Brick-finite algebras underlie classification of wide subcategories, have finite combinatorics in the τ-tilting fan, and correspond to algebras for which every wide subcategory is functorially finite and every semibrick is finite (Asai, 2016, Nasr-Isfahani, 15 Nov 2025).

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References:

• (Sentieri, 2020) "A brick version of a theorem of Auslander"
• (Mousavand et al., 15 Aug 2025) "On the bricks (Schur representations) of finite dimensional algebras"
• (Nasr-Isfahani, 15 Nov 2025) "Finiteness of semibricks and brick-finite algebras"
• (Mousavand et al., 2024) "Hom-orthogonal modules and brick-Brauer-Thrall conjectures"
• (Mousavand, 2019) "$M$6-tilting finiteness of biserial algebras"
• (Mousavand et al., 2022) "Biserial algebras and generic bricks"
• (Sengupta et al., 2024) "Characterisation of band bricks over certain string algebras and a variant of perfectly clustering words"
• (Asai et al., 16 Jun 2025) "Brick-splitting Torsion Pairs and Trim Lattices"
• (Mousavand et al., 2021) "Minimal ($M$7-)tilting infinite algebras"
• (Bautista et al., 2024) "On generic bricks over tame algebras"
• (Asai, 2016) "Semibricks"