[Paper Review] A $ au$-tilting approach to the first Brauer-Thrall conjecture

Sibylle Schroll, Hipolito Treffinger|Kölner Universitäts PublikationsServer (Universität zu Köln)|Apr 29, 2020

Algebraic structures and combinatorial models24 references4 citations

TL;DR

This paper establishes the first $τ$-Brauer-Thrall conjecture by proving that a finite-dimensional algebra is $τ$-tilting finite if all its finite-dimensional bricks have bounded composition length. Using $τ$-tilting theory, $g$- and $c$-vectors, and sign-coherence, the authors show that $τ$-tilting infinite algebras must have bricks of arbitrarily large length, thereby providing a new proof of the classical first Brauer-Thrall conjecture for such algebras.

ABSTRACT

In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded.

Motivation & Objective

• To establish a $τ$-tilting version of the first Brauer-Thrall conjecture, extending classical representation theory to new invariants.
• To investigate the relationship between bounded composition length of bricks and the finiteness of $τ$-tilting pairs in finite-dimensional algebras.
• To prove that $τ$-tilting infinite algebras must contain bricks of arbitrarily large composition length.
• To provide a new homological proof of the classical first Brauer-Thrall conjecture using $τ$-tilting theory and $c$-vector sign-coherence.

Proposed method

• Utilizes $τ$-tilting theory to relate $τ$-tilting pairs to $2$-term silting complexes and functorially finite torsion classes.
• Constructs $n$ distinct $τ$-rigid pairs from a given $τ$-tilting pair by removing one indecomposable summand from $M$ or $P$, and identifies a unique brick in each associated subcategory.
• Applies the $G$-matrix and $D_A$-matrix to derive the equation $D_A[B_i] = \langle g^i, [B_i]\rangle_A \mathsf{c}_i$, linking $c$-vectors to module invariants.
• Uses sign-coherence of $c$-vectors and the pigeonhole principle to show that for any $t$, there exists a $c$-vector with $\sum |c_i| \geq \delta_A t$, leading to bricks of length at least $t$.
• Leverages the fact that $D_A[B] = (\delta_1[B]_1, \dots, \delta_n[B]_n)$ and $\sum \delta_i[B]_i \geq \sum |m_t c_i| \geq \delta_A t$ to bound composition length.
• Applies the duality between $\hat{M}_i^\perp \cap {}^\perp\tau\hat{M}_i \cap \hat{P}_i^\perp$ and $g$-vector orthogonality to ensure $\langle g^j, [B_i]\rangle_A = 0$ for $j \neq i$, enabling diagonalization of the matrix product.

Experimental results

Research questions

• RQ1 Does bounded composition length of all bricks imply $τ$-tilting finiteness in a finite-dimensional algebra?
• RQ2 Can the first Brauer-Thrall conjecture be re-proven using $τ$-tilting theory and $c$-vector sign-coherence?
• RQ3 Do $τ$-tilting infinite algebras necessarily contain bricks of arbitrarily large composition length?
• RQ4 Is there a structural link between $g$-vectors, $c$-vectors, and the endomorphism structure of bricks in $τ$-tilting theory?
• RQ5 Can the $D_A$-matrix and $G$-matrix interaction uniquely determine the existence of high-length bricks in $τ$-tilting infinite algebras?

Key findings

• The first $τ$-Brauer-Thrall conjecture holds: if all finite-dimensional bricks have fewer than $n$ composition factors, then the algebra is $τ$-tilting finite.
• For any positive integer $t$, a $τ$-tilting infinite algebra admits a brick $B_t$ with at least $t$ composition factors.
• The $c$-vectors of the algebra are sign-coherent, a key property used to ensure the existence of high-length bricks via the pigeonhole principle.
• The matrix equation $G_{(M,P)}^T D_A X_{(M,P)} = D$ yields a diagonal matrix, confirming orthogonality and enabling the construction of distinct bricks from $g$-vector bases.
• The existence of a non-zero integer $\langle g^i, [B_i]\rangle_A$ ensures that $D_A[B_i]$ is non-trivial, linking module structure to $c$-vectors.
• The sum $\sum \delta_i[B_t]_i \geq \delta_A t$ implies that the total composition length of $B_t$ grows linearly with $t$, proving unbounded length in $τ$-tilting infinite algebras.