The second brick-Brauer-Thrall conjecture

Let AA be a finite-dimensional algebra over a field kk. Write brick(A,d)\operatorname{brick}(A,d) for the set of isomorphism classes of bricks of dimension dd, and call AA brick-finite if it has only finitely many bricks up to isomorphism. Second brick-Brauer-Thrall conjecture. The algebra AA is brick-finite if and only if brick(A,d)\operatorname{brick}(A,d) is a finite set for every dZ>0d\in\mathbb{Z}_{>0}. This conjecture connects brick-finiteness with the bounded-dimension behavior of bricks; it was settled for special biserial algebras and other families, but the source records it as having been resolved in that setting.

Sources & referencesView supporting material

Primary source

Kaveh Mousavand and Charles Paquette, “Hom-orthogonal modules and brick-Brauer-Thrall conjectures”, arXiv:2407.20877 (2025).

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