On band modules and \(\tau \)-tilting finiteness (Q2664686)

Let \(\mathbb{K}\) be a field and \(A\) be a finite dimensional algebra over \(\mathbb{K}\). The first Brauer-Thrall conjecture states that if \(A\) is of bounded representation type, then \(A\) has finite representation type. On the other hand, the second Brauer-Thrall conjecture gives a necessary condition for \(A\) to be representation infinite. Both Brauer-Thrall conjectures have been proved (see [\textit{A. V. Roiter}, Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1275--1282 (1968; Zbl 0167.31001)]) and [\textit{M. Auslander}, Commun. Algebra 1, 177--268 (1974; Zbl 0285.16028); Commun. Algebra 1, 269--310 (1974; Zbl 0285.16029)] for the first; [\textit{L. A. Nazarova} and \textit{A. V. Roiter}, in: Representation Theory finite Groups related Topics, Proc. Sympos. Pure Math. 21(1970), 111--115 (1971; Zbl 0258.16017)] and [\textit{R. Bautista}, Comment. Math. Helv. 60, 392--399 (1985; Zbl 0584.16017)] for the second conjecture). The \(\tau\)-Brauer-Thrall Conjectures concern the \(\tau\)-tilting finiteness of algebras. Namely, the first \(\tau\)-Brauer-Thrall Conjecture asserts that if there exists a positive integer \(n\) such that \(\dim_{\mathbb{K}}M \leq n\) for every finite dimensional \(A\)-module which is a brick, then \(A\) is \(\tau\)-tilting finite. The second \(\tau\)-Brauer-Thrall Conjecture says that if \(A\) is \(\tau\)-tilting infinite, then there is a positive integer \(d\) such that there are infinitely many bricks of dimension \(d\). It has been proved by the first and second authors that the first \(\tau\)-Brauer-Thrall conjecture holds for any finite dimensional algebra over any field [\textit{S. Schroll} and \textit{H. Treffinger}, ``A $\tau$-tilting approach to the first Brauer-Thrall conjecture'', Preprint, \url{arXiv:2004.14221}]. In the paper under review, the authors prove that a special biserial algebra \(A\) is \(\tau\)-tilting finite if and only if no band module of \(A\) is a brick. Using this result, they also give a new proof of a result by \textit{T. Adachi} et al. [Math. Z. 290, No. 1--2, 1--36 (2018; Zbl 1433.16010)] on the characterization of the \(\tau\)-tilting finiteness of Brauer graph algebras.