2024/12/18 by Wen Chang, Chang, Wen
Computer Science · #Advanced Algebra and Logic
paper · pdf · doi:10.48550/arxiv.2412.13971
It is demonstrated that any almost-tilting module over a gentle algebra is indeed partial-tilting, meaning it can be completed as a tilting module. Furthermore, such a module has at most 2n possible complements, thereby confirming a (modified) conjecture of Happel for the case of gentle algebras. Additionally, for any n≥ 3 and 1≤ m ≤ n-2, there always exists a (connected) gentle algebra with rank n and a pre-tilting module of rank m which is not partial-tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is an induction process involving surface cuts, which is hoped to be beneficial for other applications as well.