SEMIBRICKS OVER SPLIT-BY-NILPOTENT EXTENSIONS

In this paper, we prove that there is a bijection between the tau-tilting modules and the sincere left finite semibricks. We also construct (sincere) semibricks over split-by-nilpotent extensions. More precisely, let Gamma be a split-by-nilpotent extension of a finite-dimensional algebra Lambda by a nilpotent bimodule E-Lambda(Lambda), and S subset of mod Lambda. We prove that S circle times(Lambda) Gamma is a (sincere) semibrick in mod Gamma if and only if S is a semibrick in mod Lambda and Hom(Lambda)(S, S circle times(Lambda) E) = 0 (and S boolean OR S circle times(Lambda) E is sincere). As an application, we can construct tau-tilting modules and support tau-tilting modules over tau-tilting finite cluster-tilted algebras.