Semibricks

Abstract

In representation theory of finite-dimensional algebras, (semi)bricks are a generalization of (semi)simple modules, and they have long been studied. The aim of this paper is to study semibricks from the point of view of $\tau$-tilting theory. We construct canonical bijections between the set of support $\tau$-tilting modules, the set of semibricks satisfying a certain finiteness condition, and the set of 2-term simple-minded collections. In particular, we unify Koenig-Yang bijections and Ingalls-Thomas bijections generalized by Marks-\v{S}\v{t}ov\'{i}\v{c}ek, which involve several important notions in the derived categories and the module categories. We also investigate connections between our results and two kinds of reduction theorems of $\tau$-rigid modules by Jasso and Eisele-Janssens-Raedschelders. Moreover, we study semibricks over Nakayama algebras and tilted algebras in detail.