Semibricks

Abstract

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–Št’ovíček, 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 as examples.