Semibricks

Sota Asai

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arXiv · 1610.05860

Semibricks

Sota Asai

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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 $τ$-tilting theory. We construct canonical bijections between the set of support $τ$-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-Šťovíček, 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 $τ$-rigid modules by Jasso and Eisele-Janssens-Raedschelders. Moreover, we study semibricks over Nakayama algebras and tilted algebras in detail.

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BibTeX RIS

Sota Asai. 2018-06-06. Semibricks. https://arxiv.org/abs/1610.05860

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