# A construction of support $\tau$-tilting modules over $\tau$-tilting finite algebras

## TL;DR

This paper explores how to construct support τ-tilting modules over τ-tilting finite algebras by gluing semibricks through recollements, linking recent developments in τ-tilting theory with classical concepts.

## Contribution

It introduces a method to glue semibricks along recollements to construct support τ-tilting modules, extending the understanding of τ-tilting finite algebras.

## Key findings

- Gluing semibricks along recollements can produce support τ-tilting modules.
- Behavior of τ-tilting finiteness under recollements is characterized.
- Examples demonstrate the construction of support τ-tilting modules via this method.

## Abstract

The notion of (semi)bricks, regarded as a generalization of (semi)simple modules, appeared in a paper of Ringel in 1976. In recent years, there has been several new developments motivated by links to {\tau}-tilting theory studied by Demonet-Iyama-Jasso and Asai. In this paper, we will discuss how to glue semibricks along a recollement with the intermediate extension functor similar to gluing simple modules by Beilinson-Bernstein-Deligne. As an application, we investigate the behavior of {\tau}-tilting finite under recollements of module categories of algebras. Moreover, we give some examples to show the construction of support {\tau}-tilting modules over {\tau}-tilting finite algebras by gluing semibricks via recollements.