# $\tau$-tilting modules over one-point extensions by a projective module

## TL;DR

This paper investigates the relationship between support τ-tilting modules over an algebra and its one-point extension, establishing a correspondence and embedding between their module categories.

## Contribution

It proves that support τ-tilting modules extend and restrict between an algebra and its one-point extension, with an embedding of their support τ-tilting quivers.

## Key findings

- Extension of support τ-tilting modules from B to A is valid.
- Restriction of support τ-tilting modules from A to B is valid.
- There is a full embedding of quivers between the posets of support τ-tilting modules.

## Abstract

Let $A$ be the one point extension of an algebra $B$ by a projective $B$-module. We prove that the extension of a given support $\tau$-tilting $B$-module is a support $\tau$-tilting $A$-module; and, conversely, the restriction of a given support $\tau$-tilting $A$-module is a support $\tau$-tilting $B$-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support $\tau$-tilting modules.