Wall-and-chamber structures for finite-dimensional algebras and $\tau$-tilting theory

TL;DR

This paper explores the wall-and-chamber structure as a geometric invariant of finite-dimensional algebras, detailing its connection to torsion classes and $ au$-tilting theory.

Contribution

It provides a formal definition of the wall-and-chamber structure and clarifies its relationship with torsion classes and $ au$-tilting theory.

Findings

Defines the wall-and-chamber structure for finite-dimensional algebras

Establishes the relationship between wall-and-chamber structures and torsion classes

Connects wall-and-chamber structures with $ au$-tilting theory

Abstract

The wall-and-chamber structure is a geometric invariant that can be associated to any algebra. In this notes we give the definition of this object and we explain its relationship with torsion classes and $\tau$-tilting theory.