Brick chain filtrations

TL;DR

This paper surveys the concept of brick chain filtrations in module categories over artin algebras, highlighting recent developments and their significance in understanding the directedness of module categories without relying on $ au$-tilting theory.

Contribution

It provides a self-contained, elementary overview of brick chains and their role in module categories, emphasizing recent research and clarifying their importance beyond $ au$-tilting theory.

Findings

Brick chains reveal directedness in module categories.

The survey consolidates recent research on bricks and filtrations.

It offers an elementary treatment independent of $ au$-tilting theory.

Abstract

We deal with the category of finitely generated modules over an artin algebra $A$. Recall that an object in an abelian category is said to be a brick provided its endomorphism ring is a division ring. Simple modules are, of course, bricks, but in case $A$ is connected and not local, there do exist bricks which are not simple. The aim of this survey is to focus the attention to filtrations of modules where all factors are bricks, with bricks being ordered in some definite way. In general, a module category will have many oriented cycles. Recently, Demonet has proposed to look at so-called brick chains in order to deal with a very interesting directedness feature of a module category. These are the orderings of bricks which we will use. This is a survey which relies on recent investigations by a quite large group of mathematicians. We have singled out some important observations and have reordered them in order to obtain a completely self-contained (and elementary) treatment of the relevance of bricks in a module category. (Most of the papers we rely on are devoted to what is called $\tau$-tilting theory, but for the results we are interested in, there is no need to deal with $\tau$-tilting, or even with the Auslander-Reiten translation $\tau$).