On Riedtmann's well-behaved functors and applications to composites of irreducible morphisms

TL;DR

This survey explores Riedtmann's well-behaved functors connecting mesh categories and module categories, applying these to analyze compositions of irreducible morphisms in finite-dimensional algebras.

Contribution

It summarizes key results on mesh categories and demonstrates how Riedtmann's functors can be used to understand the composition of irreducible morphisms.

Findings

Properties of well-behaved functors facilitate the study of morphism compositions.

Criteria for non-zero compositions of irreducible morphisms are established.

Applications to the structure of module categories are discussed.

Abstract

In this survey, we summarize some results in the literature involving the mesh category, which is a combinatorial representation of the category of modules over a finite-dimensional associative algebra. We discuss Riedtmann's well-behaved functors, which compare the mesh category with the module category, and discuss how the properties of these functors can be applied to study the problem of composing irreducible morphisms, which is the problem of deciding when the composition of n irreducible morphisms is non-zero and lies on the (n+1)-th power of the radical.