The size of a stratifying system can be arbitrarily large

TL;DR

This paper constructs examples demonstrating that the size of stratifying systems in module categories of algebras can be arbitrarily large, both in infinite and finite cases, using established homological algebra results.

Contribution

It provides the first known examples showing that stratifying systems can have arbitrarily large size relative to simple modules.

Findings

Existence of infinite size stratifying systems in module categories.

Finite stratifying systems can be arbitrarily large compared to simple modules.

Examples are constructed using well-known homological algebra results.

Abstract

In this short note we construct two families of examples of large stratifying systems in module categories of algebras. The first examples consists on stratifying systems of infinite size in the module category of an algebra $A$. In the second family of examples we show that the size of a finite stratifying system in the module category of a finite dimensional algebra $A$ can be arbitrarily large in comparison to the number of isomorphism classes of simple $A$-modules. We note that both families of examples are built using well-established results in higher homological algebra.