Stratifications of algebras with two simple modules

TL;DR

This paper investigates the structure of finite-dimensional algebras with two simple modules, showing that certain stratifications imply the algebra is derived equivalent to a quasi-hereditary algebra, especially over algebraically closed fields with finite global dimension.

Contribution

It establishes a link between stratifications of derived categories and quasi-hereditary algebras for algebras with two simple modules, extending understanding of their derived equivalences.

Findings

Algebras with stratifications having simple factors are derived equivalent to quasi-hereditary algebras.

Over algebraically closed fields with finite global dimension, such algebras are either derived simple or quasi-hereditary.

The results classify the derived categories of these algebras under specific stratification conditions.

Abstract

Let $A$ be a finite-dimensional algebra with two simple modules. It is shown that if the derived category of $A$ admits a stratification with simple factors being the base field $k$, then $A$ is derived equivalent to a quasi-hereditary algebra. As a consequence, if further $k$ is algebraically closed and $A$ has finite global dimension, then $A$ is either derived simple or derived equivalent to a quasi-hereditary algebra