# Strongly quasi-hereditary algebras and rejective subcategories

## TL;DR

This paper characterizes right-strongly quasi-hereditary algebras using heredity chains and rejective subcategories, and explores their properties, including their relation to algebras of global dimension two and Nakayama algebras.

## Contribution

It provides new characterizations of right-strongly quasi-hereditary algebras and establishes conditions under which Auslander algebras are strongly quasi-hereditary.

## Key findings

- Any artin algebra with global dimension ≤ 2 is right-strongly quasi-hereditary.
- The Auslander algebra of a representation-finite algebra is strongly quasi-hereditary iff the algebra is Nakayama.
- Characterizations of these algebras via heredity chains and rejective subcategories.

## Abstract

Ringel's right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline-Parshall-Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.