Recollements of abelian categories and ideals in heredity chains - a recursive approach to quasi-hereditary algebras

TL;DR

This paper introduces a recursive homological method using recollements of abelian categories to characterize quasi-hereditary algebras and their ideals, providing new insights into their structure and applications.

Contribution

It presents a novel recursive approach to quasi-hereditary algebras via recollements, offering a homological proof of key characterizations and extending to related algebra classes.

Findings

Homological proof of Dlab and Ringel's characterization of idempotent ideals

Recursive characterization of quasi-hereditary algebras

Applications to hereditary and Morita context rings

Abstract

Recollements of abelian categories are used as a basis of a homological and recursive approach to quasi-hereditary algebras. This yields a homological proof of Dlab and Ringel's characterisation of idempotent ideals occuring in heredity chains, which in turn characterises quasi-hereditary algebras recursively. Further applications are given to hereditary algebras and to Morita context rings.