# Ringel duality as an instance of Koszul duality

## TL;DR

This paper explores the relationship between Ringel duality and Koszul duality, providing a combinatorial framework for understanding the duality of quasi-hereditary algebras and their Ext-algebras.

## Contribution

It introduces a combinatorial description of algebras and corings whose right algebra is the Ringel dual, extending the understanding of dualities in quasi-hereditary algebras.

## Key findings

- Provided a combinatorial description of the Ringel dual algebra
- Applied results to restrict the A-infinity structure of Ext-algebras
- Analyzed examples related to birational morphisms of smooth surfaces

## Abstract

In their previous work, S. Koenig, S. Ovsienko and the second author showed that every quasi-hereditary algebra is Morita equivalent to the right algebra, i.e. the opposite algebra of the left dual, of a coring. Let $A$ be an associative algebra and $V$ an $A$-coring whose right algebra $R$ is quasi-hereditary. In this paper, we give a combinatorial description of an associative algebra $B$ and a $B$-coring $W$ whose right algebra is the Ringel dual of $R$. We apply our results in small examples to obtain restrictions on the $A_\infty$-structure of the $\textrm{Ext}$-algebra of standard modules over a class of quasi-hereditary algebras related to birational morphisms of smooth surfaces.