Ringel duality as an instance of Koszul duality
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.