Koszul duality for stratified algebras I. Quasi-hereditary algebras

We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations in the bounded derived category of graded modules.