Two-step tilting for standardly stratified algebras

We study the class of standardly stratified algebras introduced by Cline, Parshall and Scott, and its subclass of the so-called weakly properly stratified algebras, which generalizes the class of properly stratified algebras introduced by Dlab. We characterize when the Ringel dual of a standardly stratified algebra is weakly properly stratified and show the existence of a two-step tilting module. This allows us to calculate the finitistic dimension of such algebras. Finally, we also give a construction showing that each finite partially pre-ordered set gives rise to a weakly properly stratified algebras with a simple preserving duality.